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[ "Secure and Trustworthy Artificial Intelligence-Extended Reality (AI-XR) for Metaverses", "Secure and Trustworthy Artificial Intelligence-Extended Reality (AI-XR) for Metaverses", "Secure and Trustworthy Artificial Intelligence-Extended Reality (AI-XR) for Metaverses", "Secure and Trustworthy Artificial Intelligence-Extended Reality (AI-XR) for Metaverses" ]
[ "Adnan Qayyum \nInformation Technology University (ITU)\nPunjab, LahorePakistan\n", "Muhammad Atif Butt \nInformation Technology University (ITU)\nPunjab, LahorePakistan\n", "Hassan Ali \nInformation Technology University (ITU)\nPunjab, LahorePakistan\n", "Muhammad Usman \nGlasgow Caledonian University\nGlasgowUnited Kingdom\n", "Osama Halabi \nQatar University\nDohaQatar\n", "Ala Al-Fuqaha \nHamad Bin Khalifa University\nDohaQatar\n", "Qammer H Abbasi ", "Muhammad Ali Imran ", "Junaid Qadir \nQatar University\nDohaQatar\n", "James Watt ", "\nSchool of Engineering\nUniversity of Glasgow\nGlasgowUnited Kingdom\n", "Adnan Qayyum \nInformation Technology University (ITU)\nPunjab, LahorePakistan\n", "Muhammad Atif Butt \nInformation Technology University (ITU)\nPunjab, LahorePakistan\n", "Hassan Ali \nInformation Technology University (ITU)\nPunjab, LahorePakistan\n", "Muhammad Usman \nGlasgow Caledonian University\nGlasgowUnited Kingdom\n", "Osama Halabi \nQatar University\nDohaQatar\n", "Ala Al-Fuqaha \nHamad Bin Khalifa University\nDohaQatar\n", "Qammer H Abbasi ", "Muhammad Ali Imran ", "Junaid Qadir \nQatar University\nDohaQatar\n", "James Watt ", "\nSchool of Engineering\nUniversity of Glasgow\nGlasgowUnited Kingdom\n" ]
[ "Information Technology University (ITU)\nPunjab, LahorePakistan", "Information Technology University (ITU)\nPunjab, LahorePakistan", "Information Technology University (ITU)\nPunjab, LahorePakistan", "Glasgow Caledonian University\nGlasgowUnited Kingdom", "Qatar University\nDohaQatar", "Hamad Bin Khalifa University\nDohaQatar", "Qatar University\nDohaQatar", "School of Engineering\nUniversity of Glasgow\nGlasgowUnited Kingdom", "Information Technology University (ITU)\nPunjab, LahorePakistan", "Information Technology University (ITU)\nPunjab, LahorePakistan", "Information Technology University (ITU)\nPunjab, LahorePakistan", "Glasgow Caledonian University\nGlasgowUnited Kingdom", "Qatar University\nDohaQatar", "Hamad Bin Khalifa University\nDohaQatar", "Qatar University\nDohaQatar", "School of Engineering\nUniversity of Glasgow\nGlasgowUnited Kingdom" ]
[]
Metaverse is expected to emerge as a new paradigm for the next-generation Internet, providing fully immersive and personalised experiences to socialize, work, and play in selfsustaining and hyper-spatio-temporal virtual world(s). The advancements in different technologies like augmented reality, virtual reality, extended reality (XR), artificial intelligence (AI), and 5G/6G communication will be the key enablers behind the realization of AI-XR metaverse applications. While AI itself has many potential applications in the aforementioned technologies (e.g., avatar generation, network optimization, etc.), ensuring the security of AI in critical applications like AI-XR metaverse applications is profoundly crucial to avoid undesirable actions that could undermine users' privacy and safety, consequently putting their lives in danger. To this end, we attempt to analyze the security, privacy, and trustworthiness aspects associated with the use of various AI techniques in AI-XR metaverse applications. Specifically, we discuss numerous such challenges and present a taxonomy of potential solutions that could be leveraged to develop secure, private, robust, and trustworthy AI-XR applications. To highlight the real implications of AI-associated adversarial threats, we designed a metaverse-specific case study and analyzed it through the adversarial lens. Finally, we elaborate upon various open issues that require further research interest from the community.
10.48550/arxiv.2210.13289
[ "https://export.arxiv.org/pdf/2210.13289v1.pdf" ]
253,098,427
2210.13289
dd35797f32051afab84aeab0d67c16aa76b346da
Secure and Trustworthy Artificial Intelligence-Extended Reality (AI-XR) for Metaverses Adnan Qayyum Information Technology University (ITU) Punjab, LahorePakistan Muhammad Atif Butt Information Technology University (ITU) Punjab, LahorePakistan Hassan Ali Information Technology University (ITU) Punjab, LahorePakistan Muhammad Usman Glasgow Caledonian University GlasgowUnited Kingdom Osama Halabi Qatar University DohaQatar Ala Al-Fuqaha Hamad Bin Khalifa University DohaQatar Qammer H Abbasi Muhammad Ali Imran Junaid Qadir Qatar University DohaQatar James Watt School of Engineering University of Glasgow GlasgowUnited Kingdom Secure and Trustworthy Artificial Intelligence-Extended Reality (AI-XR) for Metaverses 1 Metaverse is expected to emerge as a new paradigm for the next-generation Internet, providing fully immersive and personalised experiences to socialize, work, and play in selfsustaining and hyper-spatio-temporal virtual world(s). The advancements in different technologies like augmented reality, virtual reality, extended reality (XR), artificial intelligence (AI), and 5G/6G communication will be the key enablers behind the realization of AI-XR metaverse applications. While AI itself has many potential applications in the aforementioned technologies (e.g., avatar generation, network optimization, etc.), ensuring the security of AI in critical applications like AI-XR metaverse applications is profoundly crucial to avoid undesirable actions that could undermine users' privacy and safety, consequently putting their lives in danger. To this end, we attempt to analyze the security, privacy, and trustworthiness aspects associated with the use of various AI techniques in AI-XR metaverse applications. Specifically, we discuss numerous such challenges and present a taxonomy of potential solutions that could be leveraged to develop secure, private, robust, and trustworthy AI-XR applications. To highlight the real implications of AI-associated adversarial threats, we designed a metaverse-specific case study and analyzed it through the adversarial lens. Finally, we elaborate upon various open issues that require further research interest from the community. I. INTRODUCTION In the recent era, metaverse technology is rapidly emerging and there are a lot of potential applications that can benefit from these developments such as healthcare, industry, business, etc. While there is no single agreed-upon definition of a metaverse [1], the metaverse is a convergence of physical, augmented, and virtual reality and provides a powerfully immersive experience to users by allowing them to seamlessly interact with the real and virtual (computer-simulated) world. The term "metaverse" is a combination of two terms, i.e., "meta" which means transcending, and "universe" which refers to the physical universe and the current virtual world. This is the basic definition of the term metaverse, nevertheless, the literature shows that it does not have a unified definition [1]. * Corresponding author: Junaid Qadir ([email protected]) Metaverse allows the creation of shared virtual space by connecting all virtual worlds through the Internet, where digital avatars (i.e., users) can communicate and interact with each other similar to the physical world. Key backbone technologies in the metaverse include artificial intelligence (AI) and extended reality (XR) that leverage different technological developments such as virtual reality (VR), augmented reality (AR), and mixed reality (MR). In addition, similar to the current Internet, metaverse will leverage other concomitant technologies like information and communication technologies (ICT), 5G, and 6G, but metaverse will provide a qualitatively different experience to its users by enabling real-life-like 3-D experiences through the incorporation of aforementioned technologies. Metaverse allows for the digitalization of traditional brickand-mortar institutions and businesses-it will be possible to develop virtual markets, digital lands, and digital infrastructure, which can be bought and sold using blockchain and nonfungible tokens (NFTs), which are non-interchangeable units of data stored on a blockchain. Metaverse can be a game changer in terms of the impact of its potential applications due to the greater immersion, involvement, and personalization possible due to AI-XR. This is the prime reason various corporations have shown great interest in the idealization of the metaverse and are making big bets on developing their own AI-XR-based metaverse ecosystems. Metaverse is receiving increasing traction from numerous major tech companies worldwide such as Facebook (which is recently rebranded with the name "Meta"), Microsoft, Google, and Amazon. In addition, the widespread adoption of the metaverse is evident in the infusion of billions of dollars of investments by these companies in an attempt to achieve great technological transformation. However, despite such huge traction of the metaverse and its potential to transform existing ecosystems like healthcare, there are numerous challenges associated with the use of AI in the metaverse that may hinder their seamless adoption by end users in the longer term. In addition, in the backdrop of recent technologically induced social issues, there is a palpable lack of trust and confidence Falchuk et al. [2] Privacy issues and solutions for digital footprints in metaverse games. √ √ × × × × × × ≈ × 2020 Guzman et al. [3] Analyzed privacy and security in MR from data-centric perspective. √ √ ≈ × × × × × ≈ √ 2021 Ning et al. [4] General focused survey on metaverse with partial discussion on privacy and security issues. ≈ ≈ × × × × × × √ √ 2021 Pietro et al. [5] Discussed general privacy and security issues in metaverse applications. √ √ × × × × × × ≈ √ 2022 Huynh et al. [6] Discussed potential applications of AI in various metaverse applications. ≈ ≈ × × × × × × √ ≈ 2022 Zhao et al. [7] Security & privacy issues and solutions for four dimensions: communication, user information, scenario, and goods. √ √ × × × × × × ≈ × 2022 Wang et al. [8] Presented general (non ML-associated) security and privacy related challenges for different metaverse applications. √ √ √ × × × × × √ √ This Paper ML-associated security, privacy, and trustworthiness challenges and solutions for AI-XR metaverse applications. √ √ √ √ √ √ √ √ √ √ in such technologies. Since technology can be used both ways (i.e., for good and harm), it is vital that governments, corporations, and society at large seriously consider ethical and moral issues. There are many ethical questions about privacy, security, transparency, accountability, democracy, freedom of speech, and anonymity that technology alone cannot answer. Some specific concerns related to how AI-XR-based metaverse applications will impact humanity are: (1) how AI-XR-based metaverse applications will impact and promote human values and human rights? how will AI-XR-based metaverse promote social welfare and not cause harm to society at large; (3) how can we regulate critical applications of metaverse like healthcare? (4) How do we align the commercial and technical imperatives of AI-XR metaverse applications with human values and promote a moral vision and character development? (5) How do we ensure that AI-XR metaverse application developers do not exploit or manipulate their users? Keeping in mind the aforementioned questions, in this paper, we present an analysis of the security, privacy, and trust issues associated with the use of AI-XR in metaverse applications. Contributions of this Paper: To the best of our knowledge this paper is the first attempt towards analyzing the challenges associated with the use of different AI techniques in AI-XR metaverse applications. The comparison of this paper with existing survey and review articles that are focused on analyzing security and privacy aspects of AI-XR metaverse applications is presented in Table I. In the summary, the following are the salient contributions of this paper. 1) We highlight various issues that arise with the use of AI in metaverse applications that mainly include security, privacy, and trustworthiness. 2) We present a taxonomy of different potential solutions that can be used to realize secure, robust, safe, and trustworthy AI-XR applications. 3) We identify various ML-based use cases across different layers of metaverse architecture and highlight several ML-associated vulnerabilities at each layer. 4) We present a case study to highlight the real threat of AI-based vulnerabilities by considering a prospective metaverse application design scenario. 5) We elaborate upon various open issues that require further development. Organization of this Paper: The rest of the paper is orga-nized as follows. Section II presents relevant background. The discussion of challenges related to security, safety, privacy, and trust is presented in Section III. The taxonomy of different vulnerabilities associated with the use of ML in AI-XR metaverse applications is discussed in Section IV. Different potential solutions that can ensure security, privacy, safety, and trust in ML applications are discussed in Section V. Various open issues that require further research attention are highlighted in Section VI. Finally, we conclude the paper in Section VII. The organization of the paper is depicted in Figure 1. II. BACKGROUND A. Metaverse: An Introduction Before understanding the concept of the metaverse, it is very important to understand the related concepts that are described below. • Virtual Reality (VR): In VR, the users achieve an immersive experience by donning a VR headset that allows them to enter into a virtual (computer-simulated) world thus completely blanking out the real world. The key objective of enabling immersion in VR is to provide high fidelity user interaction to give him the feeling that the virtual world is real [9]. Prime examples of VR include Facebook Oculus and HTC VIVE VR headrests. VR has a wide range of applications but a VR headset is required to enter the digital world. • Augmented Reality (AR): In AR, the users obtain an immersive experience by blending the virtual (digital) and the real world and projecting digital content (text, images, and sounds) onto the real world. Unlike VR, AR can be realized without special equipment (like a headset) through the use of smartphones, implants, glasses, or contact lenses that are used to overlay digital content on top of the real world. • Mixed Reality (MR): MR is a hybrid term that is used to refer to the conjunction of virtual and real worlds to produce new environments and experiences, where physical and digital objects co-exist and interact in realtime (it is an enhanced form of AR). Microsoft HoloLens headrest is an example MR headset. • Extended Reality (XR): XR is an umbrella term that encompasses and subsumes VR, AR, and MR. It covers all the future realities that might emerge from these technologies. XR is predicted to become a $209 billion market by 2022. Immersive first-person experiences are one of the most significant aspects of XR, VR, and AR. The Metaverse takes this to the next level, allowing large groups of individuals to share an immersive first-person experience while maintaining a strong sense of mutual presence. Although the term "metaverse" is often associated with virtual reality, according to Rosenberg, there are two types of metaverses: a "virtual metaverse" in which people are avatars and an "augmented metaverse" in which layers of virtual content are overlaid on the real world and experienced by real people. Figure 2 depicts XR, VR, AR, and the virtual and augmented metaverse, as well as their interaction. Metaverse is the next generation of the Internet that will surround us both graphically and socially. A historical overview of developments regarding metaverse is shown in Figure 3 and different applications of XR in metaverse along with their enabling technologies are illustrated in Figure 4. B. Metaverse: An Architectural Overview The architecture of the metaverse expands from the people's experiences to the underlying enabling technologies and has seven layers that include: (1) Experience; (2) Discovery; (3) Creator; (4) Spatial Mapping; (5) Decentralization; (6) Human Interface; and (7) Infrastructure, which is illustrated in Figure 5 and is briefly described below. -Layer 1: Experience: It is the topmost layer in the metaverse, which is mainly concerned with the experiences of the users and it provides different services to them, e.g., games, E-sports, social interactions, events, festivals, shopping, co-working, etc. -Layer 2: Discovery: It is like a push and pull service that introduces people to new experiences in the metaverse such as virtual stores, advertising networks, ratings, social curation avatars, chatbots, etc. It will involve both inbound (i.e., users are actively seeking information regarding experiences) and outbound (i.e., an advertisement that is not explicitly requested by the user). This layer is mainly driven by metaverse service providers and content creators to inform and motivate users regarding new features and services. -Layer 3: Creator: This layer is sometimes also referred to as the creator economy. Like the previous layer, it is mainly driven by the content creator and service providers, who leverage different technologies to create content or experiences for metaverse users such as asset markets, E-commerce, design tools, and workflow. -Layer 4: Spatial Mapping: This layer provides a bridge between the digital world and the physical world and provides immersive experiences to metaverse users. It consists of different technologies like geospatial mapping, object and speech recognition, 3D engines (for enabling animations), VR, AR, XR, multitasking, and integration of user interfaces and heterogeneous sensor data (e.g., from IoT and wearable devices). It can be assumed as the backbone of the creator layer [10]. -Layer 5: Decentralization: Decentralization is very crucial in the metaverse and ideally it should not be controlled by a single entity. It provides a scalable ecosystem to developers in terms of providing online capabilities and reliability to the users at the same time. This layer will consist of multiple technologies that include edge computing, blockchain, microservices, and AI agents. -Layer 6: Human Interface: This layer is mainly concerned with the interfacing of the physical world with the digital and from the digital to the physical world. For example, let's consider the example of metaverse services that require data collected from humans using different sensors such as smartwatches, smartphones, smart glasses, wearable IoT devices, biosensors, and head-mounted displays, just to mention a few. -Layer 7: Infrastructure: This layer is responsible for connecting different enabling devices and technologies to the network for content delivery in the metaverse. Different ICT technologies will act as a backbone in the infrastructure layer of the metaverse. For example, 5G/6Gbased communication has huge potential to drastically improve the performance of metaverse applications while reducing latency and speeding up content delivery. In addition, this layer will also involve major data processing capabilities like data centers, the cloud, CPUs, GPUs, and even quantum computers. C. Metaverse: Applications Metaverse applications that incorporate different technologies like VR, AR, or XR have various potential applications in education, healthcare, industry, and scientific research, just to name a few. A detailed taxonomy of various potential metaverse applications is illustrated in Figure 6. Metaverse allows moving from text-focused Internet that supports 2D images to a 3D or even a 4D world (in which we may travel in time (forward or backward)). One of the promising applications will be social VR, which will be the enhanced version of current social media. As metaverse can leverage both VR and AR, numerous applications (e.g., voice recognition, gesture recognition, and speech translation) can benefit [11]. Over the past few years, VR and AR technologies have become very mature and nowadays their equipment is relatively cheap and readily available. This is a long way from the modest beginning of AR and VR, which were ignited in the 1960s by Ivan Sutherland in his pioneering work on the first responsive head-mounted wearable devices, which were admittedly primitive by modern standards. Modern VR headsets have become accessible (e.g., Facebook's Oculus Quest 2 is available for $300) with the price expected to go down as technology continues to advance. There are numerous AR applications such as Heads-Up-Display (HUD) features on modern luxury cars, the use of face filters in apps such as Snapchat, and games such as the addictive Pokémon GO game, where players could "see" Pokémon characters on the street. Modern mobile phones supporting Lidar technology now can support AR with new software development kits emerging such as Google's AR development platform ARCore, which provides the ability to track motion, understand the environment, and estimate light-three capabilities essential for AR. AR pioneer Louis Rosenberg predicts that within 10 years, most people will be spending more than 2 hours every day in VR, and augmented reality interfaces will replace mobile phones as our primary interface with digital content. D. Role of AI and ML in Metaverse Different AI techniques including machine learning (ML) and deep learning (DL) have many potential applications in the metaverse (as shown in Figure 7). For example, one of the most fascinating features of the metaverse will be voicebased commands that will utilize different voice recognition and analysis techniques for language processing and understanding human commands. In addition, metaverse will use different ML/DL-based regression and classification for data management and decision making. Also, to provide immersive experiences to the users it will use different generative models to generate photo-realistic avatars and for 3D reconstruction of objects from 2D images. We discuss the potential applications of ML/DL in the metaverse across five dimensions, which are described next. 1) Applications of ML in Natural Language Processing: Natural Language Processing (NLP) consists of different techniques that are used for automatically analyzing and understanding human languages (i.e., text and speech). There are many NLP applications that will be part of AI-XR metaverse applications, e.g., speech-to-text, text-to-speech, chatbots, and text/speech-based emotion recognition are the most prominent features of the metaverse. In particular, NLP techniques will be used for the recognition and understanding of complicated human conversations and commands. A key driving force behind the success of NLP methods is the advancement in ML/DL, with the development of new techniques such as recurrent neural networks (RNN), long-short term memory (LSTM), and transformer networks [12]. 2) Applications of ML in Vision: Machine vision or computer vision will be a fundamental component of AI-XR metaverse applications. Different computer vision applications will enable various functionalities in metaverse applications, for example, processing visual data from different sensors to infer high-level visual semantics. The major tasks of the visual processing pipeline include understanding users' activities, emotion recognition, object detection, scene understanding, semantic segmentation, avatar generation, etc. In addition to these applications, the metaverse is expected to have AIempowered quality assessment capabilities, e.g., for satisfying the users' demands about viewing high-resolution videos [6]. In this regard, advanced AI methods can be used to develop quantitative and qualitative benchmarks for visual quality assessment. 3) Applications of ML in Network Communication: Metaverse is expected to simultaneously entertain a massive number of users with the metaverse services provisioned mainly through wireless networks. Over the past few years, substantial research attention has been devoted to improving the overall throughput and performance of wireless network communication and the use of different AI techniques is the main driving force behind this innovation [13]. Metaverse will mainly include real-time multimedia services that require a reliable connection, high throughput, and low latency to ensure a seamless user experience. Therefore, it is expected that the metaverse will benefit from 5G and beyond empowered communication. The potential of different AI techniques has already been demonstrated for 5G and 6G, e.g., intelligent resource allocation [14], solving resource slicing problem [15], channel state estimation [16], and network modulation [17], etc. 4) Applications of ML in Blockchain: Service providers in the metaverse will provide users with different incentives in terms of digital assists (e.g., coins) for different events, games, and creative activities. The dispersion of such assets requires a transparent way to record and track such transactions. In this regard, smart contracts empowered blockchain technology can be leveraged that allows critical information to be stored on an immutable and impenetrable ledger. The decentralized nature of blockchain imbues it with great potential to address security and privacy issues in metaverse [18]. This potential increases with AI-empowered blockchain applications [19], for example, different AI-based clustering and classification techniques can be used for data analysis stored on blockchain [20]. In addition, different AI techniques can be used for knowledge discovery and learning, efficient data management, perception, reasoning, and planning. 5) Applications of ML in Digital Twin: The term digital twin refers to the digital replica (i.e., representation) of real objects. A digital twin is capable of synchronizing regular actions, operations processes, and assets with the real world, e.g., analyzing, monitoring, predicting, and visualizing [21]. The digital twin also acts as a bridge where the actual world and digital world interact with each other through different IoT devices [22]. The digital twin will be one of the most important building sectors of the metaverse that allows users to access and use services in the virtual world while exactly depicting the real world in a virtual environment. For example, surgeons and medical experts can create a digital replica of a patient to study and understand the involved complexities before performing his surgery. III. CHALLENGES IN IMPLEMENTING SECURE AND TRUSTWORTHY AI-XR METAVERSE APPLICATIONS Despite the significant potential of different AI-XR metaverse applications, there are various challenges related to security, privacy, and lack of trust that can hinder their wide adoption. A few such challenges include privacy breaches, security invasion, user profiling, unfair AI outcomes, etc. These challenges may directly or indirectly put the users' safety at risk and can influence social acceptability [23]. Moreover, as discussed above metaverse is the integration of different modern technologies like AI, blockchain, and 5G/6G, therefore, it is likely that the inherent issues associated with these technologies get translated into the metaverse. In this section, we describe different challenges that can hinder the secure, safe, robust, and trustworthy employment of AI-XR metaverse applications. Specifically, we characterized and discuss these challenges in two dimensions, i.e., challenges associated with the use of AI techniques including ML/DLbased methods, and XR-related challenges in the metaverse. We will start by first discussing AI-related challenges. A. AI Associated Challenges in AI-XR Metaverse Modern AI techniques that include ML/DL-based models suffer from different vulnerabilities that hinder the smooth, safe, secure, and trustworthy use of these methods in critical applications like healthcare, autonomous vehicles, and AI-XR metaverse applications. Below we briefly discuss various such challenges. 1) Privacy Issues: Ensuring the privacy of the end users will be a major challenge in AI-XR metaverse applications. As these applications are designed to monitor and collect users' data at an unprecedented fine-grained level, in a bid to create a replica of the digital world [2], there is a greater danger and risk of privacy breaches [8]. For example, to create an immersive virtual scene in the metaverse, data from different sensors will be collected and analyzed using AI models, e.g., facial expressions, brain wave patterns, hand movements, eye movements, biometric, and speech data [8]. This raises obvious concerns regarding the privacy of users and opens a new horizon for digital crimes [2]. Users' sensitive information including daily routine activities, personal logs, and schedules will be stored on a server, which ultimately becomes a critical privacy challenge in a publicly distributed network. Such data include body movements, voice, reflexes, and even more critical data that include subconscious and unconscious responses such as eye movements and physiological signals. Features such as eye tracking are readily accessible using commercially available products such as the HTC Vive Pro Eye and Pico Neo VR headsets even though the XR expert Louis Rosenberg recommends banning such features and data collection in nonhealth-related applications for ethical reasons. In general, the data is collected through on-device sensors at the user site, processed at their devices or nearby local server, and logged as storage in the cloud. Considering the above-mentioned procedure, some malicious attacks can be encountered which are classified as (i) data collection, (ii) data storage, (iii) data usage, and (iv) user profiling. Furthermore, as AR technology depends on the precise localization of users in the physical world, modern smartphones use Lidar sensors and Simultaneous Localization and Mapping (SLAM) algorithms to precisely locate the user in the real world. This opens up the possibility of privacy violations as sensitive information may be exploited for anti-social purposes. Some important concerns related to the privacy of data are described next. a) Data Collection: Generally, the data in AI-XR metaverse applications is collected through users' input either by voice command or textual input, and multi-modal sensors including camera, microphone, textual commands, gesture sensors, and wearable devices. These devices are expected to frequently collect personal information such as daily routine activities, voice and biometric data, and personal preferences including shopping items, TV shows, and preferred food choices. In addition, the private data will be used for the creation of avatars for a digital representation of a real human in the metaverse, which also raises privacy issues. For example, the built-in location sensor in the Oculus headset can be used for tracking users' presence in the real environment with a precise accuracy [8]. b) Data Storage: The ultimate aim of developing personalized AI-XR metaverse applications is to aid human beings and ease their daily routine activities. These applications will contain multiple sensing devices which generate a substantial amount of data. Whereas these devices will be resource constrained with small storage units, which leads to the tradeoff between data generation and storage at the edge level. To address these shortcomings, these devices upload their data along with the corresponding logs to online local or global data centers. Though, the data storage tradeoff is resolved by connecting with the servers. However, it also raises privacy concerns regarding the access permissions and data protection of the consumers. Also, if the communication channel is hacked by an attacker then the data can be manipulated to get the intended outcomes. The literature suggests that adversaries can extract information regarding the actual data even if the communication is encrypted and can track the location of the users by realizing different attacks such as advanced inference attacks [24] and differential attacks [25]. c) Data Usage and Consent: Continuous data collection through multi-modal sensors including cameras, microphones, and other sensors will be closely involved in the daily routine activities of metaverse users. These sensors collect continuous data regardless of privacy awareness, which is logged over the local or global server(s). Consequently, this procedure raises legal questions regarding the users' consent and the kind of data that is collected and shared. Moreover, metaverse service providers can also utilize this data to optimize their inference models and to make them robust and more personalized. However, it also raises privacy concerns such as the collection, disclosure, and sharing of the data without the explicit or implicit consent of users. d) User Profiling: Similar to the current social media (in which users are considered as the product), everything will be a product alike in the metaverse. Metaverse will act as a meta-platform for different entities (such as users, developers, content creators, businesses, etc.), and this raises questions about data collection and its utilization for user profiling [5]. Also, the provisioning of the metaverse services requires that the users should be uniquely identifiable in the metaverse. For this purpose, VR headsets/glasses or any other such device can likely be used for illegally tracking users in real life [26]. Moreover, such devices can be attacked by malicious actors and can be exploited to track users' real locations for possible digital and real-world crimes. Guzman et al. [3] presented a data-centric perspective to avoid unprecedented privacy challenges related to data collection and its usage in MR. 2) Lack of Trustworthiness: According to the definition of Trust, expressed by Lee and See [27], in the perspective of automated systems, "Trust is an attitude that an agent will help achieve an individual's goals in a situation characterized by uncertainty and vulnerability." Metaverse is a data-driven technology in which service providers will use different automated tools to assist human beings as a recommendation engine in various domains, e.g., shopping recommendations, movie recommendations, and even recommendations regarding their health. The efficacy of such recommendation systems is highly dependent on the collection of personalized data for intelligent decision-making, e.g., an AI model is trained using the collected fine-grained health data to suggest more accurate recommendations regarding the well-being of users. Despite the huge potential of such features of AI-XR metaverse applications, it also raises many questions about which parameters have influenced the underlying AI model in producing a particular decision and the trustworthiness of such predictions and decisions. Next, we discuss such challenges across three dimensions (i) Truthful AI, (ii) Transparency, and (iii) Explainable and Interpretable AI. a) Need for Truthful AI: The evolution in technology also brings threats among them. Over the past few years, AIbased personal assistants including Siri, Alexa, Astro, and the like, have managed to gain wide social acceptance and these devices are being used by consumers daily for automating household routine tasks. Currently, misinformation, falsehood, or malfunction in AI-based text or speech analyses and command execution are not considered a matter of concern. However, it is expected that AI-enabled intelligent systems with linguistic capabilities will be a major feature of AI-XR metaverse applications. In this regard, it will be quite challenging to enforce the criterion of truthfulness in AI-based systems to ensure the safe selection of statements and behavior according to the social norms of users while interacting with human society. b) Transparency Issues: In general, practices, developing AI-based systems is about training sophisticated algorithms on large-scale data to learn an efficient and generalizable model that can be deployed in the real-world environment. However, it is well-known fact that the performance of these models is directly proportional to the transparency of training data, i.e., AI models will perform as well as their training data. Numerous factors such as input data riddled with poorly cleansed, or selection of inherently biased data, underfitting, and overfitting influence the performance of these models and can result in fairness and accountability issues (discussed later in this section). Unlike typical application development, there are no quality assurance tools available to spot bugs and evaluate the bias factor in the training data. For instance, if it was known at which stage, the model is going to infer at a perfect scale, then there would not be a need to perform training on such large-scale data. This process is all about the hit and trial procedure, which is a quite challenging task to identify the right approximations with better data, hyperparameters, and configuration settings. c) Explainable and Interpretable AI: The rapid adoption of based applications in human society has also grown the complexity of the systems, which ultimately requires system understandability to make them legitimate and trustworthy. In a critical human-facing technology such as the AI-XR metaverse, interpretable and explainable AI models are required to answer questions about accountability and transparency of their decisions and outcomes. For example, how the employed model reached the decision, and which factors influenced the models to make that decision [28]. Such questions are particularly important for human-centric applications where the potential impact of AI will be limited if it is not able to provide accurate and transparent AI predictions. Therefore, the key objective of these models is to develop a relationship of trust between human users and AI. However, one of the main challenges in developing explainable methods is the trade-off between achieving the modesty of an algorithm and ensuring the discretion of sensitive user data. In addition, it is also a challenging task to identify the right information while generating a simple yet useful explanation for users. It is worth noting that the terms interpretability and explainability are closely related and are often used interchangeably in machine learning literature, however, these terms are different in practice. Interpretability of the models is defined as the extent to which its outcomes are predictable, i.e., for a given change in the input or model parameter(s), the interpretability enables us to predict the respective change in its output. On the counter side, explainability deals with the explanation of internal processes of the ML/DL models in a humanunderstandable way. 3) ML Security Issues: Despite the state-of-the-art performance of modern AI techniques including DL-based systems, it has been shown that these models are highly vulnerable to carefully crafted adversarial perturbations known as adversarial ML attacks [29]. The threat of these attacks has been already demonstrated for many critical applications like healthcare, autonomous vehicles, etc. On a similar note, AI-XR metaverse applications are essentially critical as they involve humans, and ensuring their safety from any harm is profoundly important. On the counter side, the existence of these challenges raises many concerns about the safety, security, and robustness of AI-based metaverse applications thus hindering their practical deployment. As it is equally important that any AI-based should be equally trusted by all stakeholders involve service providers, developers, and end users. These challenges are detailed later (Section IV). 4) Lack of Fairness and Accountability: Modern AI methods like advanced DL models lack fairness and accountability in their decisions [30]. On the other hand, such questions are particularly important for critical applications like AI-XR, in which the model's decisions can have life-threatening consequences for the end users. Moreover, AI models are developed using training data, which will be mainly collected from human users in AI-XR metaverse applications for providing immersive experiences. Humans possess certain biases that will be readily reflected in the data they generate, and when this data is used for training AI models, the data bias will be directly translated into the developed AI-based system. As a result, the model will be biased towards certain samples that contain certain features (bias), and its decision will not be fair. On a similar note, the critical nature of AI-XR metaverse applications demands accountable decisions. Consequently, data bias if remained unaddressed can ultimately lead to unintended consequences [31]. 5) Identity Theft and Authentication Attacks: Users'/avatars' identities in the metaverse can be stolen or impersonated illegally leading to authentication and access control issues in the interconnected virtual worlds. Identity theft in the metaverse will be more dangerous than traditional attacks. The identity of a user once stolen will reveal everything about that person's digital assets, avatars, and social relationships. The attackers can exploit different vulnerable VR gadgets and other service authentication loopholes to realize identity theft attacks and can steal the victim's secret keys of digital assets and bank details. It has been reported that about 17 users in the OpenSea NFT marketplace were hacked through a phishing attack and flaws in the smart contract that resulted in a loss of $1.7 million. 1 Metaverse will leverage different biometrics and passworddriven technologies to authenticate users and their avatars in the virtual worlds. The attacker can evade such authentication systems to impersonate real users' identities to get control of the whole virtual world. Evading AI-based biometric systems has become easier with the advancements in adversarial ML research. Therefore, AI-empowered speech and face recognition-based biometric systems can be easily attacked to realize impersonation attacks. Once the attacker has the access to the metaverse it can exploit the data generated by the victim's devices to deceive him, committing a crime in the virtual space. On other hand, the exposure of biological data when used for authentication purposes can also lead to severe consequences [32]. Moreover, the authentication of social friends of a user using their avatars is much more challenging in the metaverse as compared to real-world identity authentication. In this regard, facial data, voice, and videos can be used to develop an AI-based avatar authentication system, however, the unsolved inherent issues of AI can still hinder its practicality. 6) The Bias Problem: Bias refers to a model making certain unethical assumptions about the data. Human bias along with its many aspects has been studied by researchers in many disciplines including law, psychology, and so forth. In [33], bias is defined as 'the prejudice or inclination of a decision made by an AI system which is in a way considered to be unfair for or against one person or group'. Bias in recommendation systems, advertising algorithms, facial recognition systems, and risk assessment tools has been widely studied in recent years. In metaverse applications, data will be collected from a heterogeneous group of people and sources having their own characteristics, stereotypes, and behaviors, which introduces different biases in the collected data. In [34]- [36], the authors discuss different kinds of bias based on the sources and the types of bias. On the base of sources, these biases have been divided into further categories: biases caused by data, biases caused by algorithms, and biases caused by user interaction. B. XR-related Challenges in Metaverse AI-XR metaverse applications are essentially human-centric and ensuring the security, privacy, security, and robustness of such applications is of utmost importance. It has been envisioned that an entirely new form of digital media will emerge from the use of VR and AR in the modern metaverse (TV, print media, and the web). In recent years, there has been immense discussion regarding the concerns about surveillance capitalism, which is happening on the Internet in different applications. Many large tech organizations providing Internet services like Facebook, Google, Microsoft, and Amazon collect large of amount data related to the surveillance of their users, which is then used to satisfy the needs of advertisers [37]. The pioneers of VR and AR such as Jaron Lanier [38]and Louis Rosenberg 2 have predicted that the concerns about surveillance are expected to rise in Metaverse. For example, it has been shown how reconfiguring AR in Pokémon Go (an AR mobile game) drew unexpected audiences to museums and public spaces like trains to fill in the space thus creating a form of virtual trespassing. It has been reported that people were putting their lives in danger to pursue virtual characters. This highlights that safety concerns may arise when such immersive technologies are engineered for gaming and experiences. Below we discuss the key challenges that are hindering metaverse applications in general and we will later discuss the specific challenges that arise with the use of different AI techniques in metaverse applications (Section IV). 1) Safety Issues: There are various concerns regarding the mental and physical safety of metaverse users. There are several reported incidents of digital harassment, theft, and bullying in XR applications [39]. The report on "Immersive and Addictive Technologies" highlights rampant incidents of sexual harassment, cyberbullying, and grooming online [40]. Ensuring the safety of users is a major challenge for AI-XR metaverse applications because of the fact that such incidents have real damage and harm to users despite being experienced in the virtual world. The avatars generated using recent advancements in AI techniques, in particular, generative models can appear more realistic in AI-XR metaverse applications and can engage users in promotional conversation thus providing a false sense of a real human behind the avatar. The avatars in such a promotional are fueled with more personalized data (such as your vitals, emotions, expressions, etc.) to look more realistic. Also, these sales avatars can pitch products to you more persuasively than any real salesman or even a recommendation system due to their access to rich cyber-physical data about you. The research in deep fake technology and photorealistic avatars is already on the stage where computer-generated content is indistinguishable from the real. Such advancements can be leveraged to realize an adversarial attack on AI-XR metaverse applications to get the intended behavior and outcomes. 2) Potential Antisocial Aspects: There are various opinions regarding the antisocial aspects of AI-XR metaverse applications, many people think that introducing AI-XR metaverse may detract the users (humans) from their purposes and may have a somatic effect. In the literature, it has been shown that extended times online can result in users demonstrating post-VR sadness and detachment from reality. For instance, Aldous Huxley in 1932 wrote in his social science dystopian fiction novel that using technology can lead to self-inflicted harm that can lead people to be diverted from their higher priorities and become more prone to being influenced by other interests. As a result of such a quest for technological utopia, the human psyche and society as a whole are greatly afflicted. Social critics have long argued that various digital media, such as television, the Internet, and the Web, make people docile and less connected to the real world. For example, Jerry Mander in 1978 wrote in his book, "Four Arguments for the Elimination of Television" that TV removes the sense of reality from people, promotes capitalism, TV can be used as a scapegoat, and all these three factors work together negatively. The modern technological disruptions including the web and social networking services have created a filter bubble detaching people from the real world and the truth. Due to these reasons, the current era is also referred to as a post-truth era [41]. We may reasonably expect that alienation from the real world will exacerbate with the increasing adoption of VR, AR, and AI-XR metaverse applications, which aim at changing the human perspective of the world. This argument can be supported by the fact that in 2018, the World Health Organization formally included "gaming disorder" in its International Classification of Diseases following research that shows that technology can promote addictive behavior in people. Moreover, the literature focused on analyzing the social implications of metaverse argues to understand and identify potential psychological problems that can arise in metaverse [42]. 3) Ethical Aspects: Any technological intervention involving humans suffers from some serious ethical issues, especially the one that contains intelligence. The Institute of Electrical and Electronics Engineers (IEEE) has recently published a report on Ethically Aligned Design that mainly focuses on the Ethics of Autonomous and Intelligent Systems [43]. This report emphasizes the need of developing ethical autonomous and intelligent systems (A/IS) that promotes human wellbeing and protects human rights through transparent and accountable A/IS and the prevention of the misuse of AI. This report is the collective effort of hundreds of researchers having diverse backgrounds and expertise in important areas like governance, technology, civil society, and policy-making. This report has a dedicated section on XR and interestingly, IEEE also has a Global Initiative on Ethics of XR. 3 In this report, various ethical issues related to XR have been highlighted including users' preference for virtual life over the real world and complete disengagement with society. In addition, the reports conclude with the following remark regarding XR: "The nature of XR environments fosters unique legal and ethical challenges that can directly affect users' privacy, identity, and rights. Society will need to rethink notions of privacy, accessibility, and governance across public and private spaces. New laws or regulations regarding data ownership, free use, universal access, and adaptive accessibility within XR environments may need to be developed." 4) Regulatory Challenges: The big tech companies have resisted regulation decrying the fact that regulation will slow down innovation. However, there are various ethics researchers and social scientists who are arguing for much greater regulation to ensure that consumer rights are protected. In this regard, the EU General Data Protection Regulation (GDPR) has paved the way for many countries and regions attempting to develop similar regulatory laws to protect Internet users. These regulations mainly emphasize the importance of the non-profiling of users (i.e., limiting the storage of tracking data) and for better transparency (i.e., online services and applications should specify why and what information is being stored). On a similar note, AI-XR metaverse applications are subject to the requirement of being transparent in terms of data collection and utilization and should also be subject to the informed consent of users. Also, the development of such applications requires thoughtful deliberation from a regulatory perspective. For instance. is worth considering banning nonmedical applications to collect vital biomedical statistics due to the high risks of being exploited maliciously. To mitigate the risk of users being manipulated deceptively, metaverse operators may be bounded to transparently declare the staging of virtual products and experiences in the metaverse. Rosenberg, one of the pioneers of VR and AR, has already started to argue about the need for regulation for metaverse applications [44]. For instance, he suggested leveraging the arguments regarding the regulation of social media for developing a legal and philosophical basis for metaverse regulation. As the metaverse can be deemed as an evolutionary expansion of similar services. Rosenberg argued that the only solution to eliminate ethical and privacy-related concerns associated with metaverse is to shift from an advertising-based to a subscription-based business model in which users pay a subscription fee for accessing the metaverse platform. This eliminates the service providers' need to monitor their user base to a greater extent, however, this is not a feasible solution as it is difficult to say whether or not users will pay for a safer metaverse. IV. ANALYZING SECURITY AND TRUSTWORTHINESS ASPECTS OF AI-XR In this section, we will discuss the challenges associated with the use of different AI techniques (in particular, ML/DLbased models) that hinder the safe, secure, and trustworthy deployment of these methods in metaverse applications. We start by first providing a broad overview of AI security in the metaverse. A. Security of ML in Metaverse The impact of metaverse applications will be social and economic and these applications will be more susceptible to undesirable adversarial action(s). AI will be the fundamental driving force behind the success of the metaverse, there are numerous applications of AI in different layers of the metaverse. On the other hand, the use of AI algorithms in AI-XR metaverse applications also opens them up to different adversarial attacks. In Figure 8, we highlight the threat of different security and privacy attacks that can be realized in different applications in almost every layer of the metaverse. The figure also highlights that there are various common ML security issues and attack surfaces that get shared across the architectural landscape of the metaverse across different AI applications at each level. In this section, we discuss different AI-associated security and privacy attacks on AI-XR metaverse applications. B. Potential Attacks on AI-based Metaverse Applications The threat of adversarial ML attacks has already been shown to be successful in compromising the integrity of AI techniques in many critical tasks, e.g., connected and autonomous vehicles [45], computer vision [46], and healthcare [30], just to name a few. Furthermore, AI-XR algorithms could be biased either due to data imbalance or adversarial subversion. Many of the ethical dilemmas and social harms such as distraction, narcissism, disinformation, outrage, and polarization stem from the economic model of surveillance capitalism in which service providers give the customers everything and anything that makes the company money. In this way, these companies pander to the base animal desires of people and exploit their cognitive biases effectively downgrading humans and manipulating them for ulterior selfish purposes. In the adversarial ML literature, an adversarial example is defined as the input to the deployed AI model crafted by an adversary by introducing imperceptible noise into the legitimate sample to get the intended outcomes. In general, there are two types of adversarial ML attacks: (1) poisoning attacks that aim at altering the training process of the AI model; and (2) evasion attacks that are focused on evading the deployed AI model by making inferences (they are also known as inference time attacks). In poisoning attacks, the adversary mainly modifies the training data to tamper with the learning of the AI model [47]. In contrast, test data is manipulated in evasion attacks to get the desired predictions from the model [48]. Recent works have shown that AI models are vulnerable to attacks at both training and inference stages [49]. Training stage attacks typically corrupt a small subset (typically ∼1%) of the training data samples to achieve malicious goals during AI model training [50], [51]. On the other hand, the inference stage attacks cause a trained model to misbehave on adversarially crafted test inputs [52], [53]. Attacks on AI models are generally carried out by first defining a threat model. A threat model is a set of assumptions regarding the attackers' abilities to access and affect a typical AI model training pipeline. Broadly, there are two main threat models-the poisoning threat model (i.e., realizing poisoning attacks), and the adversarial threat model (realizing evasion attacks). A poisoning threat model assumes an attacker who can control a small set of the training dataset to adversely affect the training of the model. An adversarial threat model assumes an attacker who can access and, to a certain extent, perturb the inputs to an already trained AI model. In the following, we highlight major security threats associated with the use of AI techniques in metaverse applications that include computer vision, natural language processing (MLP), network communication, authentication, and recognition systems. 1) ML Associated Security Issues in Computer Vision: Computer vision is one of the central building blocks in the foundation of the metaverse. In recent years, DL algorithms have enabled major advances in computer vision ranging from image classification [54], [55] to scene understanding [56], [57] and generating realistic images [58]. However, the discovery of the adversarial vulnerabilities of DL-based image processing models by Szegedy et al. [29] sparked a growing concern regarding the reliability and security of these deep models [59]- [62]. Numerous works have analyzed these adversarial vulnerabilities in greater depth under different threat models [49]. In general, adversarial attacks work by optimizing the perturbation, ∆x, to an input image, x, such that the output of the model, F, is significantly changed, maximize ||F(x) − F(x + ∆x)||. ∆x is typically optimized based on the gradients which are either computed directly (white-box scenarios) or estimated by introducing random noise (black-box scenarios). Summary of various adversarial ML attacks on different computer vision applications can be seen in Table II. 2) ML Associated Security Issues in NLP: Similar to the vulnerabilities of ML models for vision applications, the literature demonstrates that the ML methods for modeling NLP tasks are also vulnerable to malicious attacks, at both the training and the inference stages of a typical ML pipeline [51], [52]. Below we discuss such attacks. Poisoning and Trojaning Attacks: Poisoning attacks and trojaning (also known as backdoor) attacks are the most widely known training stage attacks in NLP. Poisoning attacks aim to tamper with the training of the model so that it is unable to perform satisfactorily on the test inputs [47], [87], [88]. Trojaning attacks aim to insert a trojan-typically characterized by a specific pattern of words known only to the attacker in the input sequence-into a model such that the model behaves normally on natural test inputs, but malfunctions as desired by the attacker (through the use of of the trojan pattern of words [50], [89]). Similar to the case with the computer vision applications, the trojaning attacks, being more difficult to be detected as compared to the poisoning attacks, pose a greater threat to NLP applications in the metaverse. Adversarial Attacks. Although adversarial examples have been extensively studied in computer vision, they have received significantly limited attention in NLP tasks mainly due to the discrete input search space-minimal adversarial perturbations in the input are no longer feasible in NLP [28], [52]. Recently, however, there have been numerous works highlighting the adversarial vulnerabilities of the NLP-based ML models. Notable adversarial attacks include Text-bugger, Text-fooler, PWWS, Ali et al. [53] propose an adaptive adversarial attack to generate perturbations against statistical defenses. Kaggle Fake news 95% → 0% Jin et al. [93] identify key contributing words and replace them with synonyms while retaining the coherence. and BERT Adversarial Example (BAE). Adversarial attacks against NLP models generally follow three major steps-evaluation, perturbation, and selectionto achieve some adversarial goal-for example, targeted or untargeted misclassification-under a predefined threat model. Consider, for example, an input sequence X = {x 1 , x 2 , ..., x i , ..., x n } correctly classified by an NLP model, F, in class, F(X) = y ∈ R M . At the evaluation stage, the attacker uses some impact scoring function, to compute a set, I x , representing the impact of each word over the output. At the perturbation stage, the attacker repeatedly perturbs the most impactful words in I x using some pre-defined perturbation mechanism such that the semantic and contextual value of X remains preserved. At the selection stage, the most optimal perturbation is selected. Table III provides a summary of various adversarial ML attacks on different NLP applications. 3) ML Associated Security Issues in Networking: AI-XR metaverse applications will provide ubiquitous connectivity to a massive number of users over wireless networks. Over the last few years, many AI-based algorithms have been developed to improve the performance of wireless communication and networking systems that will be used in different layers of network architecture [13]. The use of AI in wireless communication empowers wireless devices to perform many important intelligent functions such as network composition, analyzing traffic patterns, managing content requests, analyzing wireless channel dynamics, etc. Moreover, AI-based algorithms have been used for optimizing different network constraints like high throughput and low latency for different multimedia applications. A prominent use case is to leverage intelligent proactive load management in 5G and 6G communication networks and predictive data analytics to improve network operations. Despite the significant potential of using various AI algorithms for different optimizing applications in wireless networks, recent studies have highlighted that AI application is highly susceptible to adversarial ML attacks. For instance, Usama et al. [101] used a generative adversarial network (GAN) for realizing adversarial attacks on network intrusion detection. The threat of adversarial ML attacks on network traffic classification is demonstrated in [102] and for cognitive self-driving networks is presented in [61], [62]. Similarly, the threat of adversarial ML for 5G networks is analyzed in [103]. Summary of various adversarial ML attacks on network applications is presented in Table IV. We refer interested readers to a detailed survey highlighting the threat of adversarial ML in network security [104]. In addition to the above motioned adversarial vulnerabilities associated with the use of AI techniques in many network applications, some other critical network-related issues can hinder the smooth operation of metaverse at a global level. For instance, centralized network architecture provides flexibility in terms of cost saving, simplicity, and ease in performing different operations. On the other hand, such architectures are more prone to a single point of failure (SPoF) and distributed denial of service (DDoS) attacks [108]. For example, if a powerful attacker gets control of the network, it may lead to severe challenges like SPoF and DDoS. To address such issues, the literature suggests leveraging decentralized network architecture [109]. In addition, decentralization will potentially amplify the transparency and trust of users in exchanging their virtual belongings (like digital assets and virtual currencies) among each other and across different virtual worlds in the metaverse. However, many issues arise with the use of decentralized approaches, e.g., reaching a consensus on an ambiguous operation among the huge number of entities in a dynamic metaverse. Distributed Denial of Service (DDoS): Metaverse will include a massive number of IoT devices, which can be compromised by an attacker to form a botnet to realize DDoS attacks [110]. Sybil Attacks: In a Sybil attack, the adversary pretends to have fake (or manipulated) identities of legitimate users or devices. Using such stolen identities he can take over the network. 4) Security Issues in Cloud-hosted ML Models: Outsourcing the training of ML/DL models to third-party services that offer powerful computational resources on the cloud is prevalent nowadays. These services allow ML developers to upload their data and models for training over their cloud platforms. It is expected that such services will be featured in AI-XR metaverse applications, as they provide the flexibility of developing AI models using sufficiently large training datasets while reducing the cost and time. However, the literature demonstrates that such services are vulnerable to variety of attacks such as backdoor attacks [111], exploration attacks [112], model inversion [113] and model extraction attacks [114], etc. More details about various attacks and defenses for cloud-hosted ML models can be found in [115]. Visual illustration of adversarial ML attacks on different potential applications in the AI-XR metaverse is presented in Figure 9. C. Attacks on VR The literature highlights that VR systems are vulnerable to adversarial attacks. For instance, Casey et al. [120] demonstrated that humans in VR systems can be controlled like joysticks-thus providing the adversary the ability to control the movement of VR user without his consent or getting into his knowledge. Moreover, the literature suggests that both security and privacy attacks can be realized on VR/XR systems [121]. Therefore, developing secure and robust AI-XR metaverse systems is crucial to the widespread adoption of metaverse applications that are not vulnerable to adversarial attacks or are capable to withstand such attacks and mitigating their impact. D. Analyzing Implications of ML Security, Privacy, and Trust Issues: An AI-XR Case Study In this section, we present an ML/DL-based pipeline for a potential AI-XR metaverse application use case. We then analyzed various challenges and threats that can arise at each development stage. The pipeline is developed while considering a general metaverse application-a virtual conference, in which the participants are remotely connected from different places (the pipeline is presented in Figure 10). A unique avatar is representing each participant while each avatar is expected to reflect real-time voice, facial expressions, and gestures. The voice of each participant is translated into the native language of all the participants along with generating the transcription. The pipeline depicts different ML/DL-empowered tasks: (1) 3D/4D Visual Reconstruction-responsible for generating photo-realistic avatars; (2) 3D Visual Mapping-to reflect realtime multi-modal expressions (i.e., audio, facial, and gestures, etc.); (3) Speech Recognition and Synthesis-to interpret and translate the voice of recipient into other languages; and (4) Speech-to-Text Synthesis-to generate the transcription of audio conversations of all members. As depicted in Figure 10, data acquisition is performed by collecting raw audio input through a microphone for NLP, whereas, depth cameras and laser scanners are used for 3D/4D visual reconstruction and mapping. In the next step, acquired data is pre-processed through several techniques including data denoising, deblurring, silence removal, etc. The processed data is then labeled for the training of ML/DL models in a supervised/semi-supervised learning fashion. After successful data preparation, 3D visual mesh construction and segmentation models are trained to perform 3D avatar reconstruction. On the other hand, acoustic and language models along with neural vocoders are trained to perform the multilingual translation and transcription tasks. Although in the literature, these pipelines have demonstrated significant performance in 3D reconstruction and NLP tasks, however, this pipeline is highly exposed to the various privacy and security attacks at each stage of the development pipeline (as shown in Figure 10). V. TOWARDS DEVELOPING SECURE AND TRUSTWORTHY AI-XR The development of secure, safe, and trustworthy AI-XR metaverse applications is fundamentally very important, in this section we will discuss different potential solutions that can be leveraged to address challenges associated with the use of AI in particular and for the overall system in general. An abstraction of different techniques that can be leveraged to Fig. 9. Illustration of adversarial ML attacks on different potential applications in AI-XR metaverse. Individual figure references: Image Recognition [29]; Speech Recognition [116]; Malware Detection [117]; Semantic Segmentation [118]; Speaker Identification [119]; Fake News Detection [52]; Object Detection [76]; and 3D Object Modeling [78]. address the ML-associated issues is shown in Figure 11 and these methods are described next. A. Solutions for Privacy Protection in AI-XR In the literature, privacy-preserving techniques are broadly categorized into three classes: (i) cryptographic techniques, (ii) differential privacy, and (iii) federated and distributed ML. These techniques are briefly discussed below. 1) Cryptographic Techniques: Cryptography refers to a practice of methodologies, aiming to construct and analyze communication protocols to ensure secure communication while achieving data integrity, authentication, non-repudiation, and data confidentiality. Generally, there are two common types of encryption methods: (i) symmetric encryption, and (ii) asymmetric encryption method. The symmetric encryption method is a secret-key algorithm, in which the sender and receiver must share the same key to perform encryption and decryption of the data. Whereas, asymmetric encryption method (also known as public-key cryptography) uses two keys, i.e., public and private key, associated with an entity which require to authenticate its identity electronically or encrypt data. The public key of each entity is published whereas, the corresponding private key is always kept secret to perform encryption or decryption of data. In literature, Ron Rivest, Adi Shamir and Leonard Adleman (RSA) [122], Data Encryption Standard (DES) [123], Advanced Encryption Standard (AES) [124], and Secure Hash Algorithm (SHA) [125] are a few commonly used algorithms used for data encryption. Different cryptographic techniques can be employed to convert readable information to an encrypted state, which can be later used at the receiver end after performing decryption. Below we discuss some of the most commonly used encryption methods that can be used for the development of privacy-aware AI models. a) Homomorphic Encryption: Homomorphic encryption (HE) is a computational approach that performs encryption while allowing computational tasks to be executed over encrypted data at the same time to ensure the privacy of the data. HE is defined as a public key cryptographic technique in which a pair of public and private keys is created to perform encryption and decryption operations on the data. The public key is used to encrypt the data, before sharing it with the third party for further computational tasks including training, and/or inference. Due to the homomorphic characteristics of this approach, the results can be decoded using the private key to visualize the results without showing them to thirdparty servers or unauthenticated users. In the ML literature, HE has been used for protecting the privacy of the users' data for different applications such as genome imputation [126], misinformation detection in text messages [127], etc. Specifically, the AI models are trained and inferred using encrypted training and testing data thus preserving the privacy of the sensitive data. b) Secure Multi-party Computation: Secure multi-party computation (also known as secure computation) is a type of cryptographic technique that is focused on the development of collaborative methods to perform joint computation and calculate a function over joint inputs while possessing those inputs in an isolated fashion. Contrary to the traditional cryptographic methods, where cryptography ensures confidentiality and integrity of communication or storage, while the adversary is outside the system of users, this approach protects the users' privacy from each other while performing ML-based tasks including training and inference activities. c) Garbled Circuits: The idea of garbled circuits was first proposed by Yao in 1986 to perform two-party computation [128]. Garbled circuits can be used in a scenario where multiple parties are interested in performing some computation without sharing their data. Let's assume two parties (e.g., Alice and Bob for the sake of simplicity) want to perform some computation using garbled circuits. Alice will send his input and function in the form of a garbled circuit and Bob will utilize his garbled input with the garbled circuit to get the result of the required function, once he obtains it from Alice in an oblivious fashion. In [129], garbled circuits along with HE have been used to develop privacy-aware ML models, where the authors trained three classification models namely decision tree, Naïve Bayes, and hyperplane decision classifier using the encrypted data. d) Secret Sharing: In secret sharing, multiple parties collaborate in the computation by sharing their secrets among them while holding a "share" of the individual secrets. The secret can only be reconstructed by combing all the individual shares kept by participating parties, otherwise, it will be useless. In the literature, the secret sharing technique has been successfully used for training AI models in a privacypreserving way. For instance, Bonawitz et al. [130] used the secret sharing technique to train an ML model by aggregat-ing model updates from multiple parties in a privacy-aware way. In a similar study [131], authors used this technique for the development of a privacy-aware ML-based emotion recognition system leveraging client-server architecture. In their proposed framework, the secret sharing technique was used for the communication of audio-visual data from the client side to the server, where an ensemble model based on a sparse autoencoder and a CNN model was used for the feature extraction from the collected data. The SVM classifier was then trained using the extracted features for the emotion recognition task. A secret sharing-based parallelized variant of principal component analysis (PCA) for preserving data privacy is presented in [132]. e) Secure Processors: Secure processors were pioneered by rogue software to protect sensitive code from being accessed by malicious actors at higher privilege levels. Secure processors are being used in different processors now to perform privacy-preserving operations, e.g., the Intel SGX processor. In [133], SGX processors were used to developing a data oblivious system for different ML techniques that include SVM, decision tree, matrix factorization, and k-mean clustering. The primary goal was to facilitate collaboration between multiple data proprietors performing the ML task on an SGX-empowered data center. 2) Differential Privacy: The idea of differential privacy is based on introducing noise in the data to protect sensitive information while ensuring the usefulness of the data after noise addition [134]. Differential privacy is defined in terms of the task-specific concept of neighbor datasets and it provides strong guarantees in ensuring the privacy of the data during algorithmic analysis [135]. Numerous differential privacy-based methods have been presented in the literature, such as differentially-private stochastic gradient descent (DP-SGD) [136], private-aggregation of teacher ensembles (PATE) [137], exponential noise based differential privacy-preserving methods to ensure privacy on large-scale data. These methods demonstrated better applicability in ML-based applications in various domains including intelligent transportation services, smart/virtual personal assistants, and smart healthcare services. 3) Federated Learning: Federated learning (FL) refers to a distributed-ML paradigm that is capable of learning global ML models without directly accessing and/or exchanging data from edge devices. Intuitively, basic FL-based methods consist of a collaborative learning framework where each participant such as an edge device, network node, and local server can independently train a model using its local data. These edge devices then share their model parameters with a server, which then performs aggregation of the parameters after receiving parameter updates from each edge device. Finally, the server updates the parameters of the global model and shares the updated parameters with all participants. The iterative process of FL is continuous until the desired criteria as been fulfilled, e.g., validation accuracy/loss or the maximum number of communication rounds. In this way, a global model is trained without requiring the actual data from the FL participants. Subsequently, this sharing mechanism allows ML-based systems to learn from large-scale diverse data and develop a global model. Such methods can demonstrate better applicability in terms of dealing with sensitive data in various human-centered applications such as AI-XR metaverse applications. Despite the success of FL in training an ML model with reliable performance while maintaining the privacy of the actual data, different attacks can be realized on the model being trained using the FL paradigm, e.g., backdoor attacks [138], label flipping attacks [139], free-riding attacks [140], and poisoning attacks [141], etc. Also, it has been demonstrated that sensitive information can be extracted from the shared parameters in FL settings [142]. B. Solutions to Combat Adversarial ML Attacks in AI-XR In the literature, adversarially robust ML models have been mainly categorized into three categories [30]: (1) Data Modification; (2) Model Modification; and (3) Using Auxiliary Model. Moreover, a few methods leverage a hybrid approach in which multiple defensive techniques are used to develop adversarially robust ML models. Below we discuss the most prominent methods in each category and we refer the interested readers for more details about these methods to recent and comprehensive surveys that are specifically focused on adversarial ML [8], [45], [46], [143]. 1) Data modification: Data modification methods work by modifying the input data during the training or inference phase to mitigate the effects of adversarial perturbation. A few famous data modification methods are briefly described below. • Adversarial Re-training: This method was proposed by Goodfellow et al. [144] and it is considered to be a basic method for mitigating the effect of adversarial perturbation in the trained model. In this method, adversarial examples are augmented in the training data, which is then used to (re)-train the model. This method has been extensively used in the literature, however, a few research studies demonstrated that the models trained using this method are not robust against multiple attacks [145]. • Feature Squeezing: Xu et al. [146] presented a feature squeezing-based approach that aims to squeeze feature space of input that may be exploited in response to an adversary. In this regard, the heterogeneous feature vectors have been collectively joined into a single space to reduce available feature space. Although, the proposed defense method achieved significant performance against small perturbations. However, it was found less effective against iterative adversarial attacks [147]. • Input Reconstruction: Input reconstruction-based defense methods have been proposed to mitigate the effect of adversarial attacks. These methods transform adversarial examples into legitimate samples by cleaning adversarial noise using an appropriate technique, e.g., using an autoencoder to clean adversarial perturbations [148]. 2) Model modification: Model modification methods aim at modifying the parameters of trained ML models to defuse the effect of adversarial attacks. The most commonly used model modification methods are described below. • Gradient Regularization: This method allows complex neural networks to bring a partial surge in training computational complexity to improve the performance of the network regardless of any prior knowledge about adversarial attacks. This idea was coined by Ross et al. [149] to improve the performance of CNN models on classification tasks. Though the proposed method achieved significant improvement in CNNs' robustness, it also increases the computational cost of models which prejudices the performance in real-world ML-based applications. • Defensive Distillation: Distillation in a neural network was initially conceptualized by Hinton et al. [150] to establish knowledge sharing from a larger network to a smaller one. Later, Papernot et al. [151] extended this notion by developing a distillation-based defense mechanism against adversarial attacks, which is known as defensive distillation. In this method, the larger model is trained over hard labels to maximize accuracy while predicting the output probabilities of the baseline smaller model. This method is successful in mitigating the effect of small adversarial perturbation and it fails in the presence of strong adversarial perturbations, e.g., adversarial examples generated using C&W attack [152]. • Network Verification: In this method, certain properties of the ML/DL model are verified, e.g., validating the output of models, produced in response to the corresponding input samples. Katz et al. [153] presented ReLU and satisfiability modulo theory (SMT) based network verification method to make complex neural networks robust against adversarial examples. In a similar study, authors have proposed a scalable quantitative verification framework for DNNs to prove formal probabilistic property against adversarial attacks [154]. 3) Using Auxiliary Model: Methods aiming to robustify ML models in this category use an additional model either for detection of adversarial examples or for clean adversarial perturbations. A few methods are described below. • Adversarial Detection: In such methods, a detector model is used to differentiate between normal and adversarial inputs, e.g., a binary classifier [155]. • Ensembling Defenses: In this defense strategy, an ensemble of different defensive techniques is created to withstand different adversarial attacks. PixelDefend is the most famous ensemble defense method that consists of two defense approaches, i.e., input reconstruction and adversarial detection [156]. • Using Generative Modeling: These types of methods leverage different ML/DL-based generative models for cleaning adversarial noise in adversarial examples to project them back to the same data manifold. C. Solutions for AI-XR Transparency and Trust Challenges The true potential of AI-based applications in AI-XR metaverse applications can only be realized when they are developed using fine-grained personal data for making personalized recommendations and predictions, which is only possible when users fully trust the underlying system. Therefore, addressing the challenges related to the trustworthiness aspects of AI-XR metaverse applications is very important. From an AI perspective trustworthiness itself requires predictability, interpretability, explainability, safety, and robustness. Below we discuss different methods that can be used to accomplish trustworthiness in AI applications. 1) Explainable and Interpretable AI: An AI model is referred to as explainable if it can explain the ability of parameters to justify the results. Explainability makes the AI models transparent which ultimately helps in evaluating and understanding the results provided by the models. In recent years, substantial research efforts have been conducted to enhance explainability, trustworthiness, and interpretability in AI models. Fairness, Accountability, and Transparency in Machine Learning (FAT-ML) [157] and Defense Advanced Research Projects Agency (DARPA), explainable AI program [158] are the two famous research groups working in this context. The literature argues that explainable models can be the first step toward converting black-box AI models into white-box models [159]. Interpretable models refer to the models that explain themselves. In simple words, an AI model is said to be interpretable, if its decision against some input is logically understandable such as which factors influenced the AI model to reach that decision. In the literature, various methods have been presented to leverage interpretability in ML models. These methods ensure that the predictions of interpretable models are unbiased, which ultimately makes it easier to trust these systems in human society. It is worth noting that the terms interpretable and explainable are interchangeably used in the literature, however, they are different in terms of domainspecific definitions, moreover, there is no exact definition of these terms [160]. A detailed taxonomy of different explainable and interpretable AI methods can be found in [159], [160]. 2) Trustworthy AI: The relevant literature emphasizes two famous sets of principles that can be used to attain trustworthy AI. One of them is developed by European Commission's AI High-Level Expert Group (HLEG) [161] and the other one is defined by Organisation for Economic Co-operation and Development (OECD) [162]. The following are the seven essential principles outlined in OECD: (1) Human agency and oversight; (2) Technical robustness and safety; (3) Privacy and data governance; (4) Transparency; (5) Diversity, nondiscrimination, and fairness; (6) Environmental and societal well-being; and (7) Accountability. Similarly, the following principles are outlined in HLEG: (1) Inclusive growth, sustainable development, and well-being; (2) Human-centred values and fairness; (3) Transparency and explainability; (4) Robustness, security, and safety; and (5) Accountability. One of the key noticeable insights from the above two principles set is that they mainly emphasized explainability, security, fairness, safety, and robustness aspects of AI. Therefore, these are the essential requirements that need to be fulfilled to develop trustworthy AI-based applications. In addition, we can see that these principles are essentially human-centric that respect ethical norms. As potential AI-XR metaverse applications will be more human-focused, therefore, the above-mentioned principles can be leveraged to develop trustworthy AI-based applications for the metaverse. D. Solutions for Ethical Challenges in AI-XR 1) Human in the Loop: The metaverse's inherent complexity raises different security issues. For instance, it can be envisioned that the metaverse administrators will have to push automation, that is, to handle more tasks with algorithms, rather than with human operators, due to the requirement of managing a large number of users, applications, and services. The generated data will be much larger than those managed by the current Web platforms. Delegating tasks to algorithms, especially those implemented with state-of-the-art AI approaches is necessary to meet high-level efficiency and scalability. However, in the current version of social media and the Internet, we have even started to realize the implications of using algorithms for managing societally relevant tasks. Despite the significant performance, these algorithms suffer from various issues. Some authors writing on the governance of metaverse have proposed the use of a modular approach for the development of metaverse applications, as it allows adapting regulations to specific scenarios and then controlling the system accordingly [163]. 2) Ethical and Responsible AI: To ensure socially desirable AI decisions, novel ways are required to be figured out to simultaneously minimize potential harms associated with the use of AI and its potential benefits. In this regard, the importance of taking an ethics-first approach towards the development of AI-based technologies becomes more plausible [164]. However, there are many challenges associated with the development of ethical AI pipelines due to distinct social norms and demographics of the human population, i.e., one ethical solution may be beneficial for a group of people but it is highly possible that it will not be suitable for another group on the same time. Therefore, customized solutions are required to address such issues that can consider the social norms of target users while making AI-based decisions. In this regard, different ethical guidelines can be leveraged that can be potentially used for the development of pro-social AI solutions. The literature shows a groundswell of interest in ensuring ethical and responsible AI [165]. E. Situational Awareness Situational awareness can be defined as the capacity to understand information perceived from the surrounding environment. The literature argues that situational awareness is a crucial and effective tool for monitoring the security of complex systems like metaverse [166]. Situational awareness can be used at the local and global levels for threat monitoring in a single metaverse or across multiple metaverses, respectively. The feasibility and potential of this tool have been extensively studied in the literature focused on XR and VR technology. For instance, Woodward et al. [166] performed a literature review that focused on the design of information presentation in AR headsets to enhance users' situational awareness. Authors in [167], performed immersive and realistic simulations to evaluate the effectiveness of audio-visual warning systems in increasing users' situational awareness in accident situations using VR. They demonstrated that VR can assist drivers to remain alert in emergency situations. F. Human Centric Approach for AI-XR Development Metaverse is essentially a human-centric application [168]. To realize the real social impact of different AI-XR metaverse applications, they should be analyzed and developed using human-centric design thinking. Metaverse service providers and developers must pay attention to key stakeholders (i.e., humans) by prioritizing and considering their social norms, i.e., dignity, justice, and rights, and supporting goals including creativity, self-efficacy, social connections, and responsibility. The aforementioned characteristics can be inherited in AI-XR metaverse applications by following three key concepts proposed in [169]. The first one is Human-centric frameworkthat guides the developers and researchers to ensure humancentric thinking about high-level two-dimensional control. Secondly, Design metaphors-which points out how two key goals of AI and social norms are both valuable. However, the stakeholders such as developers, researchers, policymakers, and business leaders must combine them both in developing metaverse applications to provide ultimate benefits to the users. Thirdly, Governance Structures-which ensures the bridge between the above-mentioned ethical principles and the practical measures needed to achieve the desired goals including reliable metaverse application development while ensuring cultural safety to increase privacy and trustworthiness of the users. VI. OPEN RESEARCH ISSUES In this section, we highlight various open research issues that are particularly associated with the use of ML/DL models in different AI-XR metaverse applications. A. Developing Generalizable Adversarial Defense Methods Over the past few years, substantial research attention has been devoted to adversarial ML. However, the literature highlights that the attention devoted to developing adversarially robust ML/DL models is significantly less as compared to developing novel attack methodologies [115]. In the literature, different defensive techniques have been proposed to withstand adversarial ML attacks (as discussed above), however, each method only works in a specific setting and fails to withstand unseen and powerful attacks (consequently, fails to generalize across a wider class of attacks). On the other hand, the literature focused on adversarial ML shows that the diversity and severity of these attacks are increasing with each passing day. Therefore, the development of hybrid and universal defensive techniques is the need of the hour. In addition, it is required that the defense techniques should be developed while considering evolvable and adaptable adversaries (who can adapt their capabilities to break defense strategy). The threat of adversarial ML can be a major hurdle in the development of secure, safe, robust, and trustworthy AI-XR metaverse applications and if it remained unaddressed, can cause unintended severe consequences to users and society. It is highly recommended to consider these aspects while developing ML/DL-empowered human-centric applications like the AI-XR metaverse. Moreover, the worst-case robustness test can be performed from an adversarial lens considering different attack surfaces in individual AI-XR metaverse application architecture. B. Investigating Robustness of Privacy-Preserving Methods As discussed above, AI-XR metaverse applications will collect fine-grained data that may include personal attributes to provide personalized services (empowered by ML/DL models). The models trained with such data can be inferred to reconstruct privacy-related information that can be exploited to get intended outcomes and incentives. Although different privacy-preserving ML techniques have been proposed in the literature that has been shown quite successful in preserving data privacy, however, the literature demonstrates that meaningful information can still be inferred even if the presence of an appropriate privacy-preserving method. For example, it has been demonstrated that homomorphic encryption (one of the widely used encryption techniques) is vulnerable to model extraction attacks [170]. Similarly, Boenisch et al. [142] showed that sensitive information can be reconstructed from the shared parameters in FL. This suggests that the investigation of vulnerabilities and limitations of existing privacypreserving methods can be a good step toward developing robust privacy-preserving methods. Ideally, it is required that the ML/DL models should be developed in such a way that they are by design privacy-aware, i.e., they should not be able to learn any privacy-related features from the data that could be compromised upon model inferences. C. Developing Generalizable Explainable and Intrepratble Techniques Another major limitation of DL models hindering their trustworthy applications in critical applications like the AI-XR metaverse is the lack of explainability and interpretability. This can also be exploited by adversarial agents to craft adversarial perturbations to realize attacks on different AI-XR metaverse applications. Although significant research interest has been devoted to the development of novel techniques to explain and interpret DL models, the literature shows that their application is limited to a certain data type or application [160]. While the AI-XR metaverse applications will have a complex architecture that will simultaneously use multi-modal data for making different intelligent decisions, existing explainability and interpretability techniques cannot be directly used for explaining and interpreting ML/DL-driven decisions. More work is required to create methods that can be generalized across different data types, models, and applications. D. Developing Ethical Data Analysis Pipelines Current ML/DL models are not capable of considering different ethical norms that are necessary for human-centric applications like AI-XR metaverse applications. On contrary, these considerations are yet very important to maximize potential benefits and minimize associated harms to ensure safe, robust, and fair data analysis. In AI-XR metaverse applications, different ML/DL models will be trained using massively large data collected by humans and their interactions with the real and virtual universe. While there is no guarantee that the AI decision will be ethically-committed because the data used for model training might contain data bias that will eventually result in biased decisions. Moreover, the outcomes of ML/DL models will be just a reflection of human behavior (including moral failures even if they are not intentionally committed). Therefore, to increase the trust of different stakeholders involved in AI-XR metaverse applications (particularly, end users) and to provide them with a sense of safety, fairness, and accountability, it is highly desirable to develop novel techniques to ensure fair and ethical data analysis empowered by various ML/DL techniques. E. Pushing AI on Edge: Embedded ML One of the feasible approaches to preserve the privacy of AI-XR metaverse users will be to deploy ML/DL models on their smart devices, e.g., smartphones, AR/VR gadgets, tablets, etc. By doing so models can be developed and inferred on their devices without requiring to transmit data to a central cloud. We envision that various AI-XR metaverse applications will potentially adopt embedded ML or edgeenabled ML due to the proliferation of different enabling gadgets and smart devices. However, numerous challenges related to underlying hardware computing capabilities will arise when sufficiently large ML/DL models will be deployed on resource-constrained devices. Also, the literature argues that the research on enabling edge AI is at its early stages of development [171]. Therefore, it is worth investigating the feasibility and potential of deploying M/DL models on embedded devices to ultimately develop secure, private, and robust systems to provide personalized services in AI-XR metaverse applications. We refer interested readers to a recent survey on analyzing the notion of edge-enabled metaverse applications for a more comprehensive discussion on the topic and various challenges [172]. F. AI-XR Metaverse Specific Security Solutions The future AI-XR metaverse will have a complex structure and will be a combination of various enabling (complex) technologies (that possess their associated challenges related to privacy and security, e.g., adversarial ML). Moreover, the massive connectivity of numerous entities (users, services providers, organizations, etc.) along with the decentralization will even worsen the enormity of security and privacy in AI-XR metaverse applications. Individual vulnerabilities associated with each technology can be exploited to realize a more powerful attack to halt or get control of some services or the entire metaverse. If such vulnerabilities are left unaddressed, they will eventually lead to novel challenges thus making it challenging to ensure the secure, safe, and robust operation of metaverse services. Therefore, it is very crucial to understand such challenges and develop customized defense solutions to protect AI-XR metaverse applications and services in general. VII. CONCLUSIONS In this paper, we have analyzed various security, privacy, and trustworthiness challenges associated with the use of different machine learning (ML) and deep learning (DL) techniques in artificial intelligence and extended reality (AI-XR) metaverse applications. Specifically, considering the layered architecture of the metaverse, we developed a pipeline and highlighted different potential ML/DL use cases along with identifying various vulnerabilities associated with their application. Furthermore, we provide a comprehensive overview of these challenges and discuss potential solutions that could be used to overcome such issues. To accentuate the implications of adversarial threats, we designed a customized case study (considering a prospective AI-XR metaverse application) and analyzed its security and privacy aspects. Finally, we discussed various open research issues that require further investigation. We envision that our work on this crucial topic will provide a one-stop solution to interested researchers who aim to develop secure, robust, and trustworthy AI-XR applications. Fig. 1 . 1Organization of the paper. Fig. 2 . 2An overview of different concepts related to metaverse that include VR, AR, MR, XR, virtual metaverse, and augmented metaverse. Fig. 3 . 3Historical evolution of technologies needed for metaverse Fig. 4. Applications of XR in metaverse. Fig. 5 . 5Illustration of different layers in metaverse. Fig. 6 . 6Various potential AI-XR metaverse applications. Fig. 7 . 7Applications of AI in metaverse. Fig. 8 . 8Overview of ML security in Metaverse. Fig. 10 . 10A prospective pipeline for developing AI-XR metaverse applications for multi-lingual communications, involving various security challenges at each stage. Fig. 11 . 11An abstraction of different ML-associated challenges along with a taxonomy of various solutions that can be used to address those challenges. TABLE I COMPARISON IOF OUR PAPER WITH EXISTING SURVEYS AND REVIEW PAPERS THAT ARE FOCUSED ON ANALYZING PRIVACY AND SECURITY OF AI-XR METAVERSE APPLICATIONS. (LEGEND: S → SECURITY; P → PRIVACY; R → REGULATORY; T → TRUSTWORTHY; √ → COVERED; × → NOT COVERED; ≈ → PARTIALLY COVERED)Year Authors Focused Area General Issues ML Related Issues & Solutions Background & Applications Open Issues S P R S P T XAI Ethical 2018 TABLE II SUMMARY IIOF DIFFERENT ADVERSARIAL ATTACKS ON VARIOUS COMPUTER VISION APPLICATIONS (THAT ARE EXPECTED TO BE POTENTIAL METAVERSE APPLICATIONS).Application Authors Methodology Datasets Before → After Face Authentication Goswami et al. [63] Studied how different architectures affect adversarial vulnerabilities. MEDS, PaSC 89.3% → 41.6% Sharif et al. [64] Developed adversarial glasses to fool face recognition systems. Celebrity Face 98.95% → 0% Shen et al. [65] Developed black-box attack for face recognition sys- tems using visible light. CusFace, LFW 100% → 7.9% Chatzikyriakidis et al. [66] Perturbed facial images to fool automatic face recogni- tion to secure a person's identity. CelebA 97.8% → 4% Dabouei et al. [67] Studied the vulnerability of face recognition systems against geometrically perturbed faces. VGGFace2 100% → 0.14% Zhong et al. [68] Used dropout and feature-level attacks to improve the transferability of adversarial inputs. VGGFace2 100% → 3.24% Dong et al. [69] Used evolutionary algorithm to find adversarial inputs against the models' decisions. LFW 100% → 0% Wenger et al. [70] Proposed improved physically-realizable attack against face recognition. VGGFace 100% → 10% Ali et al. [51] Proposed multi-trigger backdoor attack against back- door defenses. Celebrity Face 88% → 8% Xue et al. [71] Exploit hidden facial features as triggers of the back- door attack. VGGFace 100% → 0.02% Object Detection Zhang et al. [72] proposed generalizable contextual adversarial perturba- tions against object detectors. PascalVOC, COCO 78.8% → 1.6% Lee et al. [73] Showed that non-overlapping physical patches can fool object detectors. COCO 55.4% → 0.05% Xie et al. [74] Proposed multi-targeted adversarial attacks to fool ob- ject detectors. PascalVOC 72.07% → 3.36% Xie et al. [74] Showed that multi-targeted adversarial attacks against object detectors are transferable. PascalVOC 54.87% → 37.9% Wei et al. [75] Utilized generative methods to efficiently obtain trans- ferable adversarial inputs. PascalVOC 43% → 3% Wang et al. [76] Utilized position and label information to attack black- box object detectors. PascalVOC 100% → 16% Wu et al. [77] Leveraged natural rotations to insert a backdoor into the object detectors. PascalVOC 89.5% → 4.45% 3D-Object Modelling Wang et al. [78] Optimally generates adversarial perturbations against 3D-Object detectors. KITTI 84% → 0% Xiang et al. [79] Generated 3D adversarial point clouds against Point- Net model. ModelNet40 93% → 0% Hamdi et al. [80] Exploited an auto-encoder to generate transferable 3D adversarial perturbations to point cloud. ModelNet40 93% → 5% Meloni et al. [81] Used off-the-shelf 3D surrogates to transfer attack on 3D object models. N/A 100% → 0% Li et al. [82] Proposed a novel formulation to develop backdoor triggers against 3D point cloud models. ShapeNetPart 98.4% → 0.5% Semantic Segmentation Arnab et al. [83] Performed an in-depth study of adversarial vulnerabil- ities of semantic segmentation models. Cityscapes 77.1% → 19.3% Xie et al. [74] Proposed multi-targeted adversarial attacks to fool se- mantic segmentation models. PascalVOC 72.07% → 3.36% Hendrik et al. [84] Analyzed universal adversarial perturbation to fool a segmentation model for any input. Citscapes 64.8% → 12.9% Li et al. [85] Poisoned the segmentation models using object-level target class and semantic triggers. ADE20K 37.7% → 25.2% Feng et al. [86] Proposed frequency-injection backdoor attack against medical image segmentation tasks. KiTS-19 54.5% → 21.1% TABLE III SUMMARY IIIOF DIFFERENT ADVERSARIAL ML ATTACKS ON VARIOUS NLP APPLICATIONS.Application Authors Methodology Datasets Before → After Language to Language Modelling Zhand et al. [90] propose word saliency speedup local search method to attack translation machines. NIST (MT) 92.48% Degradation Boucher et al. [91] Uses invisible characters, homoglyphs and deletion control characters to fool the model WMT14 37% → 1% Fake-news Detection Li et al. [92] exploit BERT-MLM to fool a fine-tuned BERT model by generating coherent perturbations. AG news 94.2% → 10.6% TABLE IV SUMMARY IVOF DIFFERENT ADVERSARIAL ML ATTACKS ON DIFFERENT NETWORKING APPLICATIONS.Application Authors Methodology Datasets Before → After Intrusion Detection Usama et al. [101] Exploited GAN to craft adversarial examples to evade intrusion detection model. KDD99 89.12% → 56.55% Aiken et al. [105] Perturbed a few features to evade four ML classifiers trained for detecting DDoS attacks. KDD99 100% → 0% Network Traffic Classification Usama et al. [102] Crafted adversarial examples using mutual information in black-box settings. UNB-CIC Tor Data 96% → 77% Modulation Classification Usama et al. [103] Used C&W attack to evade traffic modulation classifier. RML2016.10a 85% → 15% Sadeghi et al. [106] Realized white-box and black-box attacks on VT-CNN model using a PCA-based perturbations. GNU Radio 75% → 38% Network Modulation Usama et al. [103] Realized black-box attack on channel autoencoder on unsupervised and DRL models. RML2016.10a 95% → 80% Malware Classification Usama et al. [61] Realized three SOTA adversarial ML attacks, i.e., FGSM, BIM, and JSMA. Malware Image Data [62] 98.39% → 1.87% Abnormal KPI Detection Usama et al. [62] Leveraged two SOTA attacks to evade ML-based ab- normal KPI detection classifiers. LTE network data 98.8% → 13.7% Channel State Estimation Sagduyu et al. [107] Realized three attacks: spectrum poisoning, jamming, and priority violation. 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[ "NEAR 3:2 AND 2:1 MEAN MOTION RESONANCES FORMATION IN THE SYSTEMS OBSERVED BY KEPLER", "NEAR 3:2 AND 2:1 MEAN MOTION RESONANCES FORMATION IN THE SYSTEMS OBSERVED BY KEPLER" ]
[ "Su Wang ", "Jianghui Ji " ]
[]
[]
The Kepler mission has released ∼ 4229 transiting planet candidates. There are approximately 222 candidate systems with three planets. Among them, the period ratios of planet pairs near 1.5 and 2.0 reveal that two peaks exist for which the proportions of the candidate systems are ∼ 7.0% and 18.0%, respectively. In this work, we study the formation of mean motion resonance (MMR) systems, particularly for the planetary configurations near 3:2 and 2:1 MMRs, and we concentrate on the interplay between the resonant configuration and the combination of stellar accretion rate, stellar magnetic field, speed of migration and additional planets. We perform more than 1000 runs by assuming a system with a solar-like star and three surrounding planets. From the statistical results, we find that under the formation scenario, the proportions near 1.5 and 2.0 can reach 14.5% and 26.0%, respectively. In addition,Ṁ = 0.1 × 10 −8 M ⊙ yr −1 is propitious toward the formation of 3:2 resonance, whereasṀ = 2 × 10 −8 M ⊙ yr −1 contributes to the formation of 2:1 resonance. The speed-reduction factor of type I migration f 1 ≥ 0.3 facilitates 3:2 MMRs, whereas f 1 ≥ 0.1 facilitates 2:1 MMRs. If additional planets are present in orbits within the innermost or beyond the outermost planet in a three-planet system, 3:2:1 MMRs can be formed, but the original systems trapped in 4:2:1 MMRs are not affected by the supposed planets. In summary, we conclude that this formation scenario will provide a likely explanation for Kepler candidates involved in 2:1 and 3:2 MMRs.
10.1088/0004-637x/795/1/85
[ "https://arxiv.org/pdf/1409.2946v1.pdf" ]
119,222,473
1409.2946
e3b7aec8f52c451aa6124bceeb33a6ccffb66b41
NEAR 3:2 AND 2:1 MEAN MOTION RESONANCES FORMATION IN THE SYSTEMS OBSERVED BY KEPLER 10 Sep 2014 Draft version September 11, 2014 September 11, 2014 Su Wang Jianghui Ji NEAR 3:2 AND 2:1 MEAN MOTION RESONANCES FORMATION IN THE SYSTEMS OBSERVED BY KEPLER 10 Sep 2014 Draft version September 11, 2014 September 11, 2014Preprint typeset using L A T E X style emulateapj v. 08/13/06 Draft versionSubject headings: (stars:) planetary systems-planets and satellites: formation-methods: numerical The Kepler mission has released ∼ 4229 transiting planet candidates. There are approximately 222 candidate systems with three planets. Among them, the period ratios of planet pairs near 1.5 and 2.0 reveal that two peaks exist for which the proportions of the candidate systems are ∼ 7.0% and 18.0%, respectively. In this work, we study the formation of mean motion resonance (MMR) systems, particularly for the planetary configurations near 3:2 and 2:1 MMRs, and we concentrate on the interplay between the resonant configuration and the combination of stellar accretion rate, stellar magnetic field, speed of migration and additional planets. We perform more than 1000 runs by assuming a system with a solar-like star and three surrounding planets. From the statistical results, we find that under the formation scenario, the proportions near 1.5 and 2.0 can reach 14.5% and 26.0%, respectively. In addition,Ṁ = 0.1 × 10 −8 M ⊙ yr −1 is propitious toward the formation of 3:2 resonance, whereasṀ = 2 × 10 −8 M ⊙ yr −1 contributes to the formation of 2:1 resonance. The speed-reduction factor of type I migration f 1 ≥ 0.3 facilitates 3:2 MMRs, whereas f 1 ≥ 0.1 facilitates 2:1 MMRs. If additional planets are present in orbits within the innermost or beyond the outermost planet in a three-planet system, 3:2:1 MMRs can be formed, but the original systems trapped in 4:2:1 MMRs are not affected by the supposed planets. In summary, we conclude that this formation scenario will provide a likely explanation for Kepler candidates involved in 2:1 and 3:2 MMRs. INTRODUCTION The Kepler mission has released data spanning over 16 months (Fabrycky et al. 2012;Batalha et al. 2013;Mazeh et al. 2013). The statistical results, obtained from 361 multiple-planet systems, show that there are two peaks for the period ratios of two planets in a system: approximately 1.5 and 2.0. This finding may provide evidence that there are a plenty of planet pairs near 3:2 mean motion resonances (MMRs) (with the period ratio of two planets in the range of [1.45, 1.54]) and 2:1 MMRs (with the period ratio of two planets in the range of [1.83, 2.18]) (Lissauer et al. 2011). As of July 2014, approximately 4229 planet candidates have been reported in 2804 planetary systems, most of which will be subsequently confirmed as authentic planets through follow-up observations and by double checking the data. In this population, there are 974 multiple planetary systems, including 652 two-planet systems, 222 three-planet systems and 75 four-planet systems. As previously mentioned, most of the planetary candidates in the multiple planetary systems are believed to be real planets Ciardi et al. 2013;Quillen et al. 2013). Based on current data, in Figure 1, we show the distribution of the period ratios of the planet pairs for all of the planetary candidates (the upper panel ) and for all of the three-planet systems (the lower panel ). The upper panel shows that the proportion of the period ratios for two planets near 1.5 (∼ 10.5%) and 2.0 (∼ 20.5%) appear to be much larger than the proportion of other period ratios. In contrast, the lower panel shows that two peaks are also observed for the case of three-planet systems (see Fig.1) in which the proportions of systems near 1.5 and 2.0 are approximately 7.0% and 18.0%, respectively. Furthermore, we note that two peaks occur at period ratios greater than the exact values of 1.5 and 2.0. Such a distribution may be related to planets near 4:2:1 MMRs (so-called Laplacian resonances) or 3:2:1 MMRs. These observations indicate that a great many systems are approximately involved in MMR (Figure 1). Therefore, investigation into the formation scenarios of such resonant configurations should inform planetary formation theory in several respects. Currently, several formation scenarios are proposed to explain the formation of systems near MMRs. First, it is difficult for planet candidates observed by Kepler to form in situ based on the Minimum Mass Solar Nebula (MMSN) model (Hayashi 1981). For all planetary candidates, the planetary radii are in the range of [0.24, 154] R ⊕ with an average value of 2.09 R ⊕ , whereas the radii of the planets in three-planet systems range from 0.31 to 26 R ⊕ with an average value of 1.92 R ⊕ . In addition, for three-planet systems, ∼ 75% of the planetary radii are in the range of [1, 3] R ⊕ with semi-major axes that are shorter than 1.14 AU. A recent investigation of the internal structure of terrestrial and Neptunelike planets indicates that a planet's mass radius follows the relationship R ∝ M 0.226−0.262 (Valencia et al. 2006(Valencia et al. , 2007Marcus et al. 2010;Howard 2013;Lee et al. 2013). Thus, the planets have average masses in the range of [12,18] M ⊕ . Considering an isolation mass m iso in a core-accretion model (Ida & Lin 2004), to yield a planet with the average mass and radius of a three-planet system, the enhancement factor of the MMSN should be greater than 16. Hence, it is difficult to explain the final configuration of multiple-planet systems based on an in situ-formation scenario. The most classical theory for producing such a configuration is the convergent migration scenario that occurs in the gaseous disk (Goldreich & Tremaine 1980;Lin et al. 1996). Given an appropriate orbital migration speed, two planets can easily be captured in MMR (Bryden et al. 2000;Masset & Snellgrove 2001). Systems such as GJ 876, with two planets of a four-planet system captured in a 2:1 MMR (Lee & Peale 2002;Ji et al. 2002Ji et al. , 2003Zhou et al. 2005;Zhang et al. 2010), and KOI-152, which consists of three companions in a near-Laplacian configuration (Wang et al. 2012), are well explained via this migration scenario, and the KBOs which are captured with Neptune in 3:2 or 2:1 MMRs are demonstrated by the migration of Neptune (Malhotra 1995). An alternative scenario is proposed (Petrovich et al. 2013) to elucidate the formation of near-MMR for systems with growing-mass planets. In this case, the authors show that the 3:2 resonance is the strongest firstorder resonance for a planet with a mass of 20-100 M ⊕ . However, the effects of dissipation orbital migration are not fully taken into account in the authors' work, which may play a significant role in the capture of resonant planets, especially for a 2:1 MMR. Recently, Ogihara & Kobayashi (2013), using numerical simulations, investigated the formation scenario that triggered two planets to capture a first-order MMR. In their work, the authors also considered the combination of orbital migration and gas damping. For two wellseparated planets, the planets may undergo convergent migration and may ultimately become trapped in MMR during their evolution, whereas for closely spaced planet pairs, the planets may be formed in situ. In their simulations, the authors explored simple systems with two equal-mass planets. Nevertheless, the formation scenario in a three-planet system would be quite different because of the gravitational perturbation caused by the third planet. Moreover, the planetary mass ratio acts as an alternative factor in shaping the final configuration of the system. In-depth investigation is required to understand planetary formation in packed systems. Based on our former study (Wang et al. 2012) and observational data, our main aim in this work is to determine the formation scenario that may lead to the various types of MMR that match statistical outcomes by taking three-planet systems as an example. In our simulations, we consider three planets whose masses are lower than 30 M ⊕ . These low-mass planets are believed to form at a distant region rather than at their present locations; thus, they will undergo type I migration until they reach the inner region of the gaseous disk and stop migrating. Subsequently, tidal interactions between the planets and the central star will circularize their resultant orbits. Such a formation process is mainly affected by three important factors: the speed of the type I migration, the stellar magnetic field, and the stellar accretion rate, which may shape the final configuration of the investigated systems. Moreover, the stellar magnetic field and stellar accretion rate affect the density profile of the gaseous disk (Kretke & Lin 2007;Kretke et al. 2009;Wang et al. 2012), which also plays an important role in determining each planet's final location. In this work, we focus on the configuration formation of near-MMR, particularly 3:2 and 2:1 MMRs, in threeplanet systems as a function of (1) the speed of the type I migration based on the value obtained by linear analysis, (2) the stellar accretion rate from early to later stages, (3) the stellar magnetic field, and (4) the potential sur-vival of additional planets in the systems that may break up the MMRs. In Section 2, we summarize our models, including the disk model and the migration scenario that planets may experience. Section 3 presents the numerical results obtained for two different cases of planet configuration. Section 4 presents our discussion and conclusions. MODELS Disk Model The surface density of a gas disk at a stellar distance a is described as (Pringle 1981) Σ g =Ṁ 3πα(a)c s h exp −t τ dep η,(1) where t is time, the timescale τ dep is estimated to be approximately several million years (Haisch et al 2001) and c s and h are the speeds of sound at the midplane and isothermal density scale height, respectively. Herein, M is the stellar accretion rate, which can be evaluated as (Natta et al. 2006;Vorobyov & Basu 2009) M ≃ 2.5 × 10 −8 M * M ⊙ 1.3±0.3 M ⊙ yr −1 .(2) For a star with a solar mass, the stellar accretion rate is ∼ 2.5 × 10 −8 M ⊙ yr −1 . The average value of this rate decreases as the star and the disk evolve. Thus, herein, we suppose that the stellar accretion rate varies from 0.1 × 10 −8 to 2.5 × 10 −8 M ⊙ yr −1 representing different stage of the star evolution for a solar-like system. In addition, the efficiency factors of angular momentum transport α and η are defined as α eff (a) = α dead − α mri 2 erf a − a crit 0.1a crit + 1 + α mri ,(3)η = 0.5 erf a − a mstr 0.1a mstr + 1 ,(4) where a crit and a mstr are the locations of the boundary of magneto-rotational instability (MRI) and the truncation of the magnetic field, respectively. Specifically, the two parameters are modeled as (Kretke et al. 2009;Koenigl 1991) a crit = 0.16 AU Ṁ 10 −8 M ⊙ yr −1 4/9 M * M ⊙ 1/3 × α mri 0.02 −1/5 κ D 1cm 2 g −1 ,(5) and a mstr = (1.06 × 10 −2 AU)β ′ R * R ⊙ 12/7 B * 1000G 4/7 × M * M ⊙ −1/7 Ṁ 10 −7 M ⊙ yr −1 −2/7 ,(6) where κ D and B * are the grain opacity and stellar magnetic field, respectively. With respect to spherical accretion, β ′ = 1 represents a typical Alfvén radius. The variables α dead = 0.001 and α mri = 0.01 denote the value of α in the dead zone and in the active zone at the midplane of the disk, respectively. Based on the above-described equations, we can calculate the gas density Σ g ∝ r −1 . The stellar accretion rate and stellar magnetic field are believed to be two important factors that affect the profile of the gas density (Wang et al. 2012). Therefore, in our model, we consider these parameters in the investigation of planet formation and test our simulations by varying them. Eccentricity damping and planetary migration For a planetary embryo embedded in a gas disk, mutual interactions will result in the eccentricity damping of the embryo on a timescale τ damp represented as (Cresswell & Nelson 2006) τ damp = ė e = Q e 0.78 M * m M * a 2 Σ g h r 4 Ω −1 × 1 + 1 4 e r h 3 yr,(7) where Q e = 0.1 is a normalized factor fitted to hydrodynamical simulation results and h, r, Ω, and e are the disk scale height, the distance from the central star, the Kepler angular velocity, and the eccentricity of the embryo, respectively. Additionally, the angular momentum exchange between embedded planets and the gaseous disk will trigger orbital migration of the planets. According to the coreaccretion scenario (Ida & Lin 2004), type I and type II migration are proposed to explain the formation of closein super Earths or hot Jupiters. If a planet embryo occupies a low mass (≤ 30 M ⊕ ), the angular momentum exchange between the embryo and gaseous disk can be analyzed using a linear model, and the net loss from the embryo will eventually lead to an inward migration (Goldreich & Tremaine 1979;Ward 1997;Tanaka et al. 2002). The timescale for type I migration can be expressed as τ migI = a |ȧ| = 1 f 1 (2.7 + 1.1β) M * m M * Σ g a 2 × h a 2 1 + ( er 1.3h ) 5 1 − ( er 1.1h ) 4 Ω −1 yr,(8) where e, r, h and Ω bear the same definition as in equation (7). The variable f 1 is the reduction factor. Using the gas-density profile given by equation (1), we have β = −d ln Σ g /d ln a = 1. In this work, we consider the gravitational interaction between each body in the system, type I migration (for planets with masses less than 30 M ⊕ ), and the eccentricity damping of the planets. The total acceleration of the planets with mass m i is expressed as d dt V i = − G(M * + m i ) r i 2 r i r i + N j =i Gm j (r j − r i ) |r j − r i | 3 − r j r 3 j +F damp + F migI ,(9) where M (×10 −8 M ⊙ yr −1 ) 0.1, 0.5, 2 0.1, 0.5, 2 B * (KG) 0.5, 1 0.5, 1 f 1 0.03, 0.1 0.3, 1 F damp = −2 (V i · r i )r i r 2 i τ damp , F migI = − V i 2τ migI ,(10) where r i and V i represent the position and velocity vectors of planet m i , respectively, and all of the vectors are expressed in stellar-centric coordinates. To explore the dynamical evolution of the planets, we integrate equations (9) using the Hermit scheme (Aarseth 2003), which is a time-symmetric integrator. For the numerical setup, we assume that the system contains a solar-mass central star with a surrounding gaseous disk. Initially, all of the planets should be in coplanar and near-circular orbits. The argument of the pericenter, the mean anomaly, and the longitude of ascending node are randomly generated to be between 0 0 to 360 0 . To examine the role of the stellar accretion rate, stellar magnetic field, the speed of the type I migration, and additional planets, we consider two cases with different initial parameters. For Case 1, we mainly investigate the configuration formation for three-planet systems with the various parameters we have chosen. For Case 2, we examine how additional planets affect the evolution of the system. In the following section, we will briefly introduce the main results obtained from the numerical simulations. NUMERICAL SIMULATIONS AND RESULTS Case 1 We mainly consider MMR formation in three-planet systems. In this case, we perform a total of 1020 runs. Furthermore, we choose three planets with variable masses for G1-G3, and we also change the initial locations for the planets for G4. Herein, we summarize each of the initial systems for G1-G4 that are adopted in the numerical simulations. G1: In this group, given the isolation mass, the planet mass m is proportional to a 3/4 . Therefore, the masses of the three planets are chosen to be 5, 10, and 15 M ⊕ (m1 labeled in panel (d) of Figure 2). For the first subgroup, the stellar accretion rate ranges from 0.1 × 10 −8 to 2.5 × 10 −8 M ⊙ yr −1 , and f 1 varies from 0.01 to 1, whereas B * always remains 0.5 KG in this subgroup. For the second subgroup, we fix the stellar accretion rate at 0.1 × 10 −8 M ⊙ yr −1 , whereas the stellar magnetic field ranges from 0.5 to 2.5 KG, and f 1 varies from 0.01 to 1. The third subgroup is quite similar to the second subgroup except that the stellar rate remains 0.5 × 10 −8 M ⊙ yr −1 . In summary, we carry out a total of 90 runs for G1 (including three subgroups). G2: For this group, three subgroups are also considered in the simulations. The initial parameters adopted are similar to the parameters adopted for G1 except for the planetary masses. The masses of the three planets are divided into two cases: (1) the three planets have an equal mass of 5 M ⊕ (m2 labeled in panel (d) of Figure 2) or (2) the inner two planets both bear a mass of 5 M ⊕ but the mass of the outer companion is 10 M ⊕ (m3 labeled in panel (d) of Figure 2). Additionally, the stellar accretion rate in subgroups 1, 2 and 3 is assumed to be 0.1 × 10 −8 , 0.5 × 10 −8 , 1 × 10 −8 , 2 × 10 −8 , and 2.5 × 10 −8 M ⊙ yr −1 , respectively. Herein, we perform 130 runs for G2. G3: The statistical results (see Figure 1) show that most of the Kepler planets are in near-MMRs, particularly 3:2 and 2:1 MMRs. For three-planet systems, we classify the ratios of the planetary masses into four types: 1:1:1.5, 2:1:7.5, 12:1:0.2 and 2:0.2:4. For further investigation, we then choose six subgroups for three planets depending on the variable masses such that the combinations of the planetary masses are as follows: [5, 5, 7.5 [15,15,22.5] M ⊕ (subgroup m7), [6,3,22.5] M ⊕ (subgroup m8) and [2, 0.2, 4] M ⊕ (subgroup m9). Herein, the stellar accretion rates are assumed to be 0.1 × 10 −8 , 0.5 × 10 −8 , 1 × 10 −8 , 2 × 10 −8 , and 2.5 × 10 −8 M ⊙ yr −1 , respectively, whereas other the initial parameters are similar to the parameters assumed for G1, except for the planetary masses. Therefore, we carry out 750 runs for G3. ] M ⊕ (subgroup m4), [2, 1, 7.5] M ⊕ (subgroup m5), [30, 2.5, 0.5] M ⊕ (subgroup m6), G4: To determine the likelihood that there is a high proportion of planet pairs near 3:2 MMR, we reset the initial locations of the three planets, which differ greatly from the locations considered in G1-G3. We also choose five subgroups for the stellar accretion rate (as performed for G1-G3), and the stellar magnetic field is set at 0.5, 1, 1.5, 2, and 2.5 G. The masses of the three planets are exactly equal to the masses in G1. For this group, 50 runs are carried out in the simulations. For each of the above mentioned runs, we performed the simulation over a timescale of 5 Myr. In the following section, we will concisely summarize the most significant results obtained from our numerical simulations. Statistic outcomes for G1-G3 In groups G1-G3, the three planets possess orbital periods of 100, 250, and 600 days for f 1 in the range of [0.01, 0.03], and 140, 500, and 1450 days for f 1 ≥ 0.1 initially. From the statistical results obtained for G1-G3, we observe that the proportion of planet pairs near a 2:1 MMR is approximately 36.4%, with the period ratio ranging from 1.83 to 2.18. However, the proportion of planet pairs near a 3:2 MMR is approximately 3.0%, with the period ratios in the range of [1.45, 1.54]. Figure 2 shows the statistical results obtained for period ratios near 1.5 and 2.0, and panels (a), (b), (c) and (d) present the results obtained by considering variations in the stellar magnetic field, stellar mass accretion rate, the speed of the type I migration, and the planet mass. Based on Fig-ure 2, we may conclude that it is difficult for two planets to be trapped near a 3:2 MMR under such a formation scenario. However, two planets can be easily captured in a 2:1 MMR. Hence, it is inevitable for three planets to be trapped in 4:2:1 MMR through the aforementioned formation scenario based on our model. Panel (a) of Figure 2 displays the statistics for periods near 1.5 and 2.0 with different stellar magnetic field. The red dotted line represents the results obtained near a period ratio of 2.0, whereas the line composed of black squares represents the results obtained near 1.5. Based on the results presented in panel (a), we can conclude that the proportion corresponding to approximately 2.0 is largest for B * = 2.5, whereas for B * = 0.5 and 2.5 the proportion near 1.5 is nearly the same, which is the highest value obtained among the five scenarios. The proportion changes slightly with an increase in the magnetic field, which enables us to conclude that the stellar magnetic field plays a less significant role in the formation of 2:1 or 3:2 MMRs over the evolution of the system. Panel (b) shows that, for the formation of 2:1 MMRs, Figure 2. Therefore, 2:1 MMRs can form easily during the early stages of star formation with a high mass accretion rate, whereas 3:2 MMRs can form easily during the late stages of star formation. In cases in which the stellar accretion rate remains unaltered, we also notice that several planet pairs are trapped near higher-order MMRs of approximately 5:3, 5:2 and 3:1. Furthermore, M = 0.1 × 10 −8 M ⊙ yr −1 is propitious for the formation of planet pairs near 5:2 and 3:1 MMRs. Panel (c) of Figure 2 shows the proportions of planet pairs with different migration speeds. Based on the results presented in Figure 2, we can determine that the proportion of planet pairs near a 2:1 MMR is greater than 35% for f 1 = 0.1, 0.3, and 1, and we also find that the proportion of planet pairs near a 3:2 MMR is greater than 2.5% for f 1 ≥ 0.3, and greater than 10% for f 1 = 1. Therefore, f 1 ≥ 0.1 appears to favor planetary configurations near-2:1 MMRs, which is consistent with the work of Ida & Lin (2008), Wang & Zhou (2011) and Wang et al. (2012). However, f 1 ≥ 0.3 is propitious for the formation of near-3:2 MMRs. We also note that f 1 = 0.03 facilitates the capture of two planets in near-3:1 MMRs. From panel (d) of Figure 2, we observe that the proportion of planet pairs forming near-2:1 MMRs is greater than 30.0% for the five scenarios, where subgroup m3 with the inner two planets being of equal mass bears the largest proportion. For planet pairs forming near-3:2 MMRs, the situation in which there are three equal-mass planets, demonstrates the largest proportion. These results indicate that two planets are easily involved in or are near a first-order MMR if they have equal masses. Statistical outcomes for G4 In G1-G3, the period ratios of the three planets are initially set to be greater than 2.0. From the results given in Section 3.1.1, we note that the fraction of planet pairs with period ratio near 1.5 from the simulations is lower than that from the observation. Therefore, in G4, we set the systems with a compact structure by varying the initial orbits of the three planets with the period ratio, which ranges between 1.5 and 2.0, at 140, 270, and 530 days initially. Thus, we perform 50 runs for G4 with those parameters in Table 1. In these simulations, the planetary masses are adopted to be 5, 10 and 15 M ⊕ , which are identical to those in G1. Upon the completion of the simulations, we collect the data obtained from G4 (50 simulations with the initial parameters given in Table 1) and G1 (50 simulations with identically initial parameters of the stellar magnetic field, the stellar accretion rate, and the reduce factor of type I migration as in G4, but with various initial locations), and the simulation results of the 100 simulations are plotted to compare them with the observations. The statistical results are shown in Figure 3. The black solid line in this figure denotes the distribution profile of the Kepler observations, whereas the grey dot represents the results of our simulations. In Figure 3, we note that there exist two peaks near 1.5 and 2.0 from the numerical simulations that are consistent with the observations. The proportion of planet pairs near 1.5 and 2.0 is approximately 14.5% and 26.0%, respectively. Compared with the results obtained for G1-G3 (see Figure 2), we find that the proportion near 1.5 increases, whereas the proportion near 2.0 decreases. Furthermore, we notice that a lower proportion peak emerges near 2.5 (5:2 MMR), but the planet pair is absent at 3:1 MMR due to the limited number of simulations performed. Moreover, we perform a KS test on the distributions, we find that the p-value p = 0.06 forṀ = 0.1 × 10 −8 M ⊙ yr −1 when the simulations compare with the observation data, implying that they roughly have similar distribution. Consequently, we conclude that the formation scenario proposed in this work may reproduce a subset of the planet pair distributions. 3.1.3. R1: Two pairs of planets both in 2:1 MMR In this run, the initial parameters (see Table 3) are assumed to be as follows: the masses of three planets are identically set to be 5 M ⊕ ; the stellar accretion rate is adopted to be 1×10 −8 M ⊙ yr −1 ,which indicates a middle stage for the star formation; the stellar magnetic field is 0.5 KG; and the reduction factor f 1 is 0.03, implying a slow speed of type I migration. Based on the simulation results, we find that the three planets are trapped in 4:2:1 MMRs in 9.5% of all simulations. Figure 4 shows the typical evolution leading to the formation of a Laplacian configuration. Panels (a) and (b) show the dynamical behavior of the semi-major axes and eccentricities over a timescale of 5 Myr. After the planets are captured in the resonances, the semi-major axes remain nearly unchanged while the eccentricities approach zero; here, the black, red and blue lines represent the innermost, intermediate and outermost planets, respectively. In addition, the trio's final configuration is clearly shown in panel (c), which is indicative of the relationship between the final orbits (denoted by the orbital periods) and the gas density profile. Moreover, panel (d) shows the variations in the period ratio of two planet pairs. Furthermore, in Figure 4, we note that the innermost planet reaches the maximum gas density very quickly at ∼ 0.2 Myr, and the inner two planets are locked in a 2:1 MMR at ∼ 0.6 Myr at the time the second planet arrives, but the outer two planets are in a 2:1 MMR at ∼ 1.8 Myr. Again, based on panel (c), we observe that the inner two planets cease migrating in the regime in which the gas density nearly reaches its maximum, and the outer two companions also stop moving inward, following the behavior of the inner two. Ultimately, the three planets stop migrating at orbital periods of 30.5, 61.3 and 123.6 days, from innermost to outermost. The planets' eccentricities are damped due to the high density of the gas disk. Such a configuration is reminiscent of the KOI-1426 system, which also consists of three planets with orbital periods of 38.9, 74.9 and 150.0 days, from innermost to outermost, indicating that the system may be captured into Laplacian resonance during dynamical evolution. 3.1.4. R2: The inner pair is in a 3:2 MMR, whereas the outer pair is in a 2:1 MMR In R2, the masses of the three planets are 5, 10 and 15 M ⊕ , the stellar accretion rate is 0.1×10 −8 M ⊙ yr −1 , and the stellar magnetic field is 1 KG. The speed of the type I migration is equal to unity, which is consistent with linear theory (Goldreich & Tremaine 1979;Ward 1997;Tanaka et al. 2002). Based on the statistics obtained, we find that approximately 2.2% of the total simulations are associated with the final configurations in which the inner two planets are in a 3:2 MMR, whereas the outer two are trapped in a 2:1 MMR. Figure 5 shows the outcomes for a typical case. Similar to Figure 4, panels (a) and (d) show that the inner two planets are locked in a 2:1 MMR at ∼ 0.2 Myr initially. Subsequently, when the third planet P3 approaches P2, the 2:1 MMR is disrupted, and the inner pair is then trapped in a 3:2 MMR at ∼ 0.3 Myr, while the outer two companions enter a 2:1 MMR. Furthermore, panel (c) shows that the three planets can travel across the region where the density reaches its maximum. In addition, the outermost planet simply halts its migration in the scenario in which the density is maximal. Moreover, the results show that the eccentricities of the inner pair can be excited to 0.01 when they enter a 2:1 MMR, and the outermost planet's eccentricity is slightly increased to 0.001 in the case of the arrival at resonance, which is different from the case of R1. At the end of the simulation, the three planets hold orbital periods of 1.5, 2.2 and 4.5 days. We also observe a comparable analog among the Kepler candidates KOI-584, with orbital periods of 6.5, 9.9 and 21.2 days, from innermost to outermost, by considering the orbital period ratios of the planet pairs. 3.1.5. R3: Two planet pairs both in 3:2 MMR In R3, we assume the parameters are as follows: the masses of the three planets are all equal to 5 M ⊕ , and the stellar accretion rate and stellar magnetic field share identical values with the values considered in R1. However, the speed of the type I migration is 0.3, ten times the speed considered in R1. In this case, in 0.65% of all of the simulations, both of the two-planet pairs are captured in 3:2 MMRs. Figure 6 shows that the inner pair falls into a 3:2 MMR quickly. Subsequently, when P3 arrives, the inner and outer pairs both evolve into 3:2 MMRs. We note that the innermost planet travels across the region of maximum density, whereas the other two planets remain in the region where the gas density remains high. However, we note that the planets' eccentricities are excited not far from zero, deceasing to 0.002-0.008 during their evolution. In summary, we can conclude that rapid type I migration drives the three planets toward 3:2 MMRs, with the orbital periods of the planets being 20.9, 31.5 and 47.5 days. In this case, no comparable planetary analog of the Kepler candidates is found. 3.1.6. R4: The inner pair is in a 2:1 MMR, whereas the outer pair is in a 3:2 MMR In the numerical simulations performed for R4, we consider the masses of the three planets to be 5, 5 and 7.5 M ⊕ , and the stellar accretion rate and reduction factor of the type I migration are equal to the values considered in R2. The stellar magnetic field is 1.5 KG, higher than the value considered in R2, resulting in a larger inner hole in the gas disk. In this group, we notice that 0.32% of all simulations consist of a 2:1 MMR for the inner pair and a 3:2 MMR for the outer pair. Therefore, the three companions are locked in the chain of a 3:2:1 MMR. Figure 7 shows one of the typical simulations. Figure 7 shows that the inner pair first falls into a 2:1 MMR at ∼ 0.4 Myr. When the planets are trapped in a 2:1 MMR, their eccentricities are increased to ∼ 0.1 but then are dissipated by the gaseous disk within 1 Myr. In addition, we find that the occurrence of P3 indeed triggers the capture of the two outer planets in a 3:2 MMR, thereby leading to a 3:2:1 MMR for the three companions. During their evolution, the three planets move across the region of maximum density and reach the inner hole of the system, with final orbital periods of 1.5, 3.0 and 4.7 days. This configuration bears a resemblance to the KOI-1835 system, with three planets exhibiting orbital periods of 2.2, 4.6 and 6.8 days. Case 2 Based on the results obtained for G1-G3, we note that the proportion of planet pairs near 2:1 MMRs is greater than that of planets near 3:2 MMRs when we compare the histogram of the numerical results with the observations. Thus, a question arises: Is there any likelihood that the occurrence of additional planets may contribute to the dynamical evolution of planet pairs in the existing systems over secular timescales, which could explain the difference between the simulations and observations? To address this question, we perform further numerical simulations, initially placing one or more additional planets into the previous systems that could be trapped in 2:1 and 3:2 resonances during their evolution. Based on the results obtained for Case 1, the systems involved in 2:1 and 3:2 MMRs are ultimately chosen for further investigation (see Table 2), for which it is assumed that the companions settle into either the closest or the most distant orbits from the central star. In this case, we perform a total of 24 runs in the simulation, with the systems divided into the two groups: A1 and A2. A1: For this group, we investigate how additional planets may affect the planets captured in or near 2:1 MMRs during their evolution. For the initialization of the numerical simulations, three subgroups are considered: (1) a planet with a mass of 2 or 5 M ⊕ is assumed to be inside the orbit of the innermost planet; (2) a planet with a mass of 5 or 20 M ⊕ is located outside the orbit of the outermost planet; (3) two planets are, respectively, placed inside the orbit of the innermost planet and outside the orbit of the outermost planet in the system (Table 2). A2: For this group, we explore the issue of how additional planets stir up 3:2 MMR systems. For a detailed study, we also classify the simulations into three subgroups, similar to those case of A1. In the following section, we briefly summarize the major results. 3.2.1. The results of A1-A2 Panels (a), (b) and (c) in Figure 8 show the variations in the period ratios for a typical run in A1, where E1 represents the added planet residing inside the orbit of the innermost planet and E2 denotes the planet inserted outside the orbit of the outermost planet. As previously mentioned, the adopted three-planet system can form a configuration approximating a 4:2:1 MMR in Case 1, with a stellar accretion rate of 2×10 −8 M ⊙ yr −1 , a stellar magnetic field of 0.5 KG, and a reduction factor of 0.03. As shown in Figure 8, E1 and E2 do not disrupt the 4:2:1 MMR in the original system, indicating that Laplacian resonance is very robust. Panels (a) and (b) exhibit orbital period ratios of P1/E1=1.5 and E2/P3=1.5, respectively, which suggests that E1 or E2 has a good probability of entering into a 3:2 MMR with the nearby planet (herein P1 or P3, respectively). Panel (c) shows that E1 and P1 are captured in 3:2 resonance within a very short timescale, whereas E2 and P3 are locked into 3:2 resonance over a much longer timescale, approximately ∼ 2 Myr. However, it is noteworthy that the resultant configuration of the 3:2:1 MMR is produced for the three planets E2, P3 and P2. Fortunately, we find that the system KOI-2433, consisting of four low-mass planets, bears resemblance to the configuration shown in panel (a), where their orbital periods are 10.0, 15.2, 27.9 and 56.4 days, from innermost to outermost. Again, this finding further indicates that the formation scenario may be applicable to other Kepler systems that are close to 3:2:1 MMRs. In Figure 8, panels (d), (e) and (f) show the typical period ratio in A2. In this run, a three-planet system with a stellar accretion rate of 2 × 10 −8 M ⊙ yr −1 , a stellar magnetic field of 0.5 KG, and a reduction factor of 0.3 in Case 1 can form a configuration in which both of the planet pairs are in near-3:2 MMRs. Panel (d) shows that the E1 and P1 pair is temporarily trapped in 3:2 resonance; thus, the two planets are ultimately locked into a 2:1 resonance, whereas P2 and P1 subsequently enter into a 3:2 MMR; however, the resonance is disrupted within 1 Myr, and the two outer planets P3 and P2 continue to evolve into a 3:2 resonance. Panel (e) illustrates that P2 and P1 first undergo a 3:2 resonance within 0.1 Myr, and then P3 and P2 are locked into this resonance, which is held for a short period of time. Subsequently, the 3:2 MMRs for the two pairs (P2, P1) and (P3, P2) are both disrupted owing to the existence of the additional planet E2. Finally, P2 and P1 enter into an alternative 2:1 resonance during their dynamical evolution, whereas E2 and P3 are eventually locked into a 3:2 resonance. Panel (f) shows that the three pairs of (P1, E1), (P2, P1) and (P3, P2) are temporarily in 3:2 MMRs over the course of their evolution, but ultimately, only the outer pair of (E2, P3) is captured in 3:2 resonance. Based on the above-described results, we conclude that Laplacian resonance is not easily disrupted by additional planets, and the additional planets will lead to the formation of 3:2 MMRs; however, for planetary systems that are originally in two 3:2 MMRs during their evolution, the 3:2 resonance will disintegrate due to the interplay of planets. CONCLUSIONS AND DISCUSSIONS In this work, we extensively investigated the planetary configuration formation of systems that are involved in first-order resonances (e.g., 2:1 and 3:2 MMRs) via numerical simulations. In total, we performed over 1000 runs, considering systems with various combinations of planetary mass, stellar accretion rate, stellar magnetic field, speed of type I migration and additional planets. Moreover, we also compared our numerical results with the planetary candidates released by the Kepler mission. We summarize the major conclusions of our study as follows. 1. Concerning the statistics of the observed data, the proportion of planet pairs near 2:1 MMRs is ∼ 18.0% for three-planet systems. However, our numerical simulations show that the proportion of planet pairs in 2:1 MMRs remains 36.4% for G1-G3 and 26.0% for G4. Apparently, near 2:1 MMR configuration can be formed easily (Lee & Peale 2002) through our formation scenario and it is able to yield a proportion of planet pairs captured in 2:1 MMRs that is similar to the observed proportions and should thus provide an explanation of the behavior of Kepler candidates involved in 2:1 MMRs. With respect to 3:2 MMRs, the observation results yielded by the Kepler mission show that the proportion of planet pairs in this MMR is ∼ 7.0% for three-planet systems. In contrast, this proportion can reach up to 14.5% in our simulations due to the original packed configuration of the systems. This formation scenario also sheds light on the generation of 3:2 resonant configurations in Kepler systems. From our simulations, we get two peak-trough features at 3:2 and 2:1 MMRs which can be seen from the observation data. In this work, we focus on the planets with mass smaller than 30 M ⊕ . Thus, we believe that the scenario with migrating planets is effective for the formation of near MMRs configuration for low mass planets (Libert & Tsiganis 2011). While for massive planets, with a mass larger than 20 M ⊕ , the two peak-trough features can be reproduced using in situ formation scenario with growing planets (Petrovich et al. 2013). 2. The formation of MMRs does not appear to be sensitive to the stellar magnetic field. 3. Moreover, we find thatṀ = 2 × 10 −8 M ⊙ yr −1 , corresponding to the early stages of star formation, is conducive to 2:1 MMR formation for two planets, whereaṡ M = 0.1 × 10 −8 M ⊙ yr −1 , which is indicative of the late stages of star formation, contributes greatly to 3:2 MMR formation. 4. Our simulations also suggest that f 1 ≥ 0.1 is a proper value for the speed of type I migration that may result in the formation of near-2:1 MMRs, which is consistent with our previously reported results (Wang et al. 2012), whereas f 1 ≥ 0.3 appears to favor the production of planet pairs in near-3:2 MMRs. To summarize, a slower speed of type I migration (as low as one-tenth the theoretical value) plays a vital role in the resonance formation of systems. The 1:2:4 and 1:2:3 MMRs usually formed when the first planet is trapped in the edge of the holes in the gas disk. The formation scenario is similar to that mentioned in Pierens & Nelson (2008), the MMR formed when the planet is captured in the gap of another planet. Furthermore, compared with our previous work, herein, we have not explored tidal interactions between a central star and its planets (Lithwick & Wu 2012;Batygin & Morbidelli 2013). In the present work, we merely consider planets with a mass of less than 30 M ⊕ based on a tidal timescale (Mardling & Lin 2004;Zhou & Lin 2008), under which the planets' semi-major axes and eccentricities will gradually decrease but the period ratios will change little due to tidal effects. In fact, tidal interactions would trigger the degradation of first-order resonance systems to near-resonance systems (Lee et al. 2013). In summary, we conclude that the near-MMR configurations such as those exhibited by Kepler candidates could be produced under this formation scenario. In addition, our investigation may reveal that this formation scenario is not only applicable to Kepler systems but is also suitable for other planetary systems, especially for systems composed of several short-period planets with low masses. In this respect, our work should shed new light on the planetary formation of such systems in general. (c) show the evolution of period ratio of planet pair for A1, respectively. Panels (d), (e) and (f) exhibit the cases of A2, respectively. Herein E1 represents the supposed planet inhabits inside the innermost planet, and E2 denotes the additional planet resides beyond the outermost planet. The green, black, red and pink lines show the period ratios of P1/E1 , P2/P1, P3/P2 and E2/P3, respectively. Fig. 1 . 1-The statistical results of planet pairs in Kepler systems. Upper panel shows the period ratio of two planets for all planetary candidates, whereas the lower panel represents the values simply for three-planet systems. Two grey dotted lines are associated with exact 3:2 and 2:1 MMRs. Fig. 2 . 2-The statistic results of all runs in G1-G3. Panel (a) shows the distribution of period ratio with different star magnetic field fixed and other parameters are free. The black square represents the planet pairs near 3:2 MMR and the red circle means the statistics of planet pairs near 2:1 MMR. Panel (b) shows the results with different star mass accretion rate. Panel (c) displays the results of different speed of type I migration. Panel (d) represents the fraction of the period ratio varies with the masses of the three planets. Fig. 3 . 3-The statistic results of observation of three-planet systems and simulation results from 50 runs in G1 and 50 runs in G4. The black solid line means the observation results and the grey dot represents the simulation results in G1 and G4. theṀ = 2 × 10 −8 M ⊙ yr −1 values are larger than the values observed in other cases. Thus,Ṁ = 2 × 10 −8 M ⊙ yr −1 is the most favorable value for producing 2:1 MMRs. To yield 3:2 MMRs, this proportion becomes greatest whenṀ = 0.1 × 10 −8 M ⊙ yr −1 , as shown in Fig. 4 . 4-Results for R1. Panels (a), (b) and (d) show the evolution of orbital period, eccentricity, and period ratio of planets pair, respectively. Panel (c) shows the final configuration of three planets. P1, P2, and P3 represent the innermost, the intermediate and the outermost planet, respectively. The black solid curve is the density profile of the gas disk initially. Fig. 5 . 5-Results for R2. Panels (a), (b), and (d) show the evolution of orbital period, eccentricity, and period ratio of the planet pair, respectively. Panel (c) shows the final configuration of three planets. P1, P2, and P3 denotes the innermost, the intermediate and the outermost planet, respectively. The black solid curve is the density profile of the gas disk initially. Fig. 6 . 6-Results for R3. Panels (a), (b), and (d) show the evolution of orbital period, eccentricity, and period ratio of planets pair, respectively. Panel (c) shows the final configuration of three planets. P1, P2 and P3 stand for the innermost, the intermediate and outermost planet, respectively. The black solid curve is the density profile of the gas disk initially. Results for R4. Panels (a), (b) and (d) show the dynamical evolution of orbital period, eccentricity and period ratio of the planet pairs, respectively. Panel (c) shows the final configuration of three planets. Herein P1, P2 and P3 denotes the innermost, the intermediate and the outermost planet, respectively. The black solid curve is the density profile of the gas disk initially. Fig . 8.-Evolution of period ratios for A1 and A2. Panels (a), (b) and TABLE 1 1The initial parameters of the groups in Case 1.Group 1 Group 2 Group 3 Group 4 Mass of planets (M ⊕ ) (5, 10, 15) (5, 5, 5), (5, 5, 10) (2-30, 0.2-15, 0.5-22.5) (5, 10, 15) M (×10 −8 M ⊙ yr −1 ) 0.1-2.5 0.1-2.5 0.1-2.5 0.1, 0.5 B * (KG) 0.5, 1, 1.5, 2, 2.5 0.5, 1, 1.5, 2, 2.5 0.5, 1, 1.5, 2, 2.5 0.5, 1, 1.5, 2, 2.5 f 1 0.01, 0.03, 0.1, 0.3, 1 0.01, 0.03, 0.1, 0.3, 1 0.01, 0.03, 0.1, 0.3, 1 0.01, 0.03, 0.1, 0.3, 1 TABLE 2 2The initial parameters of the cases in Case 2.A1 A2 TABLE 3 3The initial parameters of R1, R2, R3 and R4.R1 R2 R3 R4 Mass of planets (M ⊕ ) 5, 5, 5 5, 10, 15 5, 5, 5 5, 5, 7.5 M (×10 −8 M ⊙ yr −1 ) 1 0.1 1 0.1 B * (KG) 0.5 1 0.5 1.5 f 1 0.03 1 0.3 1 Key Laboratory of Planetary Sciences, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; [email protected], [email protected]. We thank the referee for the constructive comments that helped to improve the original content of this manuscript. 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[ "LOVÁSZ-TYPE THEOREMS AND GAME COMONADS", "LOVÁSZ-TYPE THEOREMS AND GAME COMONADS", "LOVÁSZ-TYPE THEOREMS AND GAME COMONADS", "LOVÁSZ-TYPE THEOREMS AND GAME COMONADS" ]
[ "Anuj Dawar ", "Tomáš Jakl ", "Luca Reggio ", "Anuj Dawar ", "Tomáš Jakl ", "Luca Reggio " ]
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Lovász (1967)showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lovász' theorem: the result by Dvořák (2010) that characterises equivalence of graphs in the k-dimensional Weisfeiler-Leman equivalence by homomorphism counts from graphs of treewidth at most k, and the result of Grohe (2020) characterising equivalence with respect to first-order logic with counting and quantifier depth k in terms of homomorphism counts from graphs of tree-depth at most k. The connection of our categorical formulation with these results is obtained by means of the game comonads of Abramsky et al. We also present a novel application to homomorphism counts in modal logic.
10.1109/lics52264.2021.9470609
[ "https://export.arxiv.org/pdf/2105.03274v1.pdf" ]
234,094,075
2105.03274
846617cec0e9b9e18f4d7940dcec15e47361b84b
LOVÁSZ-TYPE THEOREMS AND GAME COMONADS Anuj Dawar Tomáš Jakl Luca Reggio LOVÁSZ-TYPE THEOREMS AND GAME COMONADS Lovász (1967)showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lovász' theorem: the result by Dvořák (2010) that characterises equivalence of graphs in the k-dimensional Weisfeiler-Leman equivalence by homomorphism counts from graphs of treewidth at most k, and the result of Grohe (2020) characterising equivalence with respect to first-order logic with counting and quantifier depth k in terms of homomorphism counts from graphs of tree-depth at most k. The connection of our categorical formulation with these results is obtained by means of the game comonads of Abramsky et al. We also present a novel application to homomorphism counts in modal logic. Introduction Over fifty years ago, Lovász [20] proved that two finite graphs (or, more generally, any two finite relational structures) A and B are isomorphic if, and only if, for every finite graph C, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B. While one direction of this equivalence is obvious, the other direction establishes that the isomorphism type of a finite graph A is characterised by an infinite integer vector V indexed by F, the collection of (isomorphism classes of) finite graphs, where for C ∈ F, V C is the number of homomorphisms from C to A. This seminal result has led to extensions and investigations in many different directions. One that concerns us is that by restricting the collection F to a natural sub-collection, we can often obtain characterisations of natural coarsenings of the relation of isomorphism. For any subclass F of F, say that a pair of graphs G and H is F -homomorphism equivalent if for any K in F the number of homomorphisms from K to G is the same as the number of homomorphisms from K to H. For instance, Dvořák [11] shows that for any k, if we consider the collection T k of all graphs of tree-width at most k, then a pair of graphs are T k -homomorphism equivalent if, and only if, they are equivalent with respect to the k-dimensional Weisfeiler-Leman (k-WL) equivalence. The k-WL equivalence relations are a widely studied family of approximations of the graph isomorphism relation (see [17] for a recent exposition) with many equivalent characterisations in terms of combinatorics, logic, algebra and linear optimisation. From our point of view, one important such characterisation is the fact that two graphs G and H are k-WL equivalent if, and only if, they are not distinguished by any sentence of C k+1 -first-order logic with counting quantifiers restricted to no more than k + 1 distinct variables [9]. This logical formulation of the equivalence extends naturally to finite structures over any relational vocabulary. Dvořák's result offers yet another characterisation of k-WL equivalence, this time in terms of homomorphism counts from the equally widely studied class of graphs T k . An important special case is that of k = 1. Since T 1 is the class of finite forests, and 1-WL equivalence is the same as fractional isomorphism (see [27]), the result also provides an elegant characterisation of equivalence of graphs with respect to the number of homomorphisms from trees. It is also known that two graphs are co-spectral if, and only if, they are Cyc-homomorphism equivalent, where Cyc is the class of simple cycles, and that two graphs are quantum isomorphic if, and only if, they are P-homomorphism equivalent for the class P of planar graphs [23]. Another recent addition to this collection of results is that of Grohe [14]. He shows that, if D n is the collection of graphs of tree-depth at most n, then two graphs are D n -homomorphism equivalent if, and only if, they are not distinguished by any sentence of C n -first-order logic with counting with quantifier depth at most n. This variety of results relating homomorphism counts over restricted classes to approximations of isomorphism, often proved with very different methods, calls for the development of a more general theory. In the present paper we develop a general categorical result which yields, as example applications, the results of Lovász, Dvořák and Grohe mentioned above. The connection between our categorical result and the results of Dvořák and Grohe is established using the game comonads of Abramsky et al. [1,3]. Specifically, in [1] the graded pebbling comonad P k on the category of σ-structures (for a finite relational signature σ) is introduced and it is shown that the coalgebras for this comonad correspond in a natural way with certain tree decompositions of width at most k − 1. In particular, a finite σ-structure admits a coalgebra structure for this comonad if, and only if, it has tree-width at most k − 1. At the same time, it is shown that isomorphism in the Kleisli category corresponding to the comonad (or equivalently in the Eilenberg-Moore category) is exactly indistinguishability in C k . This brings together in one categorical construction the essential elements of Dvořák's theorem. In an exactly analogous fashion, the Ehrenfeucht-Fraïssé comonad E n of [3] captures, at the level of coalgebras, the structures of tree-depth at most n and yields a notion of isomorphism corresponding to equivalence in C n . These categorical formulations of the combinatorial parameters tree-width and tree-depth on the one hand, and the logical equivalences with respect to C k and C n on the other, are what enable us to give a single result generalising the three theorems of Lovász, Dvořák and Grohe. Our general result states that in any locally finite category with pushouts and a proper factorisation system, two objects m and n are isomorphic if, and only if, the number of morphisms from k to m is the same as that from k to n for any object k. This is proved in Section 3. In Section 4, we use the game comonads to derive the theorems of Dvořák and Grohe from this general result and also a combination of the two characterising equivalence in C k n -first-order logic with counting with quantifier depth n and k variables. In Section 5, we apply the machinery we have developed to another game comonad: the graded modal comonad M k . This gives a new Lovász-style result for pointed Kripke structures, relating homomorphism counts from synchronization trees of bounded height to equivalence in a modal logic with counting. Finally, in Section 6 we draw some conclusions from this about certain normal forms for first-order logic with counting. Related work. Other categorical generalisations of Lovász' theorem have been obtained by Pultr [26] and Isbell [15]. Cf. also Lovász' paper [21]. In modern terms, Pultr works with a finitely well-powered, locally finite category with (extremal epi, mono) factorisations and Isbell requires the category to be locally finite with a special type of factorisation system (which he calls "bicategory"). Unlike the results in op. cit., our Theorem 5 does not hinge on a combinatorial counting argument, but rather on an application of the inclusion-exclusion principle, and appears to be better suited for applications to game comonads. Preliminaries on Game Comonads In this section, we recall the necessary material on (game) comonads and their coalgebras. 2.1. Categories. We briefly recall some basic notions of category theory. For a more thorough introduction, see e.g. [5] or [22]. Let A be a category, and f : A → B and g : A → C two arrows in A. The pushout of f and g, if it exists, consists of two arrows h : C → D and i : B → D such that i • f = h • g and the following universal property is satisfied: For any two arrows i : B → E and h : C → E with i • f = h • g, there is a unique arrow ξ : D → E satisfying ξ • i = i and ξ • h = h . A B C D E f g i i h h ξ The pullback of two arrows f : A → C and g : B → C in A, if it exists, is the pushout of f and g in the opposite category A op obtained by reversing the direction of the arrows in A. Pushouts and pullbacks are special instances of the concepts of colimits and limits of diagrams, respectively. For example, the pushout of f : A → B and g : A → C coincides with the colimit of the diagram B A C. f g For arbitrary colimits, we consider diagrams of any shape. A functor F : A → B is faithful if, given any two parallel arrows f, g : A → A in A, F f = F g implies f = g. Further, F is full if, for any two objects A and A of A, each arrow F A → F A is of the form F f for some f : A → A . A subcategory A of a category B is said to be full if the inclusion functor A → B is full. A fundamental notion of category theory is that of an adjunction. Let F : A → B and G : B → A be any two functors between categories. We say that F is left adjoint to G (equivalently, G is right adjoint to F ), and write F G, provided that for each object B ∈ B there is a morphism ε B : F GB → B satisfying the following universal property: For every A ∈ A and morphism f : F A → B in B, there exists a unique morphism g : A → GB in A such that ε B • F g = f . F GB B F A ε B f F g 2.2. Comonads. A comonad (in Kleisli form) on a category A is given by: • an object map C : A → A; • a morphism ε A : C(A) → A for every A ∈ A; • a coextension operation associating with any morphism f : C(A) → B a morphism f * : C(A) → C(B) . These data must satisfy the following equations for all morphisms f : C(A) → B and g : C(B) → C: ε * A = id C(A) , ε B • f * = f, (g • f * ) * = g * • f * . In particular, we can extend C to a functor A → A by setting C(f ) := (f • ε A ) * for every morphism f : A → B. 1 We recall next some examples of comonads that play an important role in this paper. Let σ be an arbitrary relational signature. The category Σ has as objects σ-structures (denoted A, B, . . . ) and as morphisms σ-homomorphisms (sometimes simply called homomorphisms), i.e. functions f : A → B such that, for all relation symbols R ∈ σ, f (R A ) ⊆ R B where R A and R B are the interpretations of R in A and B, respectively. For each positive integer n, the Ehrenfeucht-Fraïssé comonad E n on Σ (cf. [3]) is defined as follows. For any A ∈ Σ, the universe of E n (A) is the set A ≤n of all non-empty sequences of length at most n. Before defining the interpretations of the relation symbols, let us define the map ε A : A ≤n → A, ε A [a 1 , . . . , a l ] := a l . With this notation, for each R ∈ σ of arity j, we define R En(A) to be the set of those tuples (s 1 , . . . , s j ) of sequences which are pairwise comparable in the prefix order, and such that (ε A (s 1 ), . . . , ε A (s j )) ∈ R A . The coextension operation sends a σ-homomorphism f : E n (A) → B to the σ-homomorphism f * : E n (A) → E n (B), f * [a 1 , . . . , a j ] := [b 1 , . . . , b j ] where b i := f [a 1 , . . . , a i ] for all 1 ≤ i ≤ j. It is not difficult to see that these data define a comonad on the category Σ. From the viewpoint of model-comparison games, the elements of A ≤n represent the plays in the σ-structure A of length at most n, and a σ-homomorphism E n (A) → B is a strategy for Duplicator in the Ehrenfeucht-Fraïssé game with n rounds, where Spoiler plays always in A and Duplicator responds in B. For more details, cf. [3,4]. Next we recall from [1] another comonad on Σ, which models pebble games. For each positive integer k, set k := {1, . . . , k}. Given a σ-structure A, we consider the set (k × A) + of all non-empty sequences of elements of k × A. We call a pair (p, a) ∈ k × A a move. Whenever [(p 1 , a 1 ), . . . , (p l , a l )] is a sequence of moves, p i is called the pebble index of the move (p i , a i ). As with E n , define the map a 1 ), . . . , (p l , a l )] := a l sending a play to its last move. We let P k (A) be the σ-structure with universe (k × A) + and such that, for every relation R ∈ σ of arity j, its interpretation R P k (A) consists of those tuples of sequences (s 1 , . . . , s j ) such that: (i) the s i are pairwise comparable in the prefix order; (ii) whenever s i is a prefix of s i , the pebble index of the last move in s i does not appear in the suffix of s i in s i ; and (iii) (ε A (s 1 ), . . . , ε A (s j )) ∈ R A . Finally, the coextension operation is the same, mutatis mutandis, as with E n . Again, it is not difficult to see that these data define a comonad P k on Σ, called the pebbling comonad. 1 It is easy to see that, setting δ A := id * C(A) for every A ∈ A, the tuple (C, ε, δ) is a comonad in the more traditional sense, where ε is the counit and δ the comultiplication. In fact, these two formulations are equivalent. ε A : (k × A) + → A, ε A [(p 1 , The Ehrenfeucht-Fraïssé comonad E n can be "combined" with the pebbling comonad P k to obtain a new comonad P k,n on Σ, cf. [25]. For every A ∈ Σ, the universe of P k,n (A) is the set (k × A) ≤n of all non-empty sequences of length at most n (this corresponds to bounding the length of plays in pebble games). The homomorphisms ε A and the coextension operation for P k,n are defined as the restrictions of the corresponding concepts for the pebbling comonad P k . 2.3. Coalgebras. Let C be a comonad (in Kleisli form) on a category A. A coalgebra for C is a pair (A, A α − → C(A)) such that A ∈ A, α is a morphism in A, and the following diagrams commute: A C(A) A id A α ε A A C(A) C(A) C(C(A)) α α id * C(A) C(α) We refer to α as a coalgebra structure for A. In the case of the game comonads E n , P k , and P k,n , the σ-structures admitting a coalgebra structure can be characterised in an elegant way in terms of key combinatorial parameters, as we now explain. Let (F, ≤) be any poset. For all x, y ∈ F , we write x ↑ y whenever x and y are comparable, i.e. either x ≤ y or y ≤ x, and let ↓ x := {y ∈ F | y ≤ x}. A forest is a poset (F, ≤) such that, for all x ∈ F , the set ↓ x is finite and totally ordered. The height of (F, ≤) is sup x∈F |↓ x|. If G = (V, ) is a graph (where V is the set of vertices and the adjacency relation), a forest cover of G consists of a forest (F, ≤) and an injective function f : V → F such that v v entails f (v) ↑ f (v ) for all v, v ∈ V . Finally, recall that the Gaifman graph of a σ-structure A is G A := (A, ) where a a if a = a and there exist R ∈ σ and a tuple (a 1 , . . . , a l ) ∈ R A such that a = a i and a = a j for some i, j ∈ {1, . . . , l}. A σ-structure A has tree-depth at most n if there is a forest cover (F, ≤) of G A of height ≤ n. The following is a direct consequence of [3, Theorem 17]: Proposition 1. A σ-structure A admits a coalgebra structure α : A → E n (A) if, and only if, it has tree-depth at most n. In the same way as σ-structures that admit a coalgebra structure for E n can be characterised in terms of tree-depth, the existence of a coalgebra structure for the pebbling comonad P k corresponds to tree-width. We recall the relevant definitions. Note that, if (F, ≤) is a forest cover of a graph G = (V, ), with injective map f : V → F , we can assume without loss of generality that F = V and f = id V . We shall assume this for the remainder of this section. A k-pebble forest cover of a graph G = (V, ) consists of a forest cover (V, ≤) of G and a pebbling function p : V → k such that, whenever v v with v ≤ v , we have p(v) = p(w) for all v < w ≤ v . A σ-structure A has tree-width at most k − 1 if its Gaifman graph G A admits a k-pebble forest cover. 2 The following result is a consequence of [1,Proposition 22]. Proposition 2. A σ-structure A admits a coalgebra structure α : A → P k (A) if, and only if, it has tree-width at most k − 1. Remark 3. The tree-width of a graph is usually defined in terms of tree decompositions. The definition given above is an equivalent reformulation. For a proof of the fact that a finite graph admits a tree decomposition of width < k if, and only if, it admits a k-pebble forest cover, see [3,Theorem 19]. In the same spirit, in the case of the comonad P k,n we have the following characterisation, cf. [25,Theorem 2.14]. Proposition 4. A σ-structure A admits a coalgebra structure α : A → P k,n (A) if, and only if, it has a k-pebble forest cover of height ≤ n. Finally, recall that coalgebras for a comonad C on A form themselves a category EM(C), the Eilenberg-Moore category of C. The objects of EM(C) are the coalgebras (A, α) for C, and morphisms (A, α) → (B, β) in EM(C) are morphisms h : A → B in A such that C(h) • α = β • h. There is an obvious forgetful functor U C : EM(C) → A which sends a coalgebra (A, α) to A. This functor has a right adjoint F C : A → EM(C) defined as follows: for any A ∈ A, F C (A) := (C(A), δ A ) where δ A := id * C(A) . Further, if h is a morphism in A, we set F C (h) := C(h). EM(C) A U C F C A Categorical Lovász-Type Theorem A category is locally finite if there are only finitely many morphisms between any two of its objects. A locally finite category A is said to be combinatorial if, for all m, n ∈ A, m ∼ = n ⇐⇒ | hom A (k, m)| = | hom A (k, n)| ∀k ∈ A. Thus, Lovász' theorem [20] states that, for any finite relational signature σ, the category Σ f of finite σ-structures with homomorphisms is combinatorial. The aim of this section is to prove the following generalisation of Lovász' result. Theorem 5. Let A be a locally finite category. If A has pushouts and a proper factorisation system, then it is combinatorial. 3.1. Proper factorisation systems. We start by recalling the notion of weak factorisation system. Given morphisms e and m in A, we say that e has the left lifting property with respect to m, or that m has the right lifting property with respect to e if, for every commutative square as on the left-hand side below, • • • • e m • • • • e d m there is a (not necessarily unique) diagonal filler, i.e. an arrow d such that the right-hand diagram above commutes. If this is the case, we write e m. For any class F of morphisms in A, let F (respectively F ) be the class of morphisms having the left (respectively right) lifting property with respect to every morphism in F. Definition 6. A pair of classes of morphisms (E, M) in a category A is a weak factorisation system provided it satisfies the following conditions: (1) every morphism f in A can be written as f = m • e with e ∈ E and m ∈ M; (2) E = M and M = E . A proper factorisation system is a weak factorisation system (E, M) such that all morphisms in E are epimorphisms and all morphisms in M are monomorphisms. 3 We describe two different proper factorisation systems in the category Σ of σstructures, for a fixed relational signature σ. Each homomorphism of σ-structures f : A → B factors, as a function, through its set-theoretic image A: A → A → B. There are two natural ways to turn A into a σ-structure: we can equip it with the structure induced by either B, or A. In the first case, for every R ∈ σ of arity n, we set R A := R B ∩ A n and, in the second, we set R A := f (R A ) ∩ A n . These two ways of turning A into a σ-structure yield two different weak factorisation systems (E, M) in the category Σ. 4 For example, the first one yields the weak factorisation system where E = {surjective homomorphisms} and M = {strong/induced embeddings}. Both factorisation systems are proper. Moreover, since the image of a finite σ-structure under a homomorphism is also finite, these factorisation systems restrict to the locally finite category Σ f of finite σ-structures. The category Σ also has pushouts. Given two homomorphisms of σ-structures f : A → B and g : A → C, their pushout D is computed as the quotient of the disjoint sum of σ-structures B + C by the least equivalence relation ∼ such that f (a) ∼ g(a) for every a ∈ A. Equivalently, D is obtained by equipping D , the pushout of the functions f and g in the category of sets, depicted in the following diagram, A B C D f g f g with the smallest relational structure that turns f and g into homomorphisms. Using the same idea, it is not difficult to see that Σ is cocomplete, i.e. it has all colimits, cf. [6, p. 201]. By the previous description of pushouts, it is clear that the pushout in Σ of finite σ-structures is again a finite σ-structure. It follows that Σ f has all pushouts, and thus it satisfies the assumptions of Theorem 5. This justifies our claim that the latter result is a generalisation of Lovász' theorem. We state next some well known properties of weak and proper factorisation systems (cf., e.g., [12] or [28]): Moreover, if (E, M) is proper, the following hold: (d ) g • f ∈ E implies g ∈ E; (e) g • f ∈ M implies f ∈ M. 3 It is an easy observation that any proper factorisation system is an orthogonal factorisation system, meaning that the diagonal fillers are unique. 4 These are the (epi, regular mono) and (regular epi, mono) factorisation systems, respectively. Cf. [6, pp. 200-201]. Proof of Theorem 5. For the remainder of this section, we fix a category A admitting a proper factorisation system (E, M). M-morphisms will be denoted by , and E-morphisms by . An E-pushout square in A is a pushout square consisting entirely of E-morphisms. We are interested in functors A op → FinSet, where FinSet is the category of finite sets and functions between them, that turn E-pushout squares in A into pullbacks in FinSet. Recall that pullbacks in FinSet admit the following explicit description: a commutative square in FinSet is a pullback if, and only if, A B C D f g i h ∀b ∈ B, ∀c ∈ C, if i(b) = h(c) then ∃!a ∈ A. (f (a) = b and g(a) = c). If h and i are injective then, by identifying B and C with subsets of D, the pullback A can be identified with B ∩ C. The following lemma shows that the hom-functors involved in the homomorphism counting theorem send pushout squares to pullbacks. (This is a consequence of the more general fact that representable functors preserve all limits that exist in their domain, cf. [22, pp. 116-117], but we provide here also a direct elementary proof.) Recall that, for any object n of a locally finite category B, the functor Proof. Just observe that, given a pushout square in B and the corresponding diagram in FinSet, as displayed below, hom B (−, n) : B op → FinSet sends a morphism f : a → b in B to the function − • f : hom B (b, n) → hom B (a, n).a b c d g f i h hom B (d, n) hom B (b, n) hom B (c, n) hom B (a, n) −•i −•h −•f −•g if α ∈ hom B (b, n) and β ∈ hom B (c, n) are such that α • f = β • g then, by the universal property of the pushout, there is a unique γ ∈ hom B (d, n) satisfying γ • i = α and γ • h = β. Lemma 9. If a functor F : A op → FinSet sends E-pushout squares in A to pullbacks, then it sends E-morphisms to injections. Proof. Let e : n m be an E-morphism in the category A. Because e is an epimorphism, it follows directly that the square on the left-hand side below is a pushout square in A. F (m) F (m) F (m) F (n) id id F (e) F (e) Since identities are E-morphisms, the square on the right-hand side above is a pullback in FinSet. In turn, this is equivalent to F (e) being an injection. Generic and degenerate elements. The key step in the proof of the classical Lovász' theorem consists in showing that, if | hom Σ f (C, A)| = | hom Σ f (C, B)| for all C in Σ f , then also | inj Σ f (C, A)| = | inj Σ f (C, B)| for all C in Σ f , where inj Σ f (C, A) is the set of all injective homomorphisms C → A, and similarly for inj Σ f (C, B). We aim to show the analogous fact in our setting, where injective homomorphisms are replaced by M-morphisms. The intuition behind the following definition is that a σ-homomorphism f : C → A is non-injective precisely when there exist an onto non-injective σ-homomorphism e : C C and a σ-homomorphism g : C → A such that f = g • e. In terms of the hom-functor E : = hom Σ f (−, A), f ∈ E(C) is non-injective if, and only if, f = E(e)(g) for some onto non-injective σ-homomorphism e : C C and g ∈ E(C ). Definition 10. A strict quotient in A is an E-morphism which is not an isomor- phism. (Equivalently, by Lemma 7(b), a strict quotient is an arrow in E \ M.) Further, consider a functor E : A op → FinSet. For every k ∈ A, we say that s ∈ E(k) is degenerate if there exist a strict quotient f : k l and t ∈ E(l) such that E(f )(t) = s. Otherwise, s is called generic. The subset of E(k) consisting of the generic elements is denoted by E k . The next lemma shows that this definition matches our intuition for hom-functors on A. Lemma 11. Let E = hom A (−, n) for some n ∈ A. For any k ∈ A, E k is the set of all M-morphisms k → n. Proof. Let f be an arbitrary element of E(k) = hom A (k, n). Assume f is generic, and take its (E, M) factorisation: k l n f g h Then E(g)(h) = f , and so g must be an M-morphism. By Lemma 7(a), f = h • g is also an M-morphism. Conversely, suppose f is an M-morphism and pick an E-morphism g : k l and h ∈ E(l) such that E(g)(h) = f , i.e. h • g = f . By Lemma 7(e), g is an M-morphism, so it is not a strict quotient. That is, f is generic. The following is the main technical lemma of this section, and it ultimately relies on an application of the inclusion-exclusion principle. Lemma 12. Assume A has pushouts, and let E and F be functors A op → FinSet sending E-pushout squares in A to pullbacks. If E(k) ∼ = F (k) for all k ∈ A, then E k ∼ = F k for all k ∈ A. Proof. Let E, F be as in the statement, and suppose that E(k) ∼ = F (k) for all k ∈ A. We must show that, for every k ∈ A, E k ∼ = F k . Equivalently, we show that the set E k of degenerate elements of E(k) is in bijection with the set F k of degenerate elements of F (k). Observe that E k coincides with {Im(E(f )) | f : k l is a strict quotient in A} ⊆ E(k), and similarly for F k . Since E(k) and F (k) are finite, there are finite sets S 1 , S 2 of strict quotients of k such that E k = {Im(E(f )) | f ∈ S 1 }, F k = {Im(F (f )) | f ∈ S 2 }. Let S := S 1 ∪ S 2 . By the inclusion-exclusion principle, |E k | = J⊆S,J =∅ (−1) |J|+1 f ∈J Im(E(f )) and similarly for |F k |. Therefore, to prove E k ∼ = F k it suffices to show that f ∈J Im(E(f )) = f ∈J Im(F (f )) for any non-empty J ⊆ S. To this end, fix such a J and consider the wide pushout in A of the strict quotients in J, i.e. the diagram obtained by taking consecutive pushouts of the elements of J, as shown in the diagram on the left-hand side below. k l u l 3 l 2 l 1 p . . . E(k) E(l u ) E(l 3 ) E(l 2 ) E(l 1 ) E(p) . . . By Lemma 7(c), all arrows in this diagram are E-morphisms. Hence, by Lemma 9, the diagram obtained by applying the functor E, depicted on the right-hand side above, consists of injections. Since E sends E-pushouts to pullbacks, the diagram on the right is a wide pullback of injections in FinSet, and so E( Remark 13. Direct inspection of the preceding proofs shows that Theorem 5 can be slightly strengthened by requiring only the existence of pushouts of E-morphisms along E-morphisms. p) ∼ = f ∈J Im(E(f )). Similarly, F (p) ∼ = f ∈J Im(F (f )). As E(p) ∼ = F (p) by assump- tion, we get f ∈J Im(E(f )) = f ∈J Im(F (f )) , 3.3. Specialising to finite coalgebras. To conclude this section, we show that Theorem 5 can be used to lift Lovász' homomorphism counting result from Σ f to categories of finite coalgebras for comonads on Σ. This result is then applied in Sections 4-5 to obtain homomorphism counting results in finite model theory and modal logic, respectively. Given a comonad C on Σ, we say that a coalgebra (A, α) for C is finite if A is a finite σ-structure. The full subcategory of EM(C) defined by the finite coalgebras is denoted EM f (C). Corollary 14. Let C be any comonad on Σ. Then EM f (C), the category of finite coalgebras for C, is combinatorial. In order to prove the previous result, we need the following notions. Recall that, for any category A and object A ∈ A, two epimorphisms f : A → B and g : A → C are equivalent, written f ∼ g, if there exists an isomorphism h : B → C such that h • f = g. It is not difficult to see that ∼ is an equivalence relation on the collection of epimorphisms with domain A. The category A is said to be well-copowered if, for each A ∈ A, the collection of all ∼-equivalence classes of epimorphisms with domain A is a set (as opposed to a proper class). Any cocomplete category that is wellcopowered admits a proper factorisation system, see e.g. Proof of Corollary 14. We show that the category EM f (C) satisfies the hypotheses of Theorem 5. The forgetful functor U C : EM(C) → Σ is faithful and restricts to a (A, B). Because the latter set is finite, so is the former. Therefore, EM f (C) is locally finite. functor EM f (C) → Σ f . Thus, for all (A, α), (B, β) ∈ EM f (C), there is an injection hom EM f (C) ((A, α), (B, β)) → hom Σ f We prove next that EM f (C) has all pushouts. Because the inclusion EM f (C) → EM(C) is full and faithful, it suffices to prove that the pushout in EM(C) of finite coalgebras exists and is a finite coalgebra. Since Σ is cocomplete (cf. the discussion following Definition 6) and U C : EM(C) → Σ creates colimits, the category EM(C) is cocomplete and the forgetful functor U C : EM(C) → Σ preserves all colimits. In particular, the pushout of finite coalgebras exists in EM(C) and is a finite coalgebra. The functor U C : EM(C) → Σ preserves colimits, and in particular epimorphisms. Using the fact that U C is faithful and creates isomorphisms, it is not difficult to see that EM(C) is well-copowered because so is Σ. Thus, EM(C) admits a proper factorisation system (E, M). Now, consider a morphism (A, α) → (B, β) in EM f (C) and its (E, M)-factorisation in EM(C): (A, α) ( A, α) (B, β) e m Since U C preserves epimorphisms, U (e) : A → A is an epimorphism in Σ, that is a surjective homomorphism. It follows that A is a finite σ-structure and so ( A, α) ∈ EM f (C). The fact that EM f (C) admits a proper factorisation system is then a consequence of the following easy observation: Let A be any category equipped with a proper factorisation system (E, M), and B a full subcategory of A. Assume that, whenever A → B is a morphism in B and A → A → B is its (E, M)-factorisation in A, also A ∈ B. Then (E ∩ B, M ∩ B) is a proper factorisation system in B. Remark 15. If C is a comonad on Σ that restricts to a comonad C on Σ f , then EM(C ) is isomorphic to EM f (C) and thus EM(C ) is combinatorial by Corollary 14. Also, note that the same proof as for Corollary 14 applies, mutatis mutandis, if Σ is replaced by the category of pointed σ-structures, having as objects the pairs (A, a) with A ∈ Σ and a ∈ A, and as morphisms the σ-homomorphisms preserving the distinguished points. This will be needed in Section 5 for applications in the setting of modal logic. More generally, it is not difficult to see that we may replace Σ and Σ f by any category A, and full subcategory A f of A, respectively, such that: (1) A is cocomplete and well-copowered; (2) A f is locally finite, closed under finite colimits in A and, if A → B is an epimorphism in A with A ∈ A f , then B ∈ A f . Applications to Finite Model Theory In this section, we apply Corollary 14 to the game comonads introduced in Section 2 to give new proofs of results of Grohe and Dvořák connecting homomorphism counts and indistinguishability in appropriate logic fragments. We start by establishing a general result (Theorem 16) and then proceed to show how Grohe's and Dvořák's theorems follow from it. Before stating this general result, we illustrate the underlying idea for a generic game comonad C. In Section 2 we have recalled how game comonads capture combinatorial parameters of σ-structures. On the other hand, they can also express equivalence with respect to appropriate logic fragments. For this, we need to extend the signature to account for equality in the logic. Let σ + := σ∪{I} be the relational signature obtained by adding a binary relation symbol I to σ, and let Σ + be the category of σ + -structures with homomorphisms. There is an embedding, i.e. a full and faithful functor, J : Σ → Σ + which sends a σ-structure A to the σ + -structure J(A) where I is interpreted as the identity. As the comonad C on Σ is defined for arbitrary relational signatures, we have a corresponding comonad C + on Σ + . Typically, it turns out that, for an appropriate logic fragment L, A ≡ L B ⇐⇒ F C + (J(A)) ∼ = F C + (J(B)) for all A, B ∈ Σ f , where U C + F C + : Σ + → EM(C + ) is the adjunction associated with C + , cf. Section 2.3. Provided that C + sends finite structures to finite structures, we can apply Corollary 14 to translate the isomorphism F C + (J(A)) ∼ = F C + (J(B)) into a statement about homomorphism counts for J(A) and J(B) in Σ + f . It then remains to go back to σ-structures to obtain a statement about homomorphism counts for A and B in Σ f . To this end, note that the functor J : Σ → Σ + has a left adjoint H : Σ + → Σ sending a σ + -structure D to D − /∼, where D − is the σ-reduct of D and ∼ is the equivalence relation generated by I D (for a proof of this fact, see Lemma 25). We now state the general result (whose proof is deferred to Section 4.3) from which Grohe's and Dvořák's theorems will be derived in Sections 4.1 and 4.2, respectively. Let us write im(U C ) for the full subcategory of Σ consisting of the objects of the form U C (A, α) = A for (A, α) ∈ EM(C), and similarly for im(U C + ). Theorem 16. Let C and C + be comonads on Σ and Σ + , respectively, and L a logic fragment. Suppose the following conditions are satisfied: (1) for all A, B ∈ Σ f , A ≡ L B if, and only if, F C + (J(A)) ∼ = F C + (J(B)); (2) C + sends finite σ + -structures to finite σ + -structures; (3) the embedding J : Σ → Σ + and its left adjoint H restrict to Σ f ∩ im(U C ) and Σ + f ∩ im(U C + ). Then, for any finite σ-structures A and B, A ≡ L B if, and only if, | hom Σ f (C, A)| = | hom Σ f (C, B)| for every finite σ-structure C in im(U C ). Remark 17. The proofs of Grohe's and Dvořák's theorems essentially reduce to showing that the assumptions of Theorem 16 are satisfied for the appropriate comonads. The "combinatorial core" of these results, which requires a specific argument for each comonad, corresponds to verifying that the functor H restricts to Σ + f ∩ im(U C + ) → Σ f ∩ im(U C ) . This amounts to saying that the operation D → D − /∼ does not increase the tree-depth or tree-width of D, and can be understood as an equality elimination result (cf. Section 6). 4.1. Bounded tree-depth. Let C be the extension of first-order logic obtained by adding, for each natural number i, a counting quantifier ∃ ≥i . The semantics of these quantifiers is the following: for any structure A, we have A |= ∃ ≥i x. ϕ if, and only if, A |= ϕ[a/x] holds for at least i distinct elements a ∈ A. Let C n be the fragment of C consisting of formulas of quantifier depth ≤ n. The aim of this section is to use Theorem 16 to give a new proof of a recent result of Grohe [14]: 5 Theorem 18. For any A, B finite σ-structures, A ≡ Cn B if, and only if, | hom Σ f (C, A)| = | hom Σ f (C, B)| for every finite σ-structure C with tree-depth at most n. Let E + n be the Ehrenfeucht-Fraïssé comonad on Σ + , and U E + n F E + n : Σ + → EM(E + n ) the associated adjunction (cf. Section 2.3). By Proposition 1, a σ-structure has tree-depth at most n precisely when it belongs to im(U En ). Thus, to prove Theorem 18, it suffices to show that the assumptions of Theorem 16 are satisfied for C = E n , C + = E + n , and L = C n . Item 1 in Theorem 16 is a consequence of [3, Theorem 12]: Proposition 19. For any two σ-structures A, B, we have A ≡ Cn B if, and only if, F E + n (J(A)) ∼ = F E + n (J(B)). Item 2 holds because the comonad E + n restricts to finite σ + -structures, and item 3 is a consequence of the following proposition-which concludes the proof of Grohe's theorem. Proposition 20. The embedding J : Σ → Σ + and its left adjoint H restrict to the full subcategories consisting of finite structures with tree-depth at most n. Proof. Clearly, if A is a (finite) σ-structure with tree-depth at most n, then J(A) has tree-depth at most n. It remains to prove that, for any D ∈ Σ + f , if D has tree-depth at most n then so does H(D). To improve readability, set C := H(D). (d ) | d ∈ [d]} ∩ ↓ f (d), which is a finite non-empty totally ordered set.) Define g : C → F, g([d]) := ξ(d). Clearly, g is well-defined and it is injective because so is f . It remains to prove that, for all d 1 , d 2 ∈ D, [d 1 ] [d 2 ] in G C entails g([d 1 ]) ↑ g([d 2 ]) , for then it follows that (F, ≤), along with the map g, is a forest cover of G C . If [d 1 ] [d 2 ], there are d 1 ∈ [d 1 ] and d 2 ∈ [d 2 ] satisfying d 1 d 2 , and so f (d 1 ) ↑ f (d 2 ). Assume f (d 1 ) ≤ f (d 2 ). Since g([d 1 ]) ≤ f (d 1 ) and g([d 2 ]) ≤ f (d 2 ), it follows that both g([d 1 ]) and g([d 2 ]) belong to ↓ f (d 2 ), which is totally or- dered. Therefore, g([d 1 ]) ↑ g([d 2 ]) . Reasoning in a similar manner, we see that f (d 2 ) ≤ f (d 1 ) implies g([d 1 ]) ↑ g([d 2 ]). 4.2. Bounded tree-width. Let C k be the k-variable fragment of C-first-order logic with counting quantifiers. In this section, we show how to derive from Theorem 16 the following variant of Dvořák's theorem [11]: 6 Theorem 21. For any A, B finite σ-structures, A ≡ C k B if, and only if, | hom Σ f (C, A)| = | hom Σ f (C, B)| for every finite σ-structure C with tree-width at most k − 1. The k-variable counting logic C k appearing in Theorem 21 is captured by the pebbling comonad P + k on Σ + . In fact, it follows directly from [1, Theorem 18] that, whenever A, B are finite σ-structures, F P + k (J(A)) ∼ = F P + k (J(B)) if, and only if, A ≡ C k B. However, the comonad P + k does not satisfy item 2 in Theorem 16, because P + k (A) is infinite even when A ∈ Σ + f . To circumvent this problem, we consider the comonads P + k,n , which do restrict to finite σ + -structures. Again by (the proof of) [1,Theorem 18], for any finite σ-structures A, B, A ≡ C k n B ⇐⇒ F P + k,n (J(A)) ∼ = F P + k,n (J(B)) where C k n consists of the formulas that are simultaneously in C n and C k . Thus, items 1 and 2 in Theorem 16 are satisfied for C = P k,n , C + = P + k,n , and L = C k n . Assume for a moment that item 3 is also satisfied. Then, combining Theorem 16 with Proposition 4, we obtain that A ≡ C k n B if, and only if, | hom Σ f (C, A)| = | hom Σ f (C, B) | for all finite σ-structures C whose Gaifman graph G C admits a k-pebble forest cover of height ≤ n. Observe next that (i) A ≡ C k B precisely when A ≡ C k n B for all n, and (ii) a finite σ-structure C has tree-width at most k − 1 if, and only if, G C admits a k-pebble forest cover of height ≤ n for some n. Therefore, A ≡ C k B if, and only if, | hom Σ f (C, A)| = | hom Σ f (C, B)| for all finite σ-structures C with tree-width at most k −1, thus settling Theorem 21. In the remainder of this section, we prove that item 3 in Theorem 16 is indeed satisfied. As mentioned already, Σ f ∩ im(U P k,n ) consists of the finite σ-structures that admit a k-pebble forest cover of height ≤ n, and similarly for Σ + f ∩ im(U P + k,n ). To improve readability, let us denote these categories by T k,n and T + k,n , respectively. Note that J : Σ f → Σ + f restricts to a functor T k,n → T + k,n . It remains to show that its left adjoint H restricts to T + k,n → T k,n . 6 Dvořák's result is for undirected graphs without loops. For a discussion of how his result can be recovered in our framework, see Remark 24. Recall from Section 2.3 that a forest cover of a σ-structure A can be identified with a forest order ≤ on A that is compatible, in the sense that a ↑ a whenever a a in G A . Suppose that A is equipped with a forest order ≤ and let p : A → k be any function. We define a relation on A as follows: a a (read as a sees a ) if a ↑ a and min(a, a ) < z ≤ max(a, a ) =⇒ p(z) = p (min(a, a )) for all z ∈ A. With this notation, a k-pebble forest cover of A can be identified with a triple (A, ≤, p) where ≤ is a forest order on A and p : A → k is such that a a implies a a for all a, a ∈ A. Just observe that ≤ is a compatible forest order since a a entails a ↑ a . Saying that H restricts to T + k,n → T k,n means that, whenever a finite σ + -structure A admits a k-pebble forest cover of height ≤ n, then so does the quotient A − /∼ where A − is the σ-reduct of A and ∼ is the least equivalence relation containing I A . The difficulty in showing this consists in defining the pebbling function on A − /∼. We do this by considering consecutive one-step quotients (inspired by equality elimination from Section 6), where at each step a new pair of distinct elements in I A is identified. As A is finite, this construction terminates after finitely many steps and yields the quotient A − /∼. Let A := (A, ≤, p) be a finite σ + -structure together with a k-pebble forest cover of height ≤ n. Suppose that there is a pair (u, v) ∈ I A such that u = v. Then u ↑ v, and we can assume without loss of generality that u < v. The one-step quotient A/ u∼v := (A , ≤ , p ) of A is defined as follows: • the carrier of A/ u∼v is the set A := A \ {v}; • the forest order ≤ is the restriction of ≤ to A ; • for each R ∈ σ ∪ {I}, the relation R A is obtained by replacing all occurrences of v by u in each tuple of R A . • the pebbling function p : A → k is defined by w →      p(u) if v ≤ w, p(w) = p(v), and u w min p(v) if v ≤ w, p(w) = p(u), and v w min p(w) otherwise where w min := min{w ∈ A | v < w ≤ w and p(w ) = p(w)}. Clearly, ≤ is a forest order of height ≤ n on A . Further, a a implies a a for all a, a ∈ A . This follows from the next lemma, which is proved by a careful case analysis. Lemma 22. Let w, w ∈ A \ {v}. Then the following hold: (1) v w in A implies u w in A/ u∼v ; (2) w w in A implies w w in A/ u∼v . Proof. (1) Suppose w ∈ A \ {v} and v w in A. There are two possibilities, either v < w or w < v. In both cases we have that w ↑ u and so we only need to check the condition on the pebbling function p . Assume that v < w. As (u, v) ∈ I A , we know that u v and so u v in A. Because p (z) = p(z) whenever v ≤ z, to show that u w in A/ u∼v we only need to check that ∀z ∈ A, v < z ≤ w =⇒ p (z) = p (u).(1) Observe that p (u) = p(u). Also, by definition of p , for any z such that v < z ≤ w the equality p (z) = p(u) is satisfied if, and only if, either p(z) = p(v) and u z min , or p(z) = p(u) and v z min . However, none of these two scenarios can occur because v w implies p(z) = p(v) and v z min . Hence, the condition in (1) is satisfied. On the other hand, assume that w < v. Then u ↑ w. If u < w then u v implies u w in A. Similarly, if w ≤ u then v w entails u w in A. Therefore, u w in A/ u∼v because p(z) = p (z) for every z such that v ≤ z. (2) Let w, w ∈ A \ {v} be such that w w in A. We can assume without loss of generality that w ≤ w . If v ≤ w then p(z) = p (z) for every z such that w ≤ z ≤ w , and so w w in A/ u∼v . Thus, suppose v ≤ w . It follows that v ↑ w and so either v < w or w < v. First we assume v < w ≤ w . Note that, for any z such that w < z ≤ w , if p (z) = p (w) and p(z) / ∈ {p(u), p(v)}, then by definition of p we must have p(z) = p(w), contradicting the fact that w w in A. So, we only have to take care of the case where either p(z) = p(u) or p(z) = p(v). Set U := {x ∈ A | v < x ≤ w and p(x) = p(u)} V := {x ∈ A | v < x ≤ w and p(x) = p(v)}. Suppose U and V are non-empty and so they admit minimal elements u and v, respectively. Then, either u < v or v < u. If u < v then any z ≥ u such that p(z) = p(u) satisfies v z min (because z min = u) and so p (z) = p(v), and similarly any z ≥ u such that p(z) = p(v) satisfies u z min (because z min = v) and so p (z) = p(u). On the other hand, if v < u then p (z) = p(z) for any z ≥ v. Either way, for any z between w and w , p (z) = p (w) implies p(z) = p(w). Hence, w w in A entails w w in A/ u∼v . A similar reasoning applies, mutatis mutandis, when U = ∅ or V = ∅. Suppose next that w < v < w . Then p (w) = p(w) and, as w w in A, it follows that p(w) = p(v). If p(w) = p(u) then we conclude that w w in A/ u∼v because p (z) = p (w) entails p(z) = p(w) for any element z between w and w . To conclude the proof, suppose towards a contradiction that p(w) = p(u) and there exists a z ∈ A \ {v} such that w < z ≤ w and p (z) = p (w). Then, p (z) = p(u). Since w w in A, we get p(z) = p(u) and so v < z, p(z) = p(v) and u z min in A. As u v in A, there must exist an x such that w < v < x ≤ z min ≤ z ≤ w and p(x) = p(u), contradicting the fact that w w in A. Applying this construction iteratively, we obtain the following analogue of Proposition 20, which concludes the proof of Dvořák's theorem. Proposition 23. The functor H restricts to T + k,n → T k,n . Proof. Let A 0 = (A, ≤, p) be a finite σ + -structure together with a k-pebble forest cover of height ≤ n. Then we have a sequence A 0 , A 1 , A 2 , . . . such that A i+1 is defined as A i / u∼v for some pair of distinct elements (u, v) ∈ I Ai , where A i is the σ + -structure underlying A i . This is iterated as long as such a pair (u, v) can be found. Let A n be the last element of the sequence. Note that, by construction, the σ-reduct of A n is isomorphic to H(A). Because A n admits a k-pebble forest cover of height ≤ n witnessed by A n -by repeatedly applying Lemma 22-so does H(A). Remark 24. An advantage of the categorical approach to homomorphism counting is that it specialises to any full subcategory A of Σ, provided the game comonad in question restricts to A. For example, let σ consist of a single binary relation symbol, so that Σ is the category of (directed) graphs and graph homomorphisms. As the pebbling comonad P k restricts to the full subcategory of Σ defined by undirected graphs without loops, we obtain an analogue of Theorem 21 where A, B, C range over the class of finite undirected graphs without loops. A similar observation applies to the Ehrenfeucht-Fraïssé comonad E n , yielding a variant of Theorem 18 for undirected graphs without loops. Proof of Theorem 16. For convenience of notation, let EM + f := EM f (C + ). Combining item 1 with Corollary 14 applied to the comonad C + , we see that, for all A, B ∈ Σ f , A ≡ L B if, and only if, | hom EM + f (D, F C + (J(A)))| = | hom EM + f (D, F C + (J(B)))|(2) for all finite coalgebras D ∈ EM + f . Note that here we used the fact that, by item 2, F C + (D ) is a finite coalgebra whenever D is a finite σ + -structure. In view of the adjunction U C + F C + , the condition in equation (2) is equivalent to | hom Σ + f (U C + (D), J(A))| = | hom Σ + f (U C + (D), J(B))| for all D ∈ EM + f . Finally, by item 3, an application of Lemma 27 with A := Σ f ∩ im(U C ) and A + := Σ + f ∩ im(U C + ) yields the desired statement. Homomorphism Counting in Modal Logic In this section we prove a new homomorphism counting result for (multi-)modal logic. This is derived by applying the categorical framework from Section 3 to a game comonad for modal logic defined on the category of pointed Kripke structures. Since there is no equality in the logic, this time we can dispense with the extra relation I. This considerably simplifies the proofs, compared to the previous section. Let σ be a signature consisting of relation symbols of arity at most 2. Each unary relation symbol P yields a propositional variable p, and each binary relation symbol R α yields modalities α and ♦ α . We can think of a σ-structure A as a Kripke structure for this multi-modal logic, where P A gives the valuation for the propositional variable p, and R A α gives the accessibility relation for the modalities α and ♦ α . Each formula ϕ in this modal logic admits a translation into a first-order formula ϕ x in one free variable x; this is known as the standard translation, see e.g. [7, §2.4]. We let p x := P (x) and let x commute with Boolean connectives. Further, set α ϕ x := ∀y. R α (x, y) → ϕ y and ♦ α ϕ x := ∃y. R α (x, y) ∧ ϕ y where y is a fresh variable. Consider a pointed Kripke structure, that is a pair (A, a) where A is a σ-structure and a ∈ A. Then A, a |= ϕ according to Kripke semantics if, and only if, A |= ϕ x [a/x] in the standard model-theoretic sense. Henceforth, we assume all Kripke structures under consideration are pointed. Let Kri be the category of Kripke structures and σ-homomorphisms preserving the distinguished element. For every natural number k, there is a modal comonad M k on Kri, see [3]. For any Kripke structure (A, a), the carrier of M k (A, a) is the set of all paths of length ≤ k starting from a: a R1 − − → a 1 R2 − − → a 2 → · · · Rn − − → a n where R 1 , . . . , R n are binary relations from σ. The distinguished element of the Kripke structure M k (A, a) is the trivial path (a) of length 0. If P ∈ σ is unary then P M k (A,a) is defined as the set of paths p such that the last element a n of p belongs to P A . For a binary relation symbol R ∈ σ, let R M k (A,a) be the set of pairs of paths (p, p ) such that p is obtained by extending p by one step along R. The morphism ε (A,a) : M k (A, a) → (A, a) sends a path to its last element; for the definition of the coextension operation, see [3, §2.3]. The comonad M k captures a well known combinatorial parameter of Kripke structures, as we now recall. Let us say that a Kripke structure (A, a) is rooted if, for any a ∈ A, there is a path from a to a . If we further require that this path be unique, then (A, a) is a synchronization tree. A synchronization tree (A, a) has height ≤ k if, for each a ∈ A, the length of the unique path from a to a is at most k. For a proof of the following result, see [4,Proposition 6.6]. On the other hand, the modal comonad M k characterises logical equivalence with respect to an appropriate modal logic. Let us extend the multi-modal logic introduced above by adding, for every positive integer n and binary relation symbol R α ∈ σ, graded modalities n α and ♦ n α defining A, a |= ♦ n α ϕ if there are at least n distinct R α -successors of a that satisfy ϕ (and n α ϕ = ¬♦ n α ¬ϕ). A formula in this graded modal logic has modal depth ≤ k if it contains at most k nested modalities. Let C ML k be the image of the standard translation of formulas with graded modalities that have modal depth ≤ k. The following result is a special case of [3,Theorem 16]. (A, a) ≡ C ML k (B, b) iff F M k (A, a) ∼ = F M k (B, b). We aim to exploit the two preceding propositions to obtain a characterisation of the equivalence relation ≡ C ML k over finite Kripke structures in terms of homomorphism counts from finite synchronization trees of height ≤ k (Theorem 31 below). To this end, let Kri f and RKri f be the full subcategories of Kri consisting, respectively, of the finite Kripke structures, and finite rooted Kripke structures. Note that M k restricts to a comonad on Kri f , that we denote again by M k . Further, as M k (A, a) is a finite synchronization tree whenever (A, a) is a finite Kripke structure, M k restricts to a comonad M * k on the category RKri f . Lemma 30. The following statements hold: (1) the inclusion RKri f → Kri f has a right adjoint R; (2) R lifts to a functor R : EM(M k ) → EM(M * k ) making the following diagram commute. (A, a), δ A ) is rooted and consists only of paths of length ≤ k that start at a. The equality on morphisms holds for the same reason. EM(M k ) EM(M * k ) Kri f RKri f R F M k R F M * We are now ready to prove a Lovász-type theorem for graded modal logic. = (B, b). For the purpose of this proof, write V : RKri f → Kri f for the inclusion functor and let EM * := EM(M * k ). By Proposition 28, the statement about homomorphism counts is equivalent to | hom Kri f (V (U M * k (D)), A)| = | hom Kri f (V (U M * k (D)), B)| for all coalgebras D in EM * . In turn, by virtue of the adjunctions V R and U M * k F M * k , this is equivalent to | hom EM * (D, F M * k (R(A)))| = | hom EM * (D, F M * k (R(B)))| for all D ∈ EM * . Since EM * is combinatorial by Corollary 14 and Remark 15, the last statement is equivalent to F M * k (R(A)) ∼ = F M * k (R(B)) and thus, by Lemma 30, to R(F M k (A)) ∼ = R(F M k (B)). Note that the restriction of R to the image of F M k is full and faithful. Hence, R(F M k (A)) ∼ = R(F M k (B)) if, and only if, F M k (A) ∼ = F M k (B) which, by Proposition 29, is equivalent to A ≡ C ML k B. Logical Normal Forms Let C k ∞ denote the closure of C k under infinitary conjunctions and disjunctions. This infinitary logic has been much studied in finite model theory as a means of delimiting the expressive power of fixed-point logics with counting. In this logic we can express all, and only, the properties of finite structures that are invariant under the equivalence relation ≡ C k . In fact, since it is known that each finite structure is characterised up to ≡ C k by a single sentence of C k , it follows that every class of finite structures that is invariant under ≡ C k is defined by a single infinitary disjunction of C k sentences (see [24] for details). An entirely analogous normal form holds for C n,∞ -the closure of C n under infinitary conjunctions and disjunctions (see [19,Chapter 8]). The theorems of Dvořák and Grohe provide a route to alternative normal forms for these infinitary logics. To see this, note that for any finite structure A, we can write a primitive positive sentence γ A of first-order logic such that for any B, B |= γ A if, and only if, there is a homomorphism from A to B. Here, primitive positive means that the sentence is built up from atomic formulas using only conjunctions and existential quantification. The sentence γ A is called the canonical conjunctive query of A (see [13,Chapter 6]). It is known that if A has tree-width strictly less than k, then γ A can be written using no more than k variables [18,Lemma 5.2]. Similarly, if A has tree-depth at most n, then γ A can be chosen to have quantifier depth at most n [29, Lemma 2.14]. Given a positive integer t, we can transform γ A by standard means into a sentence γ t A of C which asserts that there are at least t distinct homomorphisms from A. That is, for any B, B |= γ t A if, and only if, | hom(A, B)| ≥ t. To show this, we establish something more general. Let γ(x) be a primitive positive formula with free variables amongx. For any structure B and any interpretation ι taking the variables inx to elements of B, we write B |= γ[ι] to indicate that the formula γ is satisfied in B when the free variablesx are interpreted according to ι. Since γ is primitive positive, we have B |= γ[ι] precisely if there is a function κ mapping the existential quantifiers in γ to elements of B so that every atomic formula R(ȳ) occurring in γ is satisfied in B when each variable y inȳ is interpreted by ι(y), if this occurrence of y is free in γ, and by κ applied to the quantifier binding y otherwise. We write |B |= γ[ι]| to denote the number of distinct functions κ witnessing B |= γ[ι] and observe that |B |= γ A | = | hom(A, B)|. We now define by induction on the structure of γ a formula γ t (x) of C with the property that for any structure B and interpretation ι ofx in B, we have B |= γ t [ι] if, and only if, |B |= γ[ι]| ≥ t. For atomic γ, γ 1 is just γ and γ t is false if t > 1. For γ a conjunction γ 1 ∧ γ 2 , let T := {(t 1 , t 2 ) ∈ N 2 | t 1 · t 2 ≥ t}. Then γ t := (t1,t2)∈T (γ t1 1 ∧ γ t2 2 ). When γ is ∃x.γ 1 , let F be the collection of all finite partial functions f on N such that s∈dom(f ) sf (s) ≥ t. Then γ t := f ∈F s∈dom(f ) ∃ ≥f (s) x.γ s 1 . Note that both the quantifier rank and the total number of variables that appear in γ t A are the same as for γ A , and γ t A still has no negation symbols. It then follows from Dvořák's theorem that any sentence ϕ of C k ∞ is equivalent in the finite (i.e., over finite structures) to an infinite Boolean combination of sentences of C k of the form γ t A . Indeed, ϕ is equivalent to B∈M A∈T γ | hom(A,B)| A ∧ ¬γ | hom(A,B)|+1 A where M consists of one representative for each ≡ C k -class of structures in {B | B |= ϕ} and T consists of one representative for each isomorphism class of finite structures that have tree-width at most k − 1. Likewise, it is a consequence of Grohe's theorem that any sentence of C n,∞ is equivalent (in the finite) to an infinite Boolean combination of sentences of C n of the form γ t A ; a similar normal form was also exhibited in [14, §5]. This yields an interesting normal form for these counting logics. In particular, there is no need of universal quantifiers or of equality, as neither of these is used in the sentences γ t A . Further, if we allow dual counting quantifiers ∃ ≤i with the semantics that A |= ∃ ≤i x. ϕ if, and only if, A |= ϕ[a/x] holds for at most i distinct elements a ∈ A, then we do not need negation at all in our formulas. From our point of view, the most interesting aspect of this is the elimination of equality symbols. Consider for example the simple first-order sentence in the language of graphs: ∃x∃y.(x = y ∧ E(x, y)) asserting the existence of an edge between two distinct vertices. At first sight, it does not seem possible to express this property without using the equality symbol. However, it is possible to do so with infinitary Boolean connectives and counting quantifiers as we illustrate. First, consider the formula ∃ ≥t xy.E(x, y) asserting that there exist at least t pairs x, y for which E(x, y). Note this is just γ t when γ is the sentence ∃x∃y.E(x, y). Now, the sentence ∃x∃y.(x = y ∧ E(x, y)) is equivalent to i∈N (∃ ≥i xy.E(x, y) ∧ ∃ ≤i−1 x.E(x, x)) . We now give a direct proof, in our categorical framework, that we can eliminate equality from C k ∞ and C n,∞ . Theorem 32. Every sentence of C k ∞ is equivalent in the finite to one without equality, and every sentence of C n,∞ is equivalent in the finite to one without equality. Proof. We give the proof only for C n,∞ as the other is entirely analogous. It suffices to prove that if two finite structures A, B are not distinguished by any sentence of C n without equality, then they are not distinguished by any sentence of C n . Suppose then that A and B are not distinguished by any sentence of C n without equality. We thus have F En (A) ∼ = F En (B) in EM(E n ) and so hom(D, F En (A)) ∼ = hom(D, F En (B)) for every D in EM f (E n ). By the adjunction U En F En , we have hom(U En (D), A) ∼ = hom(U En (D), B) for every D in EM f (E n ). Since a finite structure has tree-depth at most n if, and only if, it admits an E n -coalgebra structure, this implies hom(C, A) ∼ = hom(C, B) for every finite structure C with tree-depth at most n and hence, by Theorem 18, A ≡ Cn B. Conclusion The ideas developed in this paper connect homomorphism counting results in finite model theory with the framework of game comonads recently put forward by Abramsky et al. Our categorical generalisation of Lovász' theorem allows us to give uniform proofs of results by Dvořák and Grohe. The only part of the proof specific to each of these results is the combinatorial argument underlying the fact that the adjunction between σ-structures and σ + -structures restricts to structures of bounded tree-width and bounded tree-depth, respectively. We also establish a new homomorphism counting result for modal logic which characterises the equivalence relation ≡ C ML k over finite Kripke structures in terms of homomorphism counts. As a by-product of the modularity of this general framework, we are immediately able to deduce: (i) the characterisation of the equivalence relation ≡ C k n in terms of homomorphism counts from structures admitting a k-pebble forest cover of height ≤ n, (ii) the specialisation of our results to undirected graphs without loops (cf. Remark 24), and (iii) equality elimination results for counting logics (Theorem 32). We believe that our method lays a pathway to discovering more Lovász-type theorems. In particular, any comonad on the category of σ-structures that satisfies the conditions of Theorem 16 will yield a Lovász-type theorem. The natural next step to test this theory is to apply our results to the game comonads introduced in [2] and [10]. On the other hand, there are many Lovász-type theorems that do not immediately fit into our framework. For example, quantum isomorphism [23] and cospectrality are characterised by restricting the homomorphism counts to planar graphs and cycles, respectively. Are there suitable comonads (or monads) that could allow us to recover these characterisations in a uniform manner? Another line for future investigation consists in fine-tuning the general theory introduced in this paper. For example, can we refine Theorem 31 to yield a homomorphism counting result for p-morphisms, the natural concept of morphism between Kripke structures? The ideas that we have described in this paper contribute to advancing the theory of game comonads, which connects two distinct areas of logic in computer science: on the one hand finite and algorithmic model theory, and on the other hand categorical and structural methods in semantics. Game comonads have been successfully applied, e.g., to investigate equirank-variable homomorphism preservation theorems in finite model theory [25]. We broaden the applicability of this theory by bringing these techniques to the area of Lovász-type results in finite model theory. Lemma 7 . 7Let (E, M) be a weak factorisation system in A. The following hold: (a) E and M are closed under compositions; (b) E ∩ M = {isomorphisms}; (c) the pushout in A of an E-morphism along any morphism, if it exists, is again in E. Lemma 8 . 8For any locally finite category B and n ∈ B, the functor hom B (−, n) : B op → FinSet sends all pushout squares that exist in B to pullbacks in FinSet. thus concluding the proof. We are now in a position to prove the main result of this section: Proof of Theorem 5. Fix arbitrary objects m, n ∈ A. For the non-trivial direction, assume | hom A (k, m)| = | hom A (k, n)| for all k ∈ A. Let E and F denote, respectively, the functors hom A (−, m) : A op → FinSet and hom A (−, n) : A op → FinSet. These functors send E-pushout squares in A to pullbacks by Lemma 8. Thus, by Lemma 12, E k ∼ = F k for all k ∈ A. According to Lemma 11, there is a bijection {M-morphisms k m} ∼ = {M-morphisms k n}.Setting k = m, the existence of an M-morphism m m (namely the identity) entails the existence of an M-morphism i : m n. Similarly, there exists an Mmorphism j : n m. A standard argument then shows that m ∼ = n (see e.g.[26]). We briefly sketch a proof. The set L of all M-morphisms m m is a monoid with respect to composition and contains j • i. Since all M-morphisms are monos, L satisfies the left cancellation law ab = ac ⇒ b = c. But every finite monoid satisfying the left cancellation law is a group, hence j • i has an inverse. It follows from Lemma 7(b),(d) that j is an isomorphism. recall that a functor F : A → B creates colimits if, for every diagramD in A such that F (D) admits a colimit B in B, there exists a colimit A of D in A with F (A) ∼ = B.In particular, if B is cocomplete and F creates colimits, then A is also cocomplete and the functor F preserves colimits. Further, F creates isomorphisms if, whenever h :B → F (A) is an isomorphism in B, there exists an isomorphism h : A → A in A such that F (h ) = h.It is a well known fact that, for any comonad C on A, the forgetful functor U C : EM(C) → A creates colimits and isomorphisms (see e.g.[5, Proposition 20.12] for a proof of the dual statement). Suppose (F, ≤) is a forest of height ≤ n and f : D → F an injective map such that f (d 1 ) ↑ f (d 2 ) whenever d 1 d 2 in the Gaifman graph G D . Recall from the definition of H that, at the level of sets, C = D/∼ where ∼ is the equivalence relation generated by I D . If d ∈ D, let [d] denote its ∼-equivalence class. For each d ∈ D, set ξ(d) := min {f (d ) | d ∈ [d]}. 4. 3 . 3A proof of Theorem 16. We start by showing that the functor H : Σ + → Σ sending D to D − /∼, where D − is the σ-reduct of D and ∼ is the equivalence relation generated by I D , is left adjoint to J. Lemma 25. H : Σ + → Σ is left adjoint to the embedding J. Proof. It suffices to show that for every D ∈ Σ + , the unique σ + -homomorphism e D : D → JH(D) extending the quotient map D − H(D) satisfies the following universal property: For every σ + -homomorphism j : D → J(C), there exists a unique σ-homomorphism h : H(D) → C making the following diagram commute. Proposition 28 . 28A rooted Kripke structure (A, a) admits a coalgebra structure (A, a) → M k(A, a) if, and only if, (A, a)is a synchronization tree of height ≤ k. Proposition 29 . 29For finite Kripke structures (A, a) and (B, b), k Proof. (1) For any finite Kripke structure (A, a), let R(A, a) be the substructure of (A, a) consisting of the elements accessible from a. To show that this assignment extends to a functor R that is a right adjoint to the inclusion RKri f → Kri f , it is enough to observe that, for any (A, a) ∈ Kri f , (B, b) ∈ RKri f , and homomorphismh : (B, b) → (A, a), there exists a unique homomorphism h : (B, b) → R(A, a) such that h = e • h , where e : R(A, a) → (A, a) is the substructure embedding. (2) Let R be the functor sending a coalgebra α : (A, a) → M k (A, a) to its restriction α R(A,a) : R(A, a) → M * k R(A, a). Then RF M k (A, a) coincides with F M * k R(A, a), for any (A, a) in Kri f , because the carrier of F M k (A, a) = (M k Theorem 31 . 31Let (A, a) and (B, b) be finite (pointed) Kripke structures. Then (A, a) ≡ C ML k (B, b) if, and only if, | hom Kri f ((C, c), (A, a))| = | hom Kri f ((C, c), (B, b))| for every finite synchronization tree (C, c) of height ≤ k. Proof. Let A := (A, a) and B : This minimum exists because {f (d ) | d ∈ [d]} is a connected subset of F . (Alternatively, note that ξ(d) coincides with the minimum of {f It is customary in graph theory to use k − 1, instead of k, so that trees have tree-width 1. Grohe proved this result for undirected vertex-coloured graphs, but pointed out that his proof can be extended to arbitrary σ-structures. This concludes the proof. AcknowledgmentWe acknowledge useful discussions in the Cambridge-Oxford Comonads seminar. The last-named author is grateful to André Joyal for explaining to him (private communication) the construction E( -) → E -for polyadic spaces[16], which features in Lemma 12.Pick an arbitrary σ + -homomorphism j : D → J(C). Since j preserves the relation I, d 1 ∼d 2 entails j(d 1 ) = j(d 2 ) for all d 1 , d 2 ∈ D. Hence, the following map is well definedTo conclude, note that h is unique with this property because e D is surjective.Remark 26. Since both J and H send finite structures to finite structures, they restrict to an adjunction H J : Σ f → Σ + f . Before proceeding to the proof of Theorem 16, we make the following easy observation.Lemma 27. Assume that J and H restrict to full subcategories A ⊆ Σ f and A + ⊆ Σ + f . Then, for any A, B ∈ Σ f , the following are equivalent:Proof. Assume item 1 holds. For any C ∈ A + ,Conversely, suppose item 2 holds. Then, for any C ∈ A, The pebbling comonad in finite model theory. S Abramsky, A Dawar, P Wang, 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS. S. Abramsky, A. Dawar, and P. Wang. The pebbling comonad in finite model theory. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, pages 1-12, 2017. Comonadic semantics for guarded fragments. S Abramsky, D Marsden, arXiv:2008.11094Preprint available atS. Abramsky and D. Marsden. Comonadic semantics for guarded fragments. Preprint available at arXiv:2008.11094, 2020. Relating Structure and Power: Comonadic semantics for computational resources. S Abramsky, N Shah, 27th EACSL Annual Conference on Computer Science Logic, CSL. 2S. Abramsky and N. Shah. Relating Structure and Power: Comonadic semantics for com- putational resources. In 27th EACSL Annual Conference on Computer Science Logic, CSL, pages 2:1-2:17, 2018. Relating Structure and Power: Extended version. S Abramsky, N Shah, arXiv:2010.06496Journal of Logic and Computation. preprint available atTo appear inS. Abramsky and N. Shah. Relating Structure and Power: Extended version. To appear in Journal of Logic and Computation, preprint available at arXiv:2010.06496, 2021. Abstract and concrete categories. J Adámek, H Herrlich, G E Strecker, John Wiley & Sons, IncNew YorkJ. Adámek, H. Herrlich, and G. E. Strecker. Abstract and concrete categories. John Wiley & Sons, Inc., New York, 1990. Locally presentable and accessible categories. J Adámek, J Rosický, London Mathematical Society Lecture Note Series. 189Cambridge University PressJ. Adámek and J. Rosický. 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[ "arXiv:physics/9808003v3 [physics.acc-ph] Concentrator of laser energy for thin vapour cloud production near a surface", "arXiv:physics/9808003v3 [physics.acc-ph] Concentrator of laser energy for thin vapour cloud production near a surface" ]
[ "P I Melnikov ", "B A Knyazev ", "J B Greenly ", "\nBudker Institute of Nuclear Physics\nLaboratory of Plasma Studies\nNovosibirsk State University\n630090, 630090Novosibirsk, NovosibirskRussia., Russia\n", "\nCornell University\nUpson Hall 36914853IthacaNY\n" ]
[ "Budker Institute of Nuclear Physics\nLaboratory of Plasma Studies\nNovosibirsk State University\n630090, 630090Novosibirsk, NovosibirskRussia., Russia", "Cornell University\nUpson Hall 36914853IthacaNY" ]
[]
A novel scheme is presented for production of a thin (< 1 mm) uniform vapor layer over a large surface area (> 100 cm 2 ) by pulsed laser ablation of a solid surface.Instead of dispersing the laser energy uniformly over the surface, a modified Fabry-Perot interferometer is employed to concentrate the laser energy in very narrow closely-spaced concentric rings. This approach may be optimized to minimum total laser energy for the desired vapor density. Furthermore, since the vapor is produced from a small fraction of the total surface area, the local ablation depth is large, which minimized the fraction of surface contamination in the vapor.
10.1016/s0168-9002(98)00453-7
[ "https://export.arxiv.org/pdf/physics/9808003v3.pdf" ]
119,046,261
physics/9808003
871cdbc422e038674a86457e290fd6f78fa738f7
arXiv:physics/9808003v3 [physics.acc-ph] Concentrator of laser energy for thin vapour cloud production near a surface 9 Sep 1998 P I Melnikov B A Knyazev J B Greenly Budker Institute of Nuclear Physics Laboratory of Plasma Studies Novosibirsk State University 630090, 630090Novosibirsk, NovosibirskRussia., Russia Cornell University Upson Hall 36914853IthacaNY arXiv:physics/9808003v3 [physics.acc-ph] Concentrator of laser energy for thin vapour cloud production near a surface 9 Sep 1998laser evaporation, thin gas layer formation A novel scheme is presented for production of a thin (< 1 mm) uniform vapor layer over a large surface area (> 100 cm 2 ) by pulsed laser ablation of a solid surface.Instead of dispersing the laser energy uniformly over the surface, a modified Fabry-Perot interferometer is employed to concentrate the laser energy in very narrow closely-spaced concentric rings. This approach may be optimized to minimum total laser energy for the desired vapor density. Furthermore, since the vapor is produced from a small fraction of the total surface area, the local ablation depth is large, which minimized the fraction of surface contamination in the vapor. Introduction In principle, an ideal plasma-based ion source for an ion beam would provide a perfectly uniform, very thin layer of fully ionized plasma with a single ion species, at a temperature low enough to introduce negligible beam divergence. For magnetically insulated ion diodes for light-ion inertial fusion drivers, these criteria in practice would require a plasma layer less than 1 mm thick, with areal density greater than 10 15 cm −2 , uniform to within 10% to prevent unacceptable perturbation of the plasma surface smoothness causing beam divergence, and ion temperature below 50 eV. Pulsed-laser ablation of vapor from a surface is one possible means of providing a gas layer to form such a plasma, before application of the high-voltage pulse. This gas layer can be ionized either by near-surface discharge in the applied field, or in advance, by means of photoresonance laser ionization of the vapor [1]. By this technique one could obtain space-charge-limited current from all the anode plasma surface and, consequently, to eliminate additional sources of the ion beam divergence [2]. Work at Sandia National Laboratories [3] has produced anode vapor layers in this way, by dispersing laser energy over the desired anode surface area. High uniformity and purity is very difficult to achieve by this method, especially since the areal density of vapor desired is rather small, but the area is large (> 100 cm 2 ), requiring the laser energy to be dispersed uniformly to an energy density not far above the threshold for ablation, a regime in which vapor production is a strong function of laser energy density and surface contaminants may dominate vapor production. In this paper we investigate an alternative scheme in which the laser is intentionally concentrated into a series of concentric narrow rings. Vapor is produced in these rings, which are closely enough spaced (∼ 0.5 mm) that the ablated clouds merge to an adequately uniform layer of 0.5 mm thickness. The concentration allows the evaporation to be done at a higher, optimum power density for a particular laser and anode material, so that the overall laser energy is minimized (see [4,5]). In addition, for a given total amount of vapor, the depth of ablation in the rings is larger than for uniform illumination by the ratio of total area to ring area, so surface contaminants to bulk material ratio could be much less in comparison with a case of the disperse radiation distribution at a lower power density. Principle of operation The scheme of the concentrator is shown in Fig.1. The concentrator is based on a Fabry-Perot interferometer. A laser beam is introduced by an input focusing lens into the interferometer through a small hole in the first mirror. The output lens focuses the light after the interferometer to produce concentric thin rings at the target surface. Most of the energy of the laser concentrates in these rings on the surface, except only the central part of the beam which is lost by reflection back through the hole. The device including the input lens, the interferometer, and the output lens we will refer to as a concentrator. The power density in the narrow rings is chosen to be high enough to evaporate surface material efficiently. The amount of material evaporated per unit laser energy peaks at a particular optimum value. If this value is chosen for the rings of illumination, a desired area-averaged vapor density is produced with minimum laser energy. We will use the following terms: r 1 , r 2 are reflection coefficients for first and second mirrors (r 1 ≃ 100%, r 2 < 100%); τ 1 , τ 2 are transmission coefficients (defined as τ = 1 −r); λ is the wavelength of laser generation, ∆ is the distance between mirrors. Consider a ray of the laser beam focused by the input lens with intensity I 0 on the input of the concentrator and angle θ to the axis. The intensity of this ray behind the back mirror is τ 2 I 0 . The intensity of the reflected part of the ray after the first reflection from the front mirror and passing through the back mirror is r 2 r 1 τ 2 I 0 , and after kth reflection -(r 2 r 1 ) k τ 2 I 0 . These rays interfere at the focal plane of the output lens. The electric field amplitude of the kth is equal to E k = 8π(r 1 r 2 ) k τ 2 I 0 · e (k−1)ϕi , k = 0, 1, 2... ,(1) where ϕ = 4π ∆ λ cos θ .(2) is the phase difference of subsequent rays. The number of the interfering rays N is limited by the radius R of the interferometer, and resulting the amplitude on the target is E = √ 8πτ 2 I 0 N k=0 ( √ r 1 r 2 ) k e ikϕ , or E = 8πτ 2 I 0 ( √ r 1 r 2 e iϕ ) N +1 − 1 √ r 1 r 2 e ϕi − 1 ,(3) where N = R/(2∆ tan θ). Consequently the distribution of the intensity at a limited N is I = EE * 8π = τ 2 I 0 (r 1 r 2 ) N (r 1 r 2 − 2 √ r 1 r 2 cos (N + 1)ϕ) + 1 (1 − √ r 1 r 2 ) 2 + 4 √ r 1 r 2 sin 2 (ϕ/2) .(4) For N large enough (for small θ) this expression reduces to the well known Airy-function I = I 0 τ 2 (1 − √ r 2 r 2 ) 2 + 4 √ r 1 r 2 sin 2 (ϕ/2) .(5) The intensity I in maxima increases rapidly with r 1 and r 2 close to 1. For this case √ r 2 r 1 ≃ 1 − (τ 1 + τ 2 )/2, and I = I max 1 + (16/(τ 2 + τ 1 ) 2 ) sin 2 (ϕ/2) , I max = 4τ 2 I 0 (τ 1 + τ 2 ) 2 .(6) For the distribution of laser radiation over the target surface "intensities" I and I 0 in the above written formulae have to be replaced by the radiation power densities F = dP/dS and F 0 = dP 0 /dS, which are functions of radius ρ and azimuthal angle ψ on the target (dS = ρdρdψ). In contrast to a conventional Fabry-Perot interferometer, where the power density cannot be higher then the initial power density, for the concentrator the power density in the peaks can be much higher than the incident power density. For example, if τ 1 = 0 and τ 2 = 0.04, F max = 100F 0 , e.g. the concentrator multiplies the peak power density one hundred times. There may be substantial advantages of such an energy distribution for surface gas layer production. The maximum intensity for all the rings would be equal if the initial laser beam is uniform, and the averaged power density over any area of dimension larger than the ring spacing is equal to < F >= F 0 , so that for F 0 =const the distribution of averaged power density from the concentrator would be constant. The multiplication factor K = 4/τ 2 (if τ 1 = 0) gives not only the power density multiplication in the rings but also the ratio of total area to the illuminated ring area. 3 Vapor cloud production: a practical example The concentrator described can be applied to thin vapor cloud production near an anode surface of ion accelerator. Figure 2 shows an example of using of the concentrator in a magnetically insulated ion diode. The upper diagram shows direct illumination of the anode for simplicity, while the lower diagram shows a way to remove the concentrator from the ion beam path. Let us introduce the parameters that must be involved in the calculation process. R 1 , R 2 are the inner and outer radii of the anode, δR 2 is the distance between illuminated rings on anode surface, F opt is power density needed to evaporate the anode material. The parameters of the concentrator are: f 1 , f 2 , the foci of the input and output lenses of the concentrator; α, β, the angles of the input rays that are directed by the output lens to the R 2 and R 1 radii; R the radius of the interferometer; ∆, the distance between the mirrors of the interferometer; l, the distance between the output mirror and the focus of the input lens; h, the radius of the hole; K the multiplying factor; P , the energy density on the output mirror surface. The laser parameters are: λ, the generation wavelength; δλ, the bandwidth of laser generation; t, the pulse duration; Q, the pulse energy. We now estimate the parameters needed for realizing the scheme. We choose the COBRA ion diode [6] as an example. For this diode, R 1 = 7.2 cm, R 2 = 9.5 cm. We desire δR 2 = 0.05 cm, for adequate uniformity of the vapor layer on a distance 0.05 cm , for optimum efficiency of vapor production. Experiments on vapor cloud production (see [8]) show that for every target there is a range near an optimal power density where it is possible to obtain low-ionized cold vapors without the laser-induced breakdown. It is significant also that near this value of power density the amount of evaporated material and the velocity of vapor boundary are only weakly dependent on power density [9]. Thus the vapor layer could be more uniform than the initial laser power density F 0 . We will use the value f 2 = 1.5 m for the output lens focus, which would work in the configuration of Fig.2a for the present COBRA diode. The value δR 2 defines the distance between mirrors. Actually, the maxima of power density (6) correspond to the zeros of sin(ϕ/2). So two nearby maxima differ by angle δθ when (ϕ 1 − ϕ 2 )/2 = π π = 2π ∆ λ (cos θ 1 − cos θ 2 ) = 2π ∆ λ θδθ ,(7)δR = δθf 2 = λ 2∆ f 2 θ = λ 2∆ f 2 2 R . The distance between mirrors is ∆ = λ δR 2 f 2 2 2R 2 = 1.65 cm .(8) The angles α and β are related to the R 2 and R 1 by α = R 2 /f 2 = 6.3·10 −2 , β = R 1 /f 2 = 4.8 · 10 −2 . The distance between input lens focus point and output mirror surface may now be defined l = R 2 + R 1 R 2 − R 1 ∆ = 7.26 · ∆ = 12 cm ,(9) and the hole radius is h = (l − ∆)α = 0.65 cm ,(10) The required radius of the interferometer can be determined as the radius at which the laser beam with the biggest angle α is attenuated by a factor of e −2 . R = K R 2 f 2 ∆ = 10.5 cm .(11) The number of interfered rays is N = 2/τ 2 = K/2 = 50 for this interferometer radius. We estimate the bandwidth of laser generation that is allowable to provide such narrow rings. The bandwidth must not lead to widening of the ring more than the width of the half height of the power density. From (2) and (6) we get τ 1 + τ 2 2 > 4π∆ δλ λ 2 , δλ λ < (τ 1 + τ 2 ) λ 8π∆ = λ 2πK∆ = 6.7 · 10 −8 ,(12) δλ < 4.7 · 10 −5 nm . The last value is close to a theoretical limit and impose strong requirement on the laser spectrum. The main critical parameter of the concentrator is the energy density in the center of the output mirror surface. The mirror must be of high quality to be not destroyed by the intense laser beam. P = F Al opt t f 2 l 2 K −1 = 3.6 J/cm 2 .(13) This value is not too high, but the local energy density can be 3 times more due to interference effect. So the mirror must withstand the energy density of 11 J/cm 2 . Mirrors of such quality are available [10]. The total energy of laser can be calculated from the formula Q = F Al opt tπR 2 2 K −1 = 6.5 J. Only 40% of the energy is used in the evaporation, because of noncorrespondence of the solid circular laser beam and the hollow annular anode surface. It is more efficient to use an annular laser beam. A beam with such a structure can be produced by a Nd laser with nonstable resonator. Use of 1.06 µm light leads to an increase of concentrator size (see (8,9,10,11)): ∆ = 2.52 cm, l = 7.26 · ∆ = 18.3 cm, R = 16.0 cm, h = 1 cm. But the energy density on the output mirror surface would decrease (see (13)) P = 1.6 J/cm 2 . Using the 2nd harmonic of Nd laser generation gives twice smaller size (see (8,9,10,11)): ∆ = 1.26 cm, l = 7.26 · ∆ = 9.15 cm, R = 8 cm, h = 0.5 cm. However, the energy density becomes four times higher (see (13)), and reaches a rather high value P = 6.4 J/cm 2 . To use the concentrator in such a regime would require a mirror that would not damaged by the energy density 3P = 20 J/cm 2 . This value is over the damage threshold for the best mirror coatings [10]. But it is possible that the mirror coating can withstand this energy density because averaged energy density is less 12 J/cm 2 , and the value above damage threshold is reached only in very thin rings of 20λ of thickness that would provide fast heat diffusion over the surface due to thermal conductivity. Conclusion The production of a very uniform, thin vapor layer above a solid surface by laser ablation is a difficult problem. We have suggested a method of distributing laser energy in a series of narrow, high intensity rings that could be advantageous in producing vapor efficiently, and, probably, with minimum sensitivity to surface contaminants and laser intensity variation. If this concentrated vapor production can be allowed to expand and merge into a uniform layer, these advantages might be realized. We leave aside of this paper consideration of the problem of possible small-scale non-uniformity of the gas layer produced with the concentrator as well possible anode plasma instability. Obviously, at first the technique suggested has to be verified experimentally, and the anode layer parameters must be measured. Acknowledgments This work was supported, probably in part, by U.S. Civilian Research and Development Foundation, Award RP1-239, and Russian Ministry of General and Professional Education. Figure 1 : 1Scheme of the concentrator. Figure 2 : 2Thin gas layer formation on the anode with the concentrator. from the anode surface. For evaporation by 10 ns laser pulse of an aluminium target we choose F max = F Al opt ≃ 230 MW/cm 2 [7] Fig .2b shows a possible arrangement in a focusing diode geometry. The value of f 2 may be adjusted to avoid interference of optical elements with the ion beam. We assume a ruby laser with wavelength 694.3 nm. For definiteness we put K = 100, but this figure might be varied. . H Bluhm, B A Knyazev, P I Melnikov, Tech. Phys. Lett. 23343H. Bluhm, B.A. Knyazev, P.I. Melnikov et al., Tech. Phys. Lett. 23 (1997) 343. . S A Slutz, Phys. of Fluids B. 42645S.A.Slutz, Phys. of Fluids B 4 (1992) 2645. R V Stinnet, J E Bailey, K W Bieg, Proc. of the 8th Intern. Conf. on High-Power Partical Beams. of the 8th Intern. Conf. on High-Power Partical BeamsNovosibirsk, USSR167R.V. Stinnet, J.E. Bailey, K.W. Bieg et al., Proc. of the 8th Intern. Conf. on High- Power Partical Beams, Novosibirsk, USSR, 1990, p.167. B A Knyazev, S V Lebedev, V N Snytnikov, Book of abstracts of All-Union Conf. on Low-Temperature Plasma Physics. Kiev, USSR339in RussianB.A. Knyazev, S.V. Lebedev, and V.N. Snytnikov, Book of abstracts of All-Union Conf. on Low-Temperature Plasma Physics, Kiev, USSR, 1979, p.339. (in Russian). . B A Knyazev, S V Lebedev, K I Mekler, Sov. Phys. Tech. Phys. 31773B.A. Knyazev, S.V. Lebedev, and K.I. Mekler, Sov. Phys. Tech. Phys. 31 (1986) 773. E Krastelev, B A Knyazev, F Lindholm, Proc. of the 11th Intern. Conf. on High-Power Particle Beams. of the 11th Intern. Conf. on High-Power Particle BeamsPrague, Czech Republic1143E. Krastelev, B.A. Knyazev, F. Lindholm et al., Proc. of the 11th Intern. Conf. on High-Power Particle Beams, Prague, Czech Republic, 1996, p.1143. . B A Knyazev, B I Kulikov, S V Lebedev, E P Fokin, 29Novosibirsk, USSRInstitute of Nuclear PhysicsIn RussianB.A. Knyazev, B.I. Kulikov, S.V. Lebedev, and E.P. Fokin, Preprint 80-208, Institute of Nuclear Physics, Novosibirsk, USSR, 1980. 29 p. (In Russian). Abstracts of 8th Workshop on Atomic and Molecular Physics for Ion-Driven Fusion. B A Knyazev, 67Heidelberg, GermanyB.A. Knyazev, Abstracts of 8th Workshop on Atomic and Molecular Physics for Ion-Driven Fusion, Heidelberg, Germany, P.67 (1997) . N G Basov, V A Boiko, V A Dement&apos;ev, Soviet Physics JETP. 24659N.G. Basov, V.A. Boiko, V.A. Dement'ev et al., Soviet Physics JETP 24 (1967) 659. C Fournet, B Pinot, B Geenen, Proc. SPIE 1624. SPIE 1624282C. Fournet, B. Pinot, B. Geenen et al., Proc. SPIE 1624 (1991) 282.
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[ "Multiphonon emission model of spin-dependent exciton formation in organic semiconductors", "Multiphonon emission model of spin-dependent exciton formation in organic semiconductors" ]
[ "M Wohlgenannt \nDepartment of Physics and Astronomy\nThe University of Iowa\n52242-1479Iowa CityIA\n" ]
[ "Department of Physics and Astronomy\nThe University of Iowa\n52242-1479Iowa CityIA" ]
[]
The maximum efficiency in organic light-emitting diodes (OLEDs) depends on the ratio, r = kS/kT , where kS (kT ) is the singlet (triplet) exciton formation rate. Several recent experiments found that r increases with increasing oligomer length from a value r ≈ 1 in monomers and short oligomers. Here, we model exciton formation as a multi-phonon emission process. Our model is based on two assertions: (i) More phonons are emitted in triplet formation than in singlet formation. (ii) The Huang-Rhys parameter for this phonon emission is smaller in long oligomers than in short ones. We justify these assertions based on recent experimental and theoretical data.
10.1103/physrevb.71.165111
[ "https://export.arxiv.org/pdf/cond-mat/0410458v1.pdf" ]
119,046,343
cond-mat/0410458
ae8cdfde7458f7e1c59a207064b9b13fccb1857b
Multiphonon emission model of spin-dependent exciton formation in organic semiconductors 18 Oct 2004 M Wohlgenannt Department of Physics and Astronomy The University of Iowa 52242-1479Iowa CityIA Multiphonon emission model of spin-dependent exciton formation in organic semiconductors 18 Oct 2004(Dated: January 10, 2022)APS/123-QEDnumbers: 7350Jt7350Gr7860Fi The maximum efficiency in organic light-emitting diodes (OLEDs) depends on the ratio, r = kS/kT , where kS (kT ) is the singlet (triplet) exciton formation rate. Several recent experiments found that r increases with increasing oligomer length from a value r ≈ 1 in monomers and short oligomers. Here, we model exciton formation as a multi-phonon emission process. Our model is based on two assertions: (i) More phonons are emitted in triplet formation than in singlet formation. (ii) The Huang-Rhys parameter for this phonon emission is smaller in long oligomers than in short ones. We justify these assertions based on recent experimental and theoretical data. I. INTRODUCTION The maximum possible internal quantum efficiency, η max , of fluorescent-based organic light emitting diodes (OLEDs) occurs when the probability that the injected carriers form excitons and the quantum yield for singlet emission are both unity. η max is then determined by (and identical to) the fraction, f s of injected electrons and holes (or negative and positive polarons, respectively) that pair to form emissive spin-singlet excitons, rather than nonemissive triplet excitons. If the process by which these excitons form were spin independent, then η max would be limited to 25% based on spin-degeneracy. However, recent reports indicate that η max in OLEDs ranges between 22% to 83% [1,2,3,4,5,6,7,8]. The exact value of η max and the reason for this variation, however, have remained controversial. Indeed, even the notion that η max can be larger than 25 % is currently not universally accepted [8]. A. Overview of experimental results Two entirely different experimental approaches have been employed to study spin-dependent exciton formation for OLEDs and thin films: (i) Experiments [1,2,3,5,7,8] that determine the singlet generation fraction f S directly in live OLEDs. For fluorescent devices typically only the singlet emission can be measured, information on triplet density is missing and rather involved models have to be employed to obtain f S [3]. Wilson et al. however, have recently shown [5] that in OLEDs made from organic semiconductors that exhibit spin-orbit coupling, the strong intersystem crossing implies that both singlet and triplet emission (fluorescence and phosphorescence) can be simultaneously observed. This could be used to reliably * Electronic address: [email protected] determine f S by comparing the relative intensities of fluorescence to phosphorescence for optical excitation (where initially only singlet excitons are formed) with that for electrical excitation (where both singlet and triplet excitons are formed). Importantly, they found f S = 57 % for devices made from a Platinum-containing polymer, but f S = 22 % for the corresponding monomer OLEDs. This suggests that exciton formation is spin-independent for the monomer, but that a spin-dependent formation process is effective in the polymer. (ii) Experiments [4,6,9] that measure the ratio, r = k S /k T of the spin-dependent exciton formation rates for singlet and triplet excitons, respectively. Such experiments manipulate the spin state (using electron spin resonance techniques) of the pairing polarons, and measure the effect on exciton formation rates. These experiments consider photogenerated polarons in the film and use the fact that antiparallel spin polaron pairs can either form singlet or triplet excitons, whereas parallel spin pairs can only form triplets. These optically detected magnetic resonance (ODMR) techniques are modulation experiments where the resonant µ-wave field is periodically turned on and off. Since the experiment is performed at low temperature, spin-alignment is conserved during the half-wave with µ-wave field off, and polaron recombi- Magnetic-resonance experimental data for the ratio r −1 =kT /kS of spin-dependent exciton formation rates in various polymers and oligomers as a function of the peak photon energy of the P1 transition (lower x-axis). r −1 is also shown as a function of the inverse conjugation length 1/CL (upper xaxis), which was determined from P1 (see text for discussion). The line through the data points is a linear fit. *The P1 band of this polymer does not show a clear peak in the PA spectrum, the P1 band extends to the longest wavelengths measured. **The length of this oligomer was calculated. In addition to the chemical names defined in the text, 3PE stands for the PPE trimer, PPE for poly(phenylene-ethynylene), Si-PT for silicon bridged polythiophene. For details consult original publications. nation/exciton formation obeys spin-statistics. However, during the half-wave with µ-wave field on, spin-1/2 resonance leads to rapid spin-flips of the recombining polarons. Spin alignment is therefore not conserved, and each pair may choose whether to form singlet or triplet exciton. It can easily be shown [4,6] that this leads to enhanced formation of the exciton with larger formation rate (leading to positive ODMR signal), at the expense of the more slowly forming exciton (that gives negative ODMR). In addition, the overall polaron recombination rate is enhanced, since the fast channel becomes allowed for all polaron pairs. Therefore changes occur in the photoinduced absorption (PA) from the triplet state, as well as the fluorescence from the singlet state upon magnetic resonance. In particular, from the µ-wave induced change in PA of the polaron pairs, r = k S /k T could be determined [4]. B. Polarons in π-conjugated semiconductors It is well-known that chemical doping or electrical charge injection results in the formation of polarons in πconjugated semiconductors [10]. Fig. 1 shows a comparison between different models that have been used for describing polarons in π-conjugated semiconductors. Panel (a) depicts the electron-phonon (e-p) or Su-Schrieffer-Heeger (SSH) model [10,11,12]. It predicts that the e-p coupling causes a gap between valence and conduction band. In the singly charged system two localized polaron levels appear inside the gap. Experimentally one finds two optical transitions [13] that are interpreted as the P 1 and P 2 transitions. Panel (b) depicts the molecular orbital picture where HOMO and LUMO are the highest occupied and lowest unoccupied molecular orbitals, respectively. The most generic model for polarons is the molecular crystal or Holstein model [11]; it is however not expected to be applicable in a quantitative way to π-conjugated polymers. This model yields for the "P 1 " transition [11]: P 1 = A 2 2M ω 2 E 2 1 W(1) Here A quantifies the e-p coupling strength (e.g. in eV/Å), M is the ionic mass and ω E is the Einstein phonon frequency, W is the band width before inclusion of e-p coupling. The term in the bracket is the energy, V associated with the e-p coupling [14]. Furthermore, the following two equations hold for the polaron binding energy, E b,polaron , and the deformation or relaxation energy E relax,polaron , respectively. E b,polaron = 1 3 P 1 (2) E relax,polaron = 2 3 P 1(3) Π-conjugated oligomers are often used as model compounds instead of π-conjugated polymers because they can be obtained with a well-defined chemical structure. Although the molecular weight of polymers is typically much larger than that of oligomers, nevertheless it is established that the polymer should be viewed as a string of effectively independent segments, separated by chemical or physical defects. The length of these segments is called the conjugation-length (CL). Several recent calculations found that the energy associated with the e-p coupling decreases with increasing oligomer size. In particular, Devos and Lannoo found that V = const/N in a more or less universal manner in acenes and fullerenes with various numbers, N of π-bonds [14]; and Shuai et al. [15] found that E b,polaron in oligophenyls (OP) decreases with the number of phenyl rings in a similar manner. We may therefore state that, in average, that effect of e-p coupling significantly decreases with increasing oligomer length. C. Results of magnetic resonance experiments Using ODMR, it was found that r is a monotonously increasing function of the conjugation-length (CL), and, by extrapolation, that r ≈ 1 for small molecules and monomers [6]. Electroluminescence [5] and magnetic resonance experiment therefore lead to the same qualitative conclusions, namely that exciton formation is spinindependent for the monomer, but that a spin-dependent process is effective in the polymer. Fig. 2 shows the ratio r −1 =k T /k S of spin-dependent exciton formation rates in various polymers and oligomers as a function of the peak photon energy of the P 1 transition (lower x-axis) obtained by ODMR spectroscopy [6]. In the original publication the experimental dependence r(P 1 ) was however reinterpreted in terms of the CL of the polymer films studied. This was possible, because it is known [16] that there exists a (almost) universal relationship between P 1 and the material's CL (see Fig. 3). r −1 is therefore also shown as a function of 1/CL (upper x-axis), which was determined from P 1 . D. Recent theoretical results Primary excitations in these materials are generally believed to be excitons with a binding energy in excess of kT, where k is the Boltzmann constant and T the temperature. As a result of electron-phonon and electronelectron interactions, the lowest singlet (S 1 ) and triplet (T 1 ) excitons posses both different energies (the S 1 − T 1 energy difference; the exchange energy K for the lowest excitations has been measured [18] to be K = 0.7eV in a variety of conjugated polymers) and different spatial wavefunctions (with T 1 displaying a more spatially confined character). Our model is based on recent work of Beljonne et al. [17]. They developed a theoretical model to describe intermolecular charge recombination in conjugated materials. In their treatment, Beljonne et al. found it necessary to consider two configurations of the two polymer chains involved in the exciton formation/polaron recombination process, namely cofacial and head-to-tail (see Fig. 4). They found that in the cofacial arrangement, by far the largest matrix elements are calculated for the lowest singlet S 1 and triplet T 1 excited states (see Fig. 4 a). Similar results were obtained also by Tandon et al. [19]. The situation is quite different for the head-to-tail configurations, where a number of different singlet and triplet excited states show significant electronic couplings to the polaron pair states (see Fig. 4 b). The following picture has therefore emerged: Based on the electronic coupling, polaron recombination is a direct transition predominantly to the lower lying exciton states. Such transitions, however, have to pay a high price, since the multi-phonon emission necessary to conserve energy has very low probability for reasonable values of the Huang-Rhys parameter. Therefore, unless the two chains are in exact cofacial arrangement, the exciton states formed with highest probability may not be the lowest exciton states. We may therefore adopt a picture where exciton formation has highest probability for an intermediate exciton state. E. Theoretical approach In the case of charge-recombination (CR) processes, the semiclassical expression for the CR rate writes, within the Franck-Condon approximation, as [17]: k CR = 2π | ψ i |W | ψ f | 2 1 4πλ S kT 1/2 × (4) × ν F 0ν exp − ∆G 0 + λ S + ν ω ph 2 4λ S kT Here |ψ i > and |ψ f > are the wavefunctions of the initial and final states, respectively, W is the perturbation, ω ph is the energy of the most strongly coupled (optical) phonon, F 0ν is the Franck-Condon factor of the transition with zero and ν phonons in the initial and final states, respectively, G 0 is the difference in free energy between intial and final state, and λ S is the (external) reorganization energy. Next we approximate the above expression, since the largest contribution will result from transitions that conserve energy, namely those for which ∆G 0 + λ S + ν ω ph ≈ 0. This can always be achieved, at least approximately, through emission of a number ν E of phonons: ν E = ∆G 0 + λ S ω ph(5) We therefore obtain for k CR : k CR = 2π | ψ i |W | ψ f | 2 1 4πλ S kT 1/2 F 0νE (6) The ratio, r ≡ kS kT is therefore given as: r = nS k CR,Sn S nT k CR,Tn T(7) where the sums extend over all singlet states and triplet states, respectively. For the sake of simplicity, we now replace the sums in Eq. 7 by a single "effective" state S and T , respectively. This effective state may be loosely identified with the intermediate exciton level discussed above, for which the exciton formation rate has maximum probability averaged over the ensemble of configurations. This step is justified in more detail in Appendix A. With this definition in mind we may write: r = k CR,S k CR,T = r W F 0νS F 0νT(8) where the subscript S and T denote the "effective" singlet and triplet states, and ν S and ν T denote the number of phonons required for energy conservation to form the S and T states, respectively. The definition of r W follows by comparison of Eq. 8 to Eq. 7, and is essentially the ratio of the electronic matrix elements for singlet and triplet formation, respectively. Next we discuss the calculation of F 0ν , where we closely follow the treatment by Barford et al. [20] Inter-molecular interconversion is an iso-energetic process which occurs from the lowest vibrational levels of the initial polaron pair state to the final, intra-molecular exciton states at the same energy as the initial level. In actuality, the exciton formation process involves two conformational transitions, namely the transition from the polaron lattice conformation to the exciton conformation in chain 1 and from polaron to groundstate conformation in chain 2. We therefore need to generalize our treatment in the following way: F 0νE = ν1ν2 F (1) 0ν1 F (2) 0ν2 δ(E f − E i ) (9) = ν1 F (1) 0ν1 F (2) 0(νE −ν1)(10) where F (1) 0ν1 and F (2) 0ν2 are the Franck-Condon factors associated with the vibrational wavefunction overlaps of chains 1 and 2, respectively. Next we want to write out the expression for the Franck-Condon overlaps using the displaced oscillator expression F 0ν = e −S S ν /ν!. A useful simplification to this expression arises by noting that the geometric distortions of the polarons and exciton polarons (namely the 1 1 B u or 1 3 B u states) from the ground state structure are very similar. [21] Thus, the Huang-Rhys parameter, S 1 for the 1 1 B u and 1 3 B u states relative to the positive polaron is quite small. Therefore, to the lowest approximation, only ν 1 = 0 has a non-negligible contribution, and S total ≈ S 2 ≡ S P , where the P stands for polaron conformation. In the next higher approximation, we may also include the term ν 1 = 1. As we show in Appendix B, if S 1 ≪ 1 and S 1 ≪ S P , then we may combine the two Franck-Condon factors into a single one with Huang-Rhys parameter, S = S 1 + S P . Therefore: r = r W (S 1 + S P ) −(νT −νS ) ν T ! ν S ! (11) = r W S 1 + E relax,polaron ω ph − ∆ S/T ω ph ν T ! ν S !(12) Eqs. 11 and 12 are the final result of our model. II. COMPARISON BETWEEN EXPERIMENTAL RESULTS AND MODEL Guided by the magnetic resonance experimental data shown in Fig. 2, we now study the CL dependence of Eq. 11. The CL-dependence of r W was studied previously by Beljonne et al. [17] They found that r W depends only weakly on the CL, in particular this dependence is not strong enough to account for the material dependence of r shown in Fig. 2. We must therefore study the CL dependence of the phonon emission term in Eq. 11. There is an obvious connection between our model and the experimental magnetic resonance results. These results (see Fig. 2) are given as a function of P 1 , whereas our model can be written in terms of E relax,polaron . The Holstein model relates the two quantities through Eq. 3. A similar relationship should therefore hold between P 1 and E relax,polaron in π-conjugated polymers. However, the Holstein model is formulated for the infinite system, and does not consider finite size effects. It is less than obvious that P 1 ∝ E relax,polaron holds also in oligomers. In particular, since we are interested in phonon emission, we need to distill finite size effects on the e-p coupling energy from finite size effects on the electronic energies (quantum confinement energy). Finite size effects on the e-p coupling energy have been studied by Devos and Lannoo, who found that V = const/N in a more or less universal manner in acenes and fullerenes with various numbers, N of π-bonds [14]. We therefore expect that E relax,polaron decreases strongly as the oligomer size increases. A. The conjugation-length dependence of the polaron relaxation energy To the best of our knowledge a direct measurement of E relax,polaron has not yet been performed in piconjugated polymers and oligomers. However, Bredas and coworkers have calculated [15] E relax,polaron for phenyl-capped PPV oligomers as a function of the number of rings N against which we can compare our experimental results for P 1 . Their results are shown in Fig. 5. Fig. 5 also shows a plot of the P 1 polaron transition energy in a variety of oligomers. It is seen in Fig. 5 that both experimental data for P 1 and theoretical calculation of E relax,polaron follow a very similar dependence on CL, as a matter of fact P 1 ∝ E relax,polaron to a high degree of accuracy. This is strong evidence for the notion that P 1 is indeed a measure of E relax,polaron . Therefore both experimental and theoretical data find the that E relax,polaron monotonously decreases with increasing CL. In particular (see Fig. 3), E relax,polaron = E relax,polaron,∞ + k × 1 CL(13) Through comparison of the data shown in Figs. 2 and 5 it is therefore evident that the experimental results can be brought in agreement with Eq. 11 only if ν T − ν S = 1, therefore r −1 = S 1 + E b,polaron ω ph r W ν T(14) Importantly, since the experimental relationship P 1 (CL) is universal, then it follows naturally that r(CL) is universal. B. The conjugation-length dependence of the optical Huang-Rhys parameter Our model is based on the assertion that the Huang-Rhys parameter is substantially smaller in polymers than oligomers. Thus far, we have presented three justifications for this statement: 1. The optical polaron transition energy P 1 strongly scales with CL, specifically as P 1 = P 1,∞ +k P1 ×CL. Furthermore this scaling is universal. 2. A very similar scaling law was found by calculations of E relax,polaron vs. CL in OPV performed by Bredas and coworkers. 3. The electron-phonon energy, V was found to scale as V ∝ 1/N , where N is the number of π-atoms in a large class of conjugated compounds, largely independent of the specific chemical structure. Here we now want to add another set of data to justify the main assertion on which our model is built. The lattice relaxation of the polaron state is expected to be similar to that of the lowest singlet exciton state 1 1 B u . The corresponding Huang-Rhys parameter, S 1 1 Bu can be easily measured by optical absorption and emission. We may therefore test our assertion by determining S 1Bu from absorption and emission spectra of oligomers. Fig.6 shows experimental values for the optical Huang-Rhys parameter, S 1 1 Bu , in a series of OPV and OT. The experimental data were taken from Ref. [22] and Ref. [23]. It is seen that S 1 1 Bu decreases with increasing CL, namely S 1 1 Bu = S 1 1 Bu,∞ +k × 1 CL in agreement with our expectation based on E relax,polaron . Fig. 7 shows the fit of the experimental data for r −1 using our multi-phonon emission model together with the experimentally determined optical Huang-Rhys parameter, S 1 1 Bu . Specifically we used the fit function r −1 = S 1 1 Bu r W ν T(16) It is seen that excellent agreement is achieved. III. SUMMARY We developed a model of spin-dependent exciton formation in OLEDs. We calculated r = k S /k T based on a multi-phonon emission process. The resulting equation for r therefore strongly depends on the Franck-Condon overlap integrals that are parameterized by the Huang-Rhys parameter. Guided by recent magnetic resonance experiments, we studied the dependence of Huang-Rhys parameter on the conjugation-length. We used two different approaches: 1. We relate the Huang-Rhys parameter to the polaron relaxation energy. Then we relate E relax,polaron to the optical polaron transition P 1 , which has been measured experimentally in many oligomers. This procedure leads us to conclude that S = S ∞ + k × 1 CL . 2. We relate the Huang-Rhys parameter in our model to the Huang-Rhys parameter that can be mea-sured by photoluminescence spectroscopy. Such data are available in OPV and OT. This procedure also leads us to conclude that S = S ∞ + k × 1 CL . Our work therefore leads to the following picture of exciton formation: Since the triplet exciton states lie lower in energy than singlets, more phonons must be omitted (required by energy conservation) for triplet formation than singlet formation. Since polymers have a small Huang-Rhys factor, then the emission of many phonons is unlikely, thus favoring singlet formation. In short oligomers, however, the Huang-Rhys factor is quite large, phonons are emitted easily, and singlet and triplet formation both become likely. IV. APPENDIX A The definition of a single effective state is certainly appropriate in the cofacial configuration, since only the lowest exciton states contribute to the exciton formation process. In the head-to-tail configuration, however a whole series of states contribute. Fig. 4 b) shows that the electronic couplings are roughly equal in all states that contribute, such that we may write: n 2π | ψ i |W | ψ f,n | 2 1 4πλ S kT 1/2 F 0νE,n (17) ≈ 2π | ψ i |W | ψ f,n | 2 1 4πλ S kT 1/2 n F 0νE,n Therefore the effective state in the head-totail arrangement has an electronic matrix-element ψ i |W | ψ f,n , and Franck-Condon Factor equal to n F 0νE,n , and has a binding energy ν E ω ph , such that F 0νE = n F 0νE,n . V. APPENDIX B Including only the ν 1 = 0 and ν 1 = 1 terms we then obtain F 0νE = F (1) 00 F (2) 0νE + F (1) 01 F (2) 0(νE −1)(18) = e −S1 e −SP S νE P ν E ! + e −S1 S 1 e −SP S νE −1 P (ν E − 1)! (19) = e −(S1+SP ) (S νE P + ν E S 1 S νE P ) ν E !(20) On the other hand, the Franck-Condon factor, F 0νmax for a (hypothetical) Huang-Rhys parameter, S = S 1 +S P equals F 0νE = e −(S1+SP ) (S 1 + S P ) νE ν E !(21) We see that if S 1 ≪ S P , the two expressions are equal to first order. FIG. 1 : 1Models used to describe polaron levels and optical transitions depicted here for the positive polaron. (a) Electron-phonon (SSH) model. (b) Molecular orbital picture. FIG. 2: Magnetic-resonance experimental data for the ratio r −1 =kT /kS of spin-dependent exciton formation rates in various polymers and oligomers as a function of the peak photon energy of the P1 transition (lower x-axis). r −1 is also shown as a function of the inverse conjugation length 1/CL (upper xaxis), which was determined from P1 (see text for discussion). The line through the data points is a linear fit. *The P1 band of this polymer does not show a clear peak in the PA spectrum, the P1 band extends to the longest wavelengths measured. **The length of this oligomer was calculated. In addition to the chemical names defined in the text, 3PE stands for the PPE trimer, PPE for poly(phenylene-ethynylene), Si-PT for silicon bridged polythiophene. For details consult original publications. FIG. 3 : 3The peak photon energies of the P1 polaron transition in a variety of oligomers, namely solutions of (unsubstituted) oligophenyls (OP, △, radical anion (RA)), alkylsubstituted (AS) oligophenylene-vinylenes (OPV, •, radical cation (RC)), alkoxy-substituted OPV (⊕, RC), end-capped oligothiophenes (OT, , RC), films of AS OT (⊠, PA), AS oligothienylene-vinylenes (OTV, ▽, RC). The solid line is a fit to the data excluding ⊕. FIG. 4 : 4Charge-recombination electronic couplings into singlet and triplet excited states in 6PV. Panel a) cofacial arrangement, panel b) head-to-tail arrangement. The chargetransfer state occurs at ≈ 3.7eV. The data points were taken from Ref.[17]. FIG. 5 : 5Experimental values for P1 polaron transition energy in a variety of oligomers, namely solutions of (unsubstituted) oligophenyls (OP, radical anion (RA)), alkyl-substituted (AS) oligophenylene-vinylenes (OPV, radical cation (RC)), endcapped oligothiophenes (OT, RC), films of AS OT (measured by photoinduced absorption), AS oligothienylene-vinylenes (OTV, RC) together with calculated values for the polaron relaxation energy E relax,polaron in OPV. The experimental data for P1 were taken from Ref, whereas the calculated values for E b,polaron were taken from Ref. FIG. 6 : 6Experimental values for the optical Huang-Rhys parameter, S 1 1 Bu , in a series of OPV and OT. fit of the experimental data for r −1 using our multiphonon emission model and together with the experimentally determined optical Huang-Rhys parameter, S 1 1 Bu . . M A Baldo, D F O&apos;brien, M E Thompson, S R Forrest, Phys. Rev. B. 6014422M. A. Baldo, D. F. O'Brien, M. E. Thompson, and S. R. Forrest, Phys. Rev. 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[ "Native qudit entanglement in a trapped ion quantum processor", "Native qudit entanglement in a trapped ion quantum processor" ]
[ "Pavel Hrmo [email protected] \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n", "Benjamin Wilhelm \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n", "Lukas Gerster \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n", "Martin W Van Mourik \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n", "Marcus Huber \nTechnische Universität Wien\n1020Atominstitut, ViennaAustria\n\nInstitute for Quantum Optics and Quantum Information-IQOQI Vienna\nAustrian Academy of Sciences\nBoltzmanngasse 31090ViennaAustria\n", "Rainer Blatt \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n\nInstitut für Quantenoptik und Quanteninformation\nÖsterreichische Akademie der Wissenschaften\nTechnikerstraße 21a6020InnsbruckAustria\n\nAQT\nTechnikerstraße 176020InnsbruckAustria\n", "Philipp Schindler \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n", "Thomas Monz \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n\nAQT\nTechnikerstraße 176020InnsbruckAustria\n", "Martin Ringbauer \nInstitut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria\n", "Pavel Hrmo ", "Benjamin Wilhelm " ]
[ "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "Technische Universität Wien\n1020Atominstitut, ViennaAustria", "Institute for Quantum Optics and Quantum Information-IQOQI Vienna\nAustrian Academy of Sciences\nBoltzmanngasse 31090ViennaAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "Institut für Quantenoptik und Quanteninformation\nÖsterreichische Akademie der Wissenschaften\nTechnikerstraße 21a6020InnsbruckAustria", "AQT\nTechnikerstraße 176020InnsbruckAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria", "AQT\nTechnikerstraße 176020InnsbruckAustria", "Institut für Experimentalphysik\nUniversität Innsbruck\nTechnikerstraße 25/46020InnsbruckAustria" ]
[]
Quantum information carriers, just like most physical systems, naturally occupy high-dimensional Hilbert spaces. Instead of restricting them to a twolevel subspace, these high-dimensional (qudit) quantum systems are emerging as a powerful resource for the next generation of quantum processors. Yet harnessing the potential of these systems requires efficient ways of generating the desired interaction between them. Here, we experimentally demonstrate an implementation of a native two-qudit entangling gate up to dimension 5 in a trapped-ion system. This is achieved by generalizing a recently proposed lightshift gate mechanism to generate genuine qudit entanglement in a single application of the gate. The gate seamlessly adapts to the local dimension of the system with a calibration overhead that is independent of the dimension. 1234567890():,; 1234567890():,;
10.1038/s41467-023-37375-2
[ "https://arxiv.org/pdf/2206.04104v1.pdf" ]
249,538,497
2206.04104
ee79b7a0dd28a7040652ca813dc2bdc58824e3b3
Native qudit entanglement in a trapped ion quantum processor Pavel Hrmo [email protected] Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Benjamin Wilhelm Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Lukas Gerster Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Martin W Van Mourik Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Marcus Huber Technische Universität Wien 1020Atominstitut, ViennaAustria Institute for Quantum Optics and Quantum Information-IQOQI Vienna Austrian Academy of Sciences Boltzmanngasse 31090ViennaAustria Rainer Blatt Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Institut für Quantenoptik und Quanteninformation Österreichische Akademie der Wissenschaften Technikerstraße 21a6020InnsbruckAustria AQT Technikerstraße 176020InnsbruckAustria Philipp Schindler Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Thomas Monz Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria AQT Technikerstraße 176020InnsbruckAustria Martin Ringbauer Institut für Experimentalphysik Universität Innsbruck Technikerstraße 25/46020InnsbruckAustria Pavel Hrmo Benjamin Wilhelm Native qudit entanglement in a trapped ion quantum processor 10.1038/s41467-023-37375-2Received: 18 January 2023 Accepted: 15 March 2023 Check for updatesArticle Quantum information carriers, just like most physical systems, naturally occupy high-dimensional Hilbert spaces. Instead of restricting them to a twolevel subspace, these high-dimensional (qudit) quantum systems are emerging as a powerful resource for the next generation of quantum processors. Yet harnessing the potential of these systems requires efficient ways of generating the desired interaction between them. Here, we experimentally demonstrate an implementation of a native two-qudit entangling gate up to dimension 5 in a trapped-ion system. This is achieved by generalizing a recently proposed lightshift gate mechanism to generate genuine qudit entanglement in a single application of the gate. The gate seamlessly adapts to the local dimension of the system with a calibration overhead that is independent of the dimension. 1234567890():,; 1234567890():,; Quantum computing has taken great strides in the past decades with multiple platforms demonstrating control over tens of qubits [1][2][3][4] . However, scaling these systems to a regime beyond the capabilities of classical computers remains very challenging, both in terms of increasing the size of the quantum computational Hilbert space, and in terms of increasing the depth of the computational circuits. A significant potential for tempering these daunting scaling demands, however, lies in plain sight when appreciating that the quantum systems we use are multi-level, not two-level systems. A natural way to extend the computational Hilbert space and reduce circuit complexity, without increasing the complexity of quantum devices is thus to use the full multi-level or qudit structure of existing quantum information carriers such as trapped ions, see Fig. 1. Qudit control has already been demonstrated in a number of architectures [5][6][7][8][9][10][11][12][13][14][15][16] , including trapped ions [17][18][19][20] . Qudit approaches not only enable reduced circuit complexity 21 and simplifications of virtually any quantum circuit 22 , but also benefit from more powerful quantum error correction [23][24][25][26] , and enable the implementation of optimal quantum measurements 27 as well as native quantum simulation of a range of physical systems such as lattice gauge models 28,29 , optimization problems 30 , or quantum chemistry 31 . The key to these advantages, however, is an appropriate set of quantum operations. While for qubits all entangling gates are equal up to local rotations, the same is not true for qudits, where the richer Hilbert space allows for different forms of coherence 6 and entanglement 32 . We can roughly classify qudit entangling gates by their entangling power, i.e., the amount of entanglement that is created relative to a maximally entangled state of two qubits. While in principle already the simplest gate, a qubit entangling gate embedded in a higher-dimensional Hilbert space, would suffice for universal qudit quantum computation 17,33 , having access to a range of gates with different entangling power will be crucial for unlocking the full potential of a qudit quantum processor. Not only will a diverse set of interactions enable the direct simulation of a wider range of physical systems, but it will also enable much more efficient quantum circuit decomposition. Here we describe and demonstrate a native qudit entangling gate in a trapped-ion quantum processor. Being based on differential light shifts between the ground-and excited state manifolds on an optical transition, the gate action can be made symmetric on all excited qudit states. We show that this implies that the same gate mechanism can be Results The principle behind the qudit entanglement generation is the application of light-shift (LS) gates, in which a state-dependent opticaldipole force couples the ions' electronic states to their common motion in the trap. Light-shift gates have been well studied for entangling hyperfine and Zeeman qubits with a pair of intersecting laser beams (see Fig. 2) that create a traveling wave [34][35][36][37] . The traveling wave produces a spatially modulated light shift that drives an excursion in the phase space of one of the motional modes. The ions acquire a different geometric phase depending on their electronic state, leaving them in an entangled state after completing the excursion. The beams also introduce additional, unwanted local phases as the electronic states experience different light shifts. For hyperfine or Zeeman qubits, the polarization of the optical beams can be used to null these differential light shifts on the electronic states. In practice, however, these gates are typically applied with a spin echo pulse, inserted between two halves of the LS gate pulse during which the ions complete a single loop in motional phase space. This ensures that any residual light shifts between the two electronic states due to experimental imperfections are canceled. Adding these local π pulses has the further advantage of suppressing slow qubit frequency drifts and decoupling from the optical phase of the LS gate laser. Using this technique, the LS gate can also be implemented without the requirement to intrinsically null the differential light shift. This opens the opportunity to apply the gate to qubits with an energy difference in the optical domain, without careful selection of the laser frequency. Such gates were theoretically described 38 and implemented 39 for an optical qubit formed by sub-levels of the S 1/2 and D 5/2 manifold in 40 Ca + ions. We now describe how this gate scheme can be generalized to generate genuine qudit entanglement for qudits of arbitrary dimension encoded in the Zeeman sub-levels of the S 1/2 and D 5/2 manifolds of 40 Ca + ions, see Fig. 1. Formally, the two-ion LS Hamiltonian, after adiabatic elimination of the excited states and application of the Lamb-Dicke and rotating-wave approximations, can be written in the interaction frame of the ions' center-of-mass (COM) motion as H LS = i_η 2 X j X N Δ N,j j N j N e Àiδt e iφ N a y + h:c::ð1Þ Here, Δ N,j represents the light shift on the state j ji of ion N, δ the detuning from the motional mode frequency, η the Lamb-Dicke parameter, a † the creation operator of the motional mode and φ N the phase determined by the inter-ion distance within the traveling wave of the laser. Integrating Eq. (1), we obtain the propagator U LS (t) describing a state-dependent force on the ions' motion that can be visualized as loops in motional phase space, which will periodically return to the origin after a time t g = 2π/δ. During the evolution in phase space, each initial electronic state combination j jki of the two ions will, in the most general case, pick up a different geometric phase ϕ jk after a single LS gate pulse application. Similar to the qubit case, we can now symmetrize the geometric phases for qudits, while canceling differential light shifts. We achieve this by encoding our qudits in the j0i = S 1=2,m j = À1=2 ground state and the Zeeman sub-levels of the D 5/2 manifold as jii, with i ∈ {1, 2, 3, 4}, see Fig. 1. We then choose a wavelength close to the S 1/2 ↔ P 1/2 transition, which for our choice of encoding results in Δ N,0 ≫ Δ N,j for j ≠ 0. This allows us to neglect differences in phase shifts between D 5/2 levels induced by the LS gate pulse. We interleave d applications of U LS (t g ) with two-ion cyclic permutations of the form X d = ð P dÀ1 j = 0 j j + 1 ðmod dÞih jjÞ 2 , where the populations of each level are transferred to the level with the next higher index. This ensures that each logical state spends an equal amount of time in each physical energy level. After application of the sequence G = ðX d U LS ðt g ÞÞ d we find the phases e ϕ jk imprinted on the states j jki e ϕ jk = P dÀ1 k = 0 ϕ kk if k = j P dÀ1 k = 0 P j<k ϕ jk else, (ð2Þ where ϕ jk refer to the phases from the constituent U LS (t g ) pulses. For d = 2 the sequence ðX 2 U LS ðt g ÞÞ 2 corresponds to the standard qubit light-shift gate with spin echo 39 . Equation (2) shows that after symmetrization we are left with only two different phases, one for the cases where both ions are in the same state and one where the ions are in different states, see Fig. 3. Hence, up to a global phase, the qudit light-shift gate operation G(θ) can be described for all d by GðθÞ : j jji ! j jji j jki ! expðiθÞj jki if j ≠ k:ð3Þ The operator G(θ) directly generates genuine qudit entanglement as opposed to merely embedding qubit-level entanglement in a larger Hilbert space 17 . This will enable the generation of high-fidelity qudit entanglement with a single gate operation. In the experiment θ can be chosen freely by simultaneously varying the gate detuning δ and the The interaction of Eq. (1) is generated by a pair of perpendicular laser beams, one of which is parallel and the other perpendicular to the applied magnetic field, each with a waist of approximately 45 μm, see Fig. 2. In this configuration, the difference wavevector of the two beams is parallel to the axial trap direction to only couple to the ions' axial motion. The two beams are derived from a single frequencydoubled Titanium-Sapphire laser at a wavelength of λ ≈ 401.2 nm, approximately 8.1 THz red-detuned from the S 1/2 ↔ P 1/2 transition. This results in low scattering errors on the order of 10 −4 for an LS gate pulse with a duration of t g~3 5 μs. The LS force on the state j0i is maximized by choosing both beams to be vertically polarized. The detuning between the beams is chosen as (ω COM + δ) to couple primarily to the axial COM motion. Maximizing the differential light shift for a given beam intensity requires the inter-ion distance to be an integer or half-integer multiple of the period of the traveling wave pattern created by the LS gate beams 36 . Imperfect spacing decreases the phase difference between equal and unequal states during the application of U LS on the two ions, thus reducing the achievable gate speed for a given beam intensity. Since most error sources scale with the gate duration, correctly choosing the spacing is crucial for achieving low error rates. Experimentally, we adjust the inter-ion spacing by varying the voltages on the trap electrodes, which create the confinement in the axial direction. The ion spacing is calibrated by initializing the ions in j00i and applying a resonant (δ = 0) LS gate pulse with variable time. If the spacing is set to a half-integer multiple of the standing wave, the breathing mode is excited to a coherent state, whereas the motion of the COM mode remains unaffected. For an integer spacing the relation between the motional modes is inverted. The motional state is read out by measuring the excitation when shelving the ions on the respective red sideband of the S 1/2 ↔ D 5/2 transition. By observing excitation of only the breathing mode, we infer that the inter-ion distance is set appropriately, and the unwanted phase accumulation of the equal states is minimized. After Doppler cooling, the ions' axial motional modes are cooled to around 0.1 quanta by resolved sideband cooling. We then initialize the ions in the j00i state via optical pumping. We create an equal superposition of all qudit states by applying the operator P = Y dÀ1 j = 1 R 0,j ðϑ j ,0Þ,ð4Þ with rotation angle ϑ j = 2 arcsinð1= ffiffiffiffiffiffiffiffi j + 1 p Þ and R j,k ðϑ, ϕÞ = exp Ài ϑ 2 σ j,k 1 ðϕÞ + σ j,k 2 ðϕÞ ð5Þ where σ j,k N ðϕÞ = cosðϕÞσ j,k x + sinðϕÞσ j,k y denotes the rotation on ion N on the transition j ji $ jki for Pauli matrices σ x ,σ y . Each rotation R j,k (ϑ, ϕ) is implemented by a resonant 729 nm laser pulse, where ϕ is determined by the laser phase and ϑ by the pulse area. We then apply the sequence ðX d U LS ðt g ÞÞ d , where the generalized spin echo X d is implemented by a sequence of π-pulses on the j0i $ j ji transitions. For d > 2 the phase of the first laser pulse of each X d is shifted by π to decrease the errors due to over-rotation of the local pulses, leading to a sequence of X d = Y dÀ1 j = 1 R 0,j ðπ, δ j1 πÞð6Þ for the permutation operator, where δ ij denotes the Kronecker delta. After the final permutation, we apply the conjugate P † of the initial preparation sequence. Up to dimension d = 4, this leaves the system in a maximally entangled state of the form jΨ d i = P dÀ1 j j jji= ffiffiffi d p , whereas for d = 5 the sequence will result in the state jΨ 5 i = ð3j00i + 2 P 4 j = 1 j jjiÞ=5. This is likely a consequence of there only being a single phase applied to all d 2 − d components of the state. In order to generate maximal entanglement in higher dimensions it might thus be necessary to use multiple applications of the gate, or generalize the gate action such that it imparts different phase shifts to different state components. We can directly estimate the state fidelity and the amount of entanglement of the generated states from the relative amplitudes of the components jiii, as well as their pairwise coherences. Experimentally, the population of the j00i state can be measured by driving the S 1/2 ↔ P 1/2 transition with a 397 nm laser and collecting the fluorescence photons on a photo-multiplier tube. Using additional π-pulses T j 0 = R 0,j ðπ,0Þ, the same procedure gives access to all components j jji. The coherence terms between the states j00i and j jji are estimated by applying a π/2-pulse A j 0,ϕ = R 0,j ðπ=2, ϕÞ with variable phase ϕ before performing the fluorescence readout. Applying T k 0 before A j 0,ϕ allows us to measure the coherence between the states jkki and j jji. We then extract the coherence between the two terms from the parity oscillations by Bayesian parameter estimation, which accounts for measurement statistics and guarantees that the results stay physically possible. The observed fidelity is affected by state-preparation-andmeasurement (SPAM) errors, including the pulses P, P † the transfer pulse T j 0 and analysis pulse A j 0,ϕ . In order to separate the errors from SPAM and gate G(θ) for each dimension d, we insert up to 9 applications of G(θ) between the pulses P and P † (see Fig. 4 (a)), and compute the state fidelity for each n that results in an entangled state. We then fit an exponential decay to estimate the fidelity of a single gate. Such repeated gate applications, however, are also sensitive to the presence of non-Markovian noise in our system that leads to deviations from purely exponential decay. The extracted fidelities should thus be interpreted as an estimate for the SPAM corrected average gate performance over a sequence of length n. We apply this procedure for d = 2, 3, 4, 5 and obtain fidelities of 99.6(1)%, 98.7(2)%, 97.0(3)%, 93.7(3)%, respectively. While the intrinsic limits on gate fidelity due to finite state lifetime and Raman beam scattering depend only weakly on the dimension d (see supplemental note 2), the measured gate performance degrades quadratically with dimension as seen in Fig. 4 (b). This can be understood if the total gate error is dominated by technical noise sources that do not scale linearly with d. To investigate this we construct a numeric error model that computes the expected decay data using all independently measured error sources as inputs (see supplementary note 2), reproducing the observed data with very good agreement (blue diamonds in Fig. 4 (b)). This model suggests that while for d = 2 the gate fidelity is limited by the motional coherence time of the ion and frequency noise of the gate laser, for higher dimension the dominant error sources become the gate Rabi frequency noise and slow frequency noise that causes dephasing of the local operations. We can hence conclude that the gate fidelity in higher dimensions can be significantly improved if technical noise sources such as magnetic field noise contributing to the aforementioned local operation dephasing or Rabi frequency fluctuations can be suppressed. We furthermore evaluate the entanglement properties 41 of the states generated by a single application of the gate, including the Schmidt number, Concurrence, and Entanglement of Formation, see Fig. 5 and supplementary note 3 for details. We find that the concurrence for all states with d > 2 significantly exceeds the maximal possible value for any qubit state. Crucially, while the concurrence growth is expected to slow asymptotically with dimension, the Schmidt number in each dimension is maximal, indicating the presence of genuine qudit entanglement up to d = 5. The Schmidt number has also been suggested to play a crucial role in the computational complexity of a quantum system 42 . Discussion We have demonstrated an experimental realization of a gate that directly generates native qudit entanglement between two trapped ions. The major difference between previously demonstrated qudit entangling schemes 17 and our scheme is that the gate natively couples to all transitions, whereby it creates entanglement between the multiple qudit levels in a single application of the gate rather than through repeated applications of qubit entangling operations that couple to individual transitions and thereby only generate two-level entanglement in each step. This fundamental difference dictates not only the kind of entangling dynamics that can be realized, but also the error contributions and requirements on the experimental control. As a result the calibration of our native qudit gate compares favorably to multiple applications of a pairwise entangling Mølmer-Sørensen gate. In the latter case, the gate requires careful adjustment of the entangling laser control parameters for each of the desired jSi $ jDi transitions b a Fig. 4 | Fidelity decay measured using multiple gates. a Schematic of the measurement sequence. Two ions initialized in |00i are rotated into an equal superposition of all states by applying the operator P with the 729 nm laser. After applying the gate operator G(θ) a variable number of times n a reversed preparation pulse P † is applied. The populations of the resulting state are measured by a set of transfer pulses T j 0 , which are resonant π pulses between |0i $ | j to transfer the state | j to the S 1/2 manifold, allowing us to distinguish the qudit states. An analysis pulse A j 0,ϕ consisting of a resonant π/2 pulse between |0i $ | j with variable phase ϕ is used to measure the coherence between the |0i and | j levels. Combined with the transfer pulses, all pairwise coherences can be measured. b A plot of qudit gate fidelity as a function of dimension. The average gate fidelities, shown as red circles, are extracted from fits to the fidelity decay when applying multiple gates G(θ) between P and P † . The error bars correspond to 1 standard deviation in the fit parameters. A quadratic curve has been fitted to the data to highlight the empirically observed scaling of the fidelity with dimension. The simulated fidelities from a detailed noise model are shown as blue diamonds, see supplementary note 2 for details. including compensation for light shifts and induces undesired phase shifts on spectator levels, which have to be tracked. For our qudit phase gate, increasing the dimensionality of the entangling space just requires one extra local operation per additional D 5/2 sub-level to be calibrated and the power of the gate laser to be adjusted by a known analytical ratio that is not sensitive to qubit frequency shifts from light shifts or otherwise, since the gate beam is far off-resonant. Interestingly, while all demonstrated schemes for creating genuine qudit entanglement exhibit a linear increase in gate duration with qudit dimension, this may be overcome for some interactions at the cost of a linear increase in control parameters 43 . Moreover, while we demonstrated a highly symmetrized version of the gate, exploiting different light shifts on different ground-and excited state levels allows for a wide range of gate actions, accessible through local operations alone. Data availability The data generated in this study is deposited on Zenodo at https://doi. org/10.5281/zenodo.7688595. Code availability The code used for simulation of the error budget is available upon request. Fig. 1 | 1Level scheme of the 40 Ca + ion. Encoding quantum information in the sublevels of the S 1/2 and D 5/2 manifold allows us to increase the size of the computational Hilbert space. Coherent operations between the sub-levels can be performed using 729 nm laser light, while 401 nm light is used to generate the light shift for the state-dependent force. Fig. 2 | 2Experimental setup. Two beams with λ = 401 nm and frequencies f 1 , f 2 with a relative detuning of ω COM + δ are intersected at a 90 ∘ angle to drive the statedependent force. A 729 nm laser along the axial trap direction is used to implement local operations. gate time t g or changing the laser power in order to adjust the light shifts Δ N,j . A more detailed derivation of Eq. (3) for the case of a qubit and qutrit is found in the supplementary note 1.The gate is performed on two40 Ca + ions, trapped 110 μm above the surface of a segmented surface Paul trap. The ion trap is mounted inside a liquid Helium flow-cryostat and is operated at~35 K 40 . Static potentials applied to a set of DC electrodes confine the ions along the axial direction with an axial COM mode frequency of ω COM /2π ≈ 1.1 MHz. Radio-frequency potentials create radial confinement with frequencies {ω x , ω y }/2π ≈ {3.5 MHz, 3.2 MHz}. Outside the vacuum chamber, a pair of Helmholtz coils aligned 45 ∘ with respect to the axial trap direction generates a magnetic field of~3.6 G, defining the quantization axis. The S 1=2,m j = À1=2 ground state and the sub-levels of the D 5/2 manifold are used for encoding the qudits, seeFig. 1. Singlequdit rotations between j0i and jii are implemented using a 729 nm laser with a linewidth of~10 Hz. Fig. 3 | 3Application of the gate on two qutrits. a Phase evolution of the two-qutrit state components relative to the |00i ground state during the application of the LS gate pulses. States |01i and |10i are shown in orange, |02i and |20i in blue, and |21i and |12i in green. The equal electronic states |00i,|11i,|22i in purple do not acquire any relative phases. b Corresponding pulse scheme to implement the composite qudit light-shift gate operator G(θ) for a qutrit (d = 3). Light-shift gate pulses U LS (t g ) are interlaced with cyclic permutation gates X 3 . Fig. 5 | 5Generation of genuine qudit entanglement. The measured state fidelity for d = 2, 3, 4, 5 is shown as blue data points and the corresponding concurrence as orange squares. The ideal values for maximally entangled states (the experimental target states) are shown as dashed (dotted) lines. The gray shaded bars represent the lower bound on the fidelity (blue data points) for certifying maximal Schmidt number entanglement, while the gray horizontal lines indicate the concurrence C d for a maximally entangled state in dimension d. Error bars correspond to one standard deviation of experimental shot noise. Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25/4, 6020 Innsbruck, Austria. 2 Atominstitut, Technische Universität Wien, 1020 Vienna, Austria. 3 Institute for Quantum Optics and Quantum Information-IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. 4 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria. 5 AQT, Technikerstraße 17, 6020 Innsbruck, Austria. 6 These authors contributed equally: Pavel Hrmo, Benjamin Wilhelm. e-mail: [email protected] irrespective of the qudit dimension to generate genuine qudit entanglement in the sense that the Schmidt number of the resulting state equals the qudit dimension. Crucially, this means that the experimental calibration overhead does not increase with qudit dimension. We further show that direct application of the gate can generate maximal qudit entanglement up to dimension 4. We characterize the gate dynamics and noise sources in detail, demonstrating that this gate can be a stepping stone into the world of native qudit quantum information processing with trapped ions.Received: 18 January 2023 Accepted: 15 March 2023 Check for updates 1 Nature Communications | (2023) 14:2242 © The Author(s) 2023 AcknowledgementsThis project has received funding from the European Union's Horizon 2020 research and innovation program under the MarieAdditional informationSupplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41467-023-37375-2.Correspondence and requests for materials should be addressed to Pavel Hrmo.Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. 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[ "On the spectral radius of minimally 2-(edge)-connected graphs with given size *", "On the spectral radius of minimally 2-(edge)-connected graphs with given size *", "On the spectral radius of minimally 2-(edge)-connected graphs with given size *", "On the spectral radius of minimally 2-(edge)-connected graphs with given size *" ]
[ "Zhenzhen Lou \nCollege of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina\n", "Gao Min \nCollege of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina\n", "Qiongxiang Huang \nCollege of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina\n", "Zhenzhen Lou \nCollege of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina\n", "Gao Min \nCollege of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina\n", "Qiongxiang Huang \nCollege of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina\n" ]
[ "College of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina", "College of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina", "College of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina", "College of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina", "College of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina", "College of Mathematics and Systems Science\nXinjiang University\n830046UrumqiXinjiangChina" ]
[]
A graph is minimally k-connected (k-edge-connected) if it is k-connected (kedge-connected) and deleting arbitrary chosen edge always leaves a graph which is not k-connected (k-edge-connected). A classic result of minimally k-connected graph is given by Mader who determined the extremal size of a minimally k-connected graph of high order in 1937. Naturally, for a fixed size of a minimally k-(edge)-connected graphs, what is the extremal spectral radius? In this paper, we determine the maximum spectral radius for the minimally 2-connected (2-edge-connected) graphs of given size, moreover the corresponding extremal graphs are also determined.
10.37236/11219
[ "https://export.arxiv.org/pdf/2206.07872v1.pdf" ]
249,712,335
2206.07872
0174251c32162b381bd2bf9c48f2aa201bdec2e2
On the spectral radius of minimally 2-(edge)-connected graphs with given size * Jun 2022 Zhenzhen Lou College of Mathematics and Systems Science Xinjiang University 830046UrumqiXinjiangChina Gao Min College of Mathematics and Systems Science Xinjiang University 830046UrumqiXinjiangChina Qiongxiang Huang College of Mathematics and Systems Science Xinjiang University 830046UrumqiXinjiangChina On the spectral radius of minimally 2-(edge)-connected graphs with given size * 16Jun 2022Minimally 2-(edge)-connected graphspectral radiusextremal graphdouble eigenvectors AMS Classification: 05C50 05C75 A graph is minimally k-connected (k-edge-connected) if it is k-connected (kedge-connected) and deleting arbitrary chosen edge always leaves a graph which is not k-connected (k-edge-connected). A classic result of minimally k-connected graph is given by Mader who determined the extremal size of a minimally k-connected graph of high order in 1937. Naturally, for a fixed size of a minimally k-(edge)-connected graphs, what is the extremal spectral radius? In this paper, we determine the maximum spectral radius for the minimally 2-connected (2-edge-connected) graphs of given size, moreover the corresponding extremal graphs are also determined. Introduction A graph is said to be connected if for every pair of vertices there is a path joining them. Otherwise the graph is disconnected. The connectivity (or vertex-connectivity) κ(G) of a graph G is the minimum number of vertices whose removal results in a disconnected graph or in the trivial graph. The edge-connectivity κ ′ (G) is defined analogously, only instead of vertices we remove edges. A graph is k-connected if its connectivity is at least k and k-edge-connected if its edge-connectivity is at least k. It is almost as simple to check that the minimal degree δ(G), the edge-connectivity and vertex connectivity satisfy the following inequality: δ(G) ≥ κ ′ (G) ≥ κ(G). One of the most important task for characterization of k-connected graphs is to give certain operation such that they can be produced from simple k-connected graphs by repeatedly applying this operation [1]. This goal has accomplished by Tutte [21] for 3connected graphs, by Dirac [8] and Plummer [17] for 2-connected graphs and by Slater [18] for 4-connected graphs. A graph is said to be minimally k-connected if it is kconnected but omitting any of edges the resulting graph is no longer k-connected. Clearly, a k-connected graph whose every edge is incident with one vertex of degree k is minimally k-connected, especially a k-regular and k-connected graph is minimally k-connected. Questions in extremal graph theory ask to maximize or minimize a graph invariant over a fixed family of graphs. A classic result of minimally k-connected graph is given by Mader who determined the extremal size of a minimally k-connected graph of high order in [13]. Giving a graph class G to study the bounds of spectral radius of graphs in G and to characterize the extremal graphs that achieves the bound is a famous problem in the spectral extremal graph theory [4], which attracts some authors and have produced many interesting results published in various magazines [12,14,16,20,24]. In the origin of researches, G is restricted to the graphs of order n or size m. For examples, Brualdi and Hoffman [4] gave an upper bound on spectral radius in terms of size m: if m ≤ k 2 for some integer k ≥ 1 then ρ(G) ≤ k − 1, with equality if and only if G consists of a k-clique and isolated vertices. Extending this result, Stanley [19] showed that ρ(G) ≤ √ 1+8m− 1 2 . In particular, Nosal [16] in 1970 proved that if G is a triangle-free graph with m edges then ρ(G) ≤ √ m. In the subsequent study, G is restricted to the graphs that have some combinatorial structure. For examples, in 2002, Nikiforov [14,15] showed that ρ(G) ≤ 2m(1 − 1 r ) for a graph G with given size m. Bollobás, Lee, and Letzter 2018 studied the maximizing spectral radius of subgraphs of the hypercube for giving size m [2]. Very recently, Lin, Ning and Wu [12] proved that ρ(G) ≤ √ m − 1 when G is non-bipartite and triangle-free with giving size m. Zhai, Lin and Shu [24] obtained that if G contains no pentagon or hexagon with giving size m, then ρ(G) ≤ 1 2 + m − 3 4 , with equality holds if and only if G is a book graph. In the recent works, some authors restrict G to the graphs that have connectivity. For examples, given order n of a graph, Chen and Guo in 2019 showed that K 2,n−2 attained the maximal spectral radius among all the minimally 2-(edge)-connected graphs [5]. Fan, Goryainov and Lin in 2021 proved that K 3,n−3 has the largest spectral radius over all minimally 3-connected graphs [9]. All the above studies indicate that the spectral radius of a graph are related with the parameters of graphs( such as order n and size m ), structure of graphs ( such as forbidding subgraphs ) and vertex or edge connectivity of graphs. Motivated by this researches, our paper is to study the spectral extremal problem of minimally 2-(edge)-connected graph under edge-condition restrictions. Precisely, our aim is to give an answer to the following question. Problem 1. Given size m, what is the maximum spectral radius among minimally k-(edge)-connected graphs? Denote by S K 2, m−1 2 the graph obtained from the complete bipartite graph K 2, m−1 2 by subdividing an edge once. Set K 2, m−3 2 * K 3 is the graph obtained by identifying a maximum degree of K 2, m−3 2 and a vertex of K 3 from the disjoint union of K 2, m−3 2 and K 3 . A friend graph, denoted by F t , is a graph obtained from t triangles by sharing a common vertex. In this paper, we solve Problem 1 for k = 2 and attain the unique extremal graph in the following two theorems. (G) ≤ ρ * 1 (m), where ρ * 1 (m) is the largest root of x 3 − x 2 − (m − 2)x + m − 3 = 0,(G) ≤ ρ * 2 (m), where ρ * 2 (m) is the largest root of x 4 − x 3 + (1 − m)x 2 + (m − 3)x + m − 3 = 0, the equality holds if and only if G K 2, m−3 2 * K 3 . If m = 15, then ρ(G) ≤ 1+ √ 41 2 , the equality holds if and only if G F 5 . A famous sharp lower bound of spectral radius ρ(G) ≥ 2m n giving by L. Collatz, U. Sinogowitz in [6], equality holds if and only if G is a regular graph. Thus, the m-cycle has the minimal spectral radius among all minimally 2-(edge)-connected graphs with size m. Therefore, by Theorems 1.1 and 1.2, we obtain that the spectral radius of a minimally 2-(edge)-connected graph lies the interval [2, √ m]. It means that a graph whose spectral radius out of [2, √ m] will not be minimally 2-(edge)-connected which indeed indicates the relationship of spectral radius and connectivity for a graph. The rest of the paper is organized as follows. In the next section, we will give some lemmas and some properties of a minimally 2-(edge)-connected graph. In Section 3 and 4, we will give the proof of Theorem 1.1 and 1.2, respectively. Preliminary In this section, we firstly list some symbols and then write some properties of minimally 2-(edge)-connected graphs and some useful lemmas. Let G be a graph with vertex set V(G) = {v 1 , v 2 , · · · , v n } and edge set E(G). For v ∈ V(G), A ⊂ V(G), denote by N(v) and d(v) the neighborhood and the degree of the vertex v in G, and denote N A (v) = N(v) ∩ A, d A (u) = |N A (v)|. The adjacent matrix of a graph G is defined as the n × n square matrix A(G) = (a i j ) whose entries are 1 if v i v j ∈ E(G), otherwise 0. The spectral radius of G, denote by ρ(G), is defined to be the largest eigenvalue of A(G). A chord of a graph is an edge between two vertices of a cycle that is not an edge on the cycle. If a cycle has at least one chord, then it is called a chorded cycle. A graph is 2-(edge)-connected graph if it contains a 2-vertex (edge) cut set. A graph is minimally 2-(edge)-connected, introduced in [1], if it is a 2-(edge)-connected but omitting any edge the resulting graph is no longer 2-(edge)-connected. By definition, a 2connected graph is also 2-edge-connected, but vice versa. However there exists a minimally 2-connected graph that is not minimally 2-edge-connected, for example the graph H(2, 2) shown as Fig.1. There exists a minimally 2-edge-connected graph that is not minimally 2-connected, for example the graph C n * C m . Clearly, C n is both of minimally 2-(edge)-connected. Furthermore, we will give some the properties of a minimally 2-(edge)-connected graph. Proof. Let G be a minimally 2-edge-connected graph, and H be a 2-edge-connected subgraph of G. By contrary that H is not minimal, then there exists an edge uv ∈ E(H) such that H − uv is 2-edge-connected. Since G is a minimally 2-edge-connected graph, we get that G − uv is 1-edge-connected. Thus G − uv has a cut edge, say xy, which divides V(G−uv) into two vertices sets U and V such that x ∈ U and y ∈ V, and so e G−uv (U, V) = 1. Also we have u ∈ U and v ∈ V since G is a 2-edge-connected graph. Noticed that H − uv is a subgraph of G − uv that is assumed to be 2-edge connected, we claim that H − uv has a cycle C connecting u and v which must be contained in G − uv. It is a contradiction. Thus, H is a minimally 2-edge-connected graph. Lemma 2.6. If G is a minimally 2-edge-connected graph, then no cycle of G has a chord. Proof. Suppose by contrary that the cycle of the minimally 2-edge-connected graph G has a chord. Then G contains a chorded cycle, which is not minimally 2-edge-connected. This is a contradiction from Lemma 2.5. Recall that κ(G) (κ ′ (G)) denoted the vertex connectivity (edge connectivity) of G. Lemma 2.7. If G be a minimally 2-edge-connected graph with no cut vertex, then G is minimally 2-connected. Proof. Since G is a 2-edge-connected graph and G has no cut vertex, we have 2 = κ ′ (G) ≥ κ(G) ≥ 2, and so κ(G) = 2. Note that G is minimally 2-edge connected. We have κ(G − e) ≤ κ ′ (G − e) ≤ 1 for any e ∈ E(G). Thus G is minimally 2-connected. Lemma 2.8 ( [7] ). Let G be a graph with adjacency matrix A(G), and let π be an equitable partition of G with quotient matrix B π . Then det(xI − B π ) | det(xI − A(G)). Furthermore, the largest eigenvalue of B π is just the spectral radius of G. By Lemma 2.8, we can give the bound of the spectral radius of S K 2, m−1 2 . Lemma 2.9. For odd number m > 5, we have ρ(S K 2, m−1 2 ) is the largest root of x 3 − x 2 − (m − 2)x + m − 3 = 0 and √ m − 2 < ρ(S K 2, m−1 2 )) < √ m − 1. Proof. The vertices set of S K 2, m−1 2 has equitable partition and the quotient matrix is B π =           1 1 0 1 0 m−3 2 0 2 0           . We have f (x) = det(xI 3 − B π ) = x 3 − x 2 − (m − 2)x + m − 3. By Lemma 2.8, ρ(S K 2, m−1 2 ) is the largest root of f (x) = 0. Moreover, one can verify that f ( √ m − 2) < 0, and so ρ(S K 2, m−1 2 ) > √ m − 2. Also, we have f ( √ m − 1) = √ m − 1 − 2 > 0 for m ≥ 6 and f ′ (x) = 3x 2 − 2x − (m − 2) > 0 for x ≥ √ m − 1. Thus, ρ(S K 2, m−1 2 ) < √ m − 1. Notice that if each edge of a k-connected graph is incident with at least one vertex of degree k then the graph is minimally k-(edge)-connected. Clearly, S K 2, m−1 2 is the minimally 2-(edge)-connected. Moreover, we have the following lemma. Lemma 2.10. Let G * attain the maximum spectral radius among all minimally 2-edgeconnected graph of size m ≥ 6, then ρ( G * ) > √ m − 2. Proof. If m is even, then ρ(G * ) ≥ ρ(K 2, m 2 ) = √ m > √ m − 2 since K 2, m 2 is minimally 2-edge-connected. If m is odd, then ρ(G * ) ≥ ρ(S K 2, m−1 2 ) > √ m − 2 from Lemma 2.9. Lemma 2.11 ( [10,11]). Let G and H be two graphs, and let P(G, x) be the characteristic polynomial of G. (i) If H is a proper subgraph of G, then ρ(H) < ρ(G). (ii) If P(H, λ) > P(G, λ) for λ ≥ ρ(G), then ρ(H) < ρ(G). Lemma 2.12 ( [14,15]). Let G be a C 3 -free graph of size m. Then ρ(G) ≤ √ m, the equality holds if and only if G K a,b , where ab = m. Lemma 2.13 ( [23]). Let u, v be two distinct vertices in a connected graph G, {v i | i = 1, 2, . . . , s} ⊆ N G (v) \ N G (u). X = (x 1 , x 2 , . . . , x n ) T is the Perron vector of G, where x i is corresponding to v i (1 ≤ i ≤ n). Let G ′ = G − {vv i | 1 ≤ i ≤ s} + {uv i | 1 ≤ i ≤ s}. If x u ≥ x v , then ρ(G) < ρ(G ′ ). Lemma 2.14 ( [7]). Let G be a connected graph with Perron vector X = (x 1 , x 2 , . . . , x n ) T . Let U, V and W be three disjoint subsets of V(G) such that there are no edges between V and W. Let G ′ be the graph obtained from G by deleting the edges between V and U, and adding all the edges between V and W. If u∈U x u ≤ w∈W x w , then ρ(G) < ρ(G ′ ). Lemma 2.15 ( [22] ). Let (H, v) and (K, w) be two connected rooted graphs. Then In this section, we will give the proof of Theorem 1.1. Let G * attain maximal spectral radius ρ * = ρ(G * ) among all minimally 2-connected graphs with size m. Lemma 2.1 and Lemma 2.4 indicate G * has no triangles and δ(G * ) = 2. First, by Lemma 2.12 we claim that G * K 2, m 2 if m is even. In what follows, we always assume that m is odd. Next we will consider the structure of extremal graph G * for odd size m ≥ 9. Let ρ((H, v) * (K, w)) ≤ ρ(H) 2 + ρ(K) 2 ,X = (x 1 , x 2 , . . . , x n ) T be the Perron vector of G * with coordinate x u * = max{x i | i ∈ V(G * )}. Denote by A = N(u * ) and B = V(G * ) \ (A ∪ u * ) . Since G * has no triangles, we have e(A) = 0, that is, i∈A d A (i)x i = 0, and thus ρ * 2 x u * = v∈A u∈N(v) x u = d(u * )x u * + i∈A d A (i)x i + i∈A d B (i)x i = d(u * )x u * + i∈A d B (i)x i ≤ (d(u * ) + e(A, B)) x u * = (m − e(B)) x u *(1) By Lemma 2.10, ρ * 2 > m − 2. Combining with (1), we get e(B) < 2. Thus, e(B) = 0 or 1. In the following, we will give four claims to finish our proof. Claim 3.1. d(u * ) ≥ 3. Proof. Otherwise, d(u * ) ≤ 2. Since G * is minimally 2-connected, we have d(u * ) ≥ δ(G * ) = 2= {w 1 w 2 } ∪ I, where I = {v 1 , v 2 , . . . , v t } is isolated vertices set. Moreover, we see that each v i is adjacent to u j for j = 1, 2 due to δ(G * ) = 2, and N A (w 1 ) ∩ N A (w 2 ) = ∅, d A (w 1 ) = d A (w 2 ) = 1 since G has no triangles. Without loss of generality, let N A (w 1 ) = u 1 and N A (w 2 ) = u 2 . Now G * is determined and m = 2t + 5 in this situation, where t ≥ 2. By the symmetry, we have x u 1 = x u 2 , x w 1 = x w 2 , and x u * = x v i for i = 1, 2, · · · , t. Thus from A(G * )X = ρ * X, we have          ρ * x u * = 2x u 1 , ρ * x u 1 = (t + 1)x u * + x w 1 , ρ * x w 1 = x w 2 + x u 1 = x w 1 + x u 1 . Furthermore, we get Proof. Suppose to the contrary that e(B) = 0, that is, B induces an independent set. (ρ * 2 − 2t − 2)x u * = 1 ρ * − 1 x u 1 . (2) Let g(ρ * ) = (ρ * 2 − 2t − 2)(ρ * − 1) − 1 = ρ * 3 − ρ * 2 − (2t + 2)ρ * + 2t + 1. Then g(ρ * ) = ρ * 3 − ρ * 2 − (m − 3)ρ * + m − 4. Since ρ * > √ m − 2 by Lemma 2.10 and g ′ (ρ * ) = 3ρ * 2 − 2ρ * − (m − 3) > g ′ ( √ m − 2) = 2m − 2 √ m − 2 − 3 > 0, we have g(ρ * ) > g( √ m − 2) = √ m − 2 − 2 > 0. Thus ρ * 2 − 2t − 2 > 1 ρ * −1 . Therefore, from (2) we get x u * < x u 1 , which contradicts the maximality of x u * .Recall that A is independent, for u ∈ B, it lies in a 4-cycle C = (u, v 1 , u * , v 2 ) where v 1 , v 2 ∈ A. There is at least one of v 1 and v 2 having degree three since otherwise u * will be a cut vertex. Also there is at least one of v 1 and v 2 having degree two since otherwise C contains at most one vertex of degree two, which contradicts Lemma 2.2. It implies that d(u) = 2 due to d(u * ) ≥ 3, and then we may assume that d(v 1 ) = 2 and d(v 2 ) ≥ 3. In addition, the vertex as v 2 in A is unique, because two vertices ( in A ) of degree greater ✛ ✚ ✘ ✙ ✛ ✚ ✘ ✙ ✛ ✚ ✘ ✙ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✪ ✪ ✪ ✪ ✪ ❅ ❅ ❅ ❅ ❅ ✛ ✚ ✘ ✙ ✛ ✚ ✘ ✙ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ u * u * A 1 A 2 B H H(s, t) v 1 v s v 2 u 1 u 2 u t w 1 w 2 ✈ ✈ · · · · · · · · · · · · Fig. 1: The graphs H and H(s, t), where s, t ≥ 1. than 2 must lie in a 4-cycle along with u * and in this cycle there have been three vertices of degree greater than 2. Therefore, G * is isomorphic to H showed in Fig.1. By quotient matrix of H, we have ρ * = 1 + m−1 3 < √ m − 2 for m ≥ 9, which contradicts Lemma 2.10. Thus e(B) = 1. Claim 3.3. G * [B] = K 2 . Proof. Otherwise, by Claim 3.2, we have G * [B] = {w 1 w 2 } ∪ I, where I is nonempty isolated vertex set. Choosing a vertex v ∈ I. Clearly, d(v) ≥ 2. If N(v) ⊂ A 1 = {v ∈ A | d(v) = 2}, then u * is a cut vertex, a contradiction. If N(v) ⊂ A 2 = A\A 1 , then v is included in a 4-cycle which has at most one vertex with degree two, a contradiction. So, |N A 1 (v)| = 1 and |N A 2 (v)| = 1. Let v 1 ∈ N A 1 (v), v 2 ∈ N A 2 (v) . Then u * v 1 vv 2 forms a 4-cycle. Notice that |N A (w 1 )|, |N A (w 2 )| ≥ 1. Choosing two vertices u 1 ∈ N(w 1 ) and u 2 ∈ N(w 2 ), respectively, then u * u 1 w 1 w 2 u 2 forms a 5-cycle. Moreover, the 4-cycle and 5-cycle either have a common vertex u * or have a common edge u * v 2 (u 2 coincides v 2 ). If the former occurs then u * is a cut vertex, otherwise G * contains a chorded cycle, a contradiction. By Claims 3.2 and 3.3, G * [B] = {w 1 w 2 }. Since G * has no triangles, N A (w 1 ) ∩ N A (w 2 ) = ∅. Furthermore, N A (w 1 )∪N A (w 2 ) = A. Assume |N A (w 1 )| = s, |N A (w 2 )| = t. Clearly, s, t ≥ 1 since δ(G * ) = 2. Let H(s, t) be a graph obtained from a double star graph D s,t by joining an isolated vertex u * to all its leaves vertices (see Fig.1). Now we get that G * H(s, t) for some s and t satisfying 2s + 2t + 1 = m. H(s, t)), and let N A (w 1 ) = {v 1 , · · · , v s } and N A (w 2 ) = {u 1 , · · · , u t }. By the symmetry, y v 1 = · · · = y v s and y u 1 = · · · = y u t . We have ρy v 1 = y u * + y w 1 , ρy u 1 = y u * + y w 2 , ρy w 1 = sy v 1 + y w 2 , ρy w 2 = ty u 1 + y w 1 . Then y v 1 − y u 1 = 1 ρ (y w 1 − y w 2 ) and then (ρ + 1)(y w 1 − y w 2 ) = sy v 1 − ty u 1 = s ρ (y w 1 − y w 2 ) + (s − t)y u 1 , which indicates that (ρ + 1 − s ρ )(y w 1 − y w 2 ) = (s − t)y u 1 ≤ 0. Clearly, ρ > s ρ since H(s, t) has K 1,s as a subgraph. It follows that y w 1 ≤ y w 2 . Note that s ≥ 2, obviously, G ′ = H(s, t) − v s w 1 + v s w 2 is also a minimally 2-connected graph. By Lemma 2.13, we have ρ(G ′ ) > ρ(G * ), which is a contradiction. Proof of Theorem 1.2 In this section, we will give the proof of Theorem 1.2. As we know, a block of a graph is a maximal 2-connected subgraph with respect to vertices. A block of a graph is called leaf block if it contains exactly one cut vertex. By Lemma 2.7, a minimally 2-edge-connected graph G without cut vertex is minimally 2-connected. Otherwise, G is made of some blocks including at least two leaf blocks, in which each block is minimally 2-connected by Lemma 2.7 and they intersect at cut vertices. In general, we write G = B(t, k) to denote a minimally 2-edge-connected graph with t cut vertices and k blocks. If t = 0 then k = 1 and G = B(0, 1) is a type of minimally 2-connected graph that is considered in Theorem 1.1. If t > 1 then k ≥ 2 and each block of G = B(t, k) has some cut vertices, in this case G = B(t, k) can be viewed as a tree if each block is regarded as an edge. Proof of Theorem 1.2 (i). We may assume that m ≥ 4 and X = (x 1 , . . . , x n ) T is the Perron eigenvector of G. Case 1. G has no cut vertices. In this case, G is minimally 2-connected. By Theorem 1.1, ρ(G) ≤ √ m and the equality holds if and only if G K 2, m 2 , which are just required. Case 2. G has some cut vertices. By definition, G = B(t, k) for some t ≥ 1 and k ≥ 2. Let B 1 , . . . , B k be its k blocks and m(B i ) = m i for i = 1, 2, . . . , k. We know that each B i is minimally 2-connected graph and m = m(B(t, k)) = k i=1 m i . By Theorem 1.1, ρ(B i ) ≤ √ m i , and the equality holds if and only if B i K 2, m i 2 for i = 1, 2, . . . , k. Notice that each B i is not a star since B i is minimally 2-connected. By Lemma 2.15, we obtain ρ(B(t, k)) < ρ 2 (B 1 ) + ρ 2 (B 2 ) + · · · + ρ 2 (B k ) ≤ √ m 1 + m 2 + · · · + m k = √ m, as desired. In what follows we will show (ii) of Theorem 1.2, and first we give some lemmas and propositions for the preparations. Clearly, if we transfer a leaf block of B(t, k) to another leaf block, one can simply verify the following result. Fig. 2: The graphs F 0 (t 1 , t 2 , t 3 ) and F 1 (t 1 , t 2 , t 3 ). · · · t t t t t t t t t t t t 1 2 t 2 . . . t t t t t t · · · t 3 1 2 F 1 (t 1 , t 2 , t 3 ) · · · t 1 t 1 t t t t t t t t 1 2 t 2 . . . t t t t t t · · · t 3 1 2 F 0 (t 1 , t 2 , t 3 ) Denote by u 1 the maximal degree vertex of the friend graph with t 1 triangles, u 2 a maximal degree vertex of K 2,t 2 , and u 3 a vertex of K 2,t 3 +1 with degree two. Let F 0 (t 1 , t 2 , t 3 ) be the graph obtained from the above three graphs by identifying u 1 , u 2 and u 3 . Denote F 1 (t 1 , t 2 , t 3 ) the graph by identifying a vertex of C 5 and the maximum degree vertex of F 0 (t 1 , t 2 , t 3 ) (see Fig.2), where t i ≥ 0. In order to give the proof of Theorem 1.2 (ii), we begin by proving the following two useful propositions. · · · t 1 t t t t t t t t t t t t w 3 w 4 w 1 w 2 u 1 u 2 u t 2 v . . . t t t t t t · · · v t 3 · · · t 1 t t t t t t t t t t t w 3 w 4 w 1 w 2 u 1 u 2 v t t t t t t · · · v t 3 t u ′ 1 u * u * v 1 v 2 v 1 v 2 G 1 G ′ 1 ✟ ✟ ✟ ✟ t t . . . u t 2 u ′ 1 Fig. 3: The vertex labels of G 1 F 1 (t 1 , t 2 , t 3 ) ∪ K 1 and G ′ 1 F 0 (t 1 + 1, t 2 + 1, t 3 ) ∪ 2K 1 . Proposition 4.1. ρ(F 1 (t 1 , t 2 , t 3 )) < ρ(F 0 (t 1 + 1, t 2 + 1, t 3 )), where F 1 (t 1 , t 2 , t 3 ) and F 0 (t 1 + 1, t 2 + 1, t 3 ) have the same number of edges m = 3t 1 + 2t 2 + 2t 3 + 7 for any t 1 ≥ 0, t 2 = 0 or ≥ 2 and t 3 ≥ 1, 3t 1 + 2t 2 + 5 for any t 1 ≥ 0, t 2 = 0 or ≥ 2 and t 3 = 0. (3) Proof. Let G 1 = F 1 (t 1 , t 2 , t 3 ) ∪ K 1 and let G ′ 1 = G 1 − (w 1 w 2 + w 1 w 3 + w 2 w 4 ) + (w 3 w 4 + u * u ′ 1 + u ′ 1 v) F 1 (t 1 + 1, t 2 + 1, t 3 ) ∪ 2K 1 , where the labels of V(G 1 ) and V(G ′ 1 ) are shown in Fig.4. Clearly, ρ(G 1 ) = ρ(F 1 (t 1 , t 2 , t 3 )), ρ(G ′ 1 ) = ρ(F 0 (t 1 + 1, t 2 + 1, t 3 )) and m(G 1 ) = m(G ′ 1 ) = m = 3t 1 + 2t 2 + 2t 3 + 7 for any t 1 ≥ 0, t 2 = 0 or ≥ 2 and t 3 ≥ 1, 3t 1 + 2t 2 + 5 for any t 1 ≥ 0, t 2 = 0 or ≥ 2 and t 3 = 0. It suffices to show ρ = ρ(G 1 ) < ρ ′ = ρ(G ′ 1 ). Let Y = (y 1 , . . . , y n ) and Z = (z 1 , . . . , z n ) be the Perron eigenvector of G 1 and G ′ 1 , respectively. Then we obtain (ρ ′ − ρ)Y T Z = Y T A(G ′ 1 )Z − Y T A(G 1 )Z = i j∈E(G ′ 1 ) (y i z j + z i y j ) − i j∈E(G 1 ) (y i z j + z i y j ) = (y w 3 z w 4 + z w 3 y w 4 ) + (y u * z u ′ 1 + z u * y u ′ 1 ) + (y u ′ 1 z v + z u ′ 1 y v ) −[(y w 1 z w 3 + z w 1 y w 3 ) + (y w 2 z w 1 + z w 2 y w 1 ) + (y w 2 z w 4 + z w 2 y w 4 )]. Notice that y u ′ 1 = 0, y w 1 = y w 2 , y w 3 = y w 4 , z w 1 = z w 2 = 0 and z w 3 = z w 4 , we have (ρ ′ − ρ)Y T Z = 2(y w 3 − y w 1 )z w 3 + y u * z u ′ 1 + y v z u ′ 1 .(4) Since C 5 is a proper subgraph of G 1 , by Lemma 2.11, we have ρ > ρ(C 5 ) = 2. By the eigen-equation ρy w 1 = y w 2 + y w 3 = y w 1 + y w 3 , we have y w 3 = (ρ − 1)y w 1 > y w 1 , and so the right of (4) is more than 0. Note that Y T Z ≥ 0. It follows that ρ ′ > ρ. Fig. 4: The vertex labels of G 2 F 0 (t 1 , t 2 , t 3 ) ∪ (t 3 + 1)K 1 and G ′ 2 F 0 (t 1 , t 2 + t 3 + 1, 0) ∪ (t 3 + 2)K 1 . . . . v ′ 1 v ′ 2 v 1 v 2 v t 3 t t t t · · · t t · · · u 1 u 2 u t 2 u ′ 1 u ′ t 3 +1 u * v · · · t t t t t t t t t t . . . v ′ 1 v ′ 2 v 1 v 2 v t 3 t t t t · · · t t · · · u 1 u 2 u t 2 u ′ 1 u ′ t 3 +1 u * v G 2 G ′ 2 Proposition 4.2. ρ(F 0 (t 1 , t 2 , t 3 )) < ρ(F 0 (t 1 , t 2 +t 3 +1, 0)), where F 0 (t 1 , t 2 , t 3 ) and F 0 (t 1 , t 2 + t 3 + 1, 0) have the same number of edges m = 3t 1 + 2t 2 + 2t 3 + 2, where t 1 ≥ 0, t 2 = 0 or ≥ 2, and t 3 ≥ 1. Proof. Let G 2 = F 0 (t 1 , t 2 , t 3 ) ∪ (t 3 + 1)K 1 , its vertices be labelled as in Fig.3 and the isolated set I 2 = {u ′ 1 , u ′ 2 , · · · , u ′ t 3 +1 }. Let G ′ 2 = G 2 − ( 2 i=1 u * v ′ i + 2 i=1 t 3 j=1 v ′ i v j ) + ( t 3 +1 i=1 (u * u ′ i + vu ′ i )) F 0 (t 1 , t 2 + t 3 + 1, 0) ∪ (t 3 + 2)K 1 . Clearly, ρ(F 0 (t 1 , t 2 , t 3 )) = ρ(G 2 ) and ρ(F 0 (t 1 , t 2 + t 3 + 1, 0)) = ρ(G ′ 2 ). Also, F 0 (t 1 , t 2 , t 3 ) and F 0 (t 1 , t 2 + t 3 + 1, 0) have the same number of edges 3t 1 + 2t 2 + 2t 3 + 2 = m. It suffices to show ρ = ρ(G 2 ) < ρ ′ = ρ(G ′ 2 ). Suppose to the contrary that ρ ≥ ρ ′ . Let Y = (y 1 , . . . , y n ) and Z = (z 1 , . . . , z n ) be the Perron eigenvector of G 2 and G ′ 2 , respectively. By symmetry, we have y u ′ 1 = 0, and z v 1 = z v ′ 1 = 0. Thus we obtain (ρ ′ − ρ)Y T Z = Y T A(G ′ 2 )Z − Y T A(G 2 )Z = i j∈E(G′ 2 ) (y i z j + z i y j ) − i j∈E(G 2 ) (y i z j + z i y j ) = (t 3 + 1)(y u * z u ′ 1 + z u * y u ′ 1 + y v z u ′ 1 + z v y u ′ 1 ) − 2(y u * z v ′ 1 + z u * y v ′ 1 ) − 2t 3 (y v 1 z v ′ 1 + z v 1 y v ′ 1 ) = (t 3 + 1)(y u * z u ′ 1 + y v z u ′ 1 ) − 2y v ′ 1 z u * .(5) By eign-equation of A(G 2 ) and A(G ′ 2 ), we have ρy v = t 2 y u 1 , ρy u 1 = y u * + y v . ρy v ′ 1 = y u * + t 3 y v 1 , ρy v 1 = 2y v ′ 1 . ρ ′ z u ′ 1 = z u * + z v , ρ ′ z v = (t 3 + t 2 + 1)z u ′ 1 . Then we obtain y v = t 2 ρ 2 − t 2 y u * , y v ′ 1 = ρ ρ 2 − 2t 3 y u * and z u ′ 1 = ρ ′ ρ ′2 − (t 3 + t 2 + 1) z u * . From (5), we get (ρ ′ − ρ)Y T Z = ( (t 3 + 1)ρ ′ ρ ′2 − (t 3 + t 2 + 1) ρ 2 ρ 2 − t 2 − 2ρ ρ 2 − 2t 3 )y u * z u * .(6) Since (t 3 +1)ρ ′ ρ ′2 −(t 3 +t 2 +1) monotonically decreases with respect to ρ ′ and ρ ≥ ρ ′ , from (6) we get (ρ ′ − ρ)Y T Z ≥ ( (t 3 + 1)ρ ρ 2 − (t 3 + t 2 + 1) ρ 2 ρ 2 − t 2 − 2ρ ρ 2 − 2t 3 )y u * z u * = ( t 3 + 1 ρ 2 − (t 3 + t 2 + 1) ρ 2 ρ 2 − t 2 − 2 ρ 2 − 2t 3 )ρy u * z u * ≥ ( t 3 + 1 ρ 2 − (t 3 + t 2 + 1) − 2 ρ 2 − 2t 3 )ρy u * z u *(7) Let g(ρ) = (t 3 + 1)(ρ 2 − 2t 3 ) − 2(ρ 2 − (t 3 + t 2 + 1)) = (t 3 − 1)ρ 2 − 2t 2 3 + 2t 2 + 2. It is clear that g(ρ) > 0 for ρ > ρ(K 2,t 3 +1 ) = √ 2t 3 + 2. One can also verify that ρ 2 > t 2 + t 3 + 1. Hence, (t 3 + 1) ρ 2 − (t 3 + t 2 + 1) − 2 ρ 2 − 2t 3 > 0. From (7) we have ρ ′ > ρ, a contradiction. Therefore, ρ < ρ ′ . Now is the time to prove (ii) of Theorem 1.2. Proof of Theorem 1.2 (ii). Let G * be the graph with the maximum spectral radius over all minimally 2-edge-connected graph of odd size m ≥ 11, and let X = (x 1 , . . . , x n ) T be the Perron eigenvector of G * with coordinate x u * = max{x i | i ∈ V(G * )}. Denote by ρ * = ρ(G * ), A = N(u * ) and B = V(G * ) \ (A ∪ u * ) . Notice that δ(G * ) = 2. Now we give Claims 4.1-4.8 to finish the proof of Theorem 1.2 (ii). Proof. Otherwise, there exists a vertex v ∈ B that is adjacent to a vertex u 2 ∈ A 2 . We may further assume that u 2 ∼ u ′ 2 ∈ A 2 . If v has no neighbor in B, then there exists a vertex u ∈ A adjacent to v due to δ(G * ) ≥ 2. It follows that          C = u * u ′ 2 u 2 vu 1 is a cycle with the chord u * u 2 if u = u 1 ∈ A 1 C = u * u ′ 2 vu 2 is a cycle with the chord u ′ 2 u 2 if u = u ′ 2 ∈ A 2 C = u * u ′ 2 u 2 vu 3 is a cycle with the chord u * 2 u 2 if u = u 3 ∈ A 2 It is impossible since any cycle of G * has no chord. So, d B (v) ≥ 1. However, in this situation, there exists a path P := vv 1 · · · v t in G * [B] such that v t is adjacent to some u ′ ∈ A since otherwise u 2 v will be a cut edge. By regarding u ′ as the above u, as similar above we can find a chorded cycle in G * , a contradiction. If A 2 ∅, then, from Claim 4.1, {u * } ∪ A 2 induces t 1 = |A 2 | 2 's triangles with a common vertex u * . Moreover, we see from Claim 4.2 that each of these triangles must be a leaf block of G * . Proof. By Claims 4.1 and 4.2, we know that A 2 induces some independent edges and N B (A 2 ) = ∅. Recall that A 1 induces some isolated vertices. Clearly, i∈A 1 d A 1 (i)x i = 0. From Lemma 2.10, ρ * > √ m − 2 ≥ 3 for m ≥ 11. By symmetry, for any i, j ∈ A 2 , we have x i = x j and so ρ * x i = x u * + x j = x u * + x i , which induces x i = x u * ρ * −1 < x u * 2 . Then we have ρ * 2 x u * = d(u * )x u * + i∈A 1 d A 1 (i)x i + i∈A 2 d A 2 (i)x i + i∈B d A (i)x i < d(u * )x u * + x u * 2 i∈A 2 d A 2 (i) + e(A, B)x u * = d(u * )x u * + x u * 2 · 2e(A 2 ) + e(A, B)x u * = (d(u * ) + e(A) + e(A, B))x u * = (m − e(B))x u * . Combining it with (m − 2)x u * < ρ * 2 x u * , we have e(B) < 2. It follows the result. If e(B) = 1, we may denote e = w * 1 w * 2 the unique edge in G * [B] in what follows. Without loss of generality, we may assume d A (w * 1 ) ≤ d A (w * 2 ). Claim 4.4. G * [{u * , w * 1 , w * 2 } ∪ N A 1 ({w * 1 , w * 2 })] C 5 . Proof. Firstly, we will show N A 1 (w * 1 ) ∩ N A 1 (w * 2 ) = ∅. Otherwise, let u 0 ∈ A 1 be the common vertex. If {u 0 , w * 1 , w * 2 } induces a 3-cycle, then u 0 w * 1 w * 2 a leaf block of G * . Thus G ′ = G * − u 0 w * 1 − u 0 w * 2 + u * w * 1 + u * w * 2 is minimally 2-edge-connected and ρ(G ′ ) > ρ(G * ) by Lemma 2.13. Therefore, there exists another vertex u 2 ∈ A 1 that is adjacent to at least one of {w * 1 , w * 2 }. It follows that C = w * 2 u 0 u * u 2 w * 1 is a cycle with the chord u 0 w * 1 if u 2 ∼ w * 1 , C = u * u 0 w * 1 w * 2 u 2 is a cycle with the chord u 0 w * 2 if u 2 ∼ w * 2 , which always leads a contradiction. Secondly, we will show d A 1 (w * 1 ) = 1 and d A 1 (w * 2 ) ≥ 1. In fact, since δ(G * ) = 2 and N A 2 (w * 1 ) = N A 2 (w * 2 ) = ∅ by Claim 4.2, we have d A 1 (w * 1 ), d A 1 (w * 2 ) ≥ 1. If d A 1 (w * 1 ), d A 1 (w * 2 ) ≥ 2, then G * contains H(2, 2) (see Fig.1) as a subgraph. We see that H(2, 2) is not minimally 2-edge-connected, which contradicts Lemma 2.5. Combining the above two facts, we have G * [{u * , w * 1 , w * 2 } ∪ N A 1 ({w * 1 , w * 2 })] H(1, t 2 ) for some positive t 2 ≥ 1, and clearly H(1, t 2 ) is a leaf block of G * . Finally, it suffices to show t 2 = 1. Suppose to the contrary that t 2 ≥ 2. Now let N A 1 (w * 1 ) = {u 0 } and N A 1 (w * 2 ) = {u 1 , u 2 , . . . , u t 2 }. Then G ′ = G * − w * 1 w * 2 + w * 1 u * is a graph obtained from G by replacing the block H(1, t 2 ) to K 2,t 2 * K 3 . Thus, G ′ is also minimally 2-edge-connected. By Lemma 2.13, we have ρ(G ′ ) > ρ(G * ), a contradiction. (w) = v * 1 . Notice that d(w) ≥ 2. We claim that N A 1 (w) = {v * 1 , v * 2 }. Since otherwise, w ∼ v ∈ A 1 \{v * 1 , v * 2 }, then u * vwv * 1 w * 1 w * 2 v * 2 is a cycle with the chord v * 1 u * , a contradiction. Thus, W = {u * , w, w * 1 , w * 2 , v * 1 , v * 2 } induces a minimally 2-edge-connected leaf block of G * . By symmetry, we also have x v * 1 = x v * 2 . Let G ′ = G * − w * 2 v * 2 + w * 2 v * 1 . We see that W also induces a leaf block of G ′ and so G ′ is also a minimally 2-edge connected graph. By Lemma 2.13, we have ρ(G ′ ) > ρ(G * ), a contradiction. Let A ′ 1 = A 1 \{v * 1 , v * 2 }. By Claim 4.5, each vertex of B 1 only joins some vertices in A ′ 1 . i.e. N A 1 (w i ) = N A ′ 1 (w i ) for any w i ∈ B 1 . Claim 4.6. |B 1 | ≥ 2 and |N A 1 (w 1 ) ∩ N A 1 (w 2 )| = 0 or 2 for any w 1 w 2 ∈ B 1 . Proof. Firstly, we show that |B 1 | ≥ 2. Otherwise, we may assume B 1 = {w 1 }, then G * [N A 1 (w 1 ) ∪ {w 1 , u * }] K 2,a 1 , where a 1 = |N A 1 (w 1 )|. By Claims 4.2, 4.4 and 4.5, we have G * F 1 (t 1 , a 2 , 0) for some positive t 1 ≥ 0, a 2 ≥ 2 and 3t 1 + 2a 2 + 7 = m(G * ) = m. By Proposition 4.1, we know that ρ(G * ) = ρ(F 1 (t 1 , a 2 , 0)) < ρ(F 0 (t 1 + 1, a 2 + 1, 0), a contradiction. Suppose that |N A 1 (w 1 ) ∩ N A 1 (w 2 )| ≥ 3, let {v 1 , v 2 , v 3 } ⊆ N A 1 (w 1 ) ∩ N A 1 (w 2 ), then G * contains a 5-cycle C 2 = v 1 w 1 v 2 w 2 v 3 with chord v 1 w 2 , a contradiction. Next we show that |N A 1 (w 1 ) ∩ N A 1 (w 2 )| 1. Otherwise, let N A 1 (w 1 ) ∩ N A 1 (w 2 ) = {v}, then G * contains a 6-cycle C 1 = u * w ′ 1 w 1 vw 2 w ′ 2 with chord u * v, where w ′ 1 ∈ N A 1 (w 1 )\v and w ′ 2 ∈ N A 1 (w 2 )\v, it is a contradiction. Thus, we have |N A 1 (w 1 ) ∩ N A 1 (w 2 )| = 0 or 2. Notice that d A 1 (w i ) ≥ 2 for any w i ∈ B 1 . By Claims 4.1-4.6 we can get the structure of G * shown as Fig.5. In particular, if e(B) = ∅, then G * contains no 5-cycle C = u * v * 1 w * 1 w * 2 v * 2 . For nonnegative integer p, q, G * [N A 1 (B 1 ) ∪ B 1 ] p i=1 K 1,r i q j=1 K 2,s j and satisfy 3t 1 + 2 p i=1 r i + 2 q i=1 s i + 2q + 5 = m (r i , s i ≥ 2). Furthermore, we will determine the values of p, q. Claim 4.7. p ≤ 1 and q ≤ 1. ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ ❵ ❵ ❵ ❵ ❵ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ t 1 v * 1 v * 2 w * 1 w * 2 r 1 r p w 1 w p · · · s 1 s 2 s q · · · · · · · · · A ′ 1 B 1 u * v 1 v 2 v 3 v 4 w ′ 1 w ′ 2 · · · · · · · · · · · · · · · Proof. We firstly show p ≤ 1. Suppose p ≥ 2, then there exists two vertices, say w 1 , w 2 in B 1 with G * [N A ′ 1 (w i ) ∪ {w i } = K 1,r i for i = 1, 2. Without loss of generality, we may assume that x w 1 ≥ x w 2 . Denote by G ′ = G * − v∈N A 1 (w 2 ) vw 2 + v∈N A 1 (w 2 ) vw 1 . By Lemma 2.13, we have ρ(G ′ ) > ρ * . Clearly w 2 is an isolated vertex of G ′ . Set G ′′ = G ′ − {w 2 }. Then G ′′ is also a minimally 2-edge-connected graph since N A 1 (w 1 ) ∪ N A 1 (w 2 ) ∪ {w 1 , u * } induces a block K 2,r 1 +r 2 in G ′′ . However ρ(G ′′ ) = ρ(G ′ ) > ρ * , a contradiction. Now we will show q ≤ 1. Otherwise, q ≥ 2. Then G * [N A 1 (B 1 )∪B 1 ] contains K 2,s 1 , K 2,s 2 (s 1 , s 2 ≥ 2) as induced subgraphs. Denote by w ′ i ∈ V(K 2,s i ) ∩ B 1 for i = 1, 2. Set N A 1 (w ′ 1 ) = {v 1 , v 2 } and N A 1 (w ′ 2 ) = {v 3 , v 4 }. Let X be the Perron vector of G * whose entry x v is labelled by vertex v. By the symmetry, x v 1 = x v 2 and x v 3 = x v 4 . Without loss of generality, we may assume that x v 1 ≥ x v 3 . Then x v 1 + x v 2 ≥ x v 3 + x v 4 . Let G ′′′ = G * − w ′ 2 v 3 − w ′ 2 v 4 + w ′ 2 v 1 + w ′ 2 v 2 , and ρ ′′′ = ρ(G ′′′ ). Clearly, G ′′′ is minimally 2-edge-connected. By Lemma 2.14, we get ρ ′′′ > ρ * , which contradicts with the maximality of ρ * . By comparing Fig. 2 and Fig. 5, we have t 2 = r 1 and t 3 = s 1 . From Claims 4.1-4.7, we know that G * has two forms: F 0 (t 1 , t 2 , t 3 ) or F 1 (t ′ 1 , t ′ 2 , t ′ 3 ), where t 1 , t 3 , t ′ 1 , t ′ 3 ≥ 0, t 2 , t ′ 2 = 0 or t 2 , t ′ 2 ≥ 2 and satisfy m =                3t 1 + 2t 2 if t 3 = 0 3t 1 + 2t 2 + 2t 3 + 2 if t 3 ≥ 1 3t ′ 1 + 2t ′ 2 + 5 if t ′ 3 = 0 3t ′ 1 + 2t ′ 2 + 2t ′ 3 + 2 + 5 if t ′ 3 ≥ 1. Clearly, t 1 ≥ 1 since otherwise m is even. Suppose that G * F 1 (t ′ 1 , t ′ 2 , t ′ 3 ), by Proposition 4.1, we have ρ(F 1 (t ′ 1 , t ′ 2 , t ′ 3 )) < ρ(F 0 (t ′ 1 + 1, t ′ 2 + 1, t ′ 3 )), which contradicts the maximality of ρ(G * ). Thus G * F 0 (t 1 , t 2 , t 3 ), where t 3 ≥ 0, t 2 = 0 or ≥ 2 and satisfy m = 3t 1 + 2t 2 if t 3 = 0 3t 1 + 2t 2 + 2t 3 + 2 if t 3 ≥ 1. Lemma 2.1 ( [8]). A minimally 2-connected graphs with more than three vertices contains no triangles. Every cycle of a minimally 2-connected graph contains at least two vertices of degree two. . If G is a minimally 2-(edge)-connected graph, then δ(G) = 2. Lemma 2. 5 . 5A 2-edge-connected subgraph of a minimally 2-edge-connected graph is also minimally 2-edge-connected. the equality holds if and only if both H and K are stars, where (H, v) * (K, w) is obtained by identifying v and w from disjoint union of H and K. 3 Proof of Theorem 1.1 by Lemma 2.4. This induces d(u * ) = 2 and so |A| = 2. We may assume A = {u 1 , u 2 }. Note that e(B) = 0 or 1. If e(B) = 0 then each vertex in B is adjacent with both of u 1 and u 2 since δ(G * ) = 2. This leads to the size of G * is even, a contradiction. Thus e(B) = 1, and we may assume B Proof. Without loss of generality, we assume s ≤ t. If s = 1, then t = m−3 2 , and so H(s, t) H(1, m−3 2 ). The result holds. Suppose G * H(s, t) for s ≥ 2. Let Y be the Perron eigenvector of H(s, t) with spectral radius ρ = ρ( Lemma 4 . 1 . 41For a minimally 2-edge-connected graph G = B(t, k), let B i be a leaf block of G and u ∈ B i be a cut vertex. For any v ∈ V(G) \ {u}, we have G ′ = G − w∈N(u)∩B i wu + w∈N(u)∩B i wv is also minimally 2-edge-connected. Claim 4. 1 . 1G * [A]is isomorphic to the union of some independent edges and isolated vertices.Proof. On the one hand, G * [A] contains no cycle. Otherwise, we assume that a cycle C l ⊂ G * [A] (l ≥ 3). Then there exists a wheel W l+1 in G * , it forms a chorded cycle in G * , which contradicts Lemma 2.6. On the other hand, G * [A] contains no P 3 . Otherwise, G * contains a chorded cycle with order 4, a contradiction.Let A 1 be the isolated vertex set of G * [A]. Then A 2 = A \ A 1 consists of some independent edges if A 2 ∅. Claim 4.2. N B (u) = ∅ for any u ∈ A 2 . Claim 4.3. e(B) = 0 or 1. Fig. 5 : 5The structure of G * , where t 1 ≥ 0, p + q i=1 s i = |B 1 | and p i=1 r i + 2q = |A ′ 1 |. Theorem 1.1. Let G be a minimally 2-connected graph of size m. (i) If m is even, then ρ(G) ≤ √ m, the equality holds if and only if G K 2, m 2 . (ii) If m is odd and m ≥ 9, then ρ the equality holds if and only if G S K 2, m−1 2 . Theorem 1.2. Let G be a minimally 2-edge-connected graph of size m. If m ≥ 11 is odd and m 15, then ρ(i) If m is even, then ρ(G) ≤ √ m, the equality holds if and only if G K 2, m 2 . (ii) Denote by B 1 the set of all isolated vertices in B. By Claim 4.3,B = B 1 ∪ {w * 1 w * 2 }. By Claim 4.4, we may assume that N A 1 (w * 1 ) = {v * 1 } and N A 1 (w * 2 ) = {v * 2 } in what follows. Claim 4.5. N B 1 ({v * 1 , v * 2 }) = ∅. Proof.Suppose to the contrary that there exists a vertex w in B 1 with neighbor v * 1 or v * 2 . Without loos of generality, we assume N A 1 t 1 t 1 · · · t t t t t t t t t t If t 3 = 0, then t 2 = m−3t12 , and thus G * F 0 (t 1 , t 2 , 0) = F 0 (t 1 , m−3t 1 2 , 0)). If t 3 ≥ 1, then t 2 + t 3 + 1 = m−3t12 . By Proposition 4.2, we haveBy the maximality of ρ(G * ) again, we get G * F 0 (t 1 , m−3t 1 2 , 0) for some t 1 ≥ 1 and m−3t 1 2 = 0 or ≥ 2. At last, we will show t 1 = 1.wherefor m ≥ 19. For 11 ≤ m ≤ 17, i.e. m = 15, we have 1+ √ 41 2 = ρ(F 5 ) > ρ(F 0 (1, 6, 0)). By the above arguments, G * F 5 for m = 15, and G * F 0 (1, m−3 2 , 0) for m ≥ 11 and m 15.It completes the proof of Theorem 1.2 (ii). Remark 1. For odd m < 11, by Claims 4.1-4.7, we get that the minimally 2-edge connected graph and the extremal graph G * is given by the following Tables.mMinimally 2-edge-connected graph G * ρ(G * ) 3 C 3 C 3 2 5 C 5 C 5 2 7 C 7 , S K 2,3 , C 3 * C 4 C 3 * C 4 2.5035 9 C 9 , S K 2,4 , C 3 * C 6 , C 3 * K 2,3 , C 4 * C 5 , F 3 F 3 3 Extremal graph theory, dover publications. 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[ "Prepared for submission to JHEP Mini-twistors and the Cotton Double Copy", "Prepared for submission to JHEP Mini-twistors and the Cotton Double Copy" ]
[ "Mariana Carrillo González [email protected] \nTheoretical Physics\nBlackett Laboratory\nImperial College\nSW7 2AZLondonUnited Kingdom\n", "William T Emond [email protected] \nInstitute of Physics\nCEICO\nCzech Academy of Sciences\nNa Slovance 2182 21Praha 8Czech Republic\n", "Nathan Moynihan [email protected] \nSchool of Physics and Astronomy\nHiggs Centre for Theoretical Physics\nThe University of Edinburgh\nEH9 3FDScotland, United Kingdom\n", "Justinas Rumbutis [email protected] \nDepartment of Physics and Center for Theoretical Physics\nNational Taiwan University\n10617TaipeiTaiwan\n", "Chris D White [email protected] \nDepartment of Physics and Astronomy\nQueen Mary University of London\n327 Mile End RoadE1 4NSLondonUnited Kingdom\n" ]
[ "Theoretical Physics\nBlackett Laboratory\nImperial College\nSW7 2AZLondonUnited Kingdom", "Institute of Physics\nCEICO\nCzech Academy of Sciences\nNa Slovance 2182 21Praha 8Czech Republic", "School of Physics and Astronomy\nHiggs Centre for Theoretical Physics\nThe University of Edinburgh\nEH9 3FDScotland, United Kingdom", "Department of Physics and Center for Theoretical Physics\nNational Taiwan University\n10617TaipeiTaiwan", "Department of Physics and Astronomy\nQueen Mary University of London\n327 Mile End RoadE1 4NSLondonUnited Kingdom" ]
[]
The double copy relates quantities in gauge, gravity and related theories. A well-known procedure for relating exact classical solutions is the Weyl double copy in four spacetime dimensions, and a three-dimensional analogue of this -the Cotton double copyhas recently been found for topologically massive gauge theory and gravity. In this paper, we use twistor methods to provide a derivation of the position-space Cotton double copy, where this is seen to arise from combining appropriate data in so-called minitwistor space. Our methods rely on a massive generalisation of the Penrose transform linking spacetime fields with cohomology classes in minitwistor space. We identify the relevant transform from the twistor literature, but also show that it naturally arises from considering scattering amplitudes in momentum space. We show that the Cotton double copy in position space is only valid for type N solutions, but that a simple twistor space double copy is possible for non-type N solutions, where we use anyons to illustrate our arguments.
10.1007/jhep03(2023)177
[ "https://export.arxiv.org/pdf/2212.04783v1.pdf" ]
254,535,579
2212.04783
22afc473c61ea1fd870c3d7510c11f37d24b9618
Prepared for submission to JHEP Mini-twistors and the Cotton Double Copy Mariana Carrillo González [email protected] Theoretical Physics Blackett Laboratory Imperial College SW7 2AZLondonUnited Kingdom William T Emond [email protected] Institute of Physics CEICO Czech Academy of Sciences Na Slovance 2182 21Praha 8Czech Republic Nathan Moynihan [email protected] School of Physics and Astronomy Higgs Centre for Theoretical Physics The University of Edinburgh EH9 3FDScotland, United Kingdom Justinas Rumbutis [email protected] Department of Physics and Center for Theoretical Physics National Taiwan University 10617TaipeiTaiwan Chris D White [email protected] Department of Physics and Astronomy Queen Mary University of London 327 Mile End RoadE1 4NSLondonUnited Kingdom Prepared for submission to JHEP Mini-twistors and the Cotton Double Copy QMUL-PH-22-38, Imperial/TP/2022/MC/06 The double copy relates quantities in gauge, gravity and related theories. A well-known procedure for relating exact classical solutions is the Weyl double copy in four spacetime dimensions, and a three-dimensional analogue of this -the Cotton double copyhas recently been found for topologically massive gauge theory and gravity. In this paper, we use twistor methods to provide a derivation of the position-space Cotton double copy, where this is seen to arise from combining appropriate data in so-called minitwistor space. Our methods rely on a massive generalisation of the Penrose transform linking spacetime fields with cohomology classes in minitwistor space. We identify the relevant transform from the twistor literature, but also show that it naturally arises from considering scattering amplitudes in momentum space. We show that the Cotton double copy in position space is only valid for type N solutions, but that a simple twistor space double copy is possible for non-type N solutions, where we use anyons to illustrate our arguments. Introduction In recent years, a correspondence known as the double copy has generated a great deal of interest. Inspired by previous work in string theory [1], its original incarnation stipulates that scattering amplitudes in gauge theory can be straightforwardly turned into gravity amplitudes [2][3][4]. To do so, one must substitute the appropriate coupling constants, as well as replace colour charge information with additional kinematic factors. This works for a wide variety of gauge and gravity theories, both with and without supersymmetry. Furthermore, one may also start with gauge amplitudes and go the other way, replacing kinematic with colour information. This is called the zeroth copy, and generates amplitudes in a scalar theory with two distinct types of colour charge, which has become known as biadjoint scalar theory. Whilst not a physical theory by itself, its dynamics is at least partially inherited by gauge and gravity theories. Furthermore, this ladder of theories includes a wide variety of examples (e.g. both with and without supersymmetry), and is itself part of a wider web of theories known to exhibit such correspondences: see e.g. refs. [5][6][7][8] for recent reviews. In the past few years, it has become increasingly recognised that the double copy applies beyond fixed-order scattering amplitudes, in particular to all-order perturbative information [9][10][11][12], exact classical solutions (see also refs. [37,38] for related work in a different context, and [39] for a recent overview of how this is related), perturbative classical solutions [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], and potential non-perturbative aspects [59][60][61][62][63][64][65][66][67][68][69][70][71]. The double copy offers new calculational tools for General Relativity and related theories, and indeed has already been used to generate new results needed for gravitational wave experiments [72,73]. However, it also offers new conceptual insights not only about gravity, but also the very foundations of field theory itself. It is then important to find explanations of where the double copy comes from, particularly in those cases in which exact statements can be made. In four spacetime dimensions, a well-known exact classical double copy is the Weyl double copy of ref. [22]. Written using the spinorial formalism of field theory (see e.g. refs. [74][75][76] for reviews), it relates spacetime fields in biadjoint, gauge and gravity theories directly in position space 1 . This is at odds with the original double copy for scattering amplitudes [2][3][4], which is naturally formulated in momentum space. Nevertheless, it is argued to be exact for certain gravity solutions, namely those vacuum solutions which are of type D in the well-known Petrov classification. Exact type N cases are also known [25], as well as other Petrov types at linearised level only [26,28]. In order to explain the above results, refs. [26,28] found a derivation of the Weyl double copy using twistor theory [78][79][80], a decades-old set of mathematical ideas linking field theory, complex analysis and algebraic geometry (see e.g. refs. [75,81,82] for pedagogical reviews). In a nutshell, twistor theory maps points in spacetime non-locally to an abstract twistor space, such that certain quantities in the former show up as mathematically convenient data in the latter. In particular, an integral formula known as the Penrose transform [83] relates certain "functions" in twistor space to spacetime solutions of the massless free field equations, where a given spin of the spacetime field translates to a given homogeneity of the twistor function under rescalings of its argument. By multiplying together functions of different homogeneity in an appropriate manner, refs. [26,28] showed that one could derive the Weyl double copy in position space. Furthermore, the twistor approach provided a geometric interpretation of certain aspects of the Weyl double copy that had previously been obscure, such as the inverse zeroth copy that takes one from biadjoint scalar to gauge theory. Applications of the twistor approach include generalising the Weyl double copy away from type D or N solutions (albeit at linearised level only), and also showing that physical properties such as multipoles can be straightforwardly mapped between different theories [29]. Despite the above successes, the twistor double copy is not without its conceptual problems. Chief among these is the fact that the quantities entering the Penrose transform are not, strictly speaking, functions. Rather, they may be redefined by equivalence transformations, which do not change the result of the Penrose transform integral. Mathematically speaking, such quantities are representatives of cohomology classes, as identified in ref. [84]. This then poses a puzzle, in that the non-linear product of twistor functions needed to obtain the Weyl double copy in position space is clearly inconsistent with the ability to first perform equivalence transformations. It thus seems that special representatives of each cohomology class must be chosen in each theory in order to make the double copy manifest, and it is not obvious a priori how to achieve this. A number of papers have subsequently addressed this point. First was ref. [85], which considered radiative spacetimes, and showed that data at past or future or null infinity could be used to pick out special cohomology representatives in twistor space. These were defined in terms of so-called Dolbeault cohomology groups, as distinct from theCech cohomology groups that enter the original formulation of the Penrose transform. Reference [30] also considered the Dolbeault language, and argued that one can use known methods in Euclidean signature [86] to pick out special twistor representatives for each spacetime field, so that a product structure is manifest in twistor space. It is not clear how the procedures of refs. [30,85] are related, if at all. More recently, ref. [87] considered the relationship between scattering amplitudes, twistor space, and classical solutions. It is known that certain classical solutions can be obtained as inverse Fourier transforms of momentum-space amplitudes. Reference [87] craftily split this inverse Fourier transform into two steps, where the first maps the amplitudes into quantities in twistor space. The second step then corresponds to the Penrose transform from twistor to position space, and it is therefore the case that scattering amplitudes themselves can be used to pick out cohomology representatives for certain classical fields in twistor space. As has been made clear elsewhere [88], the representatives picked out by the relevant amplitudes in gauge, gravity or scalar theory are precisely those entering the original twistor double copy of refs. [26,28]. Not only does this fix the cohomological ambiguities in the twistorial approach, it also establishes a very strong link between the double copies in momentum, twistor and position space. Equally important as explaining the origin of known double copies is to continue to generalise this correspondence to novel theories or situations. In this spirit, refs. [89,90] recently proposed a new exact classical double copy for topologically massive gauge theory and gravity, called the Cotton double copy. Like the Weyl double copy in four dimensions, it uses the spinorial formalism of field theory, and expresses a precise relationship between scalar, gauge and gravity fields in position space. Indeed, this relationship is analogous to its four-dimensional counterpart, although appears to hold for a more restricted class of solutions than in the Weyl case. That is, both refs. [89,90] only found Cotton double copy examples in position space of Petrov type N, rather than the more general type D, and this fact demands a further explanation. It is also natural to ask whether there is a twistorial justification for the Cotton double copy, that mirrors its four-dimensional counterpart. Constructing such an argument should itself settle the issue of how general the Cotton double copy is, and this paper will show that this is indeed possible. We will use the language of minitwistors in three spacetime dimensions, and the presence of a topological mass means that we will have to consider an alternative to the usual Penrose transform. Just such a transform has been provided before in the mathematical literature [91]. It is formulated by considering the most general possible cohomology classes in minitwistor space. This involves introducing an extra parameter in twistor space relative to the conventional four-dimensional case, whose presence corresponds to the presence of the topological mass in spacetime. Armed with this minitwistor transform, we will show explicitly that appropriately combining particular minitwistor representatives allows us to derive the position-space Cotton double copy. Our minitwistor derivation of the Cotton double copy will suffer from similar conceptual issues to its four-dimensional counterpart. Namely, the form of the double copy in minitwistor space involves products of "functions", which should properly be interpreted as representatives of cohomology classes, with an appropriate procedure for picking them. However, the ideas of ref. [87] will once again come to the rescue: we will show that they can be generalised to the three-dimensional case, such that the minitwistor double copy follows as a consequence of the known double copy for scattering amplitudes in topologically massive theories. We will explicitly consider amplitudes corresponding to point-like sources emitting gauge bosons, which correspond to type D classical solutions 2 . These will allow us to independently validate the form of the massive Penrose transform, where particular cohomology representatives are necessarily picked out. Interestingly, we will find that a simple twistor-space double copy occurs even in the type D case. However, a simple position-space double copy is restricted to type N only, as a direct consequence of the form of the massive Penrose transform. Our results provide a firm foundation for the Cotton double copy, whilst also providing an interesting counterpoint for the four-dimensional twistor double copy. This in turn suggests the use of twistor methods more widely in the study of (non-)exact classical double copies, including in higher dimensions where applicable. The structure of our paper is as follows. In section 2, we review relevant properties of topologically massive theories in three dimensions, including the Cotton double copy. We also introduce the concept of minitwistors, and their associated Penrose transform, following ref. [91]. In section 3, we provide a twistorial derivation of the Cotton double copy, emphasising the similarities and differences with the four-dimensional twistor double copy of refs. [26,28]. In section 4, we show how the ideas of ref. [87] can be adapted to three dimensions, and use anyon solutions to illustrate our general arguments. Finally, we discuss our results and conclude in section 5. Review of necessary concepts In this section, we review salient material for the rest of the paper, both in order to set up our notation and conventions, and also to make the presentation relatively self-contained. We begin by introducing the spinorial formalism for field theories. Spinors in (2+1) dimensions Our first encounter with relativistic field theories in four spacetime dimensions typically involves the use of 4-vectors and tensors. As is well-known, however, it is possible to recast all relevant field equations into an alternative language, namely that of 2-component spinors [92,93] (see e.g. refs. [74][75][76] for pedagogical reviews). Similar ideas occur in (2+1) dimensions [94][95][96], which we now briefly review. We will be concerned with (dual) spinors λ A (λ A ), whose indices A ∈ {0, 1} may be raised and lowered using the two-dimensional Levi-Civita symbol: λ B = BA λ A , λ B = BA λ A , AB = 0 −1 1 0 = − AB . (2.1) Note that, in contrast to the well-known four-dimensional case, only one type of spinor index A occurs. This is because the Lorentz group in (2+1) dimensions is covered by a single SL(2,C) group. In four dimensions, on the other hand, the Lorentz group is covered by two distinct SL(2,C) groups, leading to the presence of spinors λ A and conjugate spinors π A , where the prime is used to differentiate which SL(2,C) group acts on which index. We can convert (2+1)-dimensional tensors into spinors (and vice versa) using the Infeld-van der Waerden symbols, whose explicit form depends upon the chosen basis in spinor space. It is in fact possible to choose them to be real, so that we will adopt the SL(2, R) representation σ µ AB = 1 0 0 1 , 0 1 1 0 , 1 0 0 −1 . (2.2) A given tensor index is thus converted into a pair of spinor indices, and one may also verify the following useful identities: η µν = 1 2 σ µ AB σ νAB , σ µ AB σ µGD = −( AG BD + AD BG ). (2.3) As an example, a single 4-vector with real components has the spinorial translation p AB = p µ σ µ AB = p 0 + p 2 p 1 p 1 p 0 − p 2 , det(p µ σ µ AB ) = −p µ p µ ,(2.4) where the matrix thus obtained is referred to as a bispinor. As may be verified by direct computation, one can always decompose a bispinor in (2+1) dimensions into the outer product of two complex spinors p AB = λ (AλB) , (2.5) where the latter are given by λ A = 1 √ p 2 − p 0 p 2 − p 0 p 1 − im ,λ A = 1 √ p 2 − p 0 p 2 − p 0 p 1 + im . (2.6) In the scattering amplitudes literature, it is common to introduce a Dirac notation for (dual) spinors: |λ ≡ λ A , λ| ≡ λ A . (2.7) Then we can define spinor helicity variables in the usual way as λ i λ j ≡ AB λ i B λ j A , (2.8) where we note in particular the identities λ λ = 2im, λ|γ µ |λ = −2p µ . (2.9) Topologically massive theories and their double copy Having reviewed the language of two-spinors in (2+1) dimensions, let us now introduce the theories that we will encounter throughout the paper. First up is topologically massive Yang-Mills theory, which is described by the action S T M Y M = d 3 x − 1 4 F aµν F aµν + µνρ m 12 6A aµ ∂ ν A ρ a + g √ 2f abc A aµ A bν A cρ , (2.10) implying the equation of motion D µ F µν + m 2 ε νργ F ργ = 0 . (2.11) Physically, this describes a gauge boson with mass m and a single helicity h = m |m| . The mass term in the action is only possible in three spacetime dimensions, due to the presence of the three-dimensional Levi-Civita tensor. Furthermore, unlike conventional mass terms in arbitrary spacetime dimension, one may show that the mass term introduced here is manifestly gauge-invariant. The mass is topological in the sense that it is independent of the local metric. We will also be concerned with topologically massive gravity, whose action is 12) and leads to the equation of motion S T M G = 1 κ 2 d 3 x √ −g −R − 1 2m µνρ Γ α µσ ∂ ν Γ σ αρ + 2 3 Γ α µσ Γ σ νβ Γ β ρα ,(2.G µν + 1 m C µν = 0 , (2.13) where C µν is a symmetric tensor known as the Cotton tensor: C µν = µρσ D ρ R σ ν − 1 4 δ σ ν R . (2.14) We can think of this as a (2 + 1)-dimensional analogue of the Weyl tensor in four (or higher) dimensions, where the latter is what the Riemann curvature reduces to in the case of vacuum solutions of the Einstein equations. Like the Weyl tensor, the Cotton tensor vanishes for conformally flat spacetimes. As for the Yang-Mills case discussed above, the second term in the action of eq. (2.12) is impossible to write down in four spacetime dimensions. It is a correction to the pure Einstein-Hilbert action, and generates a mass for the graviton that is invariant under diffeomorphisms. As discussed in the introduction, it is by now very well-known that Yang-Mills theory and gravity (plus their generalisations) are related by the double copy, which applies to both scattering amplitudes and classical solutions. It was recently also conjectured that the topologically massive gauge and gravity theories considered here are related by a similar double copy [97], evidence for which has been presented in a number of non-trivial scenarios [98][99][100][101][102][103]. Important for this paper is the Cotton double copy [89,90] 3 , which directly relates classical solutions of the above equations of motion, expressed in the spinorial formalism. As reviewed in refs. [89,90], the spinorial translation of the free field equation for Abelian topologically massive gauge theory is ∂ G A Φ GB = mΦ AB ,(2.15) where ∂ AB is the spinorial translation of the partial derivative operator ∂ µ , and we have defined 4 Φ AB = 1 2 σ µ AB µνρ F νρ . (2.16) Similarly, the free-field equations of topologically massive gravity take the form ∂ E A C EBGD = mC ABGD ,(2.17) where C ABGD = C µν σ µ AB σ ν GD . (2.18) It is instructive to contrast these equations with their natural counterparts in four-dimensional gauge and gravity theory, namely the massless free field equation ∂ A 1 A Φ A 1 A 2 ...A 2n = 0. (2.19) Here Φ A 1 ...A 2n is a multi-index symmetric spinor corresponding to a single polarisation state of the field 5 , and the derivative operator is now the appropriate four-dimensional spinorial translation of the partial derivative operator in spacetime. We have written eq. (2.19) for a general spin-n, from which we may note that there are 2n spinor indices for a spin-n field. Apart from the slight difference in derivatives, we see that eqs. (2.15, 2.17) differ from eq. (2.19) due to the presence of the mass term on the right-hand side. Solutions of eq. (2.19) of different spin can be related to each other by the Weyl double copy [22], which has been shown to work for certain algebraically special spacetimes. Φ ABCD = Φ (AB Φ CD) Φ . (2.20) Here Φ is a field satisfying the massless Klein-Gordon equation in spacetime, and to clarify where this applies, we may note that a consequence of the limited range of spinor indices is that an arbitrary multi-index symmetric spinor can be decomposed in terms of single-index principal spinors, such that we have Φ ABCD = α (A β B γ C δ D) . (2.21) So-called Petrov type D solutions are those for which there are two distinct principal spinors, each of double multiplicity. Type N solutions have a single principal spinor of multiplicity four. These are the two cases of algebraically special solutions for which the Weyl double copy is known to be exact. Motivated by the Weyl double copy, refs. [89,90] considered whether a similar relation can be written for topologically massive Yang-Mills and gravity theories. Indeed it can, provided one replaces the Weyl tensor with the Cotton tensor, and instead considers Φ to be a solution of the massive Klein-Gordon equation: C ABGD = Φ (AB Φ GD) Φ . (2.22) This is the Cotton double copy formula alluded to above, and is known to apply at least for type N solutions. In order to see whether it is in fact more general than this (as is the Weyl double copy), it is fruitful to seek a more underlying explanation of where the Cotton double copy comes from. In the case of the Weyl double copy, refs. [26,28] provided a derivation of the position-space formula using the techniques of twistor theory. This suggests that similar techniques could prove useful in deriving the Cotton double copy. Before we can do this, however, we must first familiarise ourselves with twistor techniques in (2+1) dimensions. This is the subject of the following section. Minitwistor theory In this section, we give a brief introduction to the subject of twistors in three-dimensional space. Pedagogical reviews of four-dimensional twistor theory can be found in e.g. refs. [75,81,82]. The subject of three-dimensional twistor theory is less well-known, and thus our aim is to collect a number of useful results from the literature in one place [91,105]. The relevant concepts are similar to the case of the four-dimensional twistor double copy defined in refs. [26,28]: given flat spacetime, one may construct an abstract twistor space, such that points in spacetime are mapped non-locally to the latter and vice versa. Solutions of the massless free field equation of eq. (2.19) can be obtained as a certain contour integral in twistor space, which is known as the Penrose transform. In order to apply these same ideas to (2+1) dimensions, we must first define the relevant twistor space, and then arrive at the necessary Penrose transform, which must somehow take into account the presence of the mass in topological gauge theory or gravity. Let us take each of these topics in turn. Minitwistor geometry Let us first consider complexified Minkowski space M = C 3 , with line element ds 2 = −dt 2 + dx 2 + dy 2 , t, x, y ∈ C. (2.23) Using the Infeld-van-der-Waerden symbols of eq. (2.2), a point x ∈ M has a spinorial translation as a symmetric 2 × 2 matrix: x AB = −t − y −x −x −t + y . (2.24) We may then define minitwistor space MT as the two-dimensional set of null planes in M. Any such plane is defined by a null three-dimensional normal vector n µ , such that n µ x µ = u, n 2 = 0, u ∈ C. (2.25) Nullity of n µ , and the fact that it is defined only up to arbitrary scalings, implies that its spinorial translation factorises as follows: n AB ≡ n µ σ µ AB = λ A λ B , (2.26) where λ A is itself only defined up to an overall complex scale: n µ → α 2 n µ ⇒ λ A → αλ A , α ∈ C. (2.27) The first condition in eq. (2.25) then implies u = x AB λ A λ B , (2.28) such that a given point in minitwistor space (representing a particular null plane) is described by coordinates Z α = (u, λ A ) ,(2.(u, λ A ) ∼ r 2 u, rλ A , (2.30) for r ∈ C * 6 . A spinor λ A has two complex components, which reduces to one if an overall complex scale is removed. Thus, λ A defines a point on the Riemann sphere CP 1 . In general, we need two coordinate patches to cover the sphere, which we may choose as U 0 : λ A = (1, z) (2.31) U 1 : λ A = (w, 1). (2.32) Given that λ A is defined only up to rescalings, we may identify z = w −1 on the overlap U 0 ∩U 1 . The single complex coordinate u is defined at each point on the Riemann sphere, and thus we may think of MT as a fibre bundle, with CP 1 as the base space. Formally speaking, it is the holomorphic tangent bundle T CP 1 of the Riemann sphere. In particular, a given point in minitwistor space assigns a holomorphic tangent vector to each point on the Riemann sphere associated with λ A . To see this, note that a general holomorphic vector field on CP 1 may be written as f (z)∂ z = ∞ n=0 a n z n ∂ z = − ∞ n=0 a n w 2−n ∂ w ,(2.33) where we have used ∂ z = −w 2 ∂ w in the second equality. Holomorphicity in both coordinate choices (in particular the absence of poles) then implies a n = 0 for n > 2, such that a general holomorphic vector field on U 0 ∩ U 1 may be written as (a 0 + a 1 z + a 2 z 2 )∂ z . (2.34) The incidence relation of eq. (2.28) can be expanded in U 0 using eq. (2.24) as Figure 1. Points p and q in spacetime can be visualised as curves in minitwistor space, where the coordinate u is defined at each point on the Riemann spheres X p and X q corresponding to p and q. u = (−t + y)z 2 − 2xz − (y + t), (2.35) X u p q Curves associated with different spacetime points can intersect in at least two places. From eq. (2.34), this defines a holomorphic vector field u(z)∂ z (2.36) on CP 1 , as required 7 . As described above and as is hopefully clear from eq. (2.35), a fixed spacetime point x and picks out a specific vector at each point on the Riemann sphere associated with λ. Thus, the coordinate u defines a section u : CP 1 → T CP 1 (2.37) of the holomorphic tangent bundle of the Riemann sphere. Thus, a point in spacetime corresponds to a section of the holomorphic tangent bundle of a Riemann sphere X. As explained in ref. [105], we may visualise this in twistor space as shown in figure 1. The horizontal axis shows the Riemann sphere X represented as a complex line. The vertical axis then denotes the value of u at each point on X, such that we may visualise this as a curve. So far we have worked in complexified Minkowski space. If we wish to use real coordinates in Lorentzian signature, then the matrix given in eq. (2.24) will be real; that is, we choose to impose the reality condition (u, λ A ) ∼ (ū,λ A ). Now, the incidence relation of eq. (2.28) defines a real null-plane or a timelike line, depending on whether u and λ 0 /λ 1 are (non-)real. Dimensional reduction In what follows, we will have to obtain the relevant Penrose transform that converts data in minitwistor space into solutions of the topological gauge and gravity equations in spacetime. In doing so, we will find it useful to rely on some alternative ways of thinking about MT. The first of these relies on the more well-known concept of twistor space for four-dimensional complexified Minkowski spacetime M 4 = C 4 . As remarked already above, we must consider two types of spinor in four spacetime dimensions, given that the Lorentz group is covered by two distinct SL(2,C) groups, which we may refer to as SL(2,C) L and SL(2,C) R . These act on (conjugate) spinors, which carry (un-)primed indices respectively. To convert a given tensor or 4-vector into the spinorial language, we can again contract with the relevant Infeld-van-der-Waerden symbols, for which a suitable choice is σ µ AA = 1 0 0 1 , 0 1 1 0 , 1 0 0 −1 , 0 −i i 0 . (2.38) Comparing with eq. (2.2), we see that the Infeld-van-der-Waerden symbols carry a spinor index associated with each of the groups SL(2, C) L,R , and we have also appended the "missing" Pauli matrix to be the third component σ 3 AA , where now µ ∈ {0, 1, 2, 3}. With these conventions, one has x AA = −t + y x − iz x + iz −t − y , x AA = −t − y −x − iz −x + iz y − t . (2.39) The twistor space corresponding to four-dimensional Minkowski spacetime can also be identified with the space of certain null 2-planes. It turns out that these can be parameterised by twistor coordinates 8 Z α = (µ A , λ A ), (2.40) subject to the incidence relation µ A = x AA λ A . (2.41) Twistors satisfying this relation are defined only up to an overall rescaling 42) and are said to live in projective twistor space PT. We may now obtain minitwistor space of C 3 by dimensionally reducing four-dimensional twistor space. After dimensional reduction, one may choose to work with real coordinates in a specific signature by picking a reality condition as in the previous section. To see how this works note that, in our above examples, we can isolate the three-dimensional coordinates from x AA by introducing a constant vector with spinorial translation Z α ∼ rZ α , ⇒ (µ A , λ A ) ∼ (rµ A , rλ A ),(2.T BA = 0 1 −1 0 ,(2.43) and forming the combination x AA T B A = x AA A C T BC = x AB + 0 −iz iz 0 , (2.44) as follows from explicit computation. Then we may write x AB = x (A A T B)A , (2.45) whose geometric interpretation is that we are ignoring translations along the direction of T µ in spacetime. Furthermore, SL(2,C) covariance of eq. (2.45) means that we can pick any direction in spacetime in order to perform the dimensional reduction. Removing the symmetrising brackets in eq. (2.45) will generate an antisymmetric contribution on the left-hand side, which for a two-dimensional matrix must be proportional to the Levi-Civita symbol. Thus, on general grounds we may write [106][107][108] x A A T BA = x AB + b AB , (2.46) where taking the determinant of both sides can be used to infer the relation b = i x 2 4d − x 2 3d . (2.47) Using eq. (2.45), we can recover the minitwistor incidence relation in eq. (2.28) from the 4d incidence relation of eq. (2.41). To do so, one may define u ≡ µ A T AA λ A . (2.48) This can be shown to be invariant under the equivalent of translations along the vector field T µ in twistor space [108]. Also, the four-dimensional twistor scaling property of eq. (2.42) implies u → r 2 u, as required in eq. (2.30). Combining eqs. (2.41, 2.46, 2.48), we find u = (x AB λ A + bλ B )λ B ,(2.49) where the second term in the brackets corresponds to the effect of a translation in the T µ direction. This vanishes after contracting with the spinor λ B outside the brackets, but suggests that the most general equivalence relation in the three-dimensional twistor coordinates is (x AB λ A λ B , λ A ) ∼ (r 2 (x AB λ A + bλ B )λ B , rλ A ). (2.50) Indeed, this motivates another way to define coordinates on minitwistor space, as Z A = (µ A , λ A ) , µ A = x AB λ B ,(2.51) which arise more naturally from the dimensional reduction point of view. In this case, the coordinates are defined up to the following equivalence: (µ A , λ A ) ∼ (r(µ A + bλ A ), rλ A ) . (2.52) In discussing the Penrose transform we need for topological gauge and gravity theory, it is useful to discuss yet another, and rather more formal, way to describe minitwistor space. Recall that MT consists of the set of null two-planes in C 3 . Reference [91] considers Euclidean signature for the latter, and points out that the complex Euclidean group of transformations that define the space (rotations plus translations) is covered by the group ESL(2,C). If we then quotient this group by the group of isometries of null planes, we will obtain the group that acts on minitwistor space. The elements of the former group can be written as ESL(2, C) = {(A, B)|A ∈ SL(2, C), B ∈ {2×2 complex trace-free matrices}}, subject to the composition law (A, B) • (A , B ) = (AA , AB A −1 + B). (2.53) We thus see that A is an SL(2,C) transformation associated with rotations, and B is associated with translations. The relevant closed subgroup that we must quotient out is given by [91] 54) and to show that this is correct, we can simply apply the equivalence relation g ∼ gq (with g ∈ ESL(2, C), q ∈ Q), and show that this corresponds to the equivalence relation of eq. (2.52) when acting on minitwistor coordinates. One may then parametrise Q = R = r t 0 r −1 , S = b a 0 −b ; a, b, r, t ∈ C, r = 0 ,(2.g = (A AB = (λ A χ B ) , X = x AB ) ,(2.55) where χ B is an arbitrary spinor. From eq. (2.53), Q then acts on g as gq = (AR , X + ASA −1 ) . (2.56) Expanding appropriately and using eq. (2.54), one obtains the correspondence [91] g → gq : λ A → rλ A , x AB λ A → r(x AB λ A + bλ A ) . (2.57) We can then interpret λ A and µ B = x AB λ A as coordinates on minitwistor space, subject to equivalence relations which indeed match those found by dimensional reduction in eq. (2.52). The Penrose transform for massless free fields Having introduced minitwistor space in various ways, our next task is to find the appropriate Penrose transform that expresses spacetime fields as contour integrals in minitwistor space. To this end, let us first recall the Penrose transform in four-dimensional twistor theory [83]. Solutions of the four-dimensional massless free field equation of eq. (2.19) can be expressed via the following contour integral: φ A 1 A 2 ...A 2n = 1 2πi Γ λdλ λ A 1 λ A 2 . . . λ A 2n ρ x [f (Z α )], λdλ = λ E dλ E . (2.58) Here the contour Γ lies on the Riemann sphere X associated with a given spacetime point x, and λ A is the spinor that forms half of the twistor components of eq. (2.40). There is then a holomorphic function f (Z α ) of twistor coordinates, where the symbol ρ x denotes restriction to the Riemann sphere X, such that all twistors obey the incidence relation of eq. (2.41). The contour Γ must be such that it separates any poles of f (Z α ), and for there to be a non-zero answer, there must be at least two poles, one on either side of Γ. We may take the latter to correspond to the equator of the Riemann sphere without loss of generality. As is well-known [84], the "functions" f (Z α ) are not unique, but can be subjected to equivalence transformations that do not affect the result of the contour integral: f (Z α ) ∼ f (Z α ) + f N (Z α ) + f S (Z α ), (2.59) where f N (Z α ) (f S (Z α ) ) has poles only in the northern (southern) hemisphere of X respectively. Substituting eq. (2.59) into eq. (2.58), we may evaluate the additional contributions by simply closing the contour in the opposite side to where the poles are, giving rise to a zero result, as required. In more formal mathematical terms, we say that the quantities f (Z α ) are representatives of (Cech) cohomology classes, and a fuller exposition of this in the present context can be found in ref. [30]. There is, however, a further restriction on f (Z α ), arising from the fact that twistors obeying the incidence relation constitute points in projective twistor space PT, and thus are only defined up to the rescalings of eq. (2.42). If the integrand and measure in eq. (2.58) are to be invariant under Z α → rZ α , then it must be the case that f (Z) is a homogeneous function of degree (−2n − 2), for a spin-n spacetime field: f (rZ α ) = r −2n−2 f (Z α ). (2.60) Denoting holomorphic functions on minitwistor space of homogeneity N by O(N ), we would then say in formal language that the Penrose transform is an isomorphism between spacetime fields of spin n, and elements of theCech cohomology group 9 H 1 (PT, O(−2n − 2)). It is straightforward to write down a Penrose transform for solutions of the three-dimensional massless free field equation. The latter is given by ∂ A 1 B φ A 1 ···A 2n (x) = 0 ,(2.φ A 1 ...A 2n = 1 2πi Γ λdλ λ A 1 . . . λ A 2n ρ x [f (Z α )]. (2.62) This is directly analogous to eq. (2.58), where Γ is again a contour on the Riemann sphere X associated with the spacetime point x, and f (Z α ) a holomorphic function of the minitwistor coordinates of eq. (2.29). Above, we have presented the Penrose transforms for massless free fields in the language ofCech cohomology, in which they take the form of contour integrals in (mini-)twistor space. An alternative approach exists, in which twistor integrands are interpreted using differential forms, and the freedom to redefine twistor integrands is interpreted using Dolbeault cohomology (see e.g. ref. [30] for a recent detailed comparison of the two approaches). We will remain with theCech approach in what follows, which will turn out to be more convenient for our purposes. However, an obvious deficiency of eq. (2.62) is that it only works for massless free fields, and thus is inapplicable to topologically massive gauge and gravity theory. That it is possible to generalise the three-dimensional Penrose transform to incorporate (topological) mass is possible on very general grounds, which we review in the following section. The Penrose transform for massive free fields A three-dimensional Penrose transform for massive fields has been presented in the twistor literature by Tsai [91], whose starting point is to consider the above construction of minitwistor space MT as the quotient space G/H, where G=ESL(2,C) is the universal cover of the complex Euclidean group that generates all points in C 3 and H = Q is the group of isometries of null planes. To construct a Penrose transform, we must consider defining functions on MT. Functions at a point form a vector space V under addition, and we must therefore consider a mathematical structure consisting of a copy of V associated with all points in G/H, such that one has a type of fibre bundle. In fact, this structure is known as a homogeneous vector bundle, where the word "homogeneous" refers to the fact that the base space is itself a quotient space. There is a canonical way to construct homogeneous vector bundles (see e.g. ref. [109]), as follows. We can first think of constructing a conventional vector bundle on G by placing a copy of V (the "fibre") above each point of G, and then letting a representation ρ(g) of each element g ∈ G act on vectors v ∈ V . Then, given g ∈ G and v ∈ V , we may identify points in this vector bundle by asserting the equivalence (g, v) ∼ (gh, ρ(h −1 )v), h ∈ H. (2.63) The first component of this relation tells us that gh is to be identified with g, which is simply the action of quotienting out G by the closed subgroup H. The second component then implements the fact that the vectors in the fibres above g and gh must be identified 10 . Returning to the specific case of G=ESL(2,C) and H = Q, we will be considering scalar functions, which must then be acted on by one-dimensional representations of Q. From eqs. (2.53, 2.54), one finds (R 1 , S 1 ) • (R 2 , S 2 ) = r 1 r 2 * 0 (r 1 r 2 ) −1 , b 1 + b 2 * 0 −b 1 − b 2 ,(2.64) and thus one sees that the r parameters are multiplicative, whereas the b parameters are additive. Physically, this is related to the fact that the former are associated with rotations, and the latter with translations. A one-dimensional representation that embodies these properties can be easily written down as ρ((R, S)) = r −N e M b , N ∈ Z, M ∈ C. (2.65) Indeed, this represents an infinite family of one-dimensional representations, one for each combination (N, M ), and our reason for restricting N to be an integer will be clarified below. We have seen that, acting on minitwistor coordinates (µ A , λ A ), transformations h ∈ Q act according to eq. (2.52). Thus, functions acted on by the representation of eq. (2.65) must satisfyf M (r(µ A + bλ A ), rλ A ) = r N e −M bf M (µ A , λ A ) . (2.66) To make sense of this condition, we can consider the case M = 0, for which there will no b parameter in eq. (2.65). Then eq. (2.66) reduces tǒ f 0 (rµ A , rλ A ) = r Nf 0 (µ A , λ A ),(2.67) which is merely the requirement that the function f 0 be homogeneous with degree N . Such functions enter the massless Penrose transforms in three and four dimensions, where N = −2n − 2 is related to the spin n of the spacetime field. For non-zero M , the parameter b in eq.(2.66) corresponds to the additional freedom to redefine minitwistor coordinates, as in eq. (2.52). Given this more general class of functions, we can construct a generalised Penrose transform. First, by setting r = 1 in Eq. (2.66), differentiating this equation with respect to b, and evaluating it at b = 0 we get thatf M must obey: where a A is an arbitrary spinor. To be compatible with (2.66)ǧ must be homogeneous of degree N , which we will choose as above to be N = −2n − 2 (it is for this reason that we have chosen N ∈ Z in eq. (2.65)). We may then consider the contour integral λ A ∂f M (Z) ∂µ A = −Mf M (Z),(2.φ A 1 ...A 2n = 1 2πi Γ λdλ λ A 1 . . . λ A 2n ρ x [f M (Z α )],(2.70) which consists of simply replacing the "function" in the massless three-dimensional Penrose transform of eq. (2.62) with one of the more general types of function defined above. Acting on both sides with a derivative operator, we find ∇ B A 1 φ BA 2 ...A 2n (x) = − 1 2πi C λdλ 2λ B λ A 2 . . . λ A 2n λ (B ρ x ∂f m (Z) ∂µ A 1 ) = M 2πi C λdλ λ A 1 λ A 2 . . . λ A 2n ρ x [f M (Z)] = M φ A 1 ...A 2n (x) ,(2.71) where in the first line we have used . Note that our above arguments merely show that a cohomology class in mini-twistor space gives a solution of the field equations in spacetime. However, ref. [91] proves (see Proposition 2.10) that all possible solutions can be obtained in this way, so that the relationship between fields and cohomology classes is formally an isomorphism. ∇ B A 1 f = (σ µ ) B A 1 σ DC µ λ C ∂f ∂µ D ,(2. In this section, we have reviewed a particular generalised Penrose transform on minitwistor space, whose "functions" correspond to cohomology classes labelled by two parameters (N, M ). The first of these represents the homogeneity of the cohomology representative, and the second turns out to correspond to the mass in the field equations of topologically massive gauge theory and gravity in three spacetime dimensions. The latter are special cases of eq. (2.71), but we will also need the spinless case, for which one may verify that the spacetime field φ satisfies the massive Klein-Gordon equation (∂ 2 + m 2 )φ = 0. (2.75) In four spacetime dimensions, the Penrose transform may be used to show that the position-space double copy for massless free fields can be derived from a certain procedure in twistor space [26,28]. Now that we have identified the appropriate Penrose transform for minitwistor space, we can perform a similar analysis for topologically massive gauge and gravity theory in three spacetime dimensions. A minitwistor derivation of the Cotton double copy We have seen that the generalised Penrose transform of eq. (2.70) identifies solutions of the massive field equation of eq. (2.71) with holomorphic twistor "functions" (cohomology class representatives) having the form of eq. (2.69). The homogeneity of a spin-n field was found above to be N = −2n − 2, and thus a scalar, gauge and gravity field will be associated with twistor representatives of homogeneity −2, −4 and −6 respectively. Let us introduce a scalar representativef −2 (Z α ), and a pair of gauge theory representativesf (i) −4 (Z α ) (i ∈ {1, 2}): f −2 (Z α ) = e −m aµ aλ ǧ −2 (u, λ A ),f −4 (Z α ) = e −m aµ aλ ǧ −4 (u, λ A ) , (3.1) whereǧ N is an homogeneous "function" of degree N . It follows that one may construct a gravitational twistor representative by forming the producť f −6 (Z α ) =f (1) −4 (Z α )f (2) −4 (Z α ) f (Z α ) = e −m aµ aλ ǧ −6 (u, λ A ), (3.2) withǧ −6 (u, λ A ) =ǧ (1) −4 (u, λ A )ǧ (2) −4 (u, λ A ) g −2 (u, λ A ) . (3.3) In the four-dimensional case of refs. [26,28], it was argued that choosing certain representatives allows to derive the Weyl double copy in position space. We may do something very similar here, in order to obtain the Cotton double copy. To see how this works, we may first recall that for twistor representatives with at most two poles on the Riemann sphere X, a p−fold pole gives rise to a (2n − p + 1)−fold principal spinor of the corresponding spacetime field, at the point x (see e.g. ref. [75], and ref. [28] for a more recent discussion of this point). We may then consider the representativeš f −2−2n (Z α ) = e −m aµ aλ G(u, λ A ) (χ(u, λ A )) p . (3.4) Here G(u, λ A ) and χ(u, λ A ) are homogeneous and holomorphic minitwistor functions, such that χ(u, λ A ) has q ≤ 2n simple zeros, corresponding to poles inf −2−2n (Z α ) enclosed by the contour C. Furthermore, G(u, λ A ) is regular at the p-fold pole given by the zero of χ(u, λ A ). For Type N solutions in which the field has only one 2n-fold principal spinor, χ(u, λ A ) has a simple zero and p = 1. For Type D solutions which have 2 different n-fold principal spinors, χ(u, λ A ) has two simple zeros and p = n + 1. In refs. [89,90], the Cotton double copy was explicitly argued to hold in position space for type N solutions only. Thus, we will shortly show how the type N Cotton double copy can indeed be obtained from representatives of the form of eq. (3.4). Before moving on, however, some comments are in order regarding the product of twistor functions in eq. (3.2). As has been made clear repeatedly above, these are representatives of cohomology classes, and thus -in a given theory -can be subjected to equivalence transformations of the form of eq. (2.59). However, the non-linear product that is needed to generate gravitational solutions in the twistor space double copy of eq. (3.2) (likewise in the four-dimensional case of refs. [26,28]) is clearly incompatible with the ability to first perform equivalence transformations. This is not actually a problem if all one wants to do is to derive the Cotton double copy in position space: one merely regards the product as only being true for certain representatives in twistor space, such that any representatives which yield the correct double copy structure in position space (if it exists) will do. Nevertheless, it is desirable to have some motivation a priori for picking out certain representatives, where this would ideally relate to the physics of the double copy. Reference [85] was the first to consider this point, using the language of Dolbeault rather thanCech cohomology. The authors considered certain radiative solutions, and showed that data at null infinity could be used to uniquely fix twistor representatives in the various theories entering the double copy. Reference [30] took a different approach, by looking at spacetime fields in Euclidean signature, and using existing the ideas of ref. [86] to argue that upon choosing special cohomology representatives in twistor space (corresponding to harmonic differential forms), a product structure in twistor space is naturally obtained. Unfortunately, neither of these procedures is obviously related to the other, nor to the original BCJ double copy for scattering amplitudes. Reference [87], however, provided a much better motivation for the formula of eq. (3.2), at least in principle, by showing that special twistor representatives can be defined by a certain integral transform acting on momentum-space amplitudes. Indeed, ref. [88] showed that these representatives are precisely those entering the twistor double copy of refs. [26,28]. Thus, the twistor double copy can indeed be viewed as arising from the BCJ double copy for three-point scattering amplitudes. Similar arguments can be used in the present context of solutions of topologically massive gauge theory and gravity, and we return to this in section 4. Cotton double copy for Type N Let us now see how the type N Cotton double copy arises from twistor space. In line with our comments above, a type N solution should be generated by eq. (3.2), provided the gravity twistor representative has a simple pole in twistor space. We may thus choose representativeš f −2 (Z α ) = e −m aµ aλ G 0 (u, λ A ) χ 1 (u, λ A )ξ 1 (u, λ A ) ,f (1,2) −4 (Z α ) = e −m aµ aλ G 0 (u, λ A ) χ 1 (u, λ A )(ξ 1 (u, λ A )) 3 , (3.5) where χ 1 (u, λ A ) and ξ 1 (u, λ A ) are homogeneous of degree 1 and have simple zeros and G 0 (u, λ A ) is homogeneous of degree 0 and has no poles, such that one findš f −6 (Z α ) = e −m aµ aλ G 0 (u, λ A ) χ 1 (u, λ A )(ξ 1 (u, λ A )) 5 . (3.6) Upon substituting this into the Penrose transform of eq. (2.70), we may carry out the latter by choosing the patch U 0 in eq. (2.32), so that λ A = (1, z). We will assume without loss of generality that only the simple pole, which arises from ξ 1 , of each cohomology representative lies in U 0 . On general grounds, we may further define ρ x aµ aλ = q(x; z), ρ x [G 0 (u, λ A )] = G(x; z), ρ x [χ 1 (u, λ A )] = (z − z 0 ) N 1 (x) , ρ x [ξ 1 (u, λ A )] = (z − z 1 ) N 2 (x) ,(3.7) where z 0 is the position of the simple zero in χ 1 (u, λ A ), in terms of the parameter z, and the position dependence of each quantity arises upon imposing the incidence relation in eq. (2.28). Equation (2.70) then becomes φ AB...D (x) = 1 2πi Γ dz(1, z) A (1, z) B · · · (1, z) D G(x; z)e −m q(x;z) × N 1 (x) z − z 0 N 2 (x) z − z 1 2n+1 =(1, z 0 ) A (1, z 0 ) B · · · (1, z 0 ) D N 1 (x) G(x; z 0 ) N 2 (x) z 0 − z 1 2n+1 e −m q(x;z 0 ) , (3.8) where we have carried out the contour integral in the second line by assuming that q(x, z) is non-singular at z = z 0 . The fields of eq. (3.8) clearly satisfy the Cotton double copy of Eq. (2.22), when taking C ABGD = φ ABGD and Φ AB = φ AB . Thus, the Cotton double copy indeed emerges from a product in twistor space, as claimed. To give an explicit example of the above construction, let us examine pp-wave solutions, for which the following representatives in twistor space can be constructed for a spin-n field: f −2−2n = e −m aµ aλ 1 oλ 1 sλ 2n+1 g u (s A λ A ) 2 . (3.9) Here we have introduced the constant spinors a A = (1, 0), o A = (0, 1), and s A = (1, c), where c C. Comparing to eq. (3.7) we have ρ x [G 0 (u, λ A )] = ρ x g u (s A λ A ) 2 = g (−t + y)z 2 − 2xz − y − t (1 + cz) 2 ρ x [χ 1 (u, λ A )] = ρ x [ oλ ] = z, ρ x [ξ 1 (u, λ A )] = ρ x [ sλ ] = 1 + cz . (3.10) where the pole z 0 = 0 is in U 0 and z 1 = −1/c is in U 1 . Carrying out the relevant Penrose transforms as in eq.(3.8), one finds (3.11) with y ± = t ± y and α A = (1, 0) the principal spinor of the plane wave solutions. These are indeed the pp-wave solutions for topologically massive gravity, topologically massive electrodynamics, and a massive scalar field [110]. C ABCD = φ(y + , x) α A α B α C α D , f AB = φ(y + , x) α A α B , φ(y + , x) = g(y + )e −mx , Beyond type N solutions Having reproduced the Cotton double copy for type N solutions of refs. [89,90], it is natural to ask whether or not the arguments can be extended to type D solutions. The latter indeed double copy in the four-dimensional Weyl double copy, whose twistorial incarnation has been presented in refs. [26,28]. The twistor description allows us to address this directly, and in fact shows that type D solutions do not obey a simple position space double copy in general. To see this, we may write the explicit formulǎ f −2n−2 (Z α ) = e −m aµ aλ G 0 (u, λ A ) (χ 1 (u, λ A )ξ 1 (u, λ A )) n+1 , (3.12) where as before, χ 1 (u, λ A ) and ξ 1 (u, λ A ) are homogeneous of degree 1 and have simple zeros, and G 0 (u, λ A ) is homogeneous of degree 0. We can further define ρ x G 0 (u, λ A ) (χ 1 (u, λ A )ξ 1 (u, λ A )) n+1 = N (x) (z − z 0 ) n+1 (z − z 1 ) n+1 ,(3.13) where we have again imposed the incidence relation on the coordinate patch U 0 and we assume that only the pole at z 0 is in U 0 . Upon substituting this into the Penrose transform, one finds φ AB...D (x) = 1 2πi Γ dz(1, z) A (1, z) B · · · (1, z) D G(x; z)e −m q(x;z) × N (x) (z − z 0 ) n+1 (z − z 1 ) n+1 . (3.14) For type D solutions, poles of second order or higher will be present in the integrand which, upon taking residues, will generate terms involving derivatives of the combination G(x; z)e −mq(x;z) . (3.15) While we have shown this explicitly for type D solutions, this behaviour will hold for all non-type N solutions. Thus, rather than a single term in position space, one will obtain a sum of terms, such that a simple product of spacetime fields is not obtained in general. One way of simplifying matters is to only consider twistor representatives such that the function G(x; z) is constant. Indeed, all of the type D representatives considered in the four-dimensional twistor double copy of refs. [26,28] were of this form. However, this will not suffice in the present context, due to the exponential factor e −mq(x;z) , whose presence is an unavoidable consequence of considering topologically massive gauge and / or gravity theory. We therefore conclude that, unlike the case of the Weyl double copy in four spacetime dimensions, the exact position-space Cotton double copy will be restricted purely to type N solutions 11 . This is in stark contrast to the case of the Weyl double copy in four spacetime dimensions, where some of the simplest relevant solutions -consisting of simple point-like objects at the origin -are of type D. We have thus explained why refs. [89,90] only succeeded in finding Cotton double copies for type N solutions. Our results are also interesting in that they show that, even for non-type N solutions, there can still be a simple product-like double copy structure in twistor space. The lack of a double copy in position space is a consequence of the generalised Penrose transform, and thus ultimately due to the presence of the topological mass. It is instructive to illustrate the general discussion of this section with a concrete example. This is the subject of the following section. From scattering amplitudes to cohomology representatives In the previous section, we have seen that Cotton double copy follows naturally from minitwistor space, analogous to how the Weyl double copy in four spacetime dimensions can be derived using twistor methods [26,28]. Until recently, quite how the Weyl (position-space) and twistor double copies related to the BCJ double copy for scattering amplitudes remained mysterious. This was first settled in refs. [111,112], which showed that the Weyl and BCJ double copy for scattering amplitudes are equivalent, where they overlap, by using the so-called KMOC formalism [113] that expresses classical solutions as inverse on-shell Fourier transforms of scattering amplitudes. Reference [88] investigated this further, by using methods developed in ref. [87] to show that one may split the inverse Fourier transform from momentum to position space into two stages. The first takes momentum-space scattering amplitudes into twistor space, thereby picking out a particular cohomology representative. The second comprises the Penrose transform from twistor to position space, and ref. [88] thus makes clear that the amplitude, twistor and Weyl double copies are precisely equivalent where they overlap. A canonical example is that of a point mass or charge in gravity / gauge theory respectively, corresponding to the well-known Schwarzschild and Coulomb solutions. Similar solutions exist in topologically massive theories, namely gravitational and gauge theory anyons, whose double copy properties have been explored in refs. [98]. Such solutions are not type N, such that we do not expect them to possess a simple position-space double copy, according to the arguments of the previous section. However, we do expect to see a simple product-like twistor-space double copy, where the relevant cohomology representatives are picked out by scattering amplitudes in momentum space. It is interesting to confirm this by seeing what actually happens if we take the relevant scattering amplitudes, and generalise the arguments of refs. [87,88] to three-dimensional topologically massive gauge theories and gravity. Let us begin by developing the necessary ideas from the KMOC formalism of ref. [113], which must be adapted to the present context (see also refs. [90,111,112] for relevant ingredients). We will first focus on a scalar field, which we can mode expand in the usual way as φ(x) = dΦ(q) a(q)e −iq·x + a † (q)e iq·x ,(4.1) where dΦ(q) = d 3 q (2π) 3δ (q 2 + m 2 )Θ(q 0 ) (4.2) is the three-dimensional on-shell measure andδ(x) ≡ 2πδ(x). We are interested in the field generated by a static particle of mass M , which we take to be described by an initial state |ψ = dΦ(p)ψ(p) |p , (4.3) where ψ(p) is a wavefunction in momentum space, corresponding to a wavepacket sharply peaked around the classical momentum p µ = M u µ , with u µ the 4-velocity. Evolving this state into the far future using the S-matrix, the classical field is given by the expectation value ϕ(x) = ψ|S † φ(x)S|ψ , (4.4) which in turn yields ϕ(x) = dΦ(q) ψ|S † a(q)S|ψ e −iq·x + h.c. (4.5) Next, we can adopt the conventional expansion of the S-matrix: S = 1 + iT,(4.6) and note that ψ|a|ψ = 0, (4.7) given that there are no φ excitations in the initial state. We thus get ϕ(x) = 2 Re i dΦ(q)dΦ(p)dΦ(p )ψ(p) * ψ(p ) p |a(q)T |p e −iq·x = 2 Re i dΦ(q)dΦ(p)δ(2p · q + q 2 )ψ(p) * ψ(p + q) A (3) (q)e −iq·x ,(4.8) where in the second line we have introduced the three-point amplitude for the emission of the φ field by the source: p |a(q)T |p = A (3) (q)δ(p + p − q). (4.9) To understand this equation, note that the a(q) operator acts as a creation operator on the left, creating a quantum of the φ field. The expectation value of the T-matrix is then, by definition, the three-point amplitude multiplied by a momentum-conserving delta function. As shown in ref. [113], by carefully accounting for factors of (absent in natural units), one can neglect the shift by q in the wavefunction, and also the q 2 term in the delta function. One may then integrate out the momentum p by assuming that the wavefunction |ψ(p)| 2 is appropriately normalised to find ϕ(x) = 1 M Re i dΦ(q)δ(u · q) A (3) (q)e −iq·x . (4.10) In words: the classical field is obtained as an on-shell inverse Fourier transform of the three-point amplitude. Following refs. [87,88], we can split this transform into two stages as follows. First, we introduce spinor variables by appealing to eq. (2.5): q AB = ω(λ AλB + λ BλA ). (4.11) Here ω has units of energy, so that the spinors (λ A ,λ B ) are dimensionless, and defined only up to the little group scalings λ A → ξλ A ,λ B → 1 ξλ B . (4.12) Transforming to the new variables, we find that d 3 q = 2ω 2 dω| λλ | λdλ λ dλ . (4.13) Furthermore, the three-point amplitude for a scalar field emitted by a scalar source is simply given by a coupling constant, which we set to unity in what follows. Equation (4.10) then becomes ϕ(x) = Re i 2πM dω λdλ λ dλ |ω| | λλ | δ( λ|u|λ ) δ ω 2 λλ 2 + m 2 e −iω λ|x|λ , (4.14) where we have chosen to work in the rest frame where u µ = (1, 0, 0) and thus q 0 = 0, and we have used the fact that Θ(q 0 ) = Θ(0) = 1/2. The first delta function then implies that |λ ∝ u |λ , (4.15) which we may write as an equality by relying upon the little group rescaling of eq. (4.12) (see also ref. [88] for a discussion of this point). Performing theλ integral then yields φ = Re i 2πM dω λdλ |ω| | λ|u|λ | δ(ω 2 λ|u|λ 2 + m 2 )e −iω λ|xu|λ . (4.16) To make further progress, we note that in our original metric signature (−, +, +) for m ∈ R, it is impossible to simultaneously solve the dual kinematic conditions u · q = 0, q 2 + m 2 = 0. (4.17) To get around this problem, we may instead analytically continue to (+, +, −) signature, by setting (q 0 , q 2 ) = i(q 0 ,q 2 ), (t, y) = i(t,ỹ). From eq. (2.5) applied to q µ withq 0 = 0, we may rescale according to eq. (4.12) to write λ A = (1, z), z = q 1 − im iq 2 . (4.19) The analytically continued kinematic constraint q 2 1 −q 2 2 + m 2 = 0 (4.20) then implies |z| = 1, and we also have λ|u|λ → i λ|u|λ , λ|xu|λ → i λ|xu|λ ,(4.21) so that eq. (4.16) becomes φ = Re i 4πM m dω λdλ |ω| δ ω − m λ|u|λ + δ ω + m λ|u|λ e ω λ|xu|λ ) ,(4.22) where we have used the standard identity δ(x 2 − α 2 ) = 1 2|α| δ(x − α) + δ(x + α) . (4.23) It turns out that both delta function contributions are the same, such that we may take only the first with a factor of two. We then arrive at ϕ(x) = Re i M Γ λdλ 2π e m λ|xu|λ λ|u|λ λ|u|λ , (4.24) where Γ is the appropriate integration contour. Recognising λ|x = µ|, eq. (4.24) has precisely the form of the Penrose transform integrand of eq. (2.69), where in this case |a = u|λ , and a specific function of λ A occurs, due to having transformed a particular momentum-space amplitude into (mini-)twistor space. This is a highly useful validation that eq. (2.69) is the correct Penrose transform integrand for topologically massive theories. But it also forms the basis for calculating similar results for gauge and gravity anyon solutions, and examining their double copy properties. Explicit calculation of the various spinor products in eq. (4.24) yields φ(x) = Re i M Γ dz 2π e −m −2iỹz+x(1−z 2 ) 1+z 2 1 + z 2 ,(4.25) such that transforming to the variable z = e iα gives φ(x) = − 1 4πM Re π/2 −π/2 dα cos α exp imỹ − x sin α cos α ,(4.26) We can use the rotational symmetry of the scalar solution to setỹ = 0 and x = r, such that eq. (4.26) becomes φ = − 1 4πM Re π/2 −π/2 dα e −imr tan α cos α = − 1 2πM K 0 (mr), (4.27) where we have used a known integral representation for the modified Bessel function of the second kind 12 . Finally we may analytically continue back to the metric signature (−, +, +), which does not change the form of the result, but ensures that r = x 2 + y 2 . Topologically massive gauge theory In the previous section, we have seen that a classical scalar field can be obtained as an inverse on-shell Fourier transform of a three-point amplitude, such that splitting this transform into two stages allows us to recognise the Penrose transform from twistor to position space. Similar arguments apply to topologically massive gauge theory and gravity, and thus allow us to examine how to Cotton double copy does or does not work for pointlike solutions. In the gauge theory, we are concerned with the relevant curvature spinor, derived from the dual field strength in the case of topologically massive electromagnetism. This is in turn related to the three-point amplitude for emission of a photon by a source [90], which we again take to be a scalar particle of mass M . We can mode expand the dual field strength on-shell in the usual way 13 F µ (x) = 1 2 µνρ F νρ = −im dΦ(q) a(q) µ (q)e −iq·x + a † (q) * µ (q)e iq·x . (4.28) Then, analogously to the scalar case of eq. (4.4), the curvature spinor is related to the expectation value of the field strength evolved into the far future by the S-matrix ϕ AB (x) = ψ|S †F µ (x)σ µAB S|ψ , (4.29) where |ψ is the initial one-particle state defined in eq. (4.3), and where we now assume this particle is charged such that it may emit photons. The electromagnetic curvature spinor is then given by ϕ AB (x) = −im dΦ(q) ψ|S † a(q)S|ψ AB (q)e −iq·x + h.c. ,(4.30) such that upon following similar steps to those leading to eq. (4.10), we arrive at ϕ AB (x) = − m M Re dΦ(q)δ(u · q) A (3) gauge (q) AB (q)e −iq·x , (4.31) where A gauge is the appropriate three-point amplitude for the emission of a (topologically massive) gauge boson from a scalar. As in the scalar case, we may transform to spinor coordinates according to eq. (4.11). A suitable choice for the polarization vector is µ = ω λ|γ µ |λ 2m =⇒ AB = ω m λ A λ B ,(4.32) Furthermore, the amplitude A gauge can be fixed by dimensional analysis and little group scaling [97]: A (3) = 2eM (u · (q)). (4.33) Repeating similar arguments to those leading to eq. (4.24), we ultimately find ϕ AB (x) = −em Re i Γ λdλ 2π λ A λ B e m µ|u|λ λ|u|λ λ|u|λ 2 . (4.34) This looks almost identical to the scalar case of eq. (4.10), but such that the integrand now contains additional powers of spinor variables, as is appropriate for a spin-1 field. Again, we have obtained a specific example of the massive Penrose transform integrand of eq. (2.69). Equation (4.34) will be useful for examining double copy properties of anyon solutions, but let us first carry out the Penrose transform to position space. To do this, we may choose the parametrisation λ A = (1, z), and define the master integral I p,q (x, y) = Re i Γ dz 2π z p e −m[−2yz+x(1−z 2 )] 1+z 2 (1 + z 2 ) q+1 . (4.35) In terms of this integral, the scalar field of eq. (4.25) can be written as φ = 1 M I 0,0 ,(4.36) after analytically continuing back to the mostly plus metric signature, from which we find I 0,0 = AK 0 (mr), A = − 1 π . (4.37) Similarly, the field strength spinor of eq. (4.34) has the form ϕ 00 (x) = −emI 0,1 , ϕ 01 = ϕ 10 = −emI 1,1 , ϕ 11 = −emI 2,1 . Given the result of eq. (4.37), we may carry out the integrals on the right-hand side of eq. (4.38) without performing any further explicit integrals. To see how, note that we can differentiate eq. (4.35) to obtain the recurrence relations ∂ x I p,q = −m[I p,q+1 − I p+2,q+1 ], ∂ y I p,q = 2mI p+1,q+1 . (4.39) A further recurrence relation can be obtained by substituting 1 = 1 + z 2 1 + z 2 (4.40) into the integrand of eq. (4.35), yielding I p+2,q+1 = I p,q − I p,q+1 . (4.41) We thus find that all integrals entering the field strength spinor of eq. (4.38) can be expressed in terms of derivatives of eq. (4.37), leading to the explicit results I 0,1 = A 2 K 0 (mr) + x r K 1 (mr) ; I 1,1 = − Ay 2r K 1 (mr); I 2,1 = A 2 K 0 (mr) − x r K 1 (mr) . (4.42) We thus find ϕ 00 = em 2π K 0 (mr) + x r K 1 (mr) ; ϕ 01 = − em 2π y K 1 (mr) r ϕ 11 = em 2π K 0 (mr) − x r K 1 (mr) . (4.43) Topologically massive gravity A similar analysis to the previous section may be carried out for topologically massive gravity, where we may consider a scalar particle emitting (massive) gravitons. The classical result for the Cotton spinor of eq. (2.18) is then given by C ABCD (x) = ψ|S † C ABCD S|ψ ,(4.44) such that the analogue of eq. (4.31) is, following [90] C ABCD (x) = −κ m 2M Re i dω| λλ | λdλ λ dλ 2π δ( λ|u|λ )δ(ω 2 λλ 2 + m 2 )|ω| 3 × A grav. (q)λ A λ B λ C λ D e −iq·x .(4.45) In eq. (4.45), A grav. is the three-point amplitude for emission of a graviton by a scalar. To find this, we may quote a general result for a spin-s field coupled to two scalars: A (3) +s = g s M s (2u · (q)) s ,(4.46) where g s is some coupling. By similar arguments to the gauge theory case, this simply evaluates to a constant, such that the analogue of eq. (4.34) turns out to be C ABCD (x) = − κ 2 m 3 M 2 Re i Γ λdλ 2π λ A λ B λ C λ D e m µ|u|λ λ|u|λ λ|u|λ 3 . (4.47) In terms of the basis of integrals defined in eq. (4.35), we then have C ABCD = − κ 2 m 3 M 2 I n 1 (ABCD),2 ,(4.48) where n 1 (ABCD) is the number of 1 indices (rather than 0 indices) in the string ABCD. By use of the above recurrence relations we find C 0000 = κ 2 m 3 M 8π 2 − y 2 r 2 K 0 (mr) + 2x r + (r 2 − 2y 2 ) mr 3 K 1 (mr) ; C 1000 = κ 2 m 3 M 8π − xy r 2 K 0 (mr) + − y r − 2xy mr 3 K 1 (mr) ; C 1100 = κ 2 m 3 M 8π y 2 r 2 K 0 (mr) + − 1 mr + 2y 2 mr 3 K 1 (mr) ; C 1110 = κ 2 m 3 M 8π xy r 2 K 0 (mr) + − y r + 2xy mr 3 K 1 (mr) ; C 1111 = κ 2 m 3 M 8π (2r 2 − y 2 ) r 2 K 0 (mr) + − 2x r + 1 mr − 2y 2 mr 3 K 1 (mr) .(4.49) We have checked explicitly that the results of eqs. (4.43, 4.49) agree with the known anyon solutions in topologically massive gauge and gravity theory [114], once these are translated into the spinorial language, and up to an overall normalisation constant (which we have defined differently in our choice of constant amplitudes above). Double copy properties of anyon solutions In the previous sections, we have seen that the solution for spin-n field for a pointlike source, in a (topologically) massive theory, takes the general form ψ AB...D (x) = K n Γ λdλ 2π λ A λ B . . . λ D e m µ|u|λ λ|u|λ λ|u|λ n+1 ,(4.50) where K n is a normalisation constant. Each solution is a special case of the massive Penrose transform of eq. (3.14), and we thus see that there is a multiplicative double-copy structure in twistor space. That is, upon picking out a cohomology representative f n (λ A , u) for each spin-n field (scalar, gauge and gravity) by transforming amplitudes into twistor space, these representatives are related by a simple multiplicative rule f 2 (λ A , u) = f 1 (λ A , u)f 1 (λ A , u) f 0 (λ A , u) . (4.51) The same result is obtained in four spacetime dimensions [88]. However, unlike in that case, the simple multiplicative nature of the double copy in twistor space does not correspond to a simple structure in position space, as comparison of eqs. (4.43, 4.49) makes clear. This thus provides an explicit illustration of the general discussion in section 3.2, namely that the presence of the exponential factor in the massive Penrose transform disrupts the simple nature of the position-space double copy. Given that fields generated by a pointlike source are perhaps the simplest static solutions one can imagine, this bolsters the conclusions of ref. [88], that exact position space double copies are rather special and restricted in nature, and that the double copy prefers to live in momentum space. Another interesting feature of the three-dimensional solutions considered here is that, in contrast to their four-dimensional counterparts, the twistor space representatives have essential singularities, rather than poles. Again this is due to the presence of the exponential factor, and guarantees that one is able to reconstruct the relevant transcendental functions entering the spacetime field (i.e. modified Bessel functions) upon taking residues in the Penrose transform integral. Conclusion In this paper, we have examined the Cotton double copy recently presented in refs. [89,90], that relates solutions of topologically massive gauge and gravity theories. It is a three dimensional (massive) counterpart of the Weyl double copy for certain exact solutions in four spacetime dimensions [22]. However, whereas the latter is known to apply to arbitrary Petrov type D vacuum solutions, the former is apparently restricted solely to the type N case. In order to clarify this issue, we have here used twistor methods, which have previously been useful in examining the origin (and special nature) of the Weyl double copy [26,28,29,88]. In three spacetime dimensions, the relevant twistor space is called minitwistor space, and we have reviewed known results from the twistor literature [91] that provide a massive generalisation of the well-known Penrose transform relating classical fields in spacetime with cohomology classes in twistor space. Armed with this Penrose transform, one may show that although it is possible to construct gravitational cohomology representatives by combining representatives from (massive) scalar and gauge theories, this leads to a simple position-space double copy only in the case of type N solutions. We thus confirm the results of refs. [89,90], and further clarify our results by considering arguably the simplest possible static solutions, corresponding to a pointlike source. The relevant classical fields can be expressed as on-shell inverse Fourier transforms of three-point amplitudes, following the methods of ref. [113]. By splitting this transform into two stages, we first transform amplitudes into minitwistor space, revealing that the remaining to spacetime takes precisely the form of the massive Penrose transform mentioned above. Although we find a simple multiplicative double copy in twistor space, this fails to translate to a simple relationship in spacetime, thus validating our more general analysis. The emerging picture from this and similar recent studies [88] is that exact position-space double copies are rare. However, knowing that they exist -and what their limitations are -is undoubtedly useful. Methods for elucidating the landscape of exact double copies are a necessary part of this ongoing effort, and we hope that the twistor methods developed in this paper may prove of further use in this regard. The group H acts towards the left on elements of G, but towards the right on elements of V , such that the inverse ensures that the ordering of successive transformations acting on V is correct. 72) together with eq. (2.3). This tells us two things: (i) φ A 1 ...A 2n constructed in this manner satisfies the massive free field equation of topologically massive gauge theory and gravity (eqs. (2.15, 2.17)); (ii) the parameter M , which arose above in classifying the most general type of functions that can be defined on minitwistor space, can be identified with the topological mass m. We will thus make this identification in what follows.Note that a more general solution to eq. (2.66) can be constructed as a sum over different arbitrary spinors a A and homogeneous functions g i : ǧ i (λ α , µλ ) .(2.73)That this satisfies the general massive free field equation can be verified by explicit calculation, but anyway follows from linearity of the field equation.As in the massless case, the twistor functionsf m (Z) entering the Penrose transform of eq. (2.70) are not actually functions but defined only up to equivalence transformations, in this case of the form f m (Z) ∼f m (ǧ S (λ α , µλ ) , (2.74) whereǧ N,S has poles only in the northern and southern hemispheres of X respectively, and a N (S) λ = 0 in the southern (northern) hemisphere. In formal mathematical language, we would say that the quantityf M (Z α ) is a representative of a cohomology class, which is itself a member of theCech cohomology group H 1 (MT, O(N, M )), where O(N, M ) denotes the functions acted on by eq. (2.65) More specifically, it relates the Abelian versions of these theories. For explorations where both Abelian and non-Abelian solutions map to the same gravitational object see[9,77]. Strictly speaking we only consider linearized solutions, not exact solutions which are usually described within the Petrov classification. Note that the Cotton tensor has appeared in a different off-shell double copy construction in[104].4 In order to verify eq. (2.15), one must also use the relation αβ ∂ γ β ϕγα − ∂ γ α ϕ γβ = 0, which follows from the well-known Bianchi identity for the field strength tensor. The other polarisation state obeys a similar equation to eq. (2.19), but with (un)primed indices interchanged. Here and in what follows, C * denotes the set of non-zero complex numbers. To fully identify MT with the holomorphic tangent bundle of CP 1 , one must show that all possible holomorphic vector fields can be obtained from the incidence relation. We will not formally prove this here. For want of a better notation, we will use (non-)calligraphic symbols to refer to (three-) four-dimensional twistors respectively. Strictly speaking, the Penrose transform of eq. (2.58) relates to sheaf cohomology groups, where O(N ) then denotes the sheaf of holomorphic functions of homogeneity N . However,Cech cohomology provides a suitable approximation to sheaf cohomology for all practical purposes here. See e.g. ref.[81] for a pedagogical discussion of this point. To see why h −1 rather than h occurs in the second relation in eq. (2.63), one may demand that the effect of acting on both components with a group element h1h2 ∈ H is the same as acting first with h2, then with Note that we have assumed that the factor in the exponential has no poles in U0. Below we will see that for linearized solutions that can be constructed from three-point amplitudes, this is not the case. In such cases, we will again find that the position space Cotton double copy does not hold. 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Rumbutis, Cotton Double Copy for Gravitational Waves, 2202.10476. Scattering Amplitudes and The Cotton Double Copy. W T Emond, N Moynihan, 2202.10499W. T. Emond and N. Moynihan, Scattering Amplitudes and The Cotton Double Copy, 2202.10499. The Penrose transform for Einstein-Weyl and related spaces. C.-C Tsai, PhD thesisC.-c. Tsai, The Penrose transform for Einstein-Weyl and related spaces. PhD thesis, 1996. Invariants of General Relativity and the Classification of Spaces. L Witten, 10.1103/PhysRev.113.357Phys. Rev. 113L. Witten, Invariants of General Relativity and the Classification of Spaces, Phys. Rev. 113 (1959) 357-362. A Spinor approach to general relativity. R Penrose, 10.1016/0003-4916(60)90021-XAnnals Phys. 10R. Penrose, A Spinor approach to general relativity, Annals Phys. 10 (1960) 171-201. Three-dimensional spacetimes of maximal order. R Milson, L Wylleman, 10.1088/0264-9381/30/9/0950041210.6920Class. Quant. Grav. 3095004R. Milson and L. Wylleman, Three-dimensional spacetimes of maximal order, Class. Quant. Grav. 30 (2013) 095004, [1210.6920]. 3-D Spinors, Spin-Weighted Functions and their Applications. G Castillo, Progress in Mathematical Physics. Birkhäuser Boston. G. Castillo, 3-D Spinors, Spin-Weighted Functions and their Applications. Progress in Mathematical Physics. Birkhäuser Boston, 2003. Algebraic classification of the curvature of three-dimensional manifolds with indefinite metric. G F Torres Del Castillo, L F Gómez-Ceballos, http:/arxiv.org/abs/https:/aip.scitation.org/doi/pdf/10.1063/1.1592611Journal of Mathematical Physics. 44G. F. Torres del Castillo and L. F. Gómez-Ceballos, Algebraic classification of the curvature of three-dimensional manifolds with indefinite metric, Journal of Mathematical Physics 44 (2003) 4374-4380, [https://aip.scitation.org/doi/pdf/10.1063/1.1592611]. N Moynihan, Scattering Amplitudes and the Double Copy in Topologically Massive Theories. N. Moynihan, Scattering Amplitudes and the Double Copy in Topologically Massive Theories, 2006.15957. Anyons and the double copy. D J Burger, W T Emond, N Moynihan, 10.1007/JHEP01(2022)017JHEP. 01172103.10416D. J. Burger, W. T. Emond and N. Moynihan, Anyons and the double copy, JHEP 01 (2022) 017, [2103.10416]. Massive double copy in three spacetime dimensions. M C González, A Momeni, J Rumbutis, 10.1007/JHEP08(2021)1162107.00611JHEP. 08116M. C. González, A. Momeni and J. Rumbutis, Massive double copy in three spacetime dimensions, JHEP 08 (2021) 116, [2107.00611]. Massive Covariant Colour-Kinematics in 3D. N Moynihan, 2110.02209N. Moynihan, Massive Covariant Colour-Kinematics in 3D, 2110.02209. Quantization Conditions and the Double Copy. W T Emond, N Moynihan, L Wei, 2109.11531W. T. Emond, N. Moynihan and L. Wei, Quantization Conditions and the Double Copy, 2109.11531. Structure of Chern-Simons scattering amplitudes from topological equivalence theorem and double-copy. Y.-F Hang, H.-J He, C Shen, 10.1007/JHEP01(2022)1532110.05399JHEP. 01153Y.-F. Hang, H.-J. He and C. Shen, Structure of Chern-Simons scattering amplitudes from topological equivalence theorem and double-copy, JHEP 01 (2022) 153, [2110.05399]. M C González, A Momeni, J Rumbutis, 2112.08401Massive Double Copy in the High-Energy Limit. M. C. González, A. Momeni and J. Rumbutis, Massive Double Copy in the High-Energy Limit, 2112.08401. Off-Shell Color-Kinematics Duality for Chern-Simons. M Ben-Shahar, H Johansson, 2112.11452M. Ben-Shahar and H. Johansson, Off-Shell Color-Kinematics Duality for Chern-Simons, 2112.11452. Twistors in 2+1 dimensions. R S Ward, 10.1063/1.528550J. Math. Phys. 302246R. S. Ward, Twistors in 2+1 dimensions, J. Math. Phys. 30 (1989) 2246. . N J Hitchin, Geodesics, 10.1007/BF01208717Commun. Math. Phys. 83N. J. Hitchin, MONOPOLES AND GEODESICS, Commun. Math. Phys. 83 (1982) 579-602. Minitwistor spaces and einstein-weyl spaces. P E Jones, K P Tod, 10.1088/0264-9381/2/4/021Classical and Quantum Gravity. 2P. E. Jones and K. P. Tod, Minitwistor spaces and einstein-weyl spaces, Classical and Quantum Gravity 2 (jul, 1985) 565-577. Minitwistors and 3d Yang-Mills-Higgs theory. T Adamo, D Skinner, J Williams, 10.1063/1.50304171712.09604J. Math. Phys. 59122301T. Adamo, D. Skinner and J. Williams, Minitwistors and 3d Yang-Mills-Higgs theory, J. Math. Phys. 59 (2018) 122301, [1712.09604]. R S Ward, R O Wells, 10.1017/CBO9780511524493Twistor geometry and field theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press8R. S. Ward and R. O. Wells, Twistor geometry and field theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 8, 1991, 10.1017/CBO9780511524493. Classification of solutions in topologically massive gravity. D D K Chow, C N Pope, E Sezgin, 10.1088/0264-9381/27/10/1050010906.3559Class. Quant. Grav. 27105001D. D. K. Chow, C. N. Pope and E. Sezgin, Classification of solutions in topologically massive gravity, Class. Quant. Grav. 27 (2010) 105001, [0906.3559]. Classical solutions and their double copy in split signature. R Monteiro, D O&apos;connell, D Veiga, M Sergola, 10.1007/JHEP05(2021)268JHEP. 052682012.11190R. Monteiro, D. O'Connell, D. Peinador Veiga and M. Sergola, Classical solutions and their double copy in split signature, JHEP 05 (2021) 268, [2012.11190]. R Monteiro, S Nagy, D O&apos;connell, D Veiga, M Sergola, 2112.08336NS-NS Spacetimes from Amplitudes. R. Monteiro, S. Nagy, D. O'Connell, D. Peinador Veiga and M. Sergola, NS-NS Spacetimes from Amplitudes, 2112.08336. Amplitudes, Observables, and Classical Scattering. D A Kosower, B Maybee, D O&apos;connell, 10.1007/JHEP02(2019)1371811.10950JHEP. 02137D. A. Kosower, B. Maybee and D. O'Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137, [1811.10950]. . S Deser, Gravitational Anyons, 10.1103/PhysRevLett.64.611Phys. Rev. Lett. 64611S. Deser, GRAVITATIONAL ANYONS, Phys. Rev. Lett. 64 (1990) 611.
[]
[ "A New OFDM System for IIR Channels", "A New OFDM System for IIR Channels" ]
[ "Fellow, IEEEXiang-Gen Xia " ]
[]
[]
In this paper, we propose a new OFDM system for an IIR channel with the form of B(z)/A(z) for two polynomials A(z) and B(z). Different from the conventional OFDM transmission over an FIR channel, a guard interval of an OFDM symbol is added such that the corresponding part at receiver is the cyclic prefix (CP) of the received OFDM symbol. The guard interval and CP lengths are the same and not smaller than the orders of polynomials A(z) and B(z). The OFDM symbol without the guard interval is the same as the conventional OFDM symbol without the CP. At the receiver, the IIR channel is then converted to N intersymbol interference (ISI) free subchannels, where N is the number of subcarriers of an OFDM symbol.Index TermsInfinite impulse response (IIR) channel, finite impulse response (FIR) channel, intersymbol interference (ISI), orthogonal frequency division multiplexing (OFDM), resonant chamber
10.1109/lwc.2023.3244863
[ "https://export.arxiv.org/pdf/2212.04817v1.pdf" ]
254,535,741
2212.04817
2af1396b23dabfc749c809e42accc8064ea0f742
A New OFDM System for IIR Channels Fellow, IEEEXiang-Gen Xia A New OFDM System for IIR Channels arXiv:2212.04817v1 [eess.SP] 9 Dec 2022 1 In this paper, we propose a new OFDM system for an IIR channel with the form of B(z)/A(z) for two polynomials A(z) and B(z). Different from the conventional OFDM transmission over an FIR channel, a guard interval of an OFDM symbol is added such that the corresponding part at receiver is the cyclic prefix (CP) of the received OFDM symbol. The guard interval and CP lengths are the same and not smaller than the orders of polynomials A(z) and B(z). The OFDM symbol without the guard interval is the same as the conventional OFDM symbol without the CP. At the receiver, the IIR channel is then converted to N intersymbol interference (ISI) free subchannels, where N is the number of subcarriers of an OFDM symbol.Index TermsInfinite impulse response (IIR) channel, finite impulse response (FIR) channel, intersymbol interference (ISI), orthogonal frequency division multiplexing (OFDM), resonant chamber I. INTRODUCTION Orthogonal frequency division multiplexing (OFDM) has been well understood and applied in broadband communications systems these days, such as WiFi and cellular systems. It works well when a channel has finite impulse responses (FIR) and the cyclic prefix (CP) length is not shorter than the FIR channel length. When a channel is too long or infinite impulse responses (IIR), either channel shortened OFDM, see, for example, [2], [4], or time domain equalization, see, for example, [1], [3], [5], has been commonly proposed in the past. Although most wireless and wireline channels can be well approximated by FIR channels with a reasonable length, some wireless channels may not be done so. Such an example is a resonant chamber or an autonomous factory with well reflected walls [6], where the channel is usually IIR and cannot be well approximated by a short FIR channel. X.-G. Xia is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA (e-mail: [email protected]). DRAFT In this paper, we propose a new OFDM system for an IIR channel of the form B(z)/A(z) for two polynomials A(z) and B(z). We first consider the case of pure IIR channel, i.e., when B(z) = 1. In this case, if the input and the output are reversed, it would be an FIR channel and the conventional OFDM would work. To do so, the guard interval of an OFDM symbol at the transmitter over the IIR channel is designed such that the corresponding received OFDM symbol block includes a CP, i.e., the CP is automatically formed in the received signal at the receiver. Then, the guard interval of an OFDM symbol at the transmitter is determined by its two neighboring OFDM symbols and the coefficients of A(z) of the IIR channel. Thus, when the coefficients of A(z) of the channel are known at the transmitter, the guard interval of an OFDM symbol can be designed at the transmitter. Assume that the guard interval and CP lengths are the same and not smaller than the order of polynomial A(z). At the receiver, the IIR channel can then be converted to N ISI free subchannels, where N is the number of subcarriers in the OFDM. In this paper we will show how this design is done and how signal can be recovered at the receiver. Note that in our newly proposed method, the OFDM symbol without the guard interval is the same as a conventional OFDM symbol without the CP and it does not need to convolve with A(z) to compensate the IIR channel at the transmitter as one may usually do when the IIR channel is known at the transmitter. This may be preferred in application scenarios where some users may not have severe intersymbol interference (ISI). Furthermore, it is still an OFDM system. After a pure IIR channel is studied, we then generalize it to a general mixed FIR and IIR channel of the form B(z)/A(z) without too much difficulty. In this case, the guard interval and CP length is not smaller than the orders of polynomials A(z) and B(z). This paper is organized as follows. In Section II, we consider a pure IIR channel. In Section III, we consider a mixed FIR and IIR channel. In Section IV, we conclude this paper. II. OFDM TRANSMISSION OVER A PURE IIR CHANNEL We first consider a pure IIR channel H(z), i.e., H(z) = 1/A(z) for a polynomial A(z) = G k=0 a(k)z −k ,(1) where a(k), 0 ≤ k ≤ G, are constants. DRAFT Let x n and y n be a transmitted and the corresponding received sequences with their ztransforms X(z) and Y (z), respectively. Then, we have X(z) = A(z)Y (z). If we reverse the transmitter and the receiver, i.e., treat y n as the transmitted sequence and x n as the received sequence, then it is an FIR channel and the conventional OFDM transmission works. For this FIR channel A(z), without loss of generality (WLOG), we choose the CP length as G, and consider N subcarrier OFDM with N > G. Assume that two blocks of sequence y n , namely y 1 0 , · · · , y 1 N −1 and y 2 0 , · · · , y 2 N −1 , of the same length N correspond to two blocks x 1 0 , · · · , x 1 N −1 and x 2 0 , · · · , x 2 N −1 of sequence x n , respectively. The sequence y n after the CP insertion becomes · · · , y 1 0 , · · · , y 1 N −1 , y 1 0 , · · · , y 1 G−1 , y 2 0 , · · · , y 2 N −1 , y 2 0 , · · · , y 2 G−1 , · · ·(2) Corresponding to the above sequence y n , we set sequence x n as · · · , x 1 0 , · · · , x 1 N −1 ,x 0 , · · · ,x G−1 , x 2 0 , · · · , x 2 N −1 , · · ·(3) as shown in Fig. 1, wherex n is the inserted guard interval that has length G and is to be determined below. Since X(z) = A(z)Y (z), we have x n = G k=0 a(k)y n−k , for all integers n. Then, due to the CP insertion in sequence y n , we have the following cyclic convolutions as in the conventional OFDM systems x i n = a(n) ⊗ y i n , i = 1, 2, and 0 ≤ n ≤ N − 1, where ⊗ stands for the N-point cyclic convolution. Also, thex n part shown in Fig. 1 is generated by theȳ n part, which includes the CP part of y 1 n and the tail part of y 2 n of length G, shown in Fig. 1 as follows. For 0 ≤ n ≤ G − 1,x n = n k=0 a(k)y 2 n−k + G k=n+1 a(k)y 1 n+G−k .(6) With the above designs, it is not hard to check that the linear convolution of the channel, (4), indeed holds. Let X i k = FFT(x i n ) and Y i k = FFT(y i n ), 0 ≤ k ≤ N − 1, for i = 1, 2,(7) DRAFT where FFT stands for the N-point fast Fourier transform in terms of n. Let A k = G n=0 a(n)W nk N , for 0 ≤ k ≤ N − 1, and W N = e −j 2π N(8) and assume A k = 0 for 0 ≤ k ≤ N − 1. In practice, this assumption holds almost surely. Then, from (5), we have X i k = A k Y i k and Y i k = 1 A k X i k , 0 ≤ k ≤ N − 1, for i = 1, 2,(9) which are ISI free, i.e., the original IIR channel 1/A(z) is converted to N ISI free subchannels. Let X i k , 0 ≤ k ≤ N − 1 and i = 1, 2, be 2N information symbols to be transmitted. Then, y i n , 0 ≤ n ≤ N − 1 and i = 1, 2, can be solved from (9) and (7) for given X i k . With these solved y i n , 0 ≤ n ≤ G − 1 and i = 1, 2, and a(n), 0 ≤ n ≤ G − 1, the guard intervalx n , 0 ≤ n ≤ G − 1, as shown in Fig. 1, of an OFDM symbol can be obtained from (6). With the guard intervals solved above, we obtain a transmitted sequence x n as shown in Fig. 1, where x i n are the blocks each of which includes N information symbols X i k , 0 ≤ k ≤ N − 1, to send. At the receiver, after removing the CP parts from the received signal y n , we obtain blocks y i n of block length N. From (9) and (7), the information symbols X i k can be solved/demodulated. In practice, when there is additive noise in the channel, more sophisticated and standard receivers can be used for the demodulation of the OFDM signals as before. Another remark is that the guard interval length ofx n and the CP length in y n are the same and only required not smaller than the order G of A(z). From the above solution of the guard intervalsx n , we see that it depends on the channel parameters a(n). One might ask if the transmitter knows the channel parameters a(n), why the IIR channel 1/A(z) is not compensated by passing the information sequence through the FIR filter A(z) and then at the receiver it becomes the ideal channel. The answer is the following. First, different from compensating the channel at the transmitter, the above proposed method transmits information symbols directly without any intentional distortion in each OFDM block before the guard interval insertion. This may be suitable better for some applications where an IIR channel may not always exist and some users may not even have severe ISI. Second, a channel may be aged at the transmitter and/or the channel information may not be accurate enough at the transmitter, which may directly hurt the information symbols if they are filtered before transmission. Last, this paper proposes a different solution of an OFDM transmission to deal with an IIR channel, even when the IIR channel is known at the transmitter. III. OFDM TRANSMISSION OVER A MIXED IIR CHANNEL After a pure IIR channel was studied in the last section, it is much easier to study a mixed IIR channel in this section. Consider a general mixed FIR and IIR channel of the form: U(z) = B(z) A(z) X(z) = B(z)Y (z), with Y (z) = 1 A(z) X(z),(10) where the channel from X(z) to Y (z) is the pure IIR channel studied in the last section. WLOG, assume B(z) = G k=0 b(n)z −k and A(z) = G k=0 a(n)z −k .(11) Corresponding to the sequences x n composed of x i n andx n , and y n composed of y i n andȳ n presented in the last section, the channel output sequence is u n that is composed of u i n andū n , respectively. They are shown in Fig. 2 by adding sequence u n to the top of the sequences y n and x n in Fig. 1. Similar tox n in (6) from the IIR channel part A(z) in the last section shown in Fig. 1, we haveū n shown in Fig. 2 from the FIR channel part B(z) as follows. DRAFT For 0 ≤ n ≤ G − 1,ū n = n k=0 b(k)y 2 n−k + G k=n+1 b(k)y 1 n+G−k ,(12) which will be removed the same as the CP removal at the receiver in the conventional OFDM systems. After this CP removal, we obtain u i n = b(n) ⊗ y i n , 0 ≤ n ≤ N − 1, for i = 1, 2,(13) where ⊗ is the N-point cyclic convolution as before. Let U i k = FFT(u i n ) and B k = G n=0 b(n)W nk N , 0 ≤ k ≤ N − 1, for i = 1, 2.(14) Similar to the assumtion of A k = 0 made in the last section, we assume B k = 0, 0 ≤ k ≤ N − 1, as well. Then, from (9) and (13) we have U i k = B k Y i k = B k A k X i k , 0 ≤ k ≤ N − 1, for i = 1, 2,(15) and thus, the information symbols X i k can be solved as X i k = A k B k U i k , 0 ≤ k ≤ N − 1, for i = 1, 2,(16) which are ISI free, i.e., the original IIR channel B(z)/A(z) is converted to N ISI free subchannels (15) as in the last section for a pure IIR channel. Similar to what was mentioned in the last section, when there is additive noise in the channel, more sophisticated receivers can be used for the demodulation. Note that since the guard intervalx n in this section is the same as that for a pure IIR channel in the last section, it only depends on the information signals x i n to send and the coefficients a(n) in A(z) of the channel, but does not depend on B(z) of the channel. So, the transmitter does not need to know B(z). Also note that the guard interval length ofx n and the CP length in y n are the same and not smaller than the orders of A(z) and B(z). IV. CONCLUSION In this paper, a new OFDM system is proposed for an IIR channel. The key idea is to design a guard interval of an OFDM symbol at the transmitter such that the received OFDM block has a CP structure. With this property, one can reverse the IIR channel to be an FIR channel and the OFDM then works the same as the conventional OFDM. At the receiver, the IIR channel Fig. 1 . 1OFDM signal structure for a pure IIR channel Fig. 2 . 2OFDM signal structure for a mixed IIR channel can be then converted to N ISI free subchannels, where N is the number of subcarriers of the OFDM. This letter only presents the main OFDM system design but has not tested it in various situations, such as the case when the coefficients in A(z) of the channel are not accurately known but have errors at the transmitter. More of such studies will follow. Blind equalization of IIR channels using hidden Markov models and extended least squares. V Krishnamurthy, S Dey, J P Leblanc, IEEE Trans. Signal Process. 4312V. Krishnamurthy, S. Dey, and J. P. LeBlanc, "Blind equalization of IIR channels using hidden Markov models and extended least squares," IEEE Trans. Signal Process., vol. 43, no. 12, pp. 2994-3006, Dec. 1995 Irreducible BER of COFDM on an IIR channel. A G Burr, Electronics Letters. 323A. G. Burr, "Irreducible BER of COFDM on an IIR channel," Electronics Letters, vol. 32, no. 3, pp. 175-6, Feb. 1996. Multistep linear predictor based blind equalization of FIR/IIR single input multiple output channels with common zeros. J Tugnait, IEEE Trans. Signal Process. 476J. Tugnait, "Multistep linear predictor based blind equalization of FIR/IIR single input multiple output channels with common zeros," IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1689-1700, Jun. 1999. Impulse truncation for wireless OFDM systems. H Schmidt, K.-D Kammeyer, Proc. the 5th International OFDM Workshop. the 5th International OFDM WorkshopHamburg, GermanyH. Schmidt and K.-D. Kammeyer, "Impulse truncation for wireless OFDM systems," Proc. the 5th International OFDM Workshop, Hamburg, Germany, Sep. 11-12, 2000. A recursive blind adaptive equalizer for IIR channels with common zeros. M S Radenkovic, T Bose, Circuits Syst. Signal Process. 28M. S. Radenkovic and T. Bose, "A recursive blind adaptive equalizer for IIR channels with common zeros," Circuits Syst. Signal Process., vol. 28, pp. 467-486, 2009. Distinguished Lecture. T Marzetta, Newark, Delaware, USAUniversity of DelawareT. Marzetta, Distinguished Lecture, University of Delaware, Newark, Delaware, USA, Nov. 30, 2022. DRAFT
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[ "Nanoscale deformation mechanics reveal resilience in nacre of Pinna nobilis shell", "Nanoscale deformation mechanics reveal resilience in nacre of Pinna nobilis shell" ]
[ "Jiseok Gim \nDepartment of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA\n", "Noah Schnitzer \nDepartment of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA\n", "Laura M Otter \nDepartment of Earth and Planetary Sciences\nMacquarie University\nSydneyNSWAustralia\n", "Yuchi Cui \nDepartment of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA\n", "Sébastien Motreuil \nLaboratoire Biogéosciences\nUniversité de Bourgogne Franche-Comté (UBFC)\nDijonFrance\n", "Frédéric Marin \nLaboratoire Biogéosciences\nUniversité de Bourgogne Franche-Comté (UBFC)\nDijonFrance\n", "Stephan E Wolf \nDepartment of Materials Science & Engineering\nFriedrich-Alexander-University Erlangen-Nürnberg (FAU)\nErlangenGermany\n\nInterdisciplinary Center for Functional Particle Systems (FPS)\nFriedrich-Alexander University Erlangen-Nürnberg (FAU)\nErlangenGermany\n", "Dorrit E Jacob \nDepartment of Earth and Planetary Sciences\nMacquarie University\nSydneyNSWAustralia\n", "Amit Misra \nDepartment of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA\n", "Robert Hovden [email protected] \nDepartment of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA\n\nApplied Physics Program\nUniversity of Michigan\nAnn ArborMIUSA\n" ]
[ "Department of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA", "Department of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA", "Department of Earth and Planetary Sciences\nMacquarie University\nSydneyNSWAustralia", "Department of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA", "Laboratoire Biogéosciences\nUniversité de Bourgogne Franche-Comté (UBFC)\nDijonFrance", "Laboratoire Biogéosciences\nUniversité de Bourgogne Franche-Comté (UBFC)\nDijonFrance", "Department of Materials Science & Engineering\nFriedrich-Alexander-University Erlangen-Nürnberg (FAU)\nErlangenGermany", "Interdisciplinary Center for Functional Particle Systems (FPS)\nFriedrich-Alexander University Erlangen-Nürnberg (FAU)\nErlangenGermany", "Department of Earth and Planetary Sciences\nMacquarie University\nSydneyNSWAustralia", "Department of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA", "Department of Materials Science & Engineering\nUniversity of Michigan\nAnn ArborMIUSA", "Applied Physics Program\nUniversity of Michigan\nAnn ArborMIUSA" ]
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The combination of soft nanoscale organic components with inorganic nanograins hierarchically designed by natural organisms results in highly ductile structural materials that can withstand mechanical impact and exhibit high resilience on the macro-and nano-scale. Our investigation of nacre deformation reveals the underlying nanomechanics that govern the structural resilience and absorption of mechanical energy. Using high-resolution scanning/ transmission electron microscopy (S/TEM) combined with in situ indentation, we observe nanoscale recovery of heavily deformed nacre that restores its mechanical strength on external stimuli up to 80% of its yield strength. Under compression, nacre undergoes deformation of nanograins and non-destructive locking across organic interfaces such that adjacent inorganic tablets structurally join. The locked tablets respond to strain as a continuous material, yet the organic boundaries between them still restrict crack propagation. Remarkably, the completely locked interface recovers its original morphology without any noticeable deformation after compressive contact stresses as large as 1.2 GPa.
10.1038/s41467-019-12743-z
[ "https://arxiv.org/pdf/1910.11264v1.pdf" ]
204,835,733
1910.11264
2cc29ceadc1bb36fd5a4f413479d836abfe3c7e3
Nanoscale deformation mechanics reveal resilience in nacre of Pinna nobilis shell Jiseok Gim Department of Materials Science & Engineering University of Michigan Ann ArborMIUSA Noah Schnitzer Department of Materials Science & Engineering University of Michigan Ann ArborMIUSA Laura M Otter Department of Earth and Planetary Sciences Macquarie University SydneyNSWAustralia Yuchi Cui Department of Materials Science & Engineering University of Michigan Ann ArborMIUSA Sébastien Motreuil Laboratoire Biogéosciences Université de Bourgogne Franche-Comté (UBFC) DijonFrance Frédéric Marin Laboratoire Biogéosciences Université de Bourgogne Franche-Comté (UBFC) DijonFrance Stephan E Wolf Department of Materials Science & Engineering Friedrich-Alexander-University Erlangen-Nürnberg (FAU) ErlangenGermany Interdisciplinary Center for Functional Particle Systems (FPS) Friedrich-Alexander University Erlangen-Nürnberg (FAU) ErlangenGermany Dorrit E Jacob Department of Earth and Planetary Sciences Macquarie University SydneyNSWAustralia Amit Misra Department of Materials Science & Engineering University of Michigan Ann ArborMIUSA Robert Hovden [email protected] Department of Materials Science & Engineering University of Michigan Ann ArborMIUSA Applied Physics Program University of Michigan Ann ArborMIUSA Nanoscale deformation mechanics reveal resilience in nacre of Pinna nobilis shell 10.1038/s41467-019-12743-zARTICLE OPEN 1 The combination of soft nanoscale organic components with inorganic nanograins hierarchically designed by natural organisms results in highly ductile structural materials that can withstand mechanical impact and exhibit high resilience on the macro-and nano-scale. Our investigation of nacre deformation reveals the underlying nanomechanics that govern the structural resilience and absorption of mechanical energy. Using high-resolution scanning/ transmission electron microscopy (S/TEM) combined with in situ indentation, we observe nanoscale recovery of heavily deformed nacre that restores its mechanical strength on external stimuli up to 80% of its yield strength. Under compression, nacre undergoes deformation of nanograins and non-destructive locking across organic interfaces such that adjacent inorganic tablets structurally join. The locked tablets respond to strain as a continuous material, yet the organic boundaries between them still restrict crack propagation. Remarkably, the completely locked interface recovers its original morphology without any noticeable deformation after compressive contact stresses as large as 1.2 GPa. T he inherent tradeoff between strength and toughness inspires new design approaches to structural materials with high damage tolerance. While plastic deformation degrades materials' strength and performance lifetime, it is the key attribute for toughness and resistance to fracture. Thus overcoming the tradeoff between toughness, strength, and resilience remains a fundamental design challenge for structural materials 1 . Optimizing mechanical properties for predictable and non-catastrophic failure motivates novel design of modern high-performance structural materials 2,3 . Nature has optimized high-performance materials with unrivaled strength, toughness, and resilience using three-dimensional (3D) hierarchical architectures that traverse the atomic, nano-, micro-, to macro-scale with precision that human technology is yet to achieve 4 . Among the diverse set of structural biominerals-such as bone 5 , enamel 6 , and various biosilica 7 -to be mimicked for designing new synthetic structural materials, nacre is the prototypical supermaterial 1,5,[8][9][10] . After crack initiation, bulk nacre shows a 40-fold higher fracture toughness than the monolithic/single crystal calcium carbonate from which it is constructed 5,[11][12][13][14] . Thus a central focus has been placed on understanding the principle mechanisms of nacre's excellent mechanical properties to inspire new designs of next-generation high-performance structural materials 5,[12][13][14][15][16][17][18][19][20][21][22] . However, nacre's ability to undergo limited deformation and dissipate critical stresses before fracture has not yet been quantified nor correlated with nanomechanical processes. Nacre is constructed from layered interdigitated polygonal (or pseudo-hexagonal) aragonite (CaCO 3 ) platelets (0.5-1 µm thick and 10-20 µm wide), bonded by a thin (~5-30 nm thick) layer of organic material (the interlamellar membrane). Nacre platelets are either arranged into a brick-and-mortar-like architecture in the sheet nacre of bivalves or are stacked vertically as columnar nacre in gastropods [23][24][25][26] . A natural composite material, nacre is reported to consist of roughly 95-98 wt. % aragonite and 2-5 wt. % biopolymers 16,19,20,[26][27][28][29][30] . Our measurements herein confirm a 3.4 ± 1.0 wt. % organic fraction for Pinna nobilis (Mollusca, Bivalvia). The organic fraction of nacre consists of organic interlamellar membranes 31 and intracrystalline organics embedded in the mineral tablets [31][32][33] of~5-20 nm. Nacre tablets have a textured surface roughness and internal substructure that are both derived from space filling nano-granules 25,[34][35][36][37] . The surface contains nano-asperities suspected to play an important role in the prevention of tablet sliding 38 . Surface asperities between opposing nacre tablets occasionally form narrow (20-50 nm) intrinsic mineral bridges 33 without external stress (e.g., Supplementary Fig. 1) connecting across the interlamellar membrane, while wider (150-200 nm) major intrinsic mineral bridges are thought to be involved in the initial formation of new nacre tablets 26,31 . In nacre, Wegst et al. 9 have suggested that crack bridging and the resulting "pull-out" of mineral bricks is associated with controlled, yet limited, sliding of the aragonite layers over each other and is aided by visco-plastic energy dissipation in the organic layer. Li et al. observed the plastic deformation of aragonite surfaces under tensile load at the nanometer level using atomic force microscopy 18,21 . However, additional mechanisms for strengthening and toughening have been proposed: resistance from the lamellae nanoroughness 19 , the organic layer acting as a viscoelastic glue 7,16,20,22 , the presence of (pre-existing) mineral bridges 13,17,30 , and platelet interlocking at the microscopic level 22 . Direct observation is required to disambiguate the mechanism of nanomechanical deformation of nacre; however, most knowledge of the biomineral toughening process is assembled from microscale tribology 8 testing on bulk specimens. Understanding nanomechanical responses across the 3D hierarchical architectures is critical to understanding how the individual nacre components work together to create properties greater than the sum of their parts (i.e., far exceeding the rule of mixtures 9 ). Our investigation of toughening strategies in nacre reveals nanomechanical deformation of organic interfaces, nanocrystallites, and organic inclusions as key to the increased damagetolerance of nacre. High-resolution scanning/transmission electron microscopy (S/TEM) combined with in situ nanoindentation 41,42 has been adapted to biomineral systems to allow sub-nanometer resolution imaging of the nanomechanical deformation processes and provide precise assessment of when and where fracture occurs. We show that during compressive indentation nacre undergoes non-destructive locking where inorganic tablets come into contact across organic interfaces. Remarkably, the completely locked interface recovers its original morphology without any deformation after releasing compression and retains its full mechanical strength. During compression, the aragonite grains and organic inclusions reversibly rotate and deform indicating nanoscale resilience of the nacre tablets. Prior to tablet locking, strain attenuates up to 80% between the decoupled tablets. However, by 3% engineering strain of the first tablet, the tablets have locked to redistribute stress continuously across the organic interface and the strain attenuation decreases. When fracture occurs, we show the organic components restrict crack propagation both within and between tablets, sustaining the overall macroscale architecture through multiple fractures to allow further structural loading. This allows nacre to absorb significantly greater mechanical energy than monolithic aragonite. We report that nacre absorbs roughly 1-3 times more mechanical energy than geological (i.e., non-biogenic) monolithic aragonite before fracture results in structural failure under nanoindentation. This approach provides an energy dissipation measurement that is not derived from a crack-propagating force. In addition, we show that the yield strength measured under nanoscale compression along the c-axis (growth direction) of a single tablet can reach values three times higher (e.g.,~1.1-1.6 GPa) than previously reported for bulk nacre measured with microindentation 1,5,[13][14][15][16]27,38,40,43 . Results Nanoscale deformation and toughening processes. We observe non-linear elastic nanoscale deformation and toughening processes in nacre under compression using nanoindentation with 0.04-0.2 µm 2 contact areas approximately normal to the growth direction (c-axis). This surface normal is nacre's strongest direction 12,13,38 , although the monolithic aragonite from which it is comprised is stiffer along the planar direction 44 . Electron transparent cross-sections from a mature P. nobilis specimen (Fig. 1a) provided the structural stability required for indentation while allowing sub-nanometer resolution imaging (see "Methods"). S/TEM observation revealed a range of strengthening and toughening processes enabled by nacre's hierarchical structure: (i) tablet interlocking, (ii) strain damping, (iii) crack blunting, and (iv) intracrystalline deformation and rotation of nanograins and organics. Despite comprising only a few weight percent (i.e., 2-5 wt. % 16,19,20,[26][27][28][29][30] ) of the entire nacre architecture, the organic components of nacre provide a range of functions that absorb the energy of applied loads while remaining highly recoverable even after initial fracture. The ratio of high-angular annular dark-field (HAADF) STEM intensity estimates the total organic volume fraction in P. nobilis nacre to be 7.1 ± 2.2 vol. % (3.4 ± 1.0 wt. %) comprised of 2.5 ± 0.3 vol. % (1.2 ± 0.1 wt. %) interlamellar and 4.6 ± 1.9 vol. % (2.2 ± 0.9 wt. %) intracrystalline material (see Supplementary Fig. 2). Highly recoverable nacre tablet locking. Nacre's nanoscale organic boundaries and inclusions allow heavily deformed nacre to fully recover its original morphology on the nanoscale (Fig. 1, Supplementary Figs. 3 and 4). Under large compressive loads (e.g., 0.7 GPa in Fig. 1d), opposing nacre tablets interlock across the mineral-organic interface to form temporary inorganic connections through the joining of asperities. Further, the entire tablet volume compresses resulting in small but discernable deformation of organic inclusions (Fig. 1). After releasing the load, the mineral connections at the deformed organic interface and the intratablet nanostructure perfectly recover their initial morphology without any sustained deformation ( Nacre shows mechanical response regimes of high and low compression visible in the strain contours measured during in situ TEM indentation (Fig. 2). Low compressive loads applied along the growth direction generate strain contours, which propagate transversely within each tablet (Fig. 2a). Shearing of the interlamellar membranes prevent propagation longitudinally to neighboring tablets. At higher loads, tablets couple, coming into direct contact with one another and allowing strain contours to spread across tablets radially from the location of indentation (Fig. 2b). Strain along the c-axis is highest directly below the tip loading and tablet compression (tablet engineering strain) is measurable using interlamellar demarcation ( Supplementary Fig. 6). By~3% engineering strain in the first tablet, the contours redistribute continuously, and by~6% engineering strain, locking is strikingly visible between tablets. Initially, the engineering Bright-field TEM (with contrast inverted) on the cross-sectional nacreous region under low and high compressive contact stresses. Under low compressive stress, intra-tablet strain contours are generated, and strain propagates independently along each tablet. As the compressive stress is increased, nacre tablets interlock and larger inter-tablet strain contours propagate diagonally between tablets. c Tablet strain attenuation along the axis of the indentation source. The linear strain dissipation behavior indicates that the deformability of nacre is weakened as the applied stress is increased. Scale bar 200 nm strain of the first to second tablet along the axis of indentation decreases by >~80% when measured using a 0.1-μm 2 contact area. This measurement is only one component of an inhomogeneous strain field that, on average, dissipates away from the point of compression. As greater contact stress is applied, the tablets increasingly lock farther away from the tip and the strain attenuation linearly decreases-the deformability is reduced as the nacre behaves more like a monolithic material (Fig. 2c). This entire process occurs with elastic processes that are fully repeatable. We note that tablets also exhibit a limited amount of locking for indentation parallel to the tablet plane (Supplementary Fig. 5). This occurred near the indentation tip where stress is high, and the Poisson effect pushes the compressed tablet against its neighbors. Further away, unlocked tablets accommodate shear deformability at their interface and strain contours are discontinuous. Indentation parallel to the tablet plane was less resilient and typically resulted in unrecoverable brittle fracture ( Supplementary Fig. 5). Preservation of mechanical strength. During consecutive indentation tests, highly deformed nacre fully recovers under external loads up to~80% of its yield contact stress. This can be seen in Fig. 3a, where the elastic modulus remains unchanged during eight consecutive compressions (blue and red). As shown in the specimen of Fig. 3d, e, beyond~0.8 GPa, nacre begins nonlinear elastic deformation-yield is visible from the decreasing slope of the contact stress-displacement curve. However, unlike traditional plastic deformation, the initial structure is preserved after unloading. Full recovery was even observed in highly deformed nacre (e.g., >~0.8-1.1 GPa) prior to crack formation ( Fig. 1, Supplementary Figs. 3 and 4). This preservation of mechanical strength under repeated loading/unloading cycles reflects a non-linear elastic deformation process featuring nanomechanical resilience not present in traditional bulk materials, attributable at least in part to tablet interlocking. The rotation and deformation of organic inclusions and nano-granularity has also been predicted as another mechanism for viscoelasticity 21 . Although structurally recoverable locking of tablets is key to nacre's resilience, in the reported nanoindentation experiments, performed under dry conditions, the absorbed energy appears to primarily occur within the resilient deformation of nanograined tablets that constitute a significant volume fraction (~97%). This process is confirmed in our bright-field TEM data, where individual aragonite nanograins change contrast as nanograins reorient and organic inclusions slightly reshape their volume ( Supplementary Fig. 7, Supplementary Movie 1). The deformation of these nanometer-scale organic inclusions with compression of the material accommodates the load while avoiding irrevocable damage to the inorganic matrix ( Fig. 1c-e, also shown in insets of Supplementary Figs. 3, 4 and 7). Here nacre's response shows non-linear elastic deformation distinct from that expected in analogous nanocrystalline metals. Unlike nano-or micro-grained metals, which strengthen through reduced mobility of dislocations at grain boundaries 45,46 , nacre's proteinaceous organic components contain flexible molecular bonds that elastically accommodate strain and rotation of nanograins and restoratively return the system to the original state when an external stress is released. This process occurs without the introduction of dislocation pile-up and plastic deformation. Energy absorption during protein stretching/ unfolding and subsequent energy release upon refolding of the elastomeric molecules provides high resilience in nacre and similar to that found in bone 47 . In contrast, nanocrystalline or nanotwinned metals have lower resilience since they exhibit plasticity via dislocations. At failure, the organic components in nacre impede crack propagation both within and between tablets (Figs. 3b, c and 4c, Supplementary Fig. 8). The smaller organic inclusions embedded within the inorganic matrix hinder crack propagation within the tablet and were observed to blunt and deflect cracks (Fig. 3c). The interlamellar membrane likewise hampers propagation between tablets, where cracks often terminate or jump at the interface , corresponding to the non-linear elastic regime; structure remains fully recoverable-after deformation, nacre still preserves both its initial strength and structure. f Strength and elastic modulus of nacre from contact stress in nanoindentation on the thin cross-sectional specimen in this study and various types of testing-microscale tribology, tensile, compression, and bending-on bulk specimens in previous reports. Scale bar 50 nm (Fig. 3b). After each fracture event, the overall macroscale architecture is preserved and maintains its mechanical properties (Fig. 4a-c, Supplementary Fig. 9, Supplementary Movie 2). This extends the damage tolerance of the superstructure beyond a single fracture. In fracture mechanics 48 , the ability to resist fracture is quantified by a fracture toughness when a crack is present. In this complex material, local stress states can lead to a variety of mechanisms responsible for the fracture process zone. Here cracks can be under mixed mode loading conditions, which in general can lead to differences in the energy required for crack extension and make quantification of fracture toughness by nanoindentation challenging. In bulk specimens loaded in mode I, a fracture toughness of 10 Damage-tolerance of nacre's architecture. On a system level, nacre can sustain several fractures before total failure due to its hierarchical soft-hard matter design. This means the yield stress of nacre is not typically defined by crack initiation. In contrast, prismatic calcite and monolithic aragonite exhibit limited deformation before the yield stress is followed by catastrophic failure or crack runaway ( Fig. 4d-i, Supplementary Figs. 10 and 11, Supplementary Movies 3 and 4) from cone cracking at indentation. Monolithic aragonite responds to strain with stress contours radiating from the point of contact. Prismatic calcite from the P. nobilis mollusk behaved similar to monolithic aragonite; however, indentation near an organic interface showed significant attenuation into an adjacent prism (Fig. 4e). When compared to monolithic calcite materials, we clearly see nacre's interlamellar membranes reshape compressive strain fields. Both biogenic calcite from the prismatic layer of P. nobilis and geological monolithic aragonite were noticeably stiffer than nacre (Fig. 4k) and typically reached higher yield stresses than nacre (Fig. 4j). However, nacre's inorganic-organic architecture reliably absorbed 1-3 times more mechanical energy than prismatic calcite and monolithic aragonite before total failure. Integrating the applied stress over the displaced volume of the indenter contact area provides an upper bound on nacre's energy dissipation of 1.1 × 10 3 J/m 2 . Here nanoindentation provides us with true estimates of the energy required to cause fracture(s) that lead to structural degradation. A typical contact stress-displacement curve for nacre often included several intermediate failures, where cracking was halted, nanoscale morphology of nanograins and organic inclusions was preserved, and nacre could undergo further loading without a noticeable change of structure in its mechanical response. Notably in calcite and monolithic aragonite, crack runaway occasionally allowed noticeable energy absorption-however, this occurred after the maximum yield stress and resulted in the unrecoverable structural failure typically found in brittle materials. In situ nanoindentation enables mechanical behavior to be measured at the single tablet level, allowing the contributions of the toughening and resilience mechanisms across length scales to be assessed. For instance, while the elastic modulus of nacre and calcite from P. nobilis were comparable to previous reports on bulk specimens [14][15][16][17][18]27,38,[49][50][51][52][53][54] (Fig. 3f, Supplementary Fig. 12), the measured strength of nanoindented nacre reached values as high as 1.6 ± 0.2 GPa, roughly three times larger than the literature reports for bulk nacre in hydrated and dehydrated specimens 1,5,[13][14][15][16]27,38,40,43 . Dehydrated nacre has been shown to have a greater strength and elastic modulus but lower toughness than hydrated nacre due to the plasticizing of the organic matrix by water 12 . Here the P. nobilis specimen was sacrificed and dehydrated. In native conditions, the performance of nacre should be even better; we underestimate the recoverability of nacre under conditions of low pressure and low hydration and overestimate its tendency to fracture. Typically, nanoindentation in the thin cross-sectional specimens of the nacre and calcite from P. nobilis and geological monolithic aragonite also resulted in a yield strength (e.g., 1.1 ± 0.1 GPa) larger than previously reported bulk values (Fig. 3f, Supplementary Fig. 13). The high strength may be attributed to the finite size of the indentation tip and nanoscale size effects of the mechanical response. As previously observed in several materials-including gold nanowires 55 , polycrystalline thin films 56 , and multiwalled carbon nanotubes 57 -the size effects on mechanical properties of nanostructured materials deviate from bulk and necessitate the use of in situ nanomechanical testing 58,59 . Discussion The present in situ S/TEM nanoindentation study illuminates nacre's distinct non-linear elastic deformation processes that provide high resilience. We see how large forces can drive nacre into locked states that allow the material to distribute strain across tablets and recoverably absorb energy through inorganic and organic compression, nanograin reorientation, and the deformation of organic inclusions. After the load is removed, locked nacre completely recovers both its original morphology and mechanical strength. Even after fracture, failure is mitigated through barriers to crack propagation that preserve the macroscale architecture and allow nacre to retain its mechanical properties and further sustain impact. The material's structure and deformation mechanisms allow it to absorb more mechanical energy than geological monolithic aragonite and biogenic prismatic calcite. Using in situ S/TEM nanoindentation, the mechanical properties of the material were tested down to the individual tablets where yield~3 times stronger than bulk measurements were observed. This approach enables investigation of the wider range of evolutionary-optimized biominerals to reveal advantages underlying their nanomechanical design. The study of deformation and fracture under nanoindentation is a subset of the broader fracture phenomena in nacre and other biological materials, which may reveal additional nanomechanical responses to external forces such as tensile strain or shear. The observed mechanisms reported in this work may guide theoretical models of deformation behavior, and the demonstration of in situ S/TEM nanoindentation of nacre opens the possibility of other in situ S/TEM such as bending, tensile, etc., across a wider range of biological and bioinspired composites. For nacre under compression, the rich multiscale resilient deformation processes and interlamellar locking inspires new synthetic routes to complex structural materials. Sample preparation. After the bivalves were sacrificed, small shell sections were cut from the whole shell measuring 60 cm shell height 34 using a diamond wire saw. To avoid beam damage and amorphization from ion beam milling, cross-sections for S/TEM were prepared by mechanical wedge polishing 34 . This technique provided large-area, electron-transparent specimens with structural stability critical for nanotribology. Nacre samples had thicknesses of 124 ± 3 nm (Fig. 4a-c) and 98 ± 2 nm (Fig. 3a), prismatic calcite had a thickness of 102 ± 6 nm ( Fig. 4d-f), and monolithic aragonite had a thickness of 169 ± 1 nm (Fig. 4g-i). Electron microscopy. Real-time observation of the compressive nano-deformation was performed using S/TEM. Column pressure in the TEM column of the specimen was~1 × 10 −7 torr. Bright-field TEM with 60 µm (for nacre) and 120 µm (for biogenic prismatic calcite) apertures provided contrast of strain contours and performed on a 200 keV JEOL 2010F and Gatan OneView camera enabling frame rates of up to 200 frames per second. Images were captured at 50 frames (2048 × 2048 pixels) per second for nacre and at 12.5 frames (2048 × 2048 pixels) per second for biogenic prismatic calcite and monolithic aragonite. STEM was performed using a JEOL 3100R05 microscope with Cs aberration corrected STEM (300 keV, 22 mrad) and cold field emission gun. A HAADF detector with 120-150 mm camera lengths and a detector angle from 59-74 (inner) to 354-443 mrad (outer) were used to produce Z-contrast images where grayscale intensity is sensitive to the atomic number in the specimen's matrix. No change was observed in mechanical behavior measurements with beam exposure: whether the beam was blanked or the microscope was operated in lowdose STEM or TEM mode. Low-dose methods, beam shuttering, and examination of regions exposed to the beam were used to separate electron beam irradiation from intrinsic phenomena. For STEM measurements, with a typical field of view of 500 nm the electron dose was typically~150 e − /Å 2 and dose rates around~4 e − /Å 2 /s; the material was structurally preserved during imaging. However, for the same imaging conditions at higher magnifications (e.g., 30 nm field of view) the radiation dose increases to~2 × 10 5 e − /Å 2 and dose rate to~10 3 e − /Å 2 /s, which causes the material to show electron irradiation damage localized to the small field of view. Thus larger fields of view are preferred to minimize dose and provide a large area of observation where fracture may nucleate. Atomic-resolution STEM requires a small field of view, on-axis region of interest, and a static specimen, which was not achievable during in situ nanoindentation. This limits atomic imaging during nanoindentation despite the well-aligned instrument's probe-limited resolution of 1 Å. For TEM imaging, dose was minimized through use of a heavily diverged beam and a high-efficiency camera (DQE of 0.3) with single electron sensitivity and high-readout speed (up to 300 fps). Low-loss electron energy loss spectra (EELSs) were acquired at 300 kV with a Gatan Quantum Energy Filter, with 1.5 eV per channel to determine thickness of the specimen. The convergence semi-angle was 22 mrad and the collection semiangle was approximately 40 mrad. Linear combination of power laws with local background averaging was applied to analyze the spectrum image using the Cornell Spectrum Imager 60 . Thickness of the specimen is determined from plural scattering in EELS, which is defined by I 0 ¼ I ZLP Á e À t λ , where I 0 is total plural scattering in electron-loss spectrum, I ZLP is zero loss peak in the spectrum, t is the thickness of the nacre, and λ is the wavelength of the incident electron beam, as described in Egerton 61 . Relative organic concentration in nacre tablet was formulated by ratio of high-angle elastic electron scattering intensity, which is defined by I HAADF ¼ t Á P Z γ x Á ρ, where I HAADF is the HAADF intensity, t is the thickness of the nacre, x is a certain element in CaCO 3 or organic molecule, γ is elastic scattering cross-section ranging from 1.4 to 1.7, and ρ is the CaCO 3 or organic molecular density (Supplementary Fig. 2). In situ nanoindentation. Nanoindentation experiments were conducted in the TEM column (~25°C, 10 -6 Torr) using a Hysitron PI-85 PicoIndentor. Load-controlled nanoindentation was performed using cube corner (tip radius = ∼0.1 μm, half-angle = 35.26°, included angle = 90°) and conospherical (tip radius = ∼1 μm, semi-angle = 60°) diamond probe tips. Maximum loads varied from 10 to 400 µN. A piezoelectric actuator controlled the specimen position in all three dimensions. During indentation, the indenter was advanced at a rate of 5 nm/s for nacreous aragonite and prismatic calcite and at a rate of 60 nm/s for monolithic aragonite. Force-displacement information and movies were recorded during indentation, and still TEM micrographs were collected between tests. The electrostatic force constant of the transducer was calibrated such that the rootmean-square error fell <~10 −5 µN/V 2 using Z-axis calibration, which results in the measurement error in force and displacement within ±5%. The top surface of the tip was aligned vertically to the cross-sectional nacre specimen to achieve uniaxial compression without shear or bending. For all the samples, contact stress is calculated by dividing the measured load by cross-sectional area of the specimen in contact with the indenter tip. This contact area is estimated by multiplying the length of the contact region measured in real time with S/TEM images and the specimen cross-sectional thickness measured by the ratio of zero loss/total low-loss EELS. Total error of contact stress is calculated by a quadrature of the errors from the contact length measured by human vision (±10%), the specimen thickness estimated by EELS image (±11%), and the load reported by the nanoindentation software (±5%). The contact area changes through subsequent indentations (Fig. 3) due to stage drift (typically 20-60 nm). Toughness (J/m 2 ) is defined as the absorbed mechanical energy, which can be bounded by integrating the stress-displacement curve to find the energy absorbed per unit area (Fig. 4j). The tablet engineering strain along the c-axis is defined as the ratio of the reduction of the tablet width (that is, compressive deformation directly under the region of loading) to its initial width ( Supplementary Fig. 6). Strain attenuation is defined as the ratio of the measured tablet engineering strain between the first and the second tablet from the indenter tip contact location ( Supplementary Fig. 6). Triboindentation. Triboindentation experiments were carried out on bulk biological aragonite, calcite, and geological aragonite samples (5 × 5 mm 2 area, 3 mm thick) with polished surfaces to determine the elastic modulus of the materials using a Hysitron TI-950 Triboindenter. During indentation, the indenter was advanced at a rate of 20 nm/s for nacre, prismatic calcite, and monolithic aragonite with a Berkovich tip (i.e., three-sided pyramidal diamond tip). The probe area function was calibrated for the Berkovich tip, particularly in the low-depth ranges using a standard quartz sample before determining the mechanical properties accurately. To validate the tip calibration, a standard Al single crystal was used to confirm that our calibration values match the elastic modulus and hardness (69.6 ± 10% and 9.25 ± 10% GPa) provided by the manufacturer within a standard deviation of 5% using the standard Oliver-Pharr method. Reporting summary. Further information on research design is available in the Nature Research Reporting Summary linked to this article. Fig. 1e, SupplementaryFig. 3a-d). This remarkable recovery after tablet locking was reproduced and observed across all areas of interest (Supplementary Figs. 3 and 4). Fig. 1 Fig. 2 12Highly deformed and recovered nacre. a Schematic of the inner shell surface of the bivalve mollusk P. nobilis, with the investigated area marked by a purple square. b HAADF STEM overview image of cross-sectional interface of nacre tablets before compression. c High-resolution STEM image of two tablets and their organic interface before compression. d Tablets heavily interlocked under 40 µN compressive load. e After indenter is retracted, tablets and organic interface have fully recovered their initial morphology. Insets show the movement of organic inclusions due to the deformation of the tablet and their complete recovery after removing the compressive load Strain propagation confined by organic interfaces. a, b Fig. 3 3Recoverable mechanical strength of nacre and crack blunting within and between tablets. a Nine consecutive in situ TEM compression tests on the same nacreous tablet. Three different colors correspond to different contact areas during the series of the compressions. Stage drift caused changes to contact area between indentations. b ADF STEM images after the series of the indentation tests showing a crack blunted by an organic boundary. c ADF STEM image shows crack formed within tablet and blunted by an organic inclusion. d, e ADF STEM images of nacre tablet compressed by 47 µN (55% of σ Yield ) Fig. 4 4Toughening processes of nacre, prismatic calcite, and monolithic aragonite. a-c Bright-field TEM images of the cross-sectional nacreous region during in situ TEM indentation. The nacreous tablets made contact on the side of the tip (tip diameter:~100 nm). Inset in c shows crack blunting at the organic interface. d-f Bright-field TEM images of the cross-sectional prismatic calcite region during indentation. g-i Bright-field TEM images of nonbiogenic, monolithic aragonite during indentation. j Correlative compressive contact stress vs. displacement curve showing mechanical response of the nacreous, prismatic, and monolithic region. Stress herein is engineering stress calculated by dividing load by cross-sectional area contacted with tip.Total energy dissipation values (area under contact stress-displacement curves) marked. k Triboindentation on bulk specimens of nacre, prismatic calcite, and monolithic aragonite. Videos provided as Supplementary Material. (See Supplementary Movies 2| (2019) 10:4822 | https://doi.org/10.1038/s41467-019-12743-z | www.nature.com/naturecommunications Specimens of the protected Mediterranean P. nobilis (Pinnidae, Linnaeus 1758) bivalve species were live collected in the bay of Villefranche-sur-Mer, Département Alpes-Maritimes, France. All necessary permits were acquired from DDTM (Direction Départementale des Territoires et de la Mer) of Alpes-Maritimes department. P. nobilis is strongly protected by a European Directive (92/43/CEE). Specimens of the geological monolithic aragonite were mined in Sefrou, Morocco. ,12,15,16 , tensile 5,12,15,19,20,22,39 , or compression 14,19,40 MPa·m 1⁄2 has been reported for nacre, 40-fold larger than that of single crystal aragonite,~0.25 MPa·m 1⁄2 14,39 . NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-12743-z NATURE COMMUNICATIONS | (2019) 10:4822 | https://doi.org/10.1038/s41467-019-12743-z | www.nature.com/naturecommunications © The Author(s) 2019 Data availabilityThe authors declare that the source data underlying the main Figs. 1a-e, 2a-c, 3a-e, and 4a-k are provided as a Source Data file. All other relevant data supporting the findings of this paper are available from the corresponding author upon reasonable request.SupplementaryAuthor contributionsCompeting interestsThe authors declare no competing interests.Additional informationSupplementary information is available for this paper at https://doi.org/10.1038/s41467-019-12743-z.Correspondence and requests for materials should be addressed to R.H.Peer review information Nature Communications thanks Youping Chen, Horacio Espinosa, and Andrew Minor for their contributions to the peer review of this work. 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If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. The conflicts between strength and toughness. R O Ritchie, Nat. Mater. 10Ritchie, R. O. The conflicts between strength and toughness. Nat. Mater. 10, 817-822 (2011). Nanostructuring of metals by severe plastic deformation for advanced properties. R Valiev, Nat. Mater. 3Valiev, R. Nanostructuring of metals by severe plastic deformation for advanced properties. Nat. Mater. 3, 511-516 (2004). Designing metallic glass matrix composites with high toughness and tensile ductility. D C Hofmann, Nature. 451Hofmann, D. C. et al. Designing metallic glass matrix composites with high toughness and tensile ductility. Nature 451, 1085-1089 (2008). 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[ "Filtering of Continuous Time Periodically Correlated Isotropic Random Fields", "Filtering of Continuous Time Periodically Correlated Isotropic Random Fields" ]
[ "Iryna Golichenko ", "Oleksandr Masyutka ", "Mikhail Moklyachuk " ]
[]
[]
The problem of optimal linear estimation of functionals depending on the unknown values of a random field ζ(t, x), which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the unit sphere Sn with respect to spatial argument x ∈ Sn.Estimates are based on observations of the fieldis an uncorrelated with ζ(t, x) random field, which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the sphere Sn with respect to spatial argument x ∈ Sn. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.
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[ "https://arxiv.org/pdf/1606.01511v1.pdf" ]
88,522,146
1606.01511
37b60f59621fc287464f8ca718a4072ddfb6d2f9
Filtering of Continuous Time Periodically Correlated Isotropic Random Fields 5 Jun 2016 2nd April 2018 Iryna Golichenko Oleksandr Masyutka Mikhail Moklyachuk Filtering of Continuous Time Periodically Correlated Isotropic Random Fields 5 Jun 2016 2nd April 2018isotropic random fieldperiodically correlated random fieldrobust estimatemean square errorleast favourable spectral densityminimax spectral characteristic 2000 Mathematics Subject Classification: Primary: 60G6062M40Secondary: 62M2093E1093E11 The problem of optimal linear estimation of functionals depending on the unknown values of a random field ζ(t, x), which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the unit sphere Sn with respect to spatial argument x ∈ Sn.Estimates are based on observations of the fieldis an uncorrelated with ζ(t, x) random field, which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the sphere Sn with respect to spatial argument x ∈ Sn. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given. Introduction Cosmological Principle (first coined by Einstein): the Universe is, in the large, homogeneous and isotropic (J. G. Bartlett [3]). Last decades indicate growing interest to the spatio-temporal data measured on the surface of a sphere. These data includes cosmic microwave background (CMB) anisotropies (J. G. Bartlett [3], W. Hu and S. Dodelson [22], N. Kogo and N. Komatsu [30], T. Okamoto and W. Hu [49], P. Adshead and W. Hu [1]), medical imaging (R. Kakarala [26]), global and land-based temperature data (P. D. Jones [24], T. Subba Rao and G. Terdik [53]), gravitational and geomagnetic data, climate model (G. R. North and R. F. Cahalan [48]). Some basic results and references on the theory of isotropic random fields on a sphere can be found in the books by M. I. Yadrenko [58] and A. M. Yaglom [59,60]. For more recent applications and results see new books by C. Gaetan and X. Guyon [16], N. Cressie and C. K. Wikle [4], D. Marinucci and G. Peccati [32] and several papers covering a number of problems in general for spatial temporal isotropic observations (T. Subba Rao and G. Terdik [54], G. Terdik [55]). Periodically correlated processes and fields are not homogeneous but have numerous properties similar to properties of stationary processes and homogeneous fields. They describe appropriate models of numerous physical and man-made processes. A comprehensive list of the existing references up to the year 2005 on periodically correlated processes and their applications was proposed by E. Serpedin, F. Panduru, I. Sari and G. B. Giannakis [52]. See also reviews by J. Antoni [2] and A. Napolitano [47]. For more details see a survey paper by W. A. Gardner [18] and book by H. L. Hurd and A. Miamee [23]. Note, that in the literature periodically correlated processes are named in multiple different ways such as cyclostationary, periodically nonstationary or cyclic correlated processes. The mean square optimal estimation problems for periodically correlated with respect to time isotropic on a sphere random fields are natural generalization of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes and homogeneous random fields. Effective methods of solution of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes and random fields were developed under the condition of certainty where spectral densities of processes and fields are known exactly (see, for example, selected works of A. N. Kolmogorov [31], survey article by T. Kailath [25], books by Yu. A. Rozanov [51], N. Wiener [57], A. M. Yaglom [59,60], M. I. Yadrenko [58], articles by M. P. Moklyachuk and M. I. Yadrenko [44] - [45]). The classical approach to the problems of interpolation, extrapolation and filtering of stochastic processes and random fields is based on the assumption that the spectral densities of processes and fields are known. In practice, however, complete information about the spectral density is impossible in most cases. To overcome this complication one finds parametric or nonparametric estimates of the unknown spectral densities or selects these densities by other reasoning. Then applies the classical estimation method provided that the estimated or selected density is the true one. This procedure can result in a significant increasing of the value of error as K. S. Vastola and H. V. Poor [56] have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities from a certain class of admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error of estimates. Such problems arise when considering problems of automatic control theory, coding and signal processing in radar and sonar, pattern recognition problems of speech signals and images. A comprehensive survey of results up to the year 1985 in minimax (robust) methods of data processing can be found in the paper by S. A. Kassam and H. V. Poor [29]. J. Franke [14], J. Franke and H. V. Poor [15] investigated the minimax extrapolation and filtering problems for stationary sequences with the help of convex optimization methods. This approach makes it possible to find equations that determine the least favorable spectral densities for different classes of densities. The paper by Ulf Grenander [21] should be marked as the first one where the minimax approach to extrapolation problem for the functionals from stationary processes was developed. For more details see, for example, survey articles M. P. Moklyachuk [37], [38], [41] books by M. Moklyachuk [39], M. Moklyachuk and O. Masytka [43], I. I. Golichenko and M. P. Moklyachuk [20]. In papers by I. I. Dubovets'ka, O.Yu. Masyutka and M.P. Moklyachuk [5], I. I. Dubovets'ka and M. P. Moklyachuk [6] - [9] the minimax-robust estimation problems (extrapolation, interpolation and filtering) are investigated for linear functionals which depend on unknown values of periodically correlated stochastic processes. Methods of solution the minimax-robust estimation problems for time-homogeneous isotropic random fields on a sphere were developed by M. P. Moklyachuk [34] - [36]. In papers by I. I. Dubovets'ka, O.Yu. Masyutka and M.P. Moklyachuk [10] - [12] results of investigation of minimax-robust estimation problems for periodically correlated isotropic random fields are proposed. In this article we deal with the problem of mean square optimal linear estimation of the functional Aζ = ∞ 0 Sn a(t, x)ζ(−t, x) m n (dx)dt which depends on unknown values of a periodically correlated (cyclostationary with period T ) with respect to time isotropic on the unit sphere S n in Euclidean space E n random field ζ(t, x), t ≤ 0, x ∈ S n . Estimates are based on observations of the field ζ(t, x)+θ(t, x) at points (t, x), t ≤ 0, x ∈ S n , where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere S n random field. Formulas are derived for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional Aζ in the case of spectral certainty, where spectral densities of the fields are known. Formulas are proposed that determine the least favourable spectral densities and the minimax-robust spectral characteristic of the optimal estimate of the functional Aζ for concrete classes of spectral densities under the condition that spectral densities are not known exactly while classes D = D f × D g of admissible spectral densities are given. 2 Spectral properties of periodically correlated isotropic on a sphere random fields Let S n be a unit sphere in the n-dimensional Euclidean space E n , let m n (dx) be the Lebesgue measure on S n , and let S l m (x), l = 1, ..., h(m, n); m = 0, 1, ... be the orthonormal spherical harmonics of degree m, where h(m, n) is the number of orthonormal spherical harmonics (see books by A. Erdelyi et al. [13] and C. Müller [46] for more details). A mean-square continuous random field ζ(t, x), t ∈ R, x ∈ S n , ζ(t, x) ∈ H = L 2 (Ω, F , P), where L 2 (Ω, F , P) denotes the Hilbert space of random variables ζ with zero first moment, Eζ = 0, and finite second moment, E|ζ| 2 < ∞, is called periodically correlated (cyclostationary with period T ) with respect to time isotropic on the sphere S n if for all t, s ∈ R and x, y ∈ S n the following property holds true E ζ(t + T, x)ζ(s + T, y) = B (t, s, cos ϑ) , where cos ϑ = (x, y), ϑ is the angular distance between points x, y ∈ S n . The correlation function B (t, s, cos ϑ) of the mean-square continuous random field ζ(t, x) is continuous. It can be represented in the form of the series B (t, s, cos ϑ) = 1 ω n ∞ m=0 h(m, n) C (n−2)/2 m (cos ϑ) C (n−2)/2 m (1) B ζ m (t, s), where ω n = (2π) n/2 Γ(n/2), C l m (z) are the Gegenbauer polynomials (see book by M. I. Yadrenko [58]). It follows from the Karhunen theorem that the random field ζ(t, x) itself can be represented in the form of the mean square convergent series (see K. Karhunen [28], I. I. Gikhman and A. V. Skorokhod [17]) ζ(t, x) = ∞ m=0 h(m,n) l=1 S l m (x)ζ l m (t),(1) where ζ l m (t) = Sn ζ(t, x)S l m (x) m n (dx). In this representation ζ l m (t), l = 1, . . . , h(m, n); t ∈ R, m = 0, 1, . . . are mutually uncorrelated periodically correlated stochastic processes with the correlation functions B ζ m (t, s): E ζ l m (t + T )ζ v u (s + T ) = δ u m δ v l B ζ m (t, s), l, v = 1, . . . , h(m, n); m, u = 0, 1, . . . ; t, s ∈ R, where δ v l are the Kroneker delta-functions. Consider two mutually uncorrelated periodically correlated random fields ζ(t, x) and θ(t, x). We construct the following sequences of stochastic functions {ζ l m (j, u) = ζ l m (u + jT ), u ∈ [0, T ), j ∈ Z},(2){θ l m (j, u) = θ l m (u + jT ), u ∈ [0, T ), j ∈ Z}(3) which correspond to the random fields ζ(t, x) and θ(t, x). The sequences (2) and (3) R ζ m (k, j) = T 0 E[ζ l m (u+kT )ζ l m (u + jT )]du = T 0 B ζ m (u+(k−j)T, u)du = R ζ m (k−j), R θ m (k, j) = T 0 E[θ l m (u+kT )θ l m (u + jT )]du = T 0 B θ m (u+(k−j)T, u)du = R θ m (k−j). To describe properties of the stationary sequences {ζ l m (j), j ∈ Z} and {θ l m (j), j ∈ Z} we define in the space L 2 ([0, T ); R) the following orthonormal basis { e k = 1 √ T e 2πi{(−1) k [ k 2 ]}u/T , k = 1, 2, . . .}, e j , e k = δ j k . Making use of the introduced basis the stationary sequences {ζ l m (j), j ∈ Z} and {θ l m (j), j ∈ Z} can be represented as follows ζ l m (j) = ∞ k=1 ζ l mk (j) e k ,(4)ζ l mk (j) = ζ l m (j), e k = 1 √ T T 0 ζ l m (j, v)e −2πi{(−1) k [ k 2 ]}v/T dv, θ l m (j) = ∞ k=1 θ l mk (j) e k ,(5)θ l mk (j) = θ l m (j), e k = 1 √ T T 0 θ l m (j, v)e −2πi{(−1) k [ k 2 ]}v/T dv. Components of the constructed vector-valued stationary sequences {ζ l m (j) = (ζ l mk (j), k = 1, 2, . . . ), j ∈ Z} and {θ l m (j) = (θ l mk (j), k = 1, 2, . . . ), j ∈ Z} have the following properties [27], [33] Eζ l mk (j) = 0, ζ l m (j) 2 H = ∞ k=1 E|ζ l mk (j)| 2 = R ζ m (0), Eζ l mk (j 1 )ζ l mn (j 2 ) = K ζ m (j 1 −j 2 )e k , e n , Eθ l mk (j) = 0, θ l m (j) 2 H = ∞ k=1 E|θ l mk (j)| 2 = R θ m (0), Eθ l mk (j 1 )θ l mn (j 2 ) = K ζ m (j 1 −j 2 )e k , e n , where {e k , k = 1, 2, . . .} is a basis in the space ℓ 2 . The correlation func- tions K ζ m (j) and K θ m (j) of the stationary sequences {ζ l m (j) = (ζ l mk (j), k = 1, 2, . . . ), j ∈ Z} and {θ l m (j) = (θ l mk (j), k = 1, 2, . . . ), j ∈ Z} are correlation operator functions in ℓ 2 . The vector-valued stationary sequences {ζ l m (j) = (ζ l mk (j), k = 1, 2, . . . ), j ∈ Z} and {θ l m (j) = (θ l mk (j), k = 1, 2, . . . ), j ∈ Z} have the spectral density func- tions F m (λ) = f kn m (λ) ∞ k,n=1 , G m (λ) = g kn m (λ) ∞ k,n=1 , that are operator-valued functions of variable λ ∈ [−π, π) in the space ℓ 2 if their correlation functions K ζ m (j) and K θ m (j) can be represented in the form K ζ m (j)e k , e n = 1 2π π −π e ijλ F m (λ)e k , e n dλ, K θ m (j)e k , e n = 1 2π π −π e ijλ G m (λ)e k , e n dλ, For almost all λ ∈ [−π, π) the spectral densities F m (λ) and G m (λ) are kernel operators with integrable kernel norm ∞ k=1 1 2π π −π F m (λ)e k , e k dλ = ∞ k=1 K ζ m (0)e k , e k = ζ l m (j) 2 H = R ζ m (0), ∞ k=1 1 2π π −π G m (λ)e k , e k dλ = ∞ k=1 K θ m (0)e k , e k = θ l m (j) 2 H = R θ m (0). Hilbert space projection method of filtering Consider the problem of the mean square optimal linear estimation of the functional Aζ = ∞ 0 Sn a(t, x)ζ(−t, x) m n (dx)dt which depends on unknown values of a periodically correlated with respect to time isotropic on the unit sphere S n in Euclidean space E n random field ζ(t, x), t ≤ 0, x ∈ S n . Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x), t ≤ 0, x ∈ S n , where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere S n random field. It follows from representations (1) that the functional Aζ can be represented in the form Aζ = ∞ 0 Sn a(t, x)ζ(−t, x)m n (dx)dt = ∞ m=0 h(m,n) l=1 ∞ 0 a l m (t)ζ l m (−t)dt = = ∞ m=0 h(m,n) l=1 ∞ j=0 T 0 a l m (j, u)ζ l m (−j, −u)du, a l m (t) = Sn a(t, x)S l m (x) m n (dx), a l m (j, u) = a l m (u + jT ), u ∈ [0, T ), ζ l m (−j, −u) = ζ l m (−u − jT ), u ∈ [0, T ) . Taking into account the decomposition (4) of stationary sequence {ζ l m (j), j ∈ Z}, the functional Aζ can be represented in the following form Aζ = ∞ m=0 h(m,n) l=1 ∞ j=0 ∞ k=1 a l mk (j)ζ l mk (−j) = ∞ m=0 h(m,n) l=1 ∞ j=0 a l m (j) ⊤ ζ l m (−j), ζ l m (−j) = (ζ l mk (−j), k = 1, 2, . . . ) ⊤ , a l m (j) = (a l mk (j), k = 1, 2, . . . ) ⊤ = = (a l m1 (j), a l m3 (j), a l m2 (j), . . . , a l m(2k+1) (j), a l m(2k) (j), . . . ) ⊤ , a l mk (j) = a l m (j), e k = 1 √ T T 0 a l m (j, v)e −2πi{(−1) k [ k 2 ]}v/T dv. We will assume that coefficients { a l m (j), j = 0, 1, . . .} which form this representation satisfy the following conditions (λ) = h l mk ∞ k=1 , that satisfy condition ∞ m=0 h(m,n) l=1 π −π (h l m (λ)) ⊤ F m (λ)h l m (λ)dλ < ∞. We denote by L − 2 (F ) the subspace of L 2 (F ) generated by the functions e ijλ δ k , δ k = {δ n k } ∞ n=1 , k = 1, 2, . . . , j ≤ 0, where δ k k = 1, δ n k = 0, k = n. Every linear estimateÂζ of the functional Aζ from observations of the sequence {ζ l m (j) + θ l m (j), j ∈ Z} at points j ≤ 0 is defined by the spectral characteristic h(λ) ∈ L − 2 (F + G) and is of the form Aζ = ∞ m=0 h(m,n) l=1 π −π (h l m (λ)) ⊤ Z l ζ+θ m (dλ),(7) where Z l ζ+θ m (∆) = {Z l ζ+θ mk (∆)} ∞ k=1 is the orthogonal stochastic measure of sum of sequences ζ l m (j) and θ l m (j). Suppose that spectral densities of stationary sequence {ζ l m (j)}, {θ l m (j)} admit the canonical factorizations (G. Kallianpur and V. Mandrekar [27], M. P. Moklyachuk [33]) F m (λ) = ϕ m (λ)(ϕ m (λ)) * , ϕ m (λ) = ∞ u=0 ϕ m (u)e −iuλ ,(8)G m (λ) = ψ m (λ)(ψ m (λ)) * , ψ m (λ) = ∞ u=0 ψ m (u)e −iuλ ,(9)F m (λ) + G m (λ) = d m (λ)(d m (λ)) * , d m (λ) = ∞ u=0 d m (u)e −iuλ ,(10) where matrices ∆(h; F, G) = E|Aζ −Âζ| 2 = = ∞ m=0 h(m,n) l=1 ( Ψ l m a l m 2 + D m (a l m − h l m ) 2 − − Ψ m (a l m − h l m ), Ψ m a l m − Ψ m a l m , Ψ m (a l m − h l m ) ), where operators Ψ, D are defined as follows Ψ m a l m 2 = ∞ q=0 (Ψ m a l m ) q 2 , (Ψ m a l m ) q = q j=0 (ψ m (q − j)) ⊤ a l m (j), D m (a l m − h l m ) 2 = ∞ q=0 (D m (a l m − h l m )) q 2 , (D m (a l m − h l m )) q = q j=0 (d m (q − j)) ⊤ ( a l m (j) − h l m (j)), Ψ m (a l m − h l m ), Ψ m a l m = ∞ q=0 (Ψ m (a l m − h l m )) q , (Ψ m a l m ) q . The spectral characteristic h(F, G) of the optimal linear estimateÂζ of the functional minimizes the value of the mean square error ∆(F, G) = ∆(h(F, G); F, G) = = min h∈L − 2 (F +G) ∆(h; F, G) = min Aζ E|Aζ −Âζ| 2 .(11) In the case where the spectral densities G m (λ) and F m (λ) + G m (λ) admit factorizations (9) and (10), the spectral characteristic h(F, G), which is a solution of the optimization problem (11), and the mean square error ∆(F, G) of the optimal estimateÂζ are determined by formulas h l m (F, G) = A l m (λ) − (b m (λ)) ⊤ C l m (G)(λ),(12)∆(F, G) = ∞ m=0 h(m,n) l=1 Ψ m a l m 2 − B * m Ψ * m Ψ m a l m 2 ,(13)where b m (λ) = {b k mr (λ)} k=1,∞ r=1,M , b m (λ) = ∞ u=0 b m (u)e −iuλ , b m (λ)d m (λ) = I M , C l m (G)(λ) = ∞ j=0 (C l m (G)) j e −ijλ , A l m (λ) = ∞ j=0 a l m (j)e −ijλ , (C l m (G)) j = (B * Ψ * Ψa) j = ∞ q=0 b m (q)(Ψ * m Ψ m a l m ) j+q , (Ψ * m Ψ m a l m ) q = ∞ u=0 ψ m (u)(Ψ m a l m ) u+q , B * m Ψ * m Ψ m a l m 2 = ∞ q=0 (B * m Ψ * m Ψ m a l m ) q 2 . In the case where the spectral densities F m (λ) and F m (λ) + G m (λ) admit factorizations (8) and (10), the spectral characteristic h(F, G) and the mean square error ∆(F, G) of the optimal estimateÂζ are defined by formulas h l m (F, G) = (b m (λ)) ⊤ C l m (F )(λ),(14)∆(F, G) = ∞ m=0 h(m,n) l=1 Φ m a l m 2 − B * m Φ * m Φ m a l m 2 ,(15)C l m (F )(λ) = ∞ j=0 (C l m (F )) j e −ijλ , (C l m (F )) j = (B * m Φ * m Φ m a l m ) j = ∞ q=0 b m (q)(Φ * m Φ m a l m ) j+q , (Φ * m Φ m a l m ) q = ∞ u=0 ϕ m (u)(Φ m a l m ) u+q , (Φ m a l m ) q = q j=0 (ϕ m (q − j)) ⊤ a l m (j), Φ m a l m 2 = ∞ q=0 (Φ m a l m ) q 2 , B * m Φ * m Φ m a l m 2 = ∞ q=0 (B * m Φ * m Φ m a l m ) q 2 . Let us summarize our results and present them in the form of a theorem. Theorem 1. Let {ζ(t, x), t ∈ R, x ∈ S n } and {θ(t, x), t ∈ R, x ∈ S n } be mutually uncorrelated random fields, which are periodically correlated with respect to time argument t ∈ R and isotropic on the unit sphere S n with respect to spatial argument x ∈ S n . Let the stationary sequences {ζ l m (j), j ∈ Z} and {θ l m (j), j ∈ Z} constructed with the help of relations (2), (3), respectively, have spectral densities F m (λ) and G m (λ) that admit the canonical factorizations (8), (10) (or (9), (10)). Let coefficients { a l m (j), j = 0, 1, . . .} that determine the functional Aζ satisfy conditions (6). Then the spectral characteristic h(F, G) and the mean square error ∆(F, G) of the optimal estimate of the functional Aζ from observations of the field ζ(t, x) + θ(t, x) at points (t, x), t ≤ 0, x ∈ S n are given by formulas (14), (15) (or (12), (13)), respectively. The optimal estimatê Aζ of the functional Aζ is calculated by the formula (7). Minimax-robust method of filtering Formulas (12) - (15) for calculating the spectral characteristic and the mean square error of the optimal linear estimate of the functional Aζ can be applied in the case where the spectral densities F m (λ) and G m (λ) of stationary sequences {ζ l m (j), j ∈ Z} and {θ l m (j), j ∈ Z} constructed by relations (2), (3), are known. If the spectral densities are not exactly known while a set of admissible densities D = D F × D G is specified, then the minimax approach to estimation of functional of unknown values is reasonable. That is we find the estimate which minimizes the mean square error for all spectral densities from a given set D = D F × D G simultaneously. Definition 1. For a given class of spectral densities D = D F × D G the spectral densities F 0 m (λ) ∈ D F and G 0 m (λ) ∈ D G are called the least favorable in D for the optimal estimate of functional Aζ if ∆(F 0 , G 0 ) = ∆(h(F 0 , G 0 ); F 0 , G 0 ) = max (F,G)∈D ∆(h(F, G); F, G). Definition 2. For a given class of spectral densities D = D F × D G the spectral characteristic h 0 (λ) of the optimal linear estimate of the functional Aζ is called minimax-robust if the following relations hold true h 0 (λ) ∈ H D = (F,G)∈D L − 2 (F + G), min h∈HD max (F,G)∈D ∆(h; F, G) = max (F,G)∈D ∆(h 0 ; F, G). Taking into account the introduced definitions and relations (8) - (15) we can verify that the following lemma holds true. ∆(F, G) = ∞ m=0 h(m,n) l=1 Φ m a l m 2 − B * m Φ * m Φ m a l m 2 → sup, F m (λ) = ϕ m (λ)(ϕ m (λ)) * ∈ D F ,(16)G m (λ) = d m (λ)(d m (λ)) * − ϕ m (λ)(ϕ m (λ)) * ∈ D G , or the constrained optimization problem ∆(F, G) = ∞ m=0 h(m,n) l=1 Ψ m a l m 2 − B * m Ψ * m Ψ m a l m 2 → sup, G m (λ) = ψ m (λ)(ψ m (λ)) * ∈ D G ,(17)F m (λ) = d m (λ)(d m (λ)) * − ψ m (λ)(ψ m (λ)) * ∈ D F . Lemma 2. Let the spectral density F m (λ) be given and admits the factorization (8). Then the spectral density G 0 m (λ) is the least favorable in D G for the optimal estimation of the functional Aζ if F m (λ) + G 0 m (λ) = d 0 m (λ)(d 0 m (λ)) * , where d 0 m (λ) = ∞ u=0 d 0 m (u)e −B * m Φ * m Φ m a l m 2 → inf, G m (λ) = d m (λ)(d m (λ)) * − F m (λ) ∈ D G .(18) Lemma 3. Let the spectral density G m (λ) be given and admits the factorization (9). Then the spectral density F 0 m (λ) is the least favorable in D F for optimal estimation of the functional Aζ and admits canonical factorizations (8), (10) if F 0 m (λ) + G m (λ) = d 0 m (λ)(d 0 m (λ)) * , where d 0 m (λ) = ∞ u=0 d 0 m (u)e −B * m Ψ * m Ψ m a l m 2 → inf, F m (λ) = d m (λ)(d m (λ)) * − G m (λ) ∈ D F .(19) For more detailed analysis of properties of the least favorable spectral densities and the minimax-robust spectral characteristics we observe that the least favorable spectral densities F 0 m (λ) ∈ D F , G 0 m (λ) ∈ D G and the minimax spectral characteristic h 0 = h(F 0 , G 0 ) form a saddle point of the function ∆(h; F, G) on the set H D × D. The saddle point inequalities ∆(h 0 ; F, G) ≤ ∆(h 0 ; F 0 , G 0 ) ≤ ∆(h; F 0 , G 0 ), ∀h ∈ H D , ∀F ∈ D F , ∀g ∈ D G hold if h 0 = h(F 0 , G 0 ), h(F 0 , G 0 ) ∈ H D and (F 0 , G 0 ) is a solution of the constrained optimization problem ∆(h(F 0 , G 0 ); F, G) → sup, (F, G) ∈ D,(20) where the functional ∆(h(F 0 , G 0 ); F, G) = = ∞ m=0 h(m,n) l=1 1 2π π −π (C l m (G 0 )(λ) ⊤ b 0 m (λ)F m (λ) b 0 m (λ) * C l m (G 0 )(λ)dλ+ + 1 2π π −π (C l m (F 0 )(λ)) ⊤ b 0 m (λ)G m (λ) b 0 m (λ) * C l m (F 0 )(λ)dλ .(21) The constrained optimization problem (20) is equivalent to the following unconstrained optimization problem ∆ D (F, G) = −∆(h(F 0 , G 0 ); F, G) + δ((F, G)|D) → inf,(22) where δ((F, G)|D) is the indicator function of the set D. Solution (F 0 (λ), G 0 (λ)) to the extremum problem (22) is determined by the condition 0 ∈ ∂∆ D (F 0 , G 0 ) which is necessary for the point (F 0 , G 0 ) to belong to the set of minimums of a convex functional. Here ∂∆ D (F 0 , G 0 ) is a subdifferential of the convex functional ∆ D (F, G) at point (F, G) = (F 0 , G 0 ) (see R. T. Rockafellar [50], M. P. Moklyachuk [40]). The form (21) of the functional ∆(h(F 0 , G 0 ); F, G) is convenient for application the method of Lagrange multipliers for finding solution to the problem (22). Making use the method of Lagrange multipliers and the form of subdifferentials of the indicator functions δ((F, G)|D) we describe relations that determine the least favourable spectral densities in some special classes of spectral densities (see books by M. Moklyachuk [39], M. Moklyachuk and O. Masytka [43], I. I. Golichenko and M. P. Moklyachuk [20] for more details). 5 The least favorable spectral densities in the class D 0 × D U V Consider the problem of minimax estimation of the functional Aζ depending on the unknown values of the random field {ζ(t, x), t ∈ R, x ∈ S n }, which is periodically correlated with respect to the time argument t ∈ R and isotropic on the sphere S n with respect to spatial argument x ∈ S n based on observations of the random field ζ(t, x) + θ(t, x) at points (t, x) : t ≤ 0, x ∈ S n , under the condition that spectral densities F m (λ), G m (λ) of stationary sequences {ζ l m (j), j ∈ Z} and {θ l m (j), j ∈ Z} which are constructed with the help of relations (2), (3), respectively, are not known exactly while there are specified the following pairs of sets of admissible spectral densities. The first pair is D 1 0 = F (λ)| 1 2πω n ∞ m=0 h(m, n) π −π Tr F m (λ)dλ = p , D U V 1 = G(λ)|T rV m (λ) ≤ Tr G m (λ) ≤ T rU m (λ), 1 2πω n ∞ m=0 h(m, n) π −π Tr G m (λ)dλ = q . The second pair of sets of admissible spectral densities is D 2 0 = F (λ)| 1 2πω n ∞ m=0 h(m, n) π −π F kk m (λ)dλ = p k , k = 1, 2, . . . , D U V 2 = G(λ)|V kk m (λ) ≤ G kk m (λ) ≤ U kk m (λ), 1 2πω n ∞ m=0 h(m, n) π −π G kk m (λ)dλ = q k , k = 1, 2, . . . . The third pair of sets of admissible spectral densities is D 3 0 = F (λ)| 1 2πω n ∞ m=0 h(m, n) π −π B, F m (λ) dλ = p , D U V 3 = G(λ)| B 2 , V m (λ) ≤ B 2 , G m (λ) ≤ B 2 , U m (λ) , 1 2πω n ∞ m=0 h(m, n) π −π B 2 , G m (λ) dλ = q . The forth pair of sets of admissible spectral densities is D 4 0 = F (λ)| 1 2πω n ∞ m=0 h(m, n) π −π F m (λ)dλ = P , D U V 4 = G(λ)|V m (λ) ≤ G m (λ) ≤ U m (λ), 1 2πω n ∞ m=0 h(m, n) π −π G m (λ)dλ = Q . Here V m (λ), U m (λ) are given matrices of spectral densities, p, q, p k , q k , k = 1, 2, . . . are given numbers, B 1 , B 2 , P, Q are given positive-definite Hermitian matrices. From the condition 0 ∈ ∂∆ D (F 0 , G 0 ) we find the following equations which determine the least favourable spectral densities for these given sets of admissible spectral densities. For the first pair D 1 0 × D U V 1 we have equations h(m,n) l=1 C l m (G 0 )(λ)(C l m (G 0 )(λ)) * = α 2 m d 0 m (λ) ⊤ d 0 m (λ),(23)h(m,n) l=1 C l m (F 0 )(λ)(C l m (F 0 )(λ)) * = (β 2 m + γ m1 (λ) + γ m2 (λ))d 0 m (λ) ⊤ d 0 m (λ),(24) where γ m1 (λ) ≤ 0 and γ m1 (λ) = 0 if Tr G 0 m (λ) > Tr V m (λ), γ m2 (λ) ≥ 0 and γ m2 (λ) = 0 if Tr G 0 m (λ) < Tr U m (λ), and α 2 m , β 2 m are unknown Lagrange multipliers. For the second pair D 2 0 × D U V 2 we have equations h(m,n) l=1 C l m (G 0 )(λ)(C l m (G 0 )(λ)) * = d 0 m (λ) ⊤ α 2 mk δ kl ∞ k,l=1 d 0 m (λ),(25)h(m,n) l=1 C l m (F 0 )(λ)(C l m (F 0 )(λ)) * = d 0 m (λ) ⊤ (β 2 mk + γ m1k (λ) + γ m2k (λ))δ l k ∞ k,l=1 d 0 m (λ),(26) where γ m1k (λ) ≤ 0 and γ m1k (λ) = 0 if G 0kk m (λ) > V kk m (λ), γ m2k (λ) ≥ 0 and γ m2k (λ) = 0 if G 0kk m (λ) < U kk m (λ) , and α 2 mk , β 2 mk are unknown Lagrange multipliers. For the third pair D 3 0 × D U V 3 we have equations h(m,n) l=1 C l m (G 0 )(λ)(C l m (G 0 )(λ)) * = α 2 m d 0 m (λ) ⊤ B 1 d 0 m (λ),(27)h(m,n) l=1 C l m (F 0 )(λ)(C l m (F 0 )(λ)) * = (β 2 m + γ m1 (λ) + γ m2 (λ))d 0 m (λ) ⊤ B 2 d 0 m (λ); (28) where γ m1 (λ) ≤ 0 and γ m1 (λ) = 0 if B 2 , G 0 m (λ) > B 2 , V m (λ) , γ m2 (λ) ≥ 0 and γ m2 (λ) = 0 if B 2 , G 0 m (λ) < B 2 , U m (λ) , and α 2 m , β 2 m are unknown Lagrange multipliers. For the forth pair D 4 0 × D U V 4 we have equations h(m,n) l=1 C l m (G 0 )(λ)(C l m (G 0 )(λ)) * = d 0 m (λ) ⊤ α m · α m * d 0 m (λ),(29)h(m,n) l=1 C l m (F 0 )(λ)(C l m (F 0 )(λ)) * = d 0 m (λ) ⊤ ( β · β * + Γ m1 (λ) + Γ m2 (λ))d 0 m (λ). (30) where Γ m1 (λ), Γ m2 (λ) are Hermitian matrices, Γ m1 (λ) ≤ 0 and Γ m1 (λ) = 0 if G 0 m (λ) > V m (λ), Γ m2 (λ) ≥ 0 and Γ m2 (λ) = 0 if G 0 m (λ) < U m (λ) , and α m , β m are unknown Lagrange multipliers. (16) or (17), and restrictions on densities from the corresponding classes D 0 × D U V . The minimax spectral characteristic h(F 0 , G 0 ) of the optimal estimateÂζ is calculated by (14) or (12). The mean square error ∆(F 0 , G 0 ) is calculated by (15) or (13). In the case where one of spectral densities F m (λ) or G m (λ) from the corresponding classes is known we have the following corollary from the theorem. Corollary 1. If the spectral density F m (λ) ∈ D 0 is known and admits the canonical factorization (8), then the least favorable spectral densities G 0 m (λ) in the classes D U V k , k = 1, 2, 3, 4 are determined by relations (9), (10), (18), equations (24), (26), (28), (30) correspondingly to k = 1, 2, 3, 4 and by restrictions on densities from classes D U V k , k = 1, 2, 3, 4. If the spectral density G m (λ) ∈ D U V is known and admits the canonical factorization (9), then the least favorable spectral densities F 0 m (λ) in the classes D k 0 , k = 1, 2, 3, 4 are determined by relations (8), (10), (19), equations (23), (25), (27), (29)) correspondingly to k = 1, 2, 3, 4 and by restrictions on densities from classes D U V k , k = 1, 2, 3, 4. The minimax spectral characteristic h(F 0 , G 0 ) of the optimal estimateÂζ is calculated by (14) or (12). The mean square error ∆(F 0 , G 0 ) is calculated by (15) or (13). Conclusions In this paper we propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional Aζ = depending on unknown values of a mean-square continuous periodically correlated (cyclostationary with period T ) with respect to time argument and isotropic on the unit sphere S n in Euclidean space E n random field ζ(t, x), t ∈ R, x ∈ S n . Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x), t ≤ 0, x ∈ S n , where θ(t, x) is an uncorrelated with ζ(t, x) meansquare continuous periodically correlated with respect to time argument and isotropic on the sphere S n random field. The problem is investigated in the case of spectral certainty where matrices of spectral densities of random fields are known exactly and in the case of spectral uncertainty where matrices of spectral densities of random fields are not known exactly while some classes of admissible spectral density matrices are given. We derive formulas for calculation the spectral characteristic and the mean-square error of the optimal linear estimate of the functional Aζ in the case of spectral certainty, where spectral densities F m (λ), G m (λ) of the stationary sequences that generate the random fields ζ(t, x), θ(t, x) are known exactly. We propose a representation of the mean square error in the form of a linear functional in the L 1 × L 1 space with respect to spectral densities (F, G), which allows us to solve the corresponding constrained optimization problem and describe the minimax (robust) estimates of the functional Aζ for concrete classes of spectral densities under the condition that spectral densities are not known exactly while classes D = D f × D g of admissible spectral densities are given. form the L 2 ([0, T ); H)-valued stationary sequences {ζ l m (j), j ∈ Z} and {θ l m (j), j ∈ Z}, respectively, with the correlation functions l mk (j)| 2 . Under these condition the functional Aζ has finite second moment and operators defined below with the help of the coefficients { a l m (j), j = 0, 1, . . .} are compact. Denote by L 2 (F ) the Hilbert space of complex vector functions h(λ) = h l m (λ) : m = 0, 1, . . . ; l = 1, 2, . . . , h(m, n) , h l m d m (u) = {d r mk (u)} r=1,M k=1,∞ , ϕ m (u) = {ϕ r mk (u)} r=1,M1 k=1,∞ , ψ m (u) = {ψ r mk (u)} r=1,M2 k=1,∞ are coefficients of the canonical factorizations, M 1 is the multiplicity of ζ l m (j), M 2 is the multiplicity of θ l m (j) and M is the multiplicity of ζ l m (j) + θ l m (j). The mean square error ∆(h; F, G) of the linear estimateÂζ with the spectral characteristic h l m (λ) = ∞ j=0 h l m (j)e −ijλ can be represented in the form Lemma 1 . 1Spectral densities F 0 m (λ) ∈ D F and G 0 m (λ) ∈ D G which admit the canonical factorizations (8) -(10) are the least favorable in the class D = D F ×D G for the optimal linear estimation of the functional Aζ if the coefficients of factorizations define a solution of the constrained optimization problem iuλ and coefficients {d 0 m (u), u = 0, 1, . . .} are determined by solution of the constrained optimization problem iuλ and coefficients {d 0 m (u), u = 0, 1, . . .} are determined by solution of the constrained optimization problem Theorem 2 . 2The least favorable spectral densities F 0 m (λ), G 0 m (λ) in the classes D 0 × D U V for the optimal estimate of the functional Aζ are determined by relations (23),(24) for the first pair D Fast computation of first-order feature-bispectrum corrections. P Adshead, W Hu, Phys. 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[ "On closed surfaces with nonnegative curvature in the spectral sense", "On closed surfaces with nonnegative curvature in the spectral sense" ]
[ "Kai Xu " ]
[]
[]
We study closed orientable surfaces satisfying the spectral condition λ 1 (−∆ + βK) λ 0, where β is a positive constant and K is the Gauss curvature. This condition naturally arises for stable minimal surfaces in 3-manifolds with positive scalar curvature. We show isoperimetric inequalities, area growth theorems and diameter bounds for such surfaces. The validity of these inequalities are subject to certain bounds for β. Associated to a positive super-solution ∆ϕ βKϕ, the conformal metric ϕ 2/β g has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove Hölder precompactness and almost rigidity results concerning the main spectral condition.
null
[ "https://export.arxiv.org/pdf/2211.11715v2.pdf" ]
257,623,125
2211.11715
3d678e30b0509ef03575d573c27b4894bd8b10d0
On closed surfaces with nonnegative curvature in the spectral sense 16 Mar 2023 Kai Xu On closed surfaces with nonnegative curvature in the spectral sense 16 Mar 2023 We study closed orientable surfaces satisfying the spectral condition λ 1 (−∆ + βK) λ 0, where β is a positive constant and K is the Gauss curvature. This condition naturally arises for stable minimal surfaces in 3-manifolds with positive scalar curvature. We show isoperimetric inequalities, area growth theorems and diameter bounds for such surfaces. The validity of these inequalities are subject to certain bounds for β. Associated to a positive super-solution ∆ϕ βKϕ, the conformal metric ϕ 2/β g has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove Hölder precompactness and almost rigidity results concerning the main spectral condition. Introduction In this article, we study closed orientable surfaces that satisfy the spectral condition λ 1 (−∆ + βK) λ 0, (1.1) where K denotes the Gaussian curvature of Σ, and β > 0 is a fixed constant. Some equivalent descriptions are Σ |∇ϕ| 2 + βKϕ 2 dA λ Σ ϕ 2 dA, ∀ϕ ∈ C ∞ (Σ), (1.2) and there exists ϕ > 0 such that ∆ϕ (βK − λ)ϕ. (1.3) Clearly these conditions are implied by pointwise curvature bound K β −1 λ. By setting ϕ = 1 in (1.2), we obtain 2πβχ(Σ) λ|Σ|, hence Σ is topologically either a sphere or a torus. In fact, Σ is either a sphere or a flat torus (Lemma 4.2). We assume for the rest of this article that Σ is topologically a sphere, for which interesting results are obtained. One motivation for Condition (1.1) is the study of positive scalar curvature (PSC) and general relativity. Let Σ be a stable minimal surface in a 3-dimensional manifold M with scalar curvature R M R 0 0. The second variation formula gives Σ |∇ Σ ϕ| 2 + 1 2 (2K Σ − R M − |h| 2 )ϕ 2 dA Σ 0 (∀ϕ), and implies (1.2) with β = 1 and λ R 0 /2. The study of stable minimal surfaces is a key step for many important results in positive scalar curvature, such as Geroch's conjecture [34] and the positive mass theorem [35]. As black hole horizons can be mathematically described as minimal surfaces, condition (1.1) is also related to topics in general relativity such as the Bartnik mass [25,26]. For these background studies, we refer the reader to the book of Lee [23]. Many new results in positive scalar curvature are recently proved using Gromov's µ-bubbles [9,16,17,24], and we notice that a stable µ-bubble (with suitable prescribed mean curvature functions) in a 3-manifold with uniformly positive scalar curvature also satisfies (1.1) with β = 1, λ > 0. Extensive studies have been carried out for the non-compact case, see for example [2,7,10,14,18,28,33] and references therein. In higher dimensions there are two analogues of Condition (1.1), in which one replaces Gauss curvature either with scalar curvature [20,25], or with the minimal eigenvalue of the Ricci curvature [5,6,43]. Analogues of Theorem 1.2-1.4 can be obtained under the latter condition. In this paper we focus on the compact case in dimension two, for which the conformal structure is essential to many of our techniques. Geometric inequalities The initial motivation for this work is the question whether (1.1) can be understood as a global positivity condition on curvature. Notice that (1.1) gives no control on the local geometry: in [26] Mantoulidis and Schoen constructed C 1 -small but C 2 -uncontrolled conformal perturbations of the round sphere, such that max(0, −K) dA is arbitrarily large. On the other hand, λ 1 (−∆ + K) > 0 is preserved since it depends C 1 -continuously on the conformal factor. The natural question then becomes asking what to expect globally. We give positive answers in terms of geometric inequalities. It turns out that (1.1) implies isoperimetric inequality, volume comparison, and Bonnet-Myers' theorem (in the case λ > 0). In analytic aspects, it is known that the isoperimetric ratio (resp. Cheeger constant) is related to the optimal constant in the Sobolev inequality (resp. Poincaré inequality). The geometric inequalities we obtain are the following: Theorem 1.2 (isoperimetric inequality). Let Σ be a closed surface that satisfies λ 1 (−∆ + βK) 0. (1) When β > 1 2 , we have IN(Σ) (2β − 1) 2 16β 2 · |Σ| diam(Σ) 2 . (1.4) Moreover, the following local result holds: let γ be a closed loop in Σ, and N ρ be its collar neighborhood of distance ρ. Assume N ρ is compact. If the first Dirichlet eigenvalue bound λ D 1 (−∆ + βK) 0 is satisfied on N ρ , then |N ρ | 4β 2β − 1 ρ |γ|. (2) When 1 2 β > 1 4 , for any ε > 0 we have IN(Σ) C(β, ε) |Σ| diam(Σ) 2 1 4β−1 +ε . (1.5) It is not hard to see that Ch(Σ) 2 Let β > 1 4 and Σ be a closed surface with λ 1 (−∆ + βK) 0. Then there are universal constants C = C(β, |Σ| diam(Σ) 2 ) and C ′ = C ′ (β), such that Cr 2 |B(x, r)| C ′ r 2 for any geodesic ball of radius r. In particular, |Σ| C ′ (β) diam(Σ) 2 . The volume upper bound in Theorem 1.3 was essentially proved in Castillon [7], and the lower bound is a consequence of the isoperimetric inequality. Theorem 1.4 (weak Bonnet-Myers' Theorem). When β > 1 4 and λ > 0, for a complete surface Σ satisfying λ 1 (−∆ + βK) λ we have diam(Σ) 2πβ λ(4β − 1) . (1.6) In particular, such surface must be compact. The method for proving this statement is present in the literature, see [9,17,36,39]. We give two proofs in Appendix A. Gromov-Lawson [18] proved a generalized statement for β = 1 (the argument is easily generalized to β > 1 2 ), using the test function method to be introduced below. The author is informed that G. Carron and T. Richard has recently obtained the same diameter bound. Theorem 1.4 is not true for β 1 4 ; counterexamples include the hyperbolic disk for β < 1 4 (since λ 1 (−∆) = 1 4 ), and the metric g = dr 2 +e 4λr 2 dθ 2 with eigenfunction ϕ = e −λr 2 for β = 1 4 . Remark 1.5. There are two thresholds β = 1 2 and β = 1 4 in the above theorems. The former one was observed by Gromov-Lawson [18] (see Proposition 8.11,Theorem 10.2). This is the critical value of β for which local results can be obtained. By "local" we mean that (1.1) with Dirichlet condition in a domain implies geometric inequalities in the same domain. When β ∈ ( 1 4 , 1 2 ], the main geometric inequalities rely in an essential way on the closedness of Σ, and is no longer local in the above sense. The hyperbolic plane suggests that substantial change occurs to the geometry of (1.1) below β = 1 4 , and here we confirm the sharpness of β = 1 4 as a threshold. The same threshold also appeared in Castillon [7]. It is possible to obtain further results for β 1 4 under additional assumptions such as asymptotic area growth, see for example [12]. The direct approach The main theorems are derived by testing (1.1) with functions of the form ϕ = φ(d(x)), where d is the distance to a properly chosen smooth curve. Using coarea formula and integration by part, one transforms (1.1) to an integral inequality involving φ(s) and L(s) := |{d(x, γ) = s}|, see (2.5). We call this the fundamental equation. With different choice of φ applied to the fundamental equation, we obtain the main geometric inequalities. The core issue in this method is the low regularity of L(s) due to the cut locus, and this was resolved through a careful analysis on the geodesic parallel coordinates [13,19,40,41]. Still, this approach is restricted to dimension since the regularity issue is not resolved in higher dimensions. The fundamental equation is an efficient tool for studying condition (2.1). The earliest applications, to the author's knowledge, were due to Colding-Minicozzi [10] and Gromov-Lawson [18]. For some other results obtained, we refer the reader to Bèrard-Castillon [2], Castillon [7], Munteanu-Sung-Wang [28]. See Section 2 for details in the argument. Perspectives under a conformal change The main novelty of the present paper is the following conformal transformation technique, which provides us with new insights on (1.1). Let ϕ > 0 be any super-solution to −∆+βK, i.e. ∆ϕ βKϕ. Define u = 1 β log ϕ, and we consider the conformal change g = e 2u g. The Gauss curvature of g is computed to be K β| ∇u| 2 . (1.7) Qualitively this is analogous to Schoen-Yau's observation that minimal surfaces in PSC manifolds are Yamabe positive. The new observation is the interplay between the geometry of g and g. The following anti-Harnack inequality a consequence of the isoperimetric inequality for g. In particular it applies to any counterexample of Theorem 1.2(1) when β 1 2 . It is unlikely to obtain Harnack inequalities for ϕ, see Remark 4.6. Theorem 1.6. Let β > 0. Suppose Σ is a closed surface, and ϕ > 0 be any smooth function satisfying ∆ϕ βKϕ. Then max Σ (ϕ) min Σ (ϕ) diam(Σ) · Ch(Σ) −β/2 . Since (Σ, g) is a compact surface with nonnegative curvature, its geometry is controlled by the diameter-area ratio diam( g) 2 /|Σ| g . We prove in Theorem 4.3 a uniform upper bound of this ratio, which is a result of the uniform Sobolev inequality for g. Based on this we show that Fact. g is uniformly bi-Lipschitz equivalent to the round sphere after a suitable normalization. In this way g can be understood as an approximate uniformization of g. The proof relies on Weyl's embedding theorem and therefore requires closedness of Σ. Finally, we obtain uniform Hölder equivalence with the standard metric for metrics satisfying (1.1), as an implication of the theory of strong A ∞ weights [1,11]. See Remark 4.6 for further discussions in this direction. Theorem 1.7. Let β > 1 4 . Given any constants 0 < α < 1 and A 0 > 0, there exists a constant C = C(α, β, A 0 ) such that: for any closed surface Σ with diam(Σ) = 1, |Σ| A 0 and satisfying (1.1), there exists a diffeomorphism Ψ : (Σ, g) → (S 2 , g 0 ), where g 0 is the metric with constant curvature 1, such that C −1 d(x, y) 1/α d g 0 (Ψ(x), Ψ(y)) Cd(x, y) α . Both Theorem 1.3 and 1.7 implies that the space of metrics with diam = 1, Area A 0 and satisfying (1.1) is Gromov-Hausdorff precompact. Finally, we consider convergence of metrics in the situation of almost rigidity. Observe that (1.1) implies |Σ| 4πβλ −1 , and equality case implies that Σ has constant curvature λβ −1 (since ϕ = 1 must be the first eigenfunction). We show that g metrically converges to the round sphere when |Σ| approaches 4πβλ −1 . Theorem 1.8. Let β > 1 4 . For any ε > 0 there is a constant δ = δ(ε, β) > 0 such that: if Σ is topologically a sphere and satisfies λ 1 (−∆ + βK) β, |Σ| 4π − δ, then there exists a diffeomorphism Ψ : Σ → (S 2 , g 0 ) such that max x,y∈Σ d g 0 (Ψ(x), Ψ(y)) − d g (x, y) < ε. We have assumed λ = β in the theorem; the general case only differs by a scaling. The proof consists of two steps. The first step is to show that g (as defined above) bi-Lipschitz converges to the standard metric, and the second step is to show that g converges to g uniformly in the sense of distance function. See Subsection 4.3 for the detailed proof. This paper is organized as follows. In Section 2 we derive the fundamental equation (2.5) and use it to prove Theorem 1.2, 1.3. In Section 3 we prove several auxiliary lemmas to be used in the next section. In Section 4 we introduce the new perspective of conformal change mentioned above. The first subsection contains preparation works and the short proof of Theorem 1.6. The remaining two subsections 4.2, 4.3 are devoted to the proof of Theorem 1.7 and 1.8. In Appendix A we give two proofs of Theorem 1.4. In Appendix B we discuss counterexamples for 1 4 < β 1 2 . Notations. We use Σ to denote a closed surface, and K to denote its Gaussian curvature. We use | · | to denote the area or length of an object, whose dimension is understood from the context. Volume forms are often suppressed when there is no ambiguity. We assume that all surfaces are smooth, connected, closed and orientable (for non-orientable surface, one can pass to its orientable double cover). We use C to denote generic constants, which usually varies from term to term. The dependence of constants is indicated at the beginning of each section. Acknowledgements. The author would like to thank his advisor Hubert Bray for encouragements during the progress of this work, as well as Sven Hirsch for helpful comments on the previous drafts of this paper. Also, the author would like to thank Simon Brendle, Gilles Carron, Demetre Kazaras, Marcus Khuri, Chao Li, Peter McGrath and Alec Payne for helpful conversations. The Method of Test Functions Assume Σ |∇ϕ| 2 + βKϕ 2 dA λ Σ ϕ 2 dA (∀ϕ ∈ C ∞ ) (2.1) on a closed surface Σ. In the first subsection we derive the fundamental equation (2.5) from (2.1), extending the similar arguments in [7,18]. The equation is applicable to separating closed curves on closed surfaces, as suiting our needs for proving isoperimetric inequality. The condition φ ′′ 0 in [2,7] is removed. We present a detailed derivation here for the sake of completeness. In the second subsection, we apply the fundamental equation to prove the main geometric inequalities. Linear functions are sufficient or the case β > 1 2 , whereas functions with polynomial decay are used for the case 1 4 < β 1 2 . Functions similar to the latter were used in [7] to prove Euclidean volume growth for non-compact surfaces. The fundamental equation Let Σ be an oriented closed surface of Euler characterietic χ(Σ), and γ ⊂ Σ be a connected smooth closed curve. For the moment we assume that γ is separating, so that Σ \ γ = Ω + ∪ Ω − . Define the signed distance function d(x) := d(x, γ) (x ∈ Ω + ), − d(x, γ) (x ∈ Ω − ). Let −ρ − := min(d), ρ + := max(d). Define γ(s) := {d = s}, Ω(s) := {0 d < s} (s 0), {s > d 0} (s 0), χ(s) := χ(Ω(s)), (2.2) L(s) := |γ(s)|, K(s) := γ(s) K dl, G(s) := s 0 K(t) dt. (2.3) Finally, define Γ(s) := 2πχ(s) sgn(s) − G(s) + γ κ, where κ is the geodesic curvature of γ. We adopt the sign convention that the unit normal vector always points into Ω + . Whenever γ(s) is smooth, it follows from Gauss-Bonnet formula that Γ(s) = γ(s) κ. Also, L ′ (s) = Γ(s) when |s| is small. However, L(s) in general not continuous because of cut locus (see Figure 1 in [19]). The following partial regularity result can be obtained: 19,40,41]). For any smooth closed curve γ ⊂ M, γ(s) as defined above is piecewise smooth for almost every s. Moreover, L(s) is almost everywhere differentiable and we have Lemma 2.1 ([L ′ (s) Γ(s) for s > 0, L ′ (s) Γ(s) for s < 0. Finally, L(s 2 ) L(s 1 ) + s 2 s 1 L ′ (t) dt for a.e. 0 s 1 < s 2 ρ + and the inequality is reversed when s 1 < s 2 < 0. Remark 2.2. In the case where g and γ are both real analytic, Fiala [13] proved that L(s) is continuous and real analytic except at finitely many times. This lemma implies that L ′ Γ distributionally when s > 0 (and L ′ Γ when s < 0). Now let φ : [−ρ − , ρ + ] → [0, ∞] be a non-negative continuous function, smooth except at s = 0, with φ ′ 0 (∀s > 0) and φ ′ 0 (∀s < 0). We test (2.1) with function φ(d(x)), by coarea formula we have: λ ρ + −ρ − L(s)φ(s) 2 ds ρ + −ρ − L(s)φ ′ (s) 2 + βK(s)φ(s) 2 ds. (2.4) From Gauss-Bonnet formula and integration by part, we obtain ρ + −ρ − Kφ 2 ds = Gφ 2 ρ + ρ − − ρ + −ρ − 2Gφφ ′ = 2π χ(Ω + )φ(ρ + ) 2 + χ(Ω − )φ(ρ − ) 2 + 4π 0 −ρ − χφφ ′ − ρ + 0 χφφ ′ + ρ + −ρ − 2Γφφ ′ ds. Since Ω(s) is connected with more than one boundary components, we have Ω(s) 0 hence the integrals on χφφ ′ is non-positive. For the Γφφ ′ term, we claim that ρ + 0 Γf ds − ρ + 0 Lf ′ ds − L(0)f (0) for any non-positive f ∈ C ∞ ([0, ρ + ]). Since L ′ = Γ is smooth in a neighborhood of zero, it suffices to prove the identity for f supported away from zero. Choose ε ≪ δ ≪ 1 such that L is differentiable at ρ + − δ. Choose a cutoff function ψ with ψ| [0,ρ + −δ−ε] ≡ 0, ψ| [ρ + −δ+ε,ρ + ] ≡ 1, and |ψ ′ | 4ε −1 . Decompose f = f ψ+f (1−ψ). Note that Γf ψ ds → 0 as δ → 0. Since L ′ Γ weakly, we have ρ + 0 Γf (1 − ψ) ds − ρ + 0 L f (1 − ψ) ′ ds = − ρ + 0 Lf ′ ds + o δ (1) + ρ + 0 Lf ψ ′ ds. Finally, Lf ψ ′ ds → L(ρ + − δ)f (ρ + − δ) as ε → 0, and the latter is non-positive. This proves our claim. For the same reason, we have 0 −ρ − Γf ds − ρ + 0 Lf ′ ds + L(0)f (0) for all f 0. Applying f = 2φφ ′ and combining all the inequalities, we obtain (2β − 1) ρ + −ρ − L(φ ′ ) 2 ds + 2β ρ + −ρ − Lφφ ′′ ds + λ ρ + −ρ − Lφ 2 ds 2πβ χ(Ω + )φ(ρ + ) 2 + χ(Ω − )φ(ρ − ) 2 + 2β|γ|φ(0) φ ′ − (0) − φ ′ + (0) . (2.5) We summarize all the conditions into the following lemma: Lemma 2.3. Assume that Σ satisfies (2.1), and γ ⊂ Σ is a connected separating smooth curve. Suppose φ : [−ρ − , ρ + ] → [0, ∞] is Derivation of geometric inequalities Proof of Theorem 1.2. Suppose Σ = Ω + ∪ Ω − , with common smooth boundary γ = ∂Ω + = ∂Ω − . It suffices to prove the theorem when Ω − is connected. To see this, suppose Ω − = Ω 1 ∪ Ω 2 as a disjoint union. If both |Ω 1 | and |Ω 2 | are less than 1 2 |Σ|, then by induction on the number of connected components, we have |∂Ω i | |Ω i | s C s iso where C iso = IN(Σ) or Ch(Σ), and s = 1/2 or 1. Hence |∂Ω| |Ω| s C s iso , which proves the isoperimetric inequality for Ω. If |Ω 1 | 1 2 |Σ|, we have |∂Ω| |∂Ω 1 | C s iso |Σ \ Ω 1 | s C s iso |Σ \ Ω| s . Therefore, we assume that both Ω + and Ω − are connected, thus γ is connected (since π 1 (Σ) = 0). To prove (1), we test equation (2.5) with φ(s) = (ρ − + s)/ρ − (s 0), (ρ + − s)/ρ + (s 0). We obtain (2β − 1) |Ω − | (ρ − ) 2 + |Ω + | (ρ + ) 2 2β |γ| 1 ρ − + 1 ρ + We may assume ρ − ρ + . The first part of Theorem 1.2(1) follows from (2β − 1) 2 |Ω − | (ρ − ) 2 · |Σ| diam(Σ) 2 16β 2 |γ| 2 (ρ − ) 2 . For the second part of (1), we choose ρ + = ρ − = ρ, and note that φ is indeed a test function for the Dirichlet eigenvalue condition (i.e. φ vanishes on ∂N ρ ). To prove (2), we test with the following function: φ(s) =          (1 + σ)ρ − + s p (1 + σ) p (ρ − ) p (s 0), (1 + σ)ρ + − s p (1 + σ) p (ρ + ) p (s 0), (2.6) where the coefficients σ > 0, p > 1 are to be chosen later. Equation (2.5) now gives I − + I + 4πβ σ 2p (1 + σ) 2p + 2pβ 1 + σ |γ| 1 ρ − + 1 ρ + ,(2.7) where I − := (4β − 1)p 2 − 2βp 1 (1 + σ) 2p (ρ − ) 2p 0 −ρ − L (1 + σ)ρ − + s 2p−2 ds, (2.8) and I + is defined analogously. The coefficient (4β − 1)p 2 − 2βp must be positive, thus p > 2β 4β−1 . We choose p = 2β+ε 4β−1 , ε > 0. Now (2.7) (2.8) implies C 1 σ 2p−2 (1 + σ) 2p |Ω − | (ρ − ) 2 + |Ω + | (ρ + ) 2 4πβ σ 2p (1 + σ) 2p + 2pβ 1 + σ |γ| 1 ρ − + 1 ρ + , (2.9) where C 1 = C 1 (β, ε) is a constant depending only on β, ε. For convenience, we denote Z := |Ω − | (ρ − ) 2 + |Ω + | (ρ + ) 2 . Choose σ = C 1 8πβ Z, for this choice we have 2pβ 1 + σ |γ| 1 ρ − + 1 ρ + C 1 2 σ 2p−2 (1 + σ) 2p Z. (2.10) Without loss of generality, assume ρ − ρ + . Then we have |γ| ρ − C(β, ε) σ 2p−1 (1 + σ) 2p−1 √ Z C(β, ε) min 1, σ 2p−1 |Ω − | ρ − , or equivalently, |γ| 2 |Ω − | C(β, ε) min 1, Z 2p−1 C(β, ε) min 1, ( |Σ| diam(Σ) 2 ) 1+2ε 4β−1 . (2.11) Combined with total volume upper bound to be proved below, we obtain the strongest form (1.5). Proof of Theorem 1.3. The volume upper bound was essentially proved in [7, Proposition 2.2]. Here we include a proof for the reader's convenience. Let ε ≪ 1 such that the geodesic sphere γ = ∂B(x, ε) is smooth. Consider the test function φ(s) =        1 (s 0) (1 − s 2r ) p (0 s 2r) 0 (s 2r) Applying (2.5) to γ, similar calculation yields C 1 (2r) 2p 2r 0 L(2r − s) 2p−2 ds 4πβ + 2pβ|γ|r −1 . The left hand side is greater than Cr −2 |B(x, r + ε)|. Letting ε → 0 we obtain |B(x, r)| Cr 2 . In particular, |Σ| C diam(Σ) 2 . This proves the upper bound and shows that it depends only on β. The volume lower bound is a consequence of the isoperimetric inequality. let r 1 = sup{r : B(x, r) 1 2 |Σ|}. For any r r 1 such that ∂B(x, r) is piecewise smooth, Theorem 1.2 gives |∂B(x, r)| C(β, |Σ| diam(Σ) 2 ) |B(x, r)|. By coarea formula and gronwall's inequality we obtain |B(x, r)| C(β, |Σ| diam(Σ) 2 )r 2 . For r r 1 we have |∂B(x, r)| 1 2 |Σ| 1 2 |Σ| diam(Σ) 2 · r 2 . This completes the proof. Remark 2.4. In the non-compact case, the proof shows that for any δ > 0, |B(x, r)|/r 2 is uniformly bounded from above if λ D 1 (−∆ + βK) 0 in B(x, (1 + δ)r). When β > 1 2 , the linear function φ(s) = 1 − s/r can be used to show that |B(x, r)| Cr 2 whenever the metric ball B(x, r) satisfies λ D 1 (−∆ + βK) 0. 3 Auxiliary Lemmas 3.1 Collection of facts in convex geometry Lemma 3.1. (1) (Weyl's embedding theorem) Given any metric (S 2 , g) with curvature K > 0, there exists an isometric embedding into R 3 , unique up to rigid motion. The image of the embedding is the boundary of a strictly convex set. (2) Suppose Ω 1 ⊂ Ω 2 are two domains in R n , with Ω 1 smooth convex and Ω 2 piecewise smooth. Then the orthogonal projection from ∂Ω 2 to ∂Ω 1 is 1-Lipschitz. (3) For a closed surface Σ with curvature K 0, we have Ch(Σ) 1/ diam(Σ) and IN(Σ) |Σ|/ diam(Σ) 2 . Proof. (1) is a classical theorem, see Nirenberg [29] and Pogorelov [32]. (2) is proved by computing the differential of the projection map. Let Φ : ∂Ω 2 → ∂Ω 1 be the (well-defined) orthogonal projection map. Denote by N, A the outer unit normal vector and second fundamental form of ∂Ω 1 . Let h(x) = d(x, Φ(x)) 0. For any x ∈ ∂Ω 2 we have x − Φ(x) = h(x)N(Φ(x)). Differentiating this identity at a tangent vector v, we obtain v − dΦ x (v) = dh x (v)N(Φ(x)) + h(x)dN Φ(x) (dΦ x (v)). Taking inner product with dΦ x (v) we obtain |v| · |dΦ x (v)| − |dΦ x (v)| 2 v − dΦ x (v) · dΦ x (v) = h(x)A(dΦ x (v) , dΦ x (v)) 0. This shows dΦ is non-expanding. (3) Let Ω be any domain. An inequality of Burago-Zalgaller (see [4,30]) states that ρ|∂Ω| |Ω| + (π − 1 2 Ω K)ρ 2 , where ρ is the radius of the largest metric ball contained in Ω. By possibly switching between Ω and Ω c , we may assume π − 1 2 Ω K 0, hence |Ω| ρ|∂Ω|. This gives the Cheeger constant bound. By relative volume comparison we have |∂Ω| 2 |Ω| · |Ω| ρ 2 |Ω| · |Σ| diam(Σ) 2 , which gives the isoperimetric ratio bound. A precise form of Moser-Trudinger's inequality The classical Moser-Trudinger Sobolev inequality [27] states that for a domain Ω ⊂ R 2 and a function u ∈ H 1 0 (Ω) we have Ω exp 4πu 2 Ω |∇u| 2 C|Ω| (3.1) for a universal constant C. For general domains in smooth surfaces, (3.1) continue to hold with the same critical exponent 4π, while the constant C depends on the domain under consideration. When the surface has a conic singularity of cone angle θ < 2π, the critical exponent becomes 2θ [8,42]. We concern the case where only a lower bound on the isoperimetric ratio is known. In this case the round cone represents the worst control, as seen by Pólya-Szegö symmetrization. The following lemma is useful for later applications. Theorem 3.2. Let (Ω, g) be a smooth domain with non-empty boundary, such that ID(Ω) ξ. Then for any function u ∈ H 1 0 (Ω) we have Ω exp ξu 2 Ω |∇u| 2 C(ξ) · |Ω|, (3.2) where C(ξ) means a constant depending on ξ. Proof. We can replace u by |u| and therefore assume u 0. Since (3.2) is scale-invariant, we may assume |Ω| = 1. We apply a Pólya-Szegö symmetrization procedure, comparing Ω to a model space that has smaller isoperimetric ratio. Let (r, θ) be the polar coordinates on a disk D 2 . Equip D 2 with the cone metric g 0 = dr 2 + ε 2 r 2 dθ 2 (0 r L, 0 θ 2π), where ε is determined by requiring |∂Ω| 2 = ξ|Ω| for concentric cones Ω, and L is determined such that |D 2 | g 0 = 1. Consider the radial function u 0 (r) ∈ H 1 0 (D 2 ) uniquely determined by du 0 /dr 0 and the condition {y ∈ D 2 : u 0 (y) > t} g 0 = {x ∈ Σ : u(x) > t} g , ∀t 0. From isoperimetric ratio comparison we have {y ∈ D 2 : u 0 (y) = t} g 0 {x ∈ Σ : u(x) = t} g . The standard argument (see for example [37, Section III.1]), using rearrangement and the coarea formula, gives D 2 exp(pu 2 0 ) dA 0 = Ω exp(pu 2 ) dA and D 2 |∇ g 0 u 0 | 2 dA 0 Ω |∇u| 2 dA. The desired result follows from the Moser-Trudinger inequality with conic singularity on (D 2 , g 0 ) and u 0 . Proof. Combine Theorem 3.2 with Young's inequality 2u ξu 2 |∇u| 2 + 1 ξ |∇u| 2 . Corollary 3.3 also applies to functions u that are non-positive on the boundary, by replacing u with max(u, 0). Similar inequality also holds for closed manifold with Neumann isoperimetric bounds. Hence Ω e pu C(ξ, p) |Ω| exp 1 ξ Σ |∇u| 2 . (3.4) Proof. After a perturbation, we may assume that there exists a regular value b such that {u b} = {u b} = 1 2 |Σ|. Assume without loss of generality that b 0. By Chebyshev and Cheeger's inequality, 1 2 |Σ| = {u b} b −2 Σ u 2 4b −2 Ch(Σ) −2 Σ |∇u| 2 2b −2 |Σ| · IN(Σ) −1 Σ |∇u| 2 . Hence b 2 4ξ −1 |∇u| 2 . Apply Theorem 3.2 to Ω = {u b} (note that ID(Ω) IN(Σ)): Ω exp ξu 2 |∇u| 2 Ω exp ξ(u − b) 2 + ξb 2 |∇u| 2 C(ξ) |Ω|. The same conclusion holds for {u b}, therefore (3.3) holds. (3.4) follows from (3.3) and Young's inequality. The Perspective under Conformal Change Preparation and first consequences In this section we assume that Σ is a closed surface satisfying (1.1) and is topologically a sphere. Let ϕ = e βu satisfy ∆ϕ βKϕ, with β > 1 4 . We have e βu (β∆u + β 2 |∇u| 2 ) = ∆e βu βKe βu . (4.1) Consider the conformal change g = e 2u g. We may normalize ϕ so that |Σ| g = Σ e 2u dA = 1. The Gauss curvature of g is computed to be K = e −2u (K − ∆u) e −2u β|∇u| 2 = β| ∇u| 2 ,(4.2) where we denote | ∇u| 2 = | ∇u| 2 g for brevity. The two basic observations from (4.2) are K 0 and | ∇u| 2 d A 4πβ −1 . The former gives geometric control on g, while the latter provides strong relation between g and g. Proof of Theorem 1.6. Suppose max Σ (ϕ) = A min Σ (ϕ). Applying Burago-Zalgaller's isoperimetric inequality to (Σ, g), we have inf Ω⊂Σ ∂Ω ϕ 1/β dl min Ω ϕ 2/β dA, Ω c ϕ 2/β dA 1 diam(Σ, g) . Note that the left hand side is Ch(Σ)A 2/β (max Σ ϕ) −1/β , while the right hand side is (max Σ ϕ) −1/β / diam(Σ). Hence A 2/β 1 Ch(Σ) · diam(Σ) . This proves the theorem. Remark 4.1. We do not expect global semi-Harnack inequalities for ϕ > 0 satisfying ∆ϕ βKϕ, for the following reason. First we observe that no uniform bound on sup(ϕ) can be found. Let ϕ > 0 uniquely solve ∆ϕ = βϕ − δ x on the round sphere. Applying the heat operator P ε , we obtain a family of smooth functions ϕ ε with ∆ϕ ε βϕ ε . Next, we note the following interesting relation: ∆ g ϕ βK g ϕ ⇒ ∆ g ′ ϕ −1 βK g ′ ϕ −1 , where g ′ = ϕ 4/β g. This can be seen by reversing the implication from (4.1) to (4.2) with u → −u. Thus we have ∆ gε ϕ −1 ε βK gε ϕ −1 ε where g ε = ϕ 4/β ε g. Note that inf(ϕ −1 ε ) → 0 as ε → 0, whereas the diameter and area for g ε can be verified to have uniform controls. This shows that uniform lower bounds on inf(ϕ) are unlikely to be obtained. The rest of this section relies on Weyl's embedding theorem for (Σ, g), hence requires K > 0. However, (4.2) only gives K 0. With the following lemma available, we may assume after a C 2 approximation that (4.1) is a strict inequality, hence K > β| ∇u| 2 . Lemma 4.2. Let (Σ, g) be a closed surface with λ 1 (−∆ + βK) = 0, and let ϕ be the first eigenfunction. If Σ is not a flat torus, then for any ε > 0 there exists another metric g ′ such that ||g ′ − g|| C 2 (g) < ε with λ 1 (−∆ g ′ + βK ′ ) > 0. Furthermore, the first eigenfunction ϕ ′ satisfies ||ϕ ′ − ϕ|| C 2 (g) < ε. Proof. Consider a smooth family of metrics g(t) = e 2f (t) g, f (0) = 0, with initial variation df dt | t=0 = h. By [26, Lemma A.1], the first eigenfunctions ϕ t of −∆ g(t) + βK g(t) constitute a smooth family. We may normalize so that ||ϕ t || L 2 (gt) = 1. We compute dλ 1 dt t=0 = d dt t=0 Σ |∇ t ϕ t | 2 + βK t ϕ 2 t dA t = d dt t=0 Σ |∇ t ϕ| 2 + βK t ϕ 2 dA t = −β Σ ∆h · ϕ 2 dA (4.3) If (4.3) is zero for all h, then ∆(ϕ 2 ) = 0 ⇒ ϕ is constant, which implies that Σ is a flat torus. Hence (4.3) is nonzero for some h. We may change the sign of h to make dλ 1 /dt positive. The perturbed metric g(δ) and eigenfunction ϕ δ satisfies the theorem statement for small δ. Bi-Lipschitz equivalence with the round sphere Since (Σ, g) is positively curved, its geometry is controlled by the homogeneous diameterarea ratio. In this subsection we bound the diameter of g (recall that the area is normalized to be 1), and show that g is bi-Lipschitz equivalent to the round sphere, then derive Theorem 1.7 as a corollary. The generic constants in this subsection depend only on β and |Σ| g / diam(Σ, g) 2 , and may vary from line to line. In addition with Cauchy-Schwarz inequality e 2u · e −2u |Σ| 2 and the normalization e 2u dA = 1, we obtain e pu dA C(β, p)|Σ| 1−p/2 for all p > 1. Denote D = diam(Σ, g), A = |Σ| g . Let x, y ∈ Σ such that d g (x, y) = D. By coarea formula, there exists s ∈ [ 1 4 D, 3 4 D] such that γ = ∂B(x, s) is piecewise smooth and satisfies |γ| 4A/D and γ e 2u 4/D. Then Cauchy-Schwarz inequality gives |γ| g = γ e u dl 4A 1/2 /D. Let γ 1 be the unique connected component of γ that separates x and y (whose existence follows from π 1 (Σ) = 0), and denote Σ \ γ 1 = U ∪ U ′ . Assume x ∈ U, y ∈ U ′ . Theorem 1. The g-distance function d(y) = d (U, g) (y, γ 1 ) is Lipschitz, vanishing on ∂U = γ 1 , and satisfies |∇ d| g = e u almost everywhere. The L n/(n−1) Sobolev inequality gives || d|| L 2 (U,g) ID(U) −1/2 ||e u || L 1 (U,g) . Following the iteration argument in [15,Theorem 7.10], a uniform L n/(n−1) Sobolev inequality gives rise to a uniform L ∞ Sobolev inequality || d|| L ∞ C(p) ID(U) −1/2 |Σ| 1/2−1/p ||e u || L p (U,g) C, ∀p > 2. The same conclusion holds for d ′ = d (U ′ , g) (−, γ 1 ). Finally, the theorem follows from diam(Σ, g) max( d) + max( d ′ ) + |γ 1 | g . Next, we construct bi-Lipschitz maps from (Σ, g) to the round sphere. By Weyl's embedding theorem and Lemma 4.2, we assume that (Σ, g) is the boundary of a convex body in R 3 . We make the metric g implicit when there is no ambiguity. The following lemma is elementary. Lemma 4.4. Let Σ ⊂ R 3 be a convex surface, satisfying d 1 |x| d 2 for any x ∈ Σ. Then Φ : x → x |x| is a bi-Lipschitz map to the round sphere, with Lipschitz norms ||Φ|| Lip 1/d 1 , ||Φ −1 || Lip d 2 2 /d 1 . Proof. We compute dΦ x (v) = |x| 2 v − (x · v)x |x| 3 , v ∈ T x Σ, and |dΦ x (v)| 2 = |x| 2 − |x · v| 2 |x| 4 , |v| = 1. Thus |dΦ| where the embedding is isometric. First we consider the case that W 1 4πD . Let S be a largest inscribed sphere of Σ, and r be the radius of S. The width-inradius inequality gives D 2 r W 4 . A proof of the two-dimensional analogue can be found in [22, p.215], while the proof there can be directly generalized to three dimensions. Now Lemma 4.4 gives the desired bi-Lipschitz map. Next we assume W 1 4πD . We will replace the nearly flat top and bottom faces of Σ by two cones, thus increasing the inradius. We may assume that Σ is already embedded in the region {−W/2 z W/2} ⊂ R 3 . Let Q be the projection image of Σ onto the xy-plane, which is a convex planar domain. Thus Σ ⊂ Q × [−W/2, W/2]. We have 1 = |Σ| ∂(Q × [−W/2, W/2]) 2|Q| + 2W |∂Q| 2|Q| + 2πDW Thus |Q| 1 4 . Let r 0 be the inradius of Q (i.e. the radius of the largest disk contained in Q). From [38] we know r 0 |Q| 2 diam(Q) 1 8D . After a translation, let B := B(0, r 0 ) ⊂ Q ⊂ R 2 be an inscribed disk in Q. Let p ± = (0, 0, ±1). Let Λ + be the convex cone obtained by taking the union of all tangent line segments of Σ emanating from p + . We claim that the normal unit vector N Λ of Λ + satisfies |N Λ · ∂ z | r 0 √ r 2 0 +4 . To show this, let P be any plane tangent to Λ + . The intersection line l = P ∩ {z = −1} satisfies d(l, p − ) r 0 by convexity. Our claim then follows from a simple computation. We can mollify the vertex of Λ + so that it is smooth and satisfies |N Λ · ∂ z | 1 2 r 0 √ r 2 0 +4 everywhere. Let γ + be the set at which Λ + is tangent to Σ. Now γ + encloses a region Ω + in Σ. Let Q + be the projection image of Ω + onto the xy-plane. Consider the orthogonal projection map F + 1 : Ω + → Q + , F + 2 : Λ + → Q + (which are both bijective). It is not hard to prove the following general fact: if G is the projection map from a surface S to the xy-plane, then ||G −1 || Lip sup S |N S · ∂ z | −1 , where N S is the unit normal vector of S. By our claim above, the map F + = (F + 2 ) −1 • F + 1 : Ω + → Λ + has controlled bi-Lipschitz norms. Analogously, we can define a bi-Lipschitz map F − : Ω − → Λ − Let Ω m = Σ \ (Ω + ∪ Ω − ). Combining F + with F − we obtain a C 1,1 map F from Σ to Σ ′ = Ω m ∪ Λ + ∪ Λ − , with ||F || Lip C, ||F −1 || Lip C. It is not hard to see diam(Σ ′ ) < diam(Σ) + 4 and |x| > min(1/4, r 0 /4) on Σ ′ . From Lemma 4.4 we obtain a C 1,1 map Φ 0 : Σ → S 2 with controlled bi-Lipschitz norms. Finally, we mollify Φ 0 to obtain a smooth map Φ that C 1 -approximates Φ 0 and is therefore bi-Lipschitz. Proof of Theorem 1.7. We have obtained three metrics on Σ: the original metric g, the conformal metric g = e 2u g, and the pulled-back round metric Φ * g 0 which satisfies C −1 1 g Φ * g 0 C 1 g by Proposition 4.5. The constant C 1 depends on diam( g), hence depends on the lower bound A 0 of |Σ| g , under the normalizing assumptions of the theorem. Consider the metric g 1 = e −2u 1 g 0 where u 1 = u • Φ −1 . We have C −1 1 (Φ −1 ) * g g 1 C 1 (Φ −1 ) * g hence g 1 is bi-Lipschitz equivalent to g. It remains to prove that g 1 and g 0 are uniformly bi-Hölder equivalent. This directly follows from [1, Proposition 6.3] (whose notations can be found at the beginning of Section 4A). By (4.2) we have | ∇u| 2 d A 4πβ −1 , hence |∇ 0 u 1 | 2 dA 0 4πβ −1 C 2 1 is uniformly bounded. This verifies the assumption in the cited theorem. Moreover, C −1 1 A 0 C −1 1 |Σ| g |Σ| g 1 C 1 |Σ| g C by our assumptions and volume comparison, hence [1, Proposition 6.3] gives uniform bi-Hölder equivalence between g 0 and g 1 . Remark 4.6. Underlying the bi-Hölder equivalence result is the deep theory of strong A ∞ weights, initially introduced by David and Semmes [11] and developed recently by Aldana, Carron and Tapie [1] for closed manifolds. A density function e nf on an n-manifold M is called a strong A ∞ weight if the metric g ′ = e 2f g satisfies (1) volume-doubling condition: |B g (x, 2r)| g ′ C|B g (x, r)| g ′ , (2) volume-distance compatibility: y) n , as defined in [1,Definition 4.1]. The second condition seems somewhat unnatural in the sense that g-metric balls appear in the central term of the inequality. However, it is true that g ′ satisfies Euclidean volume growth, i.e. C −1 d g ′ (x, y) n B g (x, d g (x, y)) g ′ Cd g ′ (x,(C ′ ) −1 d g ′ (x, y) n B g ′ (x, d g ′ (x, y)) g ′ (C ′ )d g ′ (x, y) n . Moreover, it is shown by David and Semmes [11] that g ′ satisfies a uniform isoperimetric inequality and Sobolev inequality. It is an important fact that functions in the critical Sobolev space W k,n/k are strong A ∞ weights, see [3] for the case of R n and [1] for the case of closed manifolds. In our situation u 1 ∈ W 1,2 , therefore e −2u 1 is a strong A ∞ weight for g 0 . David and Semmes' isoperimetric inequality thus exactly corresponds to Theorem 1.2. However, this does not constitute an independent proof of theorem 1.2 since the uniform control on g is based on them. The case of almost rigidity In this subsection we prove Theorem 1.8. The Hölder constants originating from strong A ∞ weights are not sharp, so the proof requires more efforts. By assumption, the first eigenfunction ϕ > 0 satisfies ∆ϕ (βK − β)ϕ. From (1.1) one obtains |Σ| g 4π. The same conformal transformation in subsection 4.1 now gives K e −2u + β| ∇u| 2 . (4.5) This time we adopt the normalization |Σ| g = Σ e 2u dA = 4π for the purpose of matching the areas of g and g. Integrating (4.5) against the volume form of g and using the condition |Σ| g 4π − δ, we obtain Σ | ∇u| 2 d A β −1 δ. Notation. In this subsection, we use o δ (1) to denote universal constants that depends only on β, δ and converges to zero when δ → 0. For example, 4π−o δ (1) e −2u d A 4π. The generic constants C depend only on β unless explicitly indicated. Therefore, Σ exp δ −1/2 |u − u| d A Σ exp (u − u) 2 Cδ + 1 4 C d A C. Set S 1 = {|u − u| δ 1/4 } and S 2 = Σ \ S 1 . We have 4π − δ Σ e −2u d A = S 1 e −2u d A + S 2 e −2u d A 4πe −2u+2δ 1/4 + e −2u S 2 e |u−u|/ √ δ d A · exp (2 − δ −1/2 )δ 1/4 = e −2u (4π + o δ (1)) Hence u −o δ (1), and this shows |u| = o δ (1). Finally, for any fixed η > 0 we have by Chebyshev's inequality {|u| η} · e η/ √ δ e |u|/ √ δ Σ exp δ −1/2 |u − u| d A. (4.6) This shows |{|u| η}| → 0 when δ → 0. Proof. All the integrals in this proof are with respect to g, so we omit the area form for brevity. Fix η ≪ 1. By Lemma 4.7, for δ sufficiently small we have |{K 1 − η}| η. Denote S 1 = {K 1 − η}, S 2 = {K 1 + √ η}, S 3 = Σ \ (S 1 ∪ S 2 ). By Gauss-Bonnet formula we have 4π = S 1 K + S 2 K + S 3 K S 2 K + (4π − η − |S 2 |)(1 − η) S 2 K + (4π − η − S 2 K 1 + √ η )(1 − η) . Hence S 2 K C √ η. We first translate Σ so that the origin is contained in the interior. Let N be the outer unit normal of Σ, and H be the mean curvature. Let V be the volume enclosed by Σ. Minkowski' inequality Σ H 4 π|Σ| g and integration formula Σ K(x · N) = 1 2 Σ H (see [21]; x is the position vector) yields 4π 1 2 Σ H = Σ K(x · N) = S 2 K(x · N) + Σ\S 2 K(x · N) C √ η · diam(Σ, g) + (1 + √ η) Σ (x · N) = C √ η + 3(1 + √ η)V, where the last line follows from divergence formula. Hence V 4 3 π − C √ η. (4.7) Let ρ be the inradius of Σ (i.e. the radius of the largest ball enclosed by Σ). By a Bonnesen-type isoperimetric inequality, see formula (114) in [31], we have |Σ| 4πρ 2 3/2 − V 4 3 πρ 3 |Σ| 4πρ 2 1/2 − 1 3 ⇒ (1 − ρ) 3 1 − V 4 3 π . Combined with (4.7) this gives ρ 1 − Cη 1/6 . Translate Σ so that min x∈Σ |x| 1 − Cη 1/6 . It remains to show max x∈Σ |x| 1 + o η (1). Suppose x ∈ Σ with |x| R. Let Ω 1 be the solid ball with radius ρ, and Ω 2 be the cone with vertex x and is tangent to Ω 1 . Their union Ω = Ω 1 ∪ Ω 2 is a convex body. The orthogonal projection Σ → ∂Ω is 1-Lipschitz and does not increase area. Hence 4π = |Σ| |∂Ω| = ρ 2 2π(1 + cos θ) + π sin θ tan θ (1 − Cη 1/6 ) 2 4π (1 + sin 4 (θ/2) cos θ ), where θ = arccos(ρ/R). Therefore R 1 + o η (1) when η → 0. Now the proposition follows from Lemma 4.4. Proof of Theorem 1.8. Similar to the proof of Theorem 1.7, we let u 1 = u • Φ −1 and g 1 = e −2u 1 g 0 . Proposition 4.8 implies that g 1 is 1 + o δ (1) -bi-Lipschitz equivalent to g. The remaining work is showing |d g 1 (x, y) − d g 0 (x, y)| = o δ (1) uniformly for all x = y. Note that Σ |∇ 0 u 1 | 2 dA 0 Cδ and | e −2u 1 dA 0 −4π| = o δ (1) by Lipschitz equivalence between g and g 0 . Arguing as in Theorem 4.3, we have Σ e −pu 1 dA 0 C(β, p). The function d(y) = d g 1 (x, y) is Lipschitz and satisfies |∇ 0 d| g 0 = e −u 1 . By Morrey's inequality we have d g 1 (x, y) Cd g 0 (x, y) α , ∀ fixed 0 < α < 1. (4.8) We first show the more complicated direction d g 1 (x, y) d g 0 (x, y) − o δ (1). Suppose that d g 1 (x, y) d g 0 (x, y) − s for some s > 0 independent of δ and for some x = y. Let γ be a shortest g 1 -geodesic from x to y. Denote d 0 = d g 0 (x, y), a = d 0 −s d 0 −s/2 . We claim that we can assume d 0 2 3 π. Suppose d 0 2 3 π, we embed Σ as the unit sphere so that x, y has the same longitude and opposite latitude (= ±d 0 /2). Choose any z ∈ γ that lies on the equator, thus d g 0 (x, z) = d g 0 (y, z) π − d 0 /2 2 3 π. Moreover, d g 1 (x, z) + d g 1 (y, z) = d g 1 (x, y) d g 0 (x, z) + d g 0 (y, z) − s. Without loss of generality we assume d g 1 (x, z) d g 0 (x, z) − s/2. Then we can proceed with the pair of points (x, z) and new constant s/2. Let U = {z ∈ γ : e −u 1 (z) > a}, V = γ \ U, hence |U| g 0 d 0 − s/2. We would like to apply the Moser-Trudinger inequality to e −2u 1 with Dirichlet condition on V . Therefore, we need a corresponding isoperimetric inequality: We postpone its proof to the end of this section. Let b = min V (u 1 ), so e −2b a 2 π−s π−s/2 2 . Apply Theorem 3.2 to v = max b−u 1 √ δ , 0 , which satisfies Σ |∇ 0 v| 2 dA 0 C: Σ exp b − u 1 √ δ dA 0 Σ e v dA 0 Σ exp ξv 2 |∇ 0 v| 2 + 1 4ξ Σ |∇ 0 v| 2 dA 0 C, where ξ is the uniform lower bound of ID(Σ\ V, g 0 ). Let (1)). S 1 = {u 1 b−δ 1/4 }, S 2 = Σ\ S 1 . We have Σ e −2u 1 dA 0 e −2b |S 1 |e 2δ 1/4 + S 2 e 2(b−u 1 ) dA 0 π − s π − s/2 2 4πe 2δ 1/4 + exp (2 − δ −1/2 )δ 1/4 S 2 e (b−u 1 )/ √ δ dA 0 π − s π − s/2 2 4π(1 + o δThis contradicts |Σ| g 1 4π − o δ (1) when δ is sufficiently small. Therefore d g 1 (x, y) d g 0 (x, y) − o δ (1). Next we prove that d g 1 (x, y) d g 0 (x, y) + o δ (1). Assume otherwise that d g 1 (x, y) d g 0 (x, y) + s for some s > 0 independent of δ. By (4.8) this implies d g 0 (x, y) C(β, s). Note that Σ |∇ 0 u 1 | dA 0 C √ δ. By coarea formula, there exists a regular value s ∈ [u − 2δ 1/4 , u − δ 1/4 ] such that {u = s} Cδ 1/4 . The argument in (4.6) implies {u s} = o δ (1). Combining these two conditions, any geodesic segment contained in {u s} must have length o δ (1). Let γ be the shortest g 0 -geodesic joining x and y. Consider the parallel curves γ t = exp γ (tν), where ν is the nuit normal vector field of γ with any orientation. By coarea formula, there exists a value t 0 ∈ [−Cδ 1/4 , Cδ 1/4 ] such that γ t 0 ∩ {u = s} = ∅. Hence γ t 0 ⊂ {u s} when δ is sufficiently small. Let x ′ , y ′ be the endpoints of γ t 0 . We now have d g 1 (x, y) d g 1 (x, x ′ ) + d g 1 (y, y ′ ) + |γ t 0 | g 1 2Cδ α/4 + e −2s |γ t 0 | g 0 d g 0 (x, y) + o δ (1), which is a contradiction. This completes the proof. Proof of Lemma 4.9. We may assume that Ω is connected, as seen in the proof of Theorem 1.2. Let {φ, θ} be the spherical coordinate system. After a rotation we may assume x, y have coordinates (−d 0 /2, 0), (d 0 /2, 0). Let P φ : Σ → [−π/2, π/2] map a point to its latitude, and define I = P φ (V ) ∩ [−d 0 /2, d 0 /2]. Since P z is a surjective from γ to [−d 0 /2, d 0 /2], we have |I| d 0 − |P φ (U)| d 0 − |U| g 0 s. Let J = P φ (∂Ω). If J ⊃ I, then |∂Ω| g 0 |J| |I| s s √ 4π |Ω| 1/2 g 0 . Now suppose J I, then there exists φ 1 ∈ I with ∂Ω ∩ {φ = φ 1 } = ∅. Therefore, we have either {φ = φ 1 } ⊂ Ω or {φ = φ 1 } ∩ Ω = ∅. The first case cannot happen since we have assumed Ω ∩ V = ∅, hence the second case must hold. Now, Ω is contained in either the spherical cap {φ > φ 1 } or {φ < φ 1 }. Since φ 1 ⊂ [−π/3, π/3], the conclusion follows from the usual spherical isoperimetric inequality. A Two Proofs of weak Bonnet-Myers' Theorem In this section we present two proofs of Theorem 1.4, using the methods of weighted geodesics and weighted µ-bubbles. Let u > 0 satisfies ∆u β (βK − λ)u β , (A.1) equivalently, ∆u (K − λβ −1 )u + (1 − β)u −1 |∇u| 2 . (A.2) Note that the function u defined here is different than in Section 4. A.1 Proof using weighted geodesics The argument here is similar to the ones in Schoen-Yau [36] and Shen-Ye [39]. Let p, q ∈ Σ be two points with the largest distance. If Σ is non-compact, then choose p, q with large enough distance to obtain a contradiction below. Let γ : [0, L] → Σ, γ(0) = p, γ(L) = q, be a minimizer of the weighted length functional γ u dl. Parametrize γ with unit speed. The second variational formula gives the following inequality: 0 γ u(ϕ ′ ) 2 + (∆u − u ′′ )ϕ 2 − Kuϕ 2 − u −1 (u N ) 2 ϕ 2 dl (A.3) for any function ϕ : ϕ(0) = ϕ(L) = 0, where we denote f ′ := ∂f /∂γ ′ for a function f , and denote u N := ∂u/∂N. Substituting ϕ = u −1/2 ψ into (A.3) and using equation (A.2), we obtain 0 γ u − 1 2 u −3/2 u ′ ψ + u −1/2 ψ ′ 2 + u ′ (−u −2 u ′ ψ 2 + 2u −1 ψψ ′ ) − λβ −1 ψ 2 + (1 − β)u −2 (u ′ ) 2 dl γ ( 1 4 − β)u −2 (u ′ ) 2 ψ 2 + (ψ ′ ) 2 + u −1 u ′ ψψ ′ − λβ −1 ψ 2 dl γ 1 + 1 4 (β − 1 4 ) −1 (ψ ′ ) 2 − λβ −1 ψ 2 dl. Theorem 1.4 follows by letting ψ(t) = sin(πt/L). A.2 Proof using weighted µ-bubbles A µ-bubble is a hypersurface with prescribed mean curvature H = h, where h is a given function on the ambient manifold. The variational characterization of µ-bubbles is given by the energy functional (A.4). By choosing an appropriate function h, we can extract information about the ambient manifold from the stability inequality. The introduction here closely follows Chodosh-Li [9]. Let Ω + , Ω − be two disjoint domains in Σ. Let h be a Lipschitz function (whose conditions will be determined later) on Σ \ (Ω − ∪ Ω + ), such that h| ∂Ω ± = ±∞. A critical point of E(Ω) (or its boundary) is called a µ-bubble. Since h is infinite on ∂Ω ± , any µ-bubble must lie between Ω + and Ω − . Note that we have added a weight u into the functional. The unweighted version is E(Ω) = |∂Ω| − M (χ Ω − χ Ω 0 )h dA, whose critical point satisfies H = h. By geometric measure theory, a µ-bubble is always a C 2,α hypersurface. In [9] is was shown in detail that a global minimizer of (A.4) always exists. The first variation of (A.4) gives κ = h − u −1 u N (A.5) for a µ-bubble, where κ is the geodesic curvature of ∂Ω. The second variation of (A.4) at a global minimizer gives the following stability inequality: 0 γ u(ϕ ′ ) 2 − Kuϕ 2 − h 2 uϕ 2 + hu N ϕ 2 − u −1 u 2 N ϕ 2 + (∆u − u ′′ )ϕ 2 − h N uϕ 2 dl. (A.6) Testing (A.6) with ϕ = u −1 , we obtain 0 γ − βu −3 (u ′ ) 2 + − h 2 − λβ −1 + |∇h| u −1 + hu −2 u N − βu −3 u 2 N dl γ ( 1 4β − 1)h 2 − λβ −1 + |∇h| u −1 dl. For the last line we used If diam(Σ) > π/C 2 , then we can let p, q be points with maximal distance, and choose Ω + = B ε (p), Ω − = {x ∈ Σ : d(x, Ω + ) π/C 2 }. For these choices we obtain a contradiction with the stability inequality. This proves diam(M) π/C 2 . B Counterexamples for 1 4 < β 1 2 In this section we discuss the sharpness of Theorem 1.2 and 1.3 for 1 4 < β 1 2 by finding concrete counterexamples. The main piece for the counterexample is g = dr 2 + r −p dθ 2 , ϕ = r q (p > 0, r > 0). A short computation shows that ∆ϕ βKϕ is equivalent to q(q − 1) − pq + βp(p + 1) 0. we can achieve p 1. Observe the following properties of g when p 1: the end at r → 0 has infinite area, the perimeter of the end at r → ∞ has the order r −p , the end at r → ∞ has finite area (p > 1) or has area growth ∼ log r (p = 1). From g we can build the following counterexamples: Example B.1. Truncate g at a fixed large r and glue a spherical cap with a properly chosen metric and ϕ, such that ∆ϕ βKϕ is satisfied after the gluing. In this way we obtain an non-compact surface that violates Theorem 1.3. Example B.2. Truncate g at a large radius R and a fixed small radius r 0 . Glue a spherical cap at r 0 and a half catenoid-shaped metric (see construction below) at R. It is mentioned above that = 1 Definition 1 . 1 . 11Given a closed Riemannian manifold M of dimension n, the (Neumann) isoperimetric ratio is defined asIN(M) := inf Ω⊂M |∂Ω| n min |Ω|, |Ω c | n−1 .The Cheeger constant is defined asCh(M) := inf Ω⊂M |∂Ω| min |Ω|, |Ω c | .For a manifold with non-empty boundary, the (Dirichlet) isoperimetric ratio is defined as ID(M) := inf Ω⊂⊂M |∂Ω| n |Ω| n−1 . Theorem 1.3 (volume comparison). nonnegative and continuous, with φ ′ 0 when s > 0 and φ ′ 0 when s < 0. Then (2.5) holds. For notations see (2.2) (2.3). Corollary 3. 3 . 3For the same conditions as in Theorem 3.2, we have Ω e 2u C(ξ) |Ω| exp 1 ξ Ω |∇u| 2 . Theorem 3. 4 . 4Let (Σ, g) be a closed surface with IN(Σ) ξ > 0. Then for any smooth function u with Σ u = 0, Theorem 4. 3 . 3diam(Σ, g) C, where C depends on β and the lower bound of |Σ| diam(Σ) 2 . Proof. By (4.2) we have |∇u| 2 dA 4πβ −1 . Denote u = 1 |Σ| Σ u dA, by Theorem 1.2 and 3.4 we have Σ e pu dA C|Σ|e pu , Σ e −2u dA C|Σ|e −2u . 3 gives |U| |B(x, s)| CD 2 and |U ′ | |B(y, 1 4 D)| CD 2 . Then any domain Ω ⊂⊂ U satisfies |Σ \ Ω| |U ′ | CD 2 C D 2 A |Ω|, therefore ID(U) C IN(Σ). Analogously, ID(U ′ ) C IN(Σ). d 1 .. 5 . 15To control Φ −1 , we note that x · N d 1 by convexity. Hence for any |v| = 1 we have|dΦ x (v)| 2 = (x · N) Let Σ ⊂ R 3be a convex surface with unit area and (intrinsic) diameter D. Then there exists a smooth bi-Lipschitz map to the round sphere Φ : Σ → (S 2 , g 0 ), with ||Φ|| Lip , ||Φ −1 || Lip bounded in terms of D. Proof. Let W be the extrinsic width of Σ, defined as W := min d : we can embed Σ into the region {−d/2 z d/2} ⊂ R 3 , (4.4) Lemma 4. 7 . 7Denote u = 1 4π Σ u d A. Then |u| = o δ (1). Moreover, for any fixed η > 0 we have |{u η}| = o δ (1). 2u) d A ⇒ u 0. The other direction requires finite energy of u. By Theorem 1.4, 4.3 and Burago-Zalgaller's inequality, IN(Σ, g) has a uniform lower bound. Theorem 3.4 gives Σ exp (u − u) 2 Cδ d A C. Proposition 4. 8 . 8Embed (Σ, g) as the boundary of a convex body in R 3 . After an appropriate translation, the map Φ : x → x |x| satisfies ||Φ|| Lip 1 + o δ (1) and ||Φ −1 || Lip 1 + o δ (1). Lemma 4 . 9 . 49Any domain Ω ⊂ Σ with Ω ∩ V = ∅ satisfies |∂Ω| 2 g 0 ξ|Ω| g 0 for some constant ξ depending only on β and the lower bound of s. Let Ω 0 be a domain containing Ω + and disjoint from Ω − . (This set serves the role of renormalizing.) Consider the following functional acting on all open sets Ω with Ω∆Ω 0 ⊂⊂ Σ \ (Ω − ∪ Ω + ): ε) d(x, Ω + ) =: C 1 cot C 2 d(x, Ω + ) satisfies (A.8). Denote the resulting surface with boundary by Ω. Taking the double of Ω we obtain a closed surface Σ. Then diam(Σ) = O(R) as R → ∞, and the area of Ω has the order (Σ) · diam(Σ) is scale-invariant, by scaling this example we obtain closed surfaces with unit diameter and arbitratily small Cheeger's constant. This verifies the remark after Theorem 1.2. When β < 1 2 , for the largest possible p we haveIN(Σ) |∂Ω| 2 |Ω| = O(R −2p ) = O(R − 2 4β−1 ) and |Σ| diam(Σ) 2 = O(R −2 ), hence the power in Theorem 1.2(2) is almost sharp. Below is the technical construction of Example B.2. Example B.1 can be similarly constructed, and we omit the details. Assume β < 1 2 for simplicity. , r 1 = r 0 − 1 √ A π − tan −1 (c 3 ) − cos −1 (c 2 ) . 2 IN(Σ)/|Σ|, so the Cheeger constant is bounded from below as well. In case (1) we have Ch(Σ) Burago and Zalgaller's classical isoperimetric inequality Ch(Σ) 1/ diam(Σ) under pointwise condition K 0 [4]. When 1 4 < β 1 2 , we cannot bound the Cheeger constant solely by the diameter, in contrast to the the case β > 1 2 and case of pointwise curvature bounds. See Example B.2 for details.2β−1 2 √ 2β 1 diam(Σ) , which can be compared with − cos −1 (c 2 )) ≈ −0.16, while the right hand side is equal to − 2c 3 p √ β > − 4 c 3 = −0.1.This proves ϕ ′ (r + 1 ) < 0. 4β−1 is the largest achievable value of p. However, Example B.2 holds for all p > 1, therefore we set p = 2 − 2β for simplicity of expressions. Consider the metric g = dr 2 + f 2 (r)dθ 2 and function ϕ = ϕ(r) defined as follows:Therefore ∆ϕ βKϕ distributionally.Proof. The differentiability of f and ϕ at r 0 , R are straightforward from definitions. Note that f | [r 1 ,r 0 ] can be written asIt is therefore clear that f ′ (r + 1 ) = 1. The fact that ϕ > 0 follows from r 0 > 2/ √ βA. 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[ "Broadband X-ray spectral variability of the pulsing ULX NGC 1313 X-2", "Broadband X-ray spectral variability of the pulsing ULX NGC 1313 X-2" ]
[ "A Robba \nDipartimento di Fisica e Chimica\nUniversità degli Studi di Palermo\nvia Archirafi 36I-90123PalermoItaly\n\nINAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly\n", "C Pinto \nINAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly\n", "D J Walton \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n", "R Soria \nCollege of Astronomy and Space Sciences\nUniversity of the Chinese Academy of Sciences\n100049BeijingChina\n\nSydney Institute for Astronomy\nSchool of Physics A28\nThe University of Sydney\n2006SydneyNSWAustralia\n", "P Kosec \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n\nMIT Kavli Institute for Astrophysics and Space Research\n02139CambridgeMAUSA\n", "F Pintore \nINAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly\n", "T P Roberts \nDepartment of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK\n", "W N Alston \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n", "M Middleton \nDepartment of Physics and Astronomy\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUK\n", "G Cusumano \nINAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly\n", "H P Earnshaw \nCahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "F Fürst \nScience Operations Department\nEuropean Space Astronomy Centre (ESA/ESAC)\nVillanueva de la Canada28692MadridSpain\n", "R Sathyaprakash \nDepartment of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK\n\nInstitut de Ciències de l'Espai\nCarrer de Can Magrans08193Cerdanyola del Vallès, Barcelona\n", "E Kyritsis \nDepartment of Physics\nUniversity of Crete\nGR-71003HeraklionGreece\n\nInstitute of Astrophysics\nFORTH\nGR-71110HeraklionGreece\n", "A C Fabian \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n" ]
[ "Dipartimento di Fisica e Chimica\nUniversità degli Studi di Palermo\nvia Archirafi 36I-90123PalermoItaly", "INAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly", "INAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly", "Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK", "College of Astronomy and Space Sciences\nUniversity of the Chinese Academy of Sciences\n100049BeijingChina", "Sydney Institute for Astronomy\nSchool of Physics A28\nThe University of Sydney\n2006SydneyNSWAustralia", "Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK", "MIT Kavli Institute for Astrophysics and Space Research\n02139CambridgeMAUSA", "INAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly", "Department of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK", "Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK", "Department of Physics and Astronomy\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUK", "INAF/IASF Palermo\nvia Ugo La Malfa 153I-90146PalermoItaly", "Cahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Science Operations Department\nEuropean Space Astronomy Centre (ESA/ESAC)\nVillanueva de la Canada28692MadridSpain", "Department of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK", "Institut de Ciències de l'Espai\nCarrer de Can Magrans08193Cerdanyola del Vallès, Barcelona", "Department of Physics\nUniversity of Crete\nGR-71003HeraklionGreece", "Institute of Astrophysics\nFORTH\nGR-71110HeraklionGreece", "Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK" ]
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Context. It is thought that ultraluminous X-ray sources (ULXs) are mainly powered by super-Eddington accreting neutron stars or black holes as shown by the recent discovery of X-ray pulsations and relativistic winds. Aims. This work presents a follow-up study of the spectral evolution over two decades of the pulsing ULX NGC 1313 X-2 in order to understand the structure of the accretion disc. The primary objective is to determine the shape and nature of the dominant spectral components by investigating their variability with the changes in the source luminosity. Methods. We performed a spectral analysis over the canonical 0.3-10.0 keV energy band of all the high signal-to-noise XMM-Newton observations (96 % of the available data), and we tested a number of different spectral models, which should approximate super-Eddington accretion discs. The baseline model consists of two thermal blackbody components with different temperatures plus an exponential cutoff powerlaw.Results. The baseline model provides a good description of the X-ray spectra. In particular, the hotter and brighter (L X ∼ 6-9×10 39 erg s −1 ) thermal component describes the emission from the super-Eddington inner disc and the cutoff powerlaw describes the contribution from the accretion column of the neutron star. Instead, the cooler component describes the emission from the outer region of the disc close to the spherisation radius and the wind. The luminosity-temperature relation for the cool component follows a negative trend, which is not consistent with L ∝ T 4 , as is expected from a sub-Eddington thin disc of Shakura-Sunayev. This is not consistent with L ∝ T 2 either, as is expected for an advection-dominated disc. However, this would rather agree with a wind-dominated X-ray emitting region. Instead, the (L x ,T disk ) relation for the hotter component is somewhere in between the first two theoretical scenarios. Conclusions. Our findings agree with the super-Eddington scenario and provide further detail on the disc structure. The source spectral evolution is qualitatively similar to that seen in NGC 1313 X-1 and Holmberg IX X-1, indicating a common structure and evolution among archetypal ULXs.
10.1051/0004-6361/202140884
[ "https://arxiv.org/pdf/2106.04501v2.pdf" ]
235,367,832
2106.04501
1f9e909d080089205f10ddf0becddf2219214fee
Broadband X-ray spectral variability of the pulsing ULX NGC 1313 X-2 June 30, 2021 A Robba Dipartimento di Fisica e Chimica Università degli Studi di Palermo via Archirafi 36I-90123PalermoItaly INAF/IASF Palermo via Ugo La Malfa 153I-90146PalermoItaly C Pinto INAF/IASF Palermo via Ugo La Malfa 153I-90146PalermoItaly D J Walton Institute of Astronomy Madingley RoadCB3 0HACambridgeUK R Soria College of Astronomy and Space Sciences University of the Chinese Academy of Sciences 100049BeijingChina Sydney Institute for Astronomy School of Physics A28 The University of Sydney 2006SydneyNSWAustralia P Kosec Institute of Astronomy Madingley RoadCB3 0HACambridgeUK MIT Kavli Institute for Astrophysics and Space Research 02139CambridgeMAUSA F Pintore INAF/IASF Palermo via Ugo La Malfa 153I-90146PalermoItaly T P Roberts Department of Physics Centre for Extragalactic Astronomy Durham University South RoadDH1 3LEDurhamUK W N Alston Institute of Astronomy Madingley RoadCB3 0HACambridgeUK M Middleton Department of Physics and Astronomy University of Southampton HighfieldSO17 1BJSouthamptonUK G Cusumano INAF/IASF Palermo via Ugo La Malfa 153I-90146PalermoItaly H P Earnshaw Cahill Center for Astronomy and Astrophysics California Institute of Technology 91125PasadenaCAUSA F Fürst Science Operations Department European Space Astronomy Centre (ESA/ESAC) Villanueva de la Canada28692MadridSpain R Sathyaprakash Department of Physics Centre for Extragalactic Astronomy Durham University South RoadDH1 3LEDurhamUK Institut de Ciències de l'Espai Carrer de Can Magrans08193Cerdanyola del Vallès, Barcelona E Kyritsis Department of Physics University of Crete GR-71003HeraklionGreece Institute of Astrophysics FORTH GR-71110HeraklionGreece A C Fabian Institute of Astronomy Madingley RoadCB3 0HACambridgeUK Broadband X-ray spectral variability of the pulsing ULX NGC 1313 X-2 June 30, 2021Received ?; Accepted ?Astronomy & Astrophysics manuscript no. paperAccretion, accretion discs -X-rays: binaries -X-rays: individual (NGC 1313 X-2) Context. It is thought that ultraluminous X-ray sources (ULXs) are mainly powered by super-Eddington accreting neutron stars or black holes as shown by the recent discovery of X-ray pulsations and relativistic winds. Aims. This work presents a follow-up study of the spectral evolution over two decades of the pulsing ULX NGC 1313 X-2 in order to understand the structure of the accretion disc. The primary objective is to determine the shape and nature of the dominant spectral components by investigating their variability with the changes in the source luminosity. Methods. We performed a spectral analysis over the canonical 0.3-10.0 keV energy band of all the high signal-to-noise XMM-Newton observations (96 % of the available data), and we tested a number of different spectral models, which should approximate super-Eddington accretion discs. The baseline model consists of two thermal blackbody components with different temperatures plus an exponential cutoff powerlaw.Results. The baseline model provides a good description of the X-ray spectra. In particular, the hotter and brighter (L X ∼ 6-9×10 39 erg s −1 ) thermal component describes the emission from the super-Eddington inner disc and the cutoff powerlaw describes the contribution from the accretion column of the neutron star. Instead, the cooler component describes the emission from the outer region of the disc close to the spherisation radius and the wind. The luminosity-temperature relation for the cool component follows a negative trend, which is not consistent with L ∝ T 4 , as is expected from a sub-Eddington thin disc of Shakura-Sunayev. This is not consistent with L ∝ T 2 either, as is expected for an advection-dominated disc. However, this would rather agree with a wind-dominated X-ray emitting region. Instead, the (L x ,T disk ) relation for the hotter component is somewhere in between the first two theoretical scenarios. Conclusions. Our findings agree with the super-Eddington scenario and provide further detail on the disc structure. The source spectral evolution is qualitatively similar to that seen in NGC 1313 X-1 and Holmberg IX X-1, indicating a common structure and evolution among archetypal ULXs. Introduction Ultraluminous X-ray sources (ULXs) are among the best candidates for studying super-Eddington accretion in stellar-mass accreting compact objects. ULXs are the brightest off-nuclear, steady, point-like X-ray sources (> 10 39 erg s −1 ) in the Universe. They are often found in or near regions of recent star formation (Swartz et al. 2009;Kovlakas et al. 2020) and they have X-ray luminosities that exceed the isotropic Eddington luminosity for a standard black hole (BH) with a mass of M≈10 M (e.g. Kaaret et al. 2017). ULXs represent a heterogeneous sample of astronomical sources and are composed of a compact object, most [email protected] likely a BH or a NS, and a companion star, which has been found to be a red or blue supergiant in some ULXs, see for example Heida et al. (2019). In order to explain the high X-ray luminosities of these sources, several hypotheses have been proposed. A first scenario suggests that ULXs are powered by stellar-mass BHs whose radiation is preferentially beamed in our line of sight (King et al. 2001;Poutanen et al. 2007). A second scenario supposes that a BH more massive than 10 M (30−80 M ; e.g. Zampieri & Roberts 2009), which accretes at or below the Eddington limit, is the compact object of a ULX. The existence of more massive BHs was proven by the detection of gravitational waves (e.g. Abbott et al. 2016a,b;Abbott et al. 2019, with BH masses between Article number, page 1 of 18 arXiv:2106.04501v2 [astro-ph.HE] 29 Jun 2021 A&A proofs: manuscript no. paper 10 M and 80 M . In addition, in the past, other theories have suggested that some of these systems could be intermediatemass BHs (10 3−4 M , Colbert & Mushotzky 1999), accreting at sub-Eddington rates from low-mass companion stars with ESO243-49 HLX-1 being the best candidate (see Farrell et al. 2009). For many years, the mass estimation of putative BHs powering ULXs was the subject of significant debate. However, the recent detection of coherent pulsations in several sources clearly demonstrates that some ULXs are powered by NSs accreting at very high Eddington rates with luminosities up to ∼ 500 L Edd . The first pulsation in a ULX was discovered in M 82 X-2 by Bachetti et al. (2014) with NuSTAR observations and, at the moment, six ULXs are known to exhibit pulsations (Israel et al. 2017a,b;Fürst et al. 2016;Carpano et al. 2018;Sathyaprakash et al. 2019). It is not straightforward as to how to distinguish between BH and NS accretors based on the spectral analysis alone (Pintore et al. 2017;Koliopanos et al. 2017;Walton et al. 2018c). Moreover, pulsations are not always detectable; high count rates are indeed required for a low pulsed fraction or long exposure times for low count rates. This means that NSs are likely numerous among the compact objects of ULXs (see also Rodríguez Castillo et al. 2020;Wiktorowicz et al. 2019;King et al. 2017;Middleton & King 2016). One of the fundamental predictions of the super-Eddington accretion theory is that strong, relativistic winds are launched from the supercritical discs, driven by the extreme radiation pressure (see e.g. Poutanen et al. 2007). Middleton et al. (2014) suggest that the spectral residuals around 1 keV could be associated with the winds. The first discovery of powerful winds in two ULXs was achieved by Pinto et al. (2016) by detecting blueshifted absorption lines in the high-resolution soft X-ray spectra provided by the XMM-Newton Reflection Grating Spectrometers (RGS). Further confirmations in other ULXs and with different detectors were obtained by Walton et al. (2016), Pinto et al. (2017), Kosec et al. (2018a) and Kosec et al. (2018b). The exact launching mechanism of such winds is still unclear as magnetic pressure might also contribute, although Pinto et al. (2020a) show that the relation between their velocities and ionisation parameters with the ULX luminosities agrees with the radiation driving mechanism. A thorough understanding of the ULX phenomenology requires additional constraints on the nature of these winds and their link with the source appearance and, therefore, accretion rate. This involves a study of the wind properties via high-resolution X-ray spectroscopy combined with a careful study of the evolution of ULX broadband spectra. In this work we present the analysis of the X-ray spectra of the pulsating ULX X-2 in the galaxy NGC 1313. This barred galaxy (see Fig. 1) hosts a supernova remnant (SN 1978K) and two ULXs: X-1, close to the nucleus, and X-2, in the outskirts of the galaxy, which is the subject of this work. Sathyaprakash et al. (2019) discovered pulsations in NGC 1313 X-2 for the first time thanks to our deep XMM-Newton campaign. This ULX is characterised by strong variability in both luminosity and spectral shape. The high X-ray variability and spectral hardness suggest that the object is viewed at an inclination angle, which is low enough to allow for a direct view of the inner regions of its accretion flow and to detect pulsations (Middleton et al. 2015). NGC 1313 X-2 also shows evidence of winds in the form of Xray spectral features in the soft band (see, e.g. Middleton et al. 2015 andKosec et al. 2018a). Throughout this work, we assume a distance of D = 4.2 Mpc to NGC 1313 (Mendez et al. 2002;Tully et al. 2016). The only : XMM-Newton image of NGC 1313, which we obtained by combining all the data available from the 2017 EPIC-pn and MOS 1,2 observations. The 30 -radius circle around the ULX X-2 represents our default source extraction region. The red colour corresponds to 0.2-1 keV, green is for 1-2 keV, and blue is for 2-12 keV. Cepheid distance available is 4.6 Mpc, which is a little bit of an outlier from the other measurements (Qing et al. 2015). This paper is structured as follows. In Section 2, we provide details about the XMM-Newton observations used in this work and the data reduction. We show some basic time properties and the model-independent variability of the X-ray spectral shape. In Section 3, we describe the main results of the spectral and timing analysis, while in Section 4 we discuss the behaviour of the thermal components in the luminosity-temperature plane and, finally, provide our conclusions in Section 5. Data analysis Observations We analysed the public archival XMM-Newton data of all the high signal-to-noise observations between 2000 and 2017. For our analysis, we particularly benefitted from a recent deep ∼1Ms view of NGC 1313 (PI: Pinto). The observations were carried out with the EPIC-pn and EPIC-MOS detectors (Strüder et al. 2001;Turner et al. 2001). For observations 0803990301 and 0803990401, we used only the data provided by pn and MOS 2 since the source was out of the MOS1 field of view due to damage to CCD6 and CCD3 (since 2005 and 2012, respectively). We have excluded observations 0150280201 and 0150280701 from this analysis as they contain only MOS data and they were very short, therefore providing low statistics. We also excluded observation 0205230201 because of a slew failure. Table 1 lists the details of the high signal-to-noise XMM-Newton observations (96 % of the available data) that we analysed, including the date, the duration after the removal of flares, the count rates, and the filter of each observation. Data reduction The data analysis was performed using the XMM-Newton Science Analysis System (SAS) version 18.0.0 and the calibration files of January 2020. 1 Following the standard procedures, EPICpn and MOS data were reprocessed using the tasks 'epproc' and 'emproc'. The calibrated and concatenated event lists were filtered for high background epochs to acquire good time intervals (GTI) as follows. For each data set and each instrument, we extracted the high energy light curve (including events between 10-12 keV) to identify intervals of flaring particle background and we chose a suitable threshold (0.35 and 0.40 cts s −1 for EPIC-MOS and pn, respectively), which is above the low steady background, to create the corresponding filtered EPIC event list. As recommended, we selected only single and double events (PAT-TERN ≤4) for EPIC-pn, and single to quadruple events (PAT-TERN ≤12) for EPIC-MOS. We extracted EPIC MOS 1-2 and pn images in the 0.2-1/1-2/2-12 keV energy range and stacked them with the 'emosaic' task. The field of the NGC 1313 galaxy and the brightest Xray sources are shown in Fig. 1. We generally extracted source 1 https://www.cosmos.esa.int/web/xmm-newton/ccf-release-notes. spectra from circular regions with a radius of 30", except where the source was near the edge of the CCD (in these cases we used regions with a radius of 20") and the corresponding background from a larger circle in a nearby region on the same chip, free from contaminating point sources. The background region was also not generally placed in the Copper emission region (Lumb et al. 2002), with the exception of a small number of observations for which X-2 was also located in the Cu region. We used the task 'arfgen' to reproduce the effective area of the instrument and to correct instrumental factors, such as bad pixels and bad columns, using calibration information. The response matrix was generated with 'rmfgen'. Since the EPIC-MOS1 and EPIC-MOS2 spectra are consistent for each observation, we stacked data from the MOS cameras into a single spectrum with the 'epicspeccombine' routine. Fig. 2 shows an overview of the spectral properties of X-2. In particular, the right panel illustrates the shape of the observed EPIC-pn spectra of NGC 1313 X-2 for six individual exposures during the most recent campaign (2017). The spectra in-dicate substantial variability in luminosity by a factor of up to five and there is no significant spectral variability. The left panel of Fig. 2 instead shows a comparison between some spectra of a remarkably different shape and flux: a high flux spectrum (Obs.ID:0150280401), two spectra with intermediate flux (Obs.ID:0803990201 and Obs.ID:0782310101), and one with low flux (Obs.ID:0106860101). As it has been observed in most ULXs (e.g. Middleton et al. 2015), the spectrum becomes harder at higher fluxes. This behaviour disagrees with that seen in the classical Galactic X-ray binaries that accrete below Eddington limit and it is therefore considered as strong evidence in support of super-Eddington accretion. The left panel of Fig. 2 clearly shows that the spectrum is more variable in the hard X-ray band ( 1 keV), which indicates that at least two different (soft and hard) spectral components are responsible for the X-ray emission. Spectral analysis We carried the spectral analysis out over the 0.3-10.0 keV energy range with XSPEC version 12.10.1 (Arnaud 1996). We simultaneously fitted the spectra of the MOS and pn cameras and estimated the parameter uncertainties at the 68% confidence level. Spectra were grouped for a minimum of 25 counts per energy bin, so that the χ 2 statistic could be used. There is no coverage beyond 10 keV in most observations and even in the NuSTAR data, the source has low statistics above 10 keV (e.g. Bachetti et al. 2013). As the source is generally softer than X-1 in the NGC 1313 galaxy, the systematic error due to the lack of NuSTAR data is negligible. We therefore focussed on the canonical 0.3-10 keV X-ray band and assumed a simplified model with either two or three emission components. In the case, for instance, of the double blackbody disc and an exponential cutoff powerlaw model, the low-temperature component mimics the emission of the outer disc and wind, while the hotter temperature component reproduces the emission from the inner super-Eddington accretion flow. The third component is necessary to describe the contributions of the central accretion columns by the magnetic accretor, that is, the NS. Walton et al. (2018c) show that in all ULXs, this latter component is significant above 8-9 keV (see also Bachetti et al. 2013). Timing analysis In order to better understand our spectra, we extracted light curves in the whole band (0.3-10 keV) and in two different bands (i.e. soft 0.3-1.2 keV and hard 1.2-10 keV). The latter were extracted to calculate the hardness ratios (HRs), which were computed as follows: HR = Hard Rate 1.2−10 keV S o f t Rate 0.3−1.2 keV + Hard Rate 1.2−10 keV .(1) We chose to split the bands at 1.2 keV because this is the average energy at which the spectral curvature changes in the EPIC spectra (the case for Obs.ID:0803990201 is shown in Fig. 4). We plotted the full-band light curve colour-coded according to the HR in Fig. 3, where the vertical grey-dashed lines separate each observation. NGC 1313 X-2 shows high variability during the two decades of (non-contiguous) observations. As a short-term flux variability test, we adopted the normalised excess variance in the whole band (0.3-10 keV), which is useful to quantify the different amplitudes of intrinsic variability of each light curve. As reported by Nandra et al. (1997), it is defined as follows: σ 2 NXV = 1 Nx 2 N i=1 [(x i −x) 2 − σ 2 err,i ],(2) where x i and σ err,i are the count rate and its error in the i-th bin,x is the mean count rate, and N is the number of bins used to estimate σ 2 NXV . The associated error is the following: ∆σ 2 NXV = S D x 2 (N) 1/2 , S D = 1 N − 1 N i=1 {[(x i −x) 2 − σ 2 err,i ] − σ 2 NXVx 2 } 2 . The fractional root mean square (RMS) variability amplitude (F var , see Vaughan et al. 2003) is the square root of the normalised excess variance, that is F var (%) = σ 2 NXV × 100 = 1 Nx 2 N i=1 [(x i −x) 2 − σ 2 err,i ] × 100 (3) ∆σ F var (%) = 1 2 * ∆σ 2 NXV σ 2 NXV × 100,(4) where F var is a linear statistic, which gives the same information as σ 2 NXV , but in percentage terms. In order to compare the RMS and to account for the different exposure times, we split all light curves into 40ks segments, which is approximately a minimum-common-denominator segment for the long observations. Then we averaged the RMS results from the two or more segments available in longer (80-120 ks) observations and binned with ∆T of 1000 s. For the observations shorter than 40 ks, we did not estimate the RMS. Main results Spectral modelling In this section we present the spectral analysis of NGC 1313 X-2. In order to obtain an adequate description of the continuum, ULX broadband X-ray spectra require several emission components. Among the several models tested, we adopted one similar model to that used in Walton et al. (2018c). All models also include neutral absorption, which was modelled with TBABS (Wilms et al. 2000). This absorption, which is due to an interstellar and circumstellar medium in the line of sight towards the source, is necessary to partially explain the low-energy spectral curvature. We adopted a lower limit equal to the Galactic value of N H =7 × 10 20 cm −2 . Baseline models Our baseline model for the X-ray spectra of NGC 1313 X-2 consists of two thermal components, a cold disc blackbody (DISKBB) and a warmer DISKBB for the outer and the inner disc, respectively. It also consists of an exponential cutoff powerlaw component (CUTOFFPL) for the central accretion column that forms when the material flows down onto the magnetic poles. For the CUTOFFPL component, we set its spectral parameters to the average values seen from the pulsed emission of the following ULX pulsars currently known: Γ = 0.59 and E cut = 7.9 keV (Brightman et al. 2016;Walton et al. 2018b,c,a). In most spectra, there are notable residuals at ∼1 keV, related to atomic emission and absorption associated with the likely presence of an extreme outflow powered by wind Kosec et al. 2018a,b;Wang et al. 2019). Despite this, the overall shape of the X-ray spectra is well reproduced by our featureless continuum model. Indeed, as pointed by Pinto et al. (2020a) and Walton et al. (2020), the presence of the wind has no dramatic effects on the modelling of the spectral continuum and, therefore, we did not include any line emission or absorption components in our current fits. In addition, this would require the use of deep, on-axis, RGS spectra that are unavailable since the source is off-axis in most of the observations. This is valid for the optically thin ionised wind phase responsible for the spectral lines, but it might not hold in the case of further optically thick wind components. The spectral fits with the best-fitting baseline double discblackbody and the exponential cutoff powerlaw models are presented in Appendix A. As we show in Table A.1, we obtained a column density between N H = (0.095-0.32) × 10 22 cm −2 , a cold temperature in the range from 0.20 < T 1 < 0.49 keV, and a warm temperature in the 0.69 < T 2 < 1.9 keV range. The corresponding luminosities are L 1 = (1-6) × 10 39 erg s −1 and L 2 = (2-8) × 10 39 erg s −1 . The goodness of the spectral fits is indicated by the χ 2 , which are satisfactory as shown in Table A.1 (the reduced χ 2 are in the range from 0.8-1.4). Alternative models We also tested several other models to describe the spectral shape of the spectrum of X-2. We show examples of our model fits for Obs.ID:0803990201 in Fig. 4 with corresponding best-fit parameters in Table 2. Spectral modelling of ULXs using XMM-Newton data is well established with two-component models, which resemble multicolour blackbody emission (e.g. Gladstone et al. 2009 andStobbart et al. 2006). In this context, the first attempt consists of a double disc-blackbody DISKBB+DISKBB, in which the first component describes the outer accretion flow and the possible optically thick wind and the second one takes both the inner flow and any other emission closer to the NS into account. As we expected (considering the lack of the hard cutoff powerlaw component), in comparison to the best-fitting model, the values of the temperatures are larger. The column densities, on the other hand, are similar (see Table B.1). We also tested the model with the column density fixed to N H = 1.94 × 10 21 cm −2 , obtained from the weighted average of the previous fits with free N H , to evaluate the contribution of neutral absorption and systematic effects. However, we note that this assumption does not strongly influence the broadband continuum fits, so we preferred to keep N H free to vary amongst the spectra. The spectral fits with two thermal components are generally worse than the baseline three component model with the greatest discrepancy at the lowest flux, where the hard cutoff powerlaw component starts to be important (∆χ 2 = 41 and 61 for Obs.ID 0405090101 and 0782310101, respectively, for 1 additional degree of freedom). Spectral parameters are reported in Appendix B (see Table B.1). The second alternative model consists of replacing the double DISKBB continuum model with a single DISKBB modified by a SIMPL component. This empirical model of Comptonisation assumes that a fraction of the soft photons is scattered into a corona or a photosphere warmer than the disc. As shown in Table 2, although the parameters are physically acceptable, we found a worse fit with an increase of χ 2 (∆χ 2 = 140) in observation 0803990201. Column densities, N H , and cool temperature values are similar and follow the same behaviour as the baseline model with two DISKBBs plus a CUTOFFPL component. Afterwards, we tried using a DISKBB+DISKPBB model, where DISKPBB is a multiple blackbody disc model characterised by the additional free parameter p (Mineshige et al. 1994). The p parameter defines the radial dependency of the temperature, following the law T ∝ R −p (p=0.75 for a standard disc, p<0.75 for a profile affected by advection, and p=0.5 for a slim disc, see Abramowicz et al. 1988). In this case, the temperatures obtained for the cool component are similar to parameters from the best-fit (in the DISKBB+DISKBB+CUTOFFPL baseline model); whereas for the hot component, the temperature is largely unconstrained in some observations. In general, the values for parameter p (e.g. p=0.573±0.019, for Obs.ID:0803990201) indicate a regime very close to the slim disc profile (the mean value for all spectral fits is p mean ∼ 0.57). The addition of the extra free parameter (p) provides spectral fits that are statistically halfway between the two DISKBB and three component models. All the models show some residuals around 1 keV, as described in Section 3.1.1. They appear very similar and narrow (∼ 0.1 keV) in all fits, indicating that they are independent from the particular model chosen and do not affect our results. Time variability NGC 1313 X-2 shows high variability during the two decades of (non-contiguous) observations, as seen in the long-term light curve (see Fig. 3). Strong short-term (∼ hours) variability is also seen during some individual and long observations. We see that the source becomes harder when it is brighter, which is indicative of an increasingly brighter super-Eddington inner disc. This is because the hot disc is the dominant component in the 0.3-10 keV energy band. The timing properties of X-2 were also probed using the fractional variability (see Section 2.4). The average values of F var for several observations and the corresponding HR mean are presented in Table 3. We have reported only the values for the observations that have a common time baseline of exposure time (40 ks) for comparing the RMS estimated. The complete values Table 2. calculated for each segment are shown in Appendix C (see Table C.1). Also, in cases where the observed variance is less than the error associated with each bin, the excess variance value is negative. These values are excluded from our considerations. As shown in Fig. 5, the variability of the source considerably increases in the observations with a higher flux, which is in agreement with the RMS-flux variability of accretion discs. Sutton et al. (2013) suggest that the increased flux variability observed at energies above 1 keV in ULXs with hard spectra can be attributed to the obscuration of the hard central emission when observed through the clumpy edge of the outflowing wind and from the photosphere of the super-Eddington disc. Sathyaprakash et al. (2019) have found evidence of pulsations during observations Obs.ID:080399401 and Obs.ID:0803990601. This corresponds to the low-flux end of the new campaign, which suggests that the bright variable continuum is strongly affected by the inner accretion flow. In other words, at higher accretion rates the disc flux may significantly exceed the flux of the accretion column, thereby decreasing the pulse fraction. Walton et al. (2018c) also suggest this for the ensemble of ULXs observed by NuSTAR. In addition, this would imply a low-to-mild magnetic field (B 10 12 G, see e.g. King & Lasota 2020), because the magnetospheric radius (R m ) is likely smaller than the spherisation radius (R sph ) for this object, given the presence of a bubble nebula (e.g. Pakull & Mirioni 2002) that is thought to be inflated by the disc wind. Discussion We have performed a detailed spectral analysis of the ultraluminous X-ray source NGC 1313 X-2, focussing on the XMM-Newton observations, with the aim of understanding the long-term behaviour of the spectral components. Given that the compact object is now known to be a NS, the accretion rate must be highly super-Eddington, and its spectral and temporal properties are expected to diverge from the case of sub-Eddington thin accretion discs. As Vierdayanti et al. (2010) showed, the evolution of NGC 1313 X-2 appears similar to the archetypal ULX Holmberg IX X-1, characterised by strong spectral variability (see also Luangtip et al. 2016;Pintore et al. 2014;Walton et al. 2014). As Walton et al. (2017), the spectra of Holmberg IX X-1 are well fit by two thermal blackbody components, which dominate the emission below 10 keV, plus a powerlaw tail which dominates above 10 keV. As shown in Fig. 4, the spectrum is reasonably well described by a combination of two DISKBB plus a CUTOFFPL components. This model approximates super-Eddington accretion onto a NS, characterised by a standard outer disc, a thick inner disc region, and strong optically thick winds, which may contribute to the cooler component. In particular, the ∼ 1-10 keV band is mainly dominated by the hotter component, which describes the emission from the inner region, and the CUTOFFPL component representing the accretion column and the boundary layer near the NS. Instead, the emission from the outer disc or from the wind is responsible for the cooler blackbody component with a temperature around 0.3 keV. Wind is expected to be launched from the upper regions of the inner disc at accretion rates comparable to or higher than the Eddington limit. This scenario is supported by the presence of strong and narrow residuals around 1 keV that have been resolved in similar sources with the aid of high-resolution X-ray spectra . Table 3). The modelling of the spectra with high statistics shows that the column density, N H , is broadly consistent with ∼ 2 × 10 21 cm −2 (see Table A.1). Fixing the N H (see Appendix D) does not strongly influence the broadband continuum fits, so we preferred to keep N H free. With the intention of improving the analysis, we tested several spectral models, as described in Section 3.1.2. When the accretion rate is high, the structure of the disc is expected to deviate considerably from the standard Shakura-Sunayev thin disc. For this reason, as reported by Bachetti et al. (2013) . In general, for several observations, the p values are compatible with an accretion regime significantly affected by advection and they are sometimes close to the slim disk regime, which strongly argues in favour of super-Eddington accretion. In the future we will use principal component analysis (PCA) to decompose the spectrum in a model-independent way (e.g. Pinto et al. 2020b). Moreover, we will search for winds with both PCA and physical models and then we will study how they vary with the spectral shape. Luminosity-temperature plane In this section we discuss the temporal behaviour of the two thermal components, investigating how they evolve in the luminosity-temperature plane. We calculated the unabsorbed luminosities (i.e. corrected for interstellar absorption by setting the column density N H =0) for each of the thermal components individually, over the broad band from 0.001-10 keV, which pro-vides their bolometric values. The comparison between the temperature and the luminosity of each component in the case of free column density is shown in Fig. 6. We show the L-T trends for both the cool (blue) and hot (orange) thermal components in the same figure because their separation in temperature is sufficiently high to avoid confusion. It is very useful to compare the observed trends with the L-T α relationships as expected from theoretical scenarios such as a sub-Eddington thin disc of Shakura-Sunayev (L ∝ T 4 for constant emitting area, see Shakura & Sunyaev 1973) or an advection-dominated disc (L ∝ T 2 , see Watarai et al. 2000). These laws are shown as red-solid and green-dashed lines in Fig. 6, respectively. The L-T trend observed for the hot disk-blackbody component is somewhere in between the two trends above. We do indeed obtain α H = 3.0 ± 0.35 by fitting the L-T group of the hot component, which also argues in favour of super-Eddington accretion with a thicker disc. The trend followed by the cool component is instead negative (α C = −3.9 ± 1.0) and differs form the one expected from a sub-Eddington thin disc of Shakura-Sunayev (L∝T 4 ) or an advection-dominated disc (L∝T 2 ). Our results are in agreement with the interpretation of Qiu & Feng (2021), suggesting that the soft emission originates from the photosphere of the optically thick wind, driven by supercritical accretion. In this scenario, the measured blackbody luminosities of the PULXs are often higher than the Eddington limit of NSs, assuming spherical accretion (i.e. L EDD,NS = (1-3) × 10 38 erg s −1 ). Moreover, we may expect an inversion of the L-T relationship at accretion rates much higher than the Eddington limit due to the expansion of the photosphere, which is marked by an increase in the size of the emitting region of the soft component and a decrease in temperature. This effect can be interpreted as the fact that the spherisation radius, where the strong winds start to be launched, would increase with the accretion rate, yielding lower temperatures at a higher luminosity (see, e.g. Poutanen et al. 2007.) Although the nature of the compact object is unknown, the hotter disc component of NGC 1313 X-1 also exhibits a positive relation for the luminosity-temperature trend, consistent with the theoretical scenarios. Walton et al. (2020) did not seem to find a strong anti-correlation between L and T for the low-T component, but this might be due to the low number of spectra (seven) along with the more complex DISKBB+SIMPL*DISKPBB and DISKPBB+DISKBB+CUTOFFPL model. In order to check the presence of the third component, we tested the L-T trends with the simpler DISKBB+DISKBB model. As can be seen from Fig.7, the L-T trend observed for the hot disk-blackbody component shows a much more chaotic behaviour. Observations with better statistics (higher fluxes and longer durations) still sit somewhere in between the two theoretical cases outlined above, but there are considerable deviations for a cluster of points where the hotter component is characterised by high temperatures and low luminosities. Indeed, we obtain α H = 2.8 ± 0.6 by fitting the L-T group of the hot component, which also argues in favour of super-Eddington accretion with a thicker disc. The trend followed by the cool component is instead much flatter (α C = −0.13 ± 0.36) than the one expected from the theoretical scenarios. We conclude that the introduction of a third (CUTOFFPL) component significantly improves the spectral fits of some observations, which is confirmed by the detection of more regular trends in Fig. 6 and from their lower reduced χ 2 . The presence of a correlation or anti-correlation between these points (luminosity and temperature for the hot and cool components) for the baseline model was also verified by the computation of the correlation coefficients of Pearsons and Spearmans (see Table 4). They were calculated using the PYTHON routines scipy.stats.pearsonr and scipy.stats.spearmanr. The result for the cooler component indicates a negative correlation and a positive correlation for the hotter component. The coefficients obtained are not very high because, as opposed to the least squares method, they do not take the error associated with the luminosity and temperature into account. Table 4: Pearson and Spearman correlation coefficients calculated for the trends between luminosity and temperature for both the cool and the hot DISKBB components (assuming the baseline model, i.e. DISKBB+DISKBB+CUTOFFPL). Parameters Pearson Spearman L hot -T hot 0.69 0.69 L cool -T cool -0.44 -0.47 The behaviour of the soft component has also been recently examined by Gúrpide et al. (2021). As they report, the significantly softer spectra of NGC 1313 X-2 can be explained by the scenario consistent with the wind structure responsible for highly anisotropic emission, given the wide HR variability the source spans. However, they do not report a clear correlation between the temperature and luminosity. Although their model is a double DISKBB model and thus is conceptually similar to ours, they adopted a different model for the high-energy component, using a SIMPL powerlaw continuum instead. As previously discussed, this may exacerbate parameter degeneracies when only soft X-ray data (i.e. <10 keV) are available, and this may explain the different conclusion found here. Moreover, for some observations they have used only pn spectra, resulting in a lower overall signal-to-noise ratio (S/N) and thus larger uncertainties. As we also suggest, they hypothesise that the observed residuals at soft energies around 1 keV can be produced by the presence of strong outflows. Kajava & Poutanen (2009) also studied the (L,T) relation of the soft component, although much less data were available at that time (i.e. prior to our new 2017 campaign). They found that, taken together, the soft component followed a trend L so f t ∝ T −3.5 for a sample of ULXs that included NGC 1313 X-2, although the behaviour of individual sources was not considered in detail. In addition, Soria (2007) showed an anticorrelation between the disc luminosity and temperature, supporting the non-standard outer disc. As Soria (2007) showed, if we assume that the boundary between the outer and inner disc scales asṁ, the negative slope of the L-T relation depends on the T(R) function. For T ∼ R −0.5 , it is expected that the L ∼ T −4 , which is consistent with our result (L ∼ T −3.9 ). Therefore, we suggest that the outer disc is affected by the wind. About 10% of the available data have also been analysed by Pintore & Zampieri (2012). They adopted a different model where the hot DISKBB is replaced with the much broader Comptonisation component, COMPTT. They also tied the temperature of the seed photons to that of the cool blackbody. For the broad X-ray spectra of ULXs, this has the effect of lowering the contribution in terms of flux from the cooler blackbody component. They reported a weak correlation between the luminosity of the soft component with the inner temperature, L disc ∝ T 1.2±0.3 disc . Fig. 6: 0.001-10 keV (i.e. bolometric) luminosity versus temperature for both the cool DISKBB (blue points) and hot DISKBB (orange points) components with free column density, N H (model: DISKBB+DISKBB+CUTOFFPL). Radius-luminosity relation In order to test the consistency with the luminosity-temperature trends, we also estimated the mean inner radii for the DISKBB component for each of the two groups, using the normalisation factor with the formula R in = (norm * D 2 10 )/cos i, where D 10 is the distance to the source in units of 10 kpc and i is the inclination of the disc. The mean inner radii are R in,1 ∼ 1722 (cos i) −1/2 km and R in,2 ∼ 146 (cos i) −1/2 km (where subscripts 1 and 2 refer to the lower and higher temperature tracks, respectively), which correspond to ∼ 414 R S and ∼ 35 R S (where R S = 2GM/c 2 is the Schwarzshild radius, assuming M = 1.4 M ). Using the relation L = AσT 4 , we can also determine the emitting areas for each DISKBB component, as shown in Fig. 8. We note that, for the hot component, the size of the emitting area of the accretion disc is broadly constant (in units of R NS =10 km, ∆R ∼ 57 R NS ) for several observations. Instead, the emitting area for the cool component varies from R min ∼ 180 R NS to R max ∼ 1600 R NS . This would agree with the expectations from a local increase in the accretion rate (Poutanen et al. 2007). Using the formalism outlined in King & Lasota (2020) and assuming 20 <ṁ < 25 (which is reasonable for the observed luminosity), we can also estimate the spherisation radius R sph ∼ 70-87 R S . We found that the inner radius for the cool component is larger than the spherisation radius, that is R in,1 > R sph , Fig. 8: Left Y-axis: Emitting area versus luminosity for both the cool and hot DISKBB components (blue and orange points, respectively). Right Y-axis: Radius, in units of typical NS radius of 10 km, versus luminosity for both the cool and hot DISKBB components (blue and orange points, respectively). which suggests that the cool component takes both the emission from the cool disc and the contribution from the wind in account. The inner radius of the hot DISKBB is instead smaller than the spherisation radius (R in,2 < R sph ), indicating that this component is likely reproducing the super-Eddington inner accretion flow within R sph . Our approach has made use of phenomenological models in order to describe the long-term spectral evolution of the source. Future work will benefit from adopting physically motivated models which account for the thick nature of super-Eddington discs and Compton scattering through the disc atmosphere. Conclusions In this paper we have presented the analysis of the X-ray emission spectrum of NGC 1313 X-2, a pulsating ultraluminous Xray source, using all the available observations performed by XMM-Newton between 2000 and 2017. With the aim of characterising the spectral shape and its long-term variability, we have tested various spectral models for all observations. Since pulsations have been detected, we know that the compact object is a NS. In order to reach high luminosities (up to 10 40 erg s −1 in some observations), the compact object must be in a super-Eddington accretion regime. The spectral model that provides the best description of the XMM-Newton/EPIC data consists of two multi-colour disk blackbody components plus an exponential cutoff powerlaw, as in Walton et al. (2018c). Similar to previous works, we find that the hotter of the two thermal components dominates the 1-10 keV band, and the cooler one is more relevant in the energy band below 1 keV. Furthermore, we found that the inner radius of the hot component is smaller than the spherisation radius, while the characteristic radii associated with the cooler component are all larger than the spherisation radius. In the framework of super Eddington accretion, this is strong evidence that the hotter component describes the dominant emission from the innermost region of the disc. Instead, the cooler component accounts for the emission from a larger region of the disc, outside the spherisation radius and the characteristic wind launching radius. Finally, we argued that the cutoff powerlaw component comes from the accretion columns onto the NS. Most spectra show narrow residuals around ∼1 keV, which are likely associated with atomic emission and absorption lines produced by powerful outflows, as they are qualitatively similar to those seen in other ULXs that have been resolved and identified with the aid of dedicated high-resolution observations (see e.g. Pinto et al. 2020b). The luminosity and the temperature of each emission component evolves along L-T trends that deviate from those expected for some typical regimes such as the sub-Eddington thin disc of Shakura-Sunayev (L ∝ T 4 , for constant emitting area) or the advection-dominated disc (L ∝ T 2 ). In particular, we obtain power indexes of α H = 3.0 ± 0.35 and α C = −3.9 ± 1.0 for the hotter and the cooler components, respectively. This implies a super-Eddington accretion regime and suggests a geometry closer to the funnel than a thin disk for the inner regions due to intense radiation pressure. Optically thick winds are also likely responsible for the scatter seen in Fig. 6. This behaviour is qualitatively similar to the broadband spectral evolution seen in NGC 1313 X-1 and Holmberg IX X-1. The similar spectral evolution between these sources, where the flux above 1 keV increases significantly at higher luminosity, may indicate a common structure and evolution among archetypal ULXs. Fig. 1 1Fig. 1: XMM-Newton image of NGC 1313, which we obtained by combining all the data available from the 2017 EPIC-pn and MOS 1,2 observations. The 30 -radius circle around the ULX X-2 represents our default source extraction region. The red colour corresponds to 0.2-1 keV, green is for 1-2 keV, and blue is for 2-12 keV. observation identifier; (2) observation date (yyyy-mm-dd), (3) exposure time corrected for solar flares, (4) count rates in the 0.3-10 keV energy band. Fig. 2 : 2Left panel: Comparison between four spectra, from low (Obs.ID:0106860101), to intermediate (Obs.ID:0803990201 and Obs.ID:0782310101), and to high flux (Obs.ID:0150280401). Right panel: XMM-Newton/EPIC-pn spectra of the recent observations (2017 campaign). Fig. 3 : 30.3-10 keV long-term EPIC/pn light curve of NGC 1313 X-2 colour-coded according to the HR for all the XMM-Newton observations, with time bins of 1000 s. The HR was computed using the light curves in the soft [0.3-1.2 keV] and in the hard [1.2-10 keV] X-ray energy bands. Observations occur at different epochs: we removed the gaps between observations with grey-dashed lines for visual purposes. Fig. 4 : 4Left panel: XMM-Newton unfolded spectrum of observation 0803990201 of NGC 1313 X-2. The black points are EPIC-pn and the red points show the stacked EPIC-MOS1/2 data. We overlapped the best-fit DISKBB+DISKBB+CUTOFFPL model (black and red, solid lines) and its single components (dashed lines). Right panel: Residuals from a selection of models listed in Fig. 5 : 5Average HR (from the 1.2-10/0.3-10 keV energy bands) versus luminosity in linear space. The size of the markers correspond to the different values of the fractional variability (reported in Fig. 7 : 70.001-10 keV (i.e. bolometric) luminosity versus temperature for both the cool DISKBB (blue points) and hot DISKBB (orange points) components with free column density, N H (model: DISKBB+DISKBB). A. 1 : 1Best fitting spectral parameters of NGC1313 X-2 in different observations obtained with the absorbed DISKBB+DISKBB+CUTOFFPL modelParameter uncertainties were estimated at 68%. ( * ) Luminosity values (in units of 10 39 erg/s) are quoted for the unabsorbed model integrated over 0.3-10 keV. Table 1 : 1XMM-Newton observations of NGC 1313 X-2.Obs.ID (1) Date (2) Exposure time (3) (s) Count rates (4) (cts/s) Filter pn MOS1 MOS2 pn MOS1 MOS2 0106860101 2000-10-17 31638 29250 29183 0.2554 0.07654 0.07588 Medium 0150280101 2003-11-25 19173 12498 12501 0.712 0.2181 0.2584 Thin1 0150280201 2003-12-09 5648 3609 3622 - - - Thin1 0150280301 2003-12-21 10335 11971 11976 0.8365 0.2803 0.2795 Thin1 0150280401 2003-12-23 14098 15278 15293 0.9459 0.3081 0.3104 Thin1 0150280501 2003-12-25 15299 15262 15249 0.5249 0.1551 0.1510 Thin1 0150280601 2004-01-08 14757 15382 15377 0.4095 0.1277 0.1287 Thin1 0150280701 2003-12-27 16686 17779 17810 - - - Thin1 0150281101 2004-01-16 7036 8671 8676 0.3545 0.1117 0.1170 Thin1 0205230201 2004-05-01 3459 12270 12275 - - - Thin1 0205230301 2004-06-05 10036 11672 11674 0.9813 0.3181 0.32 Thin1 0205230401 2004-08-23 16137 17771 17776 0.2829 0.08781 0.09462 Thin1 0205230501 2004-11-23 14137 15769 15774 0.3190 0.09469 0.09883 Thin1 0205230601 2005-02-07 12437 14071 14074 0.9053 0.2974 0.3023 Thin1 0301860101 2006-03-06 19937 21570 21575 0.6510 0.1734 0.1963 Medium 0405090101 2006-10-15 121190 122456 122453 0.6437 0.1689 0.1576 Medium 0693850501 2012-12-16 123341 124921 124927 0.5395 0.1820 0.1819 Medium 0693851201 2012-12-22 123341 124921 124924 0.275 0.1036 0.1086 Medium 0722650101 2013-06-08 28841 30421 30426 0.1713 0.009892 0.01064 Medium 0742590301 2014-07-05 60040 61653 61624 0.3937 0.1074 0.1035 Medium 0742490101 2015-03-30 100041 98937 101625 0.07816 0.07223 0.07989 Medium 0764770101 2015-12-05 71941 73555 73524 0.2417 0.07106 0.07295 Thin1 0764770401 2016-03-23 30041 31653 31624 0.4102 0.1260 0.1261 Thin1 0782310101 2016-10-08 88041 89655 89626 0.5705 0.1863 0.1865 Medium 0794580601 2017-03-29 44542 46155 46127 0.3424 0.07281 0.1022 Medium 0803990101 2017-06-14 134142 133036 135725 0.2304 0.09635 0.09032 Medium 0803990201 2017-06-20 130841 132453 132425 0.3525 0.07631 0.08879 Medium 0803990301 2017-08-31 96686 - 91971 0.2380 - 0.07478 Medium 0803990401 2017-09-02 64008 - 63472 0.1077 - 0.09849 Medium 0803990701 2017-09-24 14500 9798 9797 0.2483 - 0.06602 Medium Table 2 : 2Best-fit parameters for the models tested in this work for Obs.ID:0803990201.TBABS*(DISKBB+DISKBB+CUTOFFPL) Model component Parameter Unit TBABS N H [10 22 cm −2 ] 0.28 ± 0.02 DISKBB T in [keV] 0.25 ± 0.02 norm 14 +7 −5 DISKBB T in [keV] 1.13 ± 0.04 norm 0.053 ± 0.007 CUTOFFPL PhoIndex 0.59 (fixed) HighECut [keV] 7.9 (fixed) norm [10 −5 ] 2.8 +0.4 −0.5 χ 2 /dof 314.91/259 TBABS*(DISKBB+DISKBB) Model component Parameter Unit TBABS N H [10 22 cm −2 ] 0.242 +0.014 −0.013 DISKBB T in [keV] 0.32±0.02 norm 4.3 +1.7 −1.2 DISKBB T in [keV] 1.36 +0.03 −0.02 norm 0.028 ± 0.002 χ 2 /dof 336.86/260 TBABS*(SIMPL*DISKBB) Model component Parameter Unit TBABS N H [10 22 cm −2 ] 0.158 +0.002 −0.004 SIMPL Γ 2.75 +0.07 −0.02 DISKBB T in [keV] 0.56 +0.01 −0.02 norm 0.62 +0.09 −0.08 χ 2 /dof 455.69/260 TBABS*(DISKBB+DISKPBB) Model component Parameter Unit TBABS N H [10 22 cm −2 ] 0.32 +0.03 −0.02 DISKBB T in [keV] 0.21 ± 0.02 norm 30 +37 −15 DISKPBB T in [keV] 1.57 +0.07 −0.05 p 0.57 ± 0.02 norm 0.008 ± 0.002 χ 2 /dof 321.32/259 Table 3 : 3Fractional variability measured using the average value of the EPIC-pn light curve segments for each observation with an exposure time of 40 ks and 1000 s bins.Obs.ID F var mean (%) HR mean 0405090101 6.06 ± 0.02 0.65 ± 0.03 0693850501 11.76 ± 0.07 0.62 ± 0.03 0693851201 0.748 ± 0.008 0.53 ± 0.04 0742590301 7.93 ± 0.07 0.61 ± 0.05 0742490101 3.26 ± 0.04 0.58 ± 0.07 0764770101 1.154 ± 0.002 0.49 ± 0.04 0782310101 12.17 ± 0.09 0.60 ± 0.07 0803990101 14.99 ± 0.04 0.60 ± 0.05 0803990201 12.93 ± 0.09 0.61 ± 0.05 0803990301 1.04 ± 0.04 0.52 ± 0.06 0803990401 1.91 ± 0.04 0.55 ± 0.06 0803990501 7.56 ± 0.16 0.55 ± 0.08 0803990601 4.79 ± 0.10 0.6 ± 0.1 reported by , we replaced the hotter DISKBB with a DISKPBB component (we tested the DISKBB+DISKPBB model). Considering only the DISKPBB component, we have obtained similar results. Comparing the same observations analysed in Bachetti et al. (Obs.ID1:0693850501 and Obs.ID2:0693851201), we found the values of p to be consistent within 2 σ (this work: p ID 1=0.62±0.03 and p ID2 =0.62 +0.18 −0.07 ; Bachetti et al.: p ID1 =0.58±0.01 and p ID2 =0.500 +0.006 * ). We also found consistent temperatures (this work: T in1 =1.52 +0.06 −0.04 and T in2 =1.15 +0.06 −0.07 ; Bachetti et al.: T in1 =1.56±0.06 and T in2 =1.27±0.05) Table Table B B.1: Best fitting spectral parameters of NGC1313 X-2 in different observations obtained with the absorbed DISKBB+DISKBB model.ObsID N H T 1 norm 1 T 2 norm 2 χ 2 /dof (10 22 cm −2 ) (keV) (keV) Article number, page 12 of 18 A. Robba et al.: Broadband X-ray spectral variability of the pulsing ULX NGC 1313 X-2 Acknowledgements. The authors would like to thank the anonymous referee, who provided useful suggestions for improving the final manuscript. 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[ "Maximum-Width Rainbow-Bisecting Empty Annulus *", "Maximum-Width Rainbow-Bisecting Empty Annulus *" ]
[ "Sang Won Bae \nDivision of Computer Science and Engineering\nKyonggi University\nKorea\n", "Sandip Banerjee \nInstitute of Informatics\nUniversity of Wrocław\nPoland\n", "‡ Arpita Baral \nDepartment of Computer Science and Engineering\nUniversity of Kalyani\nIndia\n", "Priya Ranjan ", "Sinha Mahapatra \nDepartment of Computer Science and Engineering\nUniversity of Kalyani\nIndia\n", "§ Sang ", "Duk Yoon \nDepartment of Service and Design Engineering\nSungshin Women's University\nKorea\n" ]
[ "Division of Computer Science and Engineering\nKyonggi University\nKorea", "Institute of Informatics\nUniversity of Wrocław\nPoland", "Department of Computer Science and Engineering\nUniversity of Kalyani\nIndia", "Department of Computer Science and Engineering\nUniversity of Kalyani\nIndia", "Department of Service and Design Engineering\nSungshin Women's University\nKorea" ]
[]
Given a set of n colored points with k colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axisparallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus A of a particular shape with maximum possible width such that A does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in O(n 3 ) time using O(n) space, in O(k 2 n 2 log n) time using O(n log n) space and in O(n 3 ) time using O(n 2 ) space respectively.
10.48550/arxiv.2305.09248
[ "https://export.arxiv.org/pdf/2305.09248v1.pdf" ]
258,715,242
2305.09248
11dc1b4c84af43ddfad5c2d5ec8ce3bc68aae7fb
Maximum-Width Rainbow-Bisecting Empty Annulus * Sang Won Bae Division of Computer Science and Engineering Kyonggi University Korea Sandip Banerjee Institute of Informatics University of Wrocław Poland ‡ Arpita Baral Department of Computer Science and Engineering University of Kalyani India Priya Ranjan Sinha Mahapatra Department of Computer Science and Engineering University of Kalyani India § Sang Duk Yoon Department of Service and Design Engineering Sungshin Women's University Korea Maximum-Width Rainbow-Bisecting Empty Annulus * Given a set of n colored points with k colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axisparallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus A of a particular shape with maximum possible width such that A does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in O(n 3 ) time using O(n) space, in O(k 2 n 2 log n) time using O(n log n) space and in O(n 3 ) time using O(n 2 ) space respectively. Introduction In the context of facility location most of the existing literature deals with the placement of the facilities (e.g. pipelines) among a set of customers (represented by points in R 2 ) but there are scenarios where the facilities are hazardous (e.g pipelines transporting toxic materials). In these scenarios the objective is maximizing the minimal distance between the hazardous facility and the given customers. Considering this situation, some problems have been studied in literature that computes a widest L-shaped empty corridor [8], a widest 1-corner corridor [13], a largest empty annulus [11]. For empty annulus we additionally impose another constraint that each of the two non-empty regions corresponds to two self-sustained smart cities, meaning that each of the smart cities is comprised of essential facilities of each type such as schools, hospitals, etc. This situation calls for the computation of two rainbow regions separated by an empty region where empty region contains hazardous facility and each color represents each essential facilities in each of the rainbow region. Given a set of n points in R 2 , each point colored with one of the k (k ≤ n) colors, a rainbow (or color spanning) region in R 2 contains at least one point of each color. Abellenas et al. first proposed algorithms for computing a smallest color-spanning axis-parallel rectangle [2], narrowest strip [2], circle [1] in O(n(n−k) log 2 k), O(n 2 α(k) log k) and O(kn log n) respectively. Later the time complexities to compute a smallest color-spanning rectangle and a narrowest color-spanning strip were improved to O(n(n − k) log k) and O(n 2 log n) by Das et al. [9]. Kanteimouri et al. [16] computed a smallest axis-parallel color-spanning square in O(n log 2 n) time. Hasheminejad et.al [15] proposed an O(n log n) time algorithm to compute a smallest color-spanning equilateral triangle. The shortest color-spanning interval, smallest color-spanning t squares and smallest color-spanning t circles have been studied by Banerjee et al. [7] and they have given some hardness and tractability results from parameterized complexity point of view. Acharyya et al. [3] computed a minimum-width color-spanning annulus for circle, equilateral triangle, axis-parallel square and rectangle in O(n 3 log n), O(n 2 k), O(n 3 + n 2 k log k) and O(n 4 ) time respectively. D. Báñez et al. [11] first studied the largest empty circular annulus problem and proposed an O(n 3 log n) time algorithm. They later improved the time complexity to O(n 3 ) time [12]. Bae et al. [6] proposed algorithms for computing a maximum-width empty axis-parallel square and rectangular annulus in O(n 3 ) and O(n 2 log n) time respectively. Recently Erkin et al. [5] studied a variant of a covering problem where the objective is to cover a set of points by a conflict free set of objects, where an object is said to be conflict free if it covers at most one point from each color class. In this paper, we mainly address three problems -(I) computing a maximum-width rainbowbisecting empty axis-parallel square annulus, (II) computing a maximum-width rainbow-bisecting empty axis-parallel rectangular annulus and (III) computing a maximum-width rainbow-bisecting empty circular annulus. For the last problem we also consider an additional case where the center of the empty circular annulus will lie on a given query line in the plane. Problem I is solved in O(n 3 ) time using O(n) space. Note that our Problem I generalizes the problem of computing a largest rainbow-bisecting empty L-shaped corridor. Hence, we also consider the corridor problem as a sub-case of Problem I and present an O(n log n)-time algorithm. This siginificantly improves the time complexity of the O(n 2 log n)-time algorithm for the widest empty axis-parallel L-shaped corridor problem [6]. For problems II and III, we propose algorithms that run in O(k 2 n 2 log n) time using O(n log n) space and in O(n 3 ) time using O(n 2 ) space respectively. Definition and Terminologies We are given a set P containing n points in R 2 where each point is colored with one of the given colors {1, . . . , k} and k ≤ n/2. For each color i ∈ [k], we assume that there are at least two points in P of color i. For any point p ∈ P , we denote its x-and y-coordinates, color by x(p), y(p) and α(p) respectively. For any two points p and q in P , x-and y-gap between p and q is defined as |x(p) − x(q)| and |y(p) − y(q)|, respectively. Let S be an axis-parallel square. The four sides of S are called the top, bottom, left, and right sides, respectively, according to their relative position in S. The center of S is defined as the intersection between two diagonals of S, and the radius of S is defined as half of its side length. The offset of S by a real number δ ≥ 0 is defined as a smaller axis-parallel square obtained by sliding each side of S inwards by δ. An axis-parallel square annulus (Figure 1(a)) A is the region between an axis-parallel square S out and its offset S in by a real number δ ≥ 0. We call S out , S in , and δ the outer square, inner square, and the width of the annulus, respectively. Note that S out and S in are concentric, so the center of S out and S in can be treated as the center of the annulus. We allow the outer and inner squares of an axis-parallel square annulus A to have one or more sides at infinity (±∞) which means the associated coordinate value of that side is infinite. For the rectangular annulus, we consider the definition of Mukherjee et al. [17]. An axis- Figure 1: (a) A RBSA of width δ with outer and inner squares S out and S in . (b) A RBCA with outer and inner disks C out and C in . For both annuli, c denotes the center. Figure 2: A RBRA with outer and inner rectangles R out and R in whose top-, bottom-, left-, right-widths are t, b, l, and r, respectively. parallel rectangular annulus is the region obtained by subtracting the interior of an axis-parallel rectangle R in from another axis-parallel rectangle R out such that R in ⊆ R out . We call R out and R in the outer rectangle and inner rectangle of the annulus, respectively. Consider a rectangular annulus A defined by its outer and inner rectangles, R out and R in respectively. By our definition, note that R out and R in defining annulus A do not have to be concentric, so that A may not be a symmetric shape. The top-width of A is the vertical distance between the top sides of R out and R in , and the bottom-width of A is the vertical distance between their bottom sides. Analogously, the left-width and right-width of A are defined to be the horizontal distance between the left sides of R out and R in and the right sides of R out and R in , respectively. Then, the width of A is defined to be the minimum of the four values: the top-width, bottom-width, left-width, and right-width of A. See Figure 2 for an illustration. δ S in S out (a) c C in (b) C out c A circular annulus A is the region between two concentric disks C out (outer disk) and C in (inner disk) where C in ⊆ C out . The width of a circular annulus A is defined to be the difference of the radii of its outer and inner disks (see Figure 1(b)). An annulus A is said to be rainbow-bisecting empty if it does not contain any point of P in its interior and divides P into two non-empty subsets such that each subset is a rainbow. One subset of P lies within or on the boundary of the inner square/rectangle/disk of A and the other subset of P lies outside or on the boundary of the outer square/rectangle/disk of A. We now refer to an axis-parallel square (resp. rectangle) as square (resp. rectangle) for simplicity. Any rainbow-bisecting empty square annulus, rainbow-bisecting empty rectangular annulus and rainbow-bisecting empty circular annulus are denoted by RBSA, RBRA and RBCA, respectively. Maximum-Width Rainbow-Bisecting Empty Square Annulus In this section, we compute a maximum-width RBSA from a given point set P on R 2 . Lemma 1 There is a maximum-width RBSA with the outer square S out and the inner square S in such that a pair of sides with the same relative position in S out and S in each contains a point of P , and one of the following three conditions is satisfied. • C 1 : The remaining sides of S out and S in are at infinity (Figure 3(a)). • C 2 : For the side of S out containing a point of P , one of two adjacent sides of S out contains a point of P . The remaining sides of S out are at infinity, and the corresponding sides of S in are also at infinity (Figure 3(b)). • C 3 : For the side of S out containing a point of P , the opposite side of S out contains a point of P (Figure 3(c)). Proof. We start by proving the fact that outer and inner squares of a maximum-width RBSA each contains a point of P on their boundaries. For a contradiction, we assume that there is no point on the boundaries of the outer square S out and the inner square S in of a maximum-width RBSA, A. Then, we can shrink S in until the boundary of S in hits a point in P , while keeping the center of A fixed. In this process no point goes outside from the inside of S in , so A remains empty. But the width of A increases which contradicts our assumption. Similarly, we can increase the width of A by extending S out while keeping the center of A fixed, a contradiction. So there must be at least one point of P lying on the boundary of S in and on the boundary of S out . Suppose that the sides of S in and S out that contain the points of P have different relative positions in S in and S out . Then, we can always get a RBSA with larger width by transforming S in and S out as in Figure 4. Therefore, the sides of S in and S out that contain the points of P have the same relative position. Without loss of generality, we assume that both top sides of S in and S out contain the points of P . S out S in S out S in S out S in (a) (b) (c) Next, we enlarge S in and S out . During the process, the top sides of S in and S out keep the points of P on the sides, and the width of A is fixed. The enlarging process will be finished when S out encounters a point of P . If there is no such a point, then A belongs to C 1 configuration. If the enlarging process is finished with a point of P on the bottom side of S out , then A belongs to C 3 configuration. If the enlarging process is finished with a point of P on the left (resp. right) side of S out , we can continue the enlarging process with the bottom and the right (resp. left) sides of S in and S out . If the enlarging process is finished with another point of P , then A belongs to C 3 configuration. If there is no such a point, then A belongs to C 2 configuration. Thus to find a maximum-width RBSA from a given point set P , we solve the above three configurations and propose different algorithms for handling each of them. As a part of preprocessing, we sort the points in P using their x-and y-coordinates, respectively. Note that any RBSA belonging to configuration C 1 maps to an empty strip such that regions on both side of the strip are rainbow. We call such RBSA a rainbow-bisecting empty strip, RBES. Now we discuss the solution technique for C 2 configuration. Observe solving this configuration corresponds to the problem of finding a widest rainbow-bisecting empty axis-parallel L-shaped corridor (RBLC). A RBLC is an axis-parallel empty L-shaped corridor which partitions the input point set P into two non-empty rainbow subsets. Note that one can solve the problem by constructing n 2 grid points for given n points and have the following trivial result. We design a non-trivial algorithm for computing a maximum-width RBLC. Without loss of generality, we only consider the case where the RBLC is pointing down and right, and its width is determined from the y-gap of two points in P (Figure 3(b)). Observe that we can handle all other cases analogously. Let P = {p 1 , p 2 , . . . , p n } be the given set of points, sorted in the ascending order of their y-coordinates, that is, y(p 1 ) ≤ y(p 2 ) ≤ · · · ≤ y(p n ). We imagine sweeping a horizontal line H upwards over the plane. It starts from the position y = y(p 1 ) and gradually moves upwards until all points in P are visited. When the line H sweeps through a point p i ∈ P , we decide the existence of a RBLC L such that the horizontal part of L is right above H, and its width is larger than the width of the widest RBLC found so far. To find such RBLC L, we use the following queries. Let Q a be the set of points above y(p i ) and Q b be the set of remaining points P \ Q a . (i) Boundary points query: Let w be the width of the widest RBLC found so far. First, we find the lowest point p j in Q a such that the y-gap between p i and p j is larger than w. Then y-gap between p i and p j becomes the width of L. Among the points whose y-coordinates are in between y(p i ) and y(p j ), we find the point p k with the largest x-coordinate. Observe that the horizontal part of L should be contained in the region bounded by p i , p j , and p k from below, above, and left, respectively ( Figure 5(a)). (ii) Rainbow range query: Let S b and S t are two staircase structures such that any empty axisparallel L-shaped corridor whose two corners are lying in the region above S b and below S t is a RBLC ( Figure 5(b)). After the boundary points query, we compute the intersection I t (resp. I b ) between the horizontal line y = y(p j ) and S t (resp. y = y(p i ) and S b ). The left (resp. right) side of the vertical part of L should be on the right (resp. left) of the intersection. (iii) Maximum x-gap query: After the rainbow range query, we find the maximum x-gap be- tween two consecutive points of Q b in range [max(x(I t ), x(p k )), x(I b )] , when the points of Q b are sorted with respect to the x-coordinates. If the maximum x-gap is larger than y-gap between p i and p j , then L exists. To answer the above queries we use the following data structures : • The point p j can be found in O(log n) time without any additional data structure. Let T be a one dimensional range tree built on the y-coordinate values of the points in P . Also each node v ∈ T stores the point with the maximum x-coordinate value among the points in the canonical subset of v. The tree T can be constructed in O(n) time using O(n) space [10], and finding the points with the maximum x-coordinate values can be done in O(n) time in total. Using this data structure, we can find the point p k in O(log n) time. • The corners of S b and S t are maintained in two balanced binary search trees T b and T t built on their y-coordinate values, respectively. We compute S b as follows, and S t can be computed similarly. Let c be a point at (−∞, −∞), and consider the region R c which is the intersection between two half spaces x ≥ x(c) and y ≤ y(c). Now we move c upward until R c becomes a rainbow set. Next, we move c to the right while R c is a rainbow set. When c is stopped, there should be a point p on the left side of R c , and we move c upward again until R c meets a point q such that α(p) = α(q). Then, we move c to the right again, while R c is a rainbow set. By repeating this process, we can get S b from the trace of c. As P is sorted with respect to the y-coordinates, the next point to be contained in R c (as c moves upward) can be found in O(1) time. When adding a new point to R c , we spend O(log n) time to keep the points in R c sorted with respect to the x-coordinates. Then we can find the next point to be removed from R c (as c moves to the right) in O(log n) time. In total, we can compute T b that stores the corners of S b in O(n log n) time using O(n) space. From T b and T t , rainbow range query can be answered in O(log n) time. • Let T i be a one dimensional range tree build on the x-coordinate values of the points in Q b . Also each node v ∈ T i stores the maximum x-gap of two consecutive points in the canonical subset of v. The structure T i can be computed from [a, b] such that a = (max(x(p i ), x(p j )) − r, y( )) and b = min(x(p i ), x(p j )) + r, y( )), where y( ) = (y(p i ) + y(p j ))/2 and r = (y(p j ) − y(p i ))/2. for a given range can be answered in O(log n) time by comparing O(log n) values including every gap between two consecutive canonical subsets. With the above data structures in hand, we have the following result. Lemma 3 The existence of a RBLC L can be determined (during the sweeping process) in O(log n) time such that the horizontal part of L is right above the sweeping line and it's width is larger than the width of the widest RBLC found so far. Our algorithm uses the result of Lemma 3 to obtain a maximum-width RBLC from P . Theorem 1 Given n points in R 2 where each one is colored with one of the k colors, a maximumwidth RBLC can be found in O(n log n) time using O(n) space. Proof. The sweeping line H starts from the position y = y(p 1 ) and gradually moves upwards. When the sweeping line H passes through a point p i , we determine the existence of a RBLC L such that the horizontal part of L is right above H and its width is larger than the width of the widest RBLC found so far. We repeat this until there is no such RBLC, and move H to y = y(p i+1 ). The width of L is determined by the y-gap between two points p i and p j where 1 ≤ i < j ≤ n, and both indices i and j do not decrease during the sweeping process. Therefore, the number of RBLCs we find during the sweeping process is O(n), and we can find a maximumwidth RBLC in O(n log n) time using O(n) space. Here we mention that our algorithm improves the time complexity for computing a widest empty axis parallel L corridor for a given set of n points in plane from O(n 2 log n) (Theorem 1 of [6]) to O(n log n). C 3 configuration: Any RBSA here is defined by two input points on both two opposite sides of its outer square. Assume that each of the top and bottom sides of the outer square of a RBSA contain a point of P . We design an algorithm to compute a maximum-width RBSA with the the top and bottom sides of its outer square containing points p i and p j , respectively, from all pairs of indices (i, j) with 1 ≤ i < j − 1 < n. We start this case with the following observation. Observation 1 For fixed points p i and p j which define the bottom and top sides of the outer square S respectively, the potential locations of the center of S lie on the line such that y( ) = (y(p i ) + y(p j ))/2 (Ref. Figure 6). Note that the outer square of a potential candidate annulus for fixed points p i and p j lies inside a rectangle R (Ref. Figure 6 ) such that x(left)= max{x(p i ), x(p j )} − 2r and x(right)= min{x(p i ), x(p j )} + 2r, where r = (y(p j ) − y(p i ))/2. We maintain color counter vectors to maintain the number of points of each color present outside and inside the horizontal strip defined by points p i and p j . Since the target annulus is a RBSA, therefore rectangle R must be a rainbow rectangle. We only consider such rectangles. For a fixed outer square S(c) with center c lies on [a, b] ⊂ , we compute the radius of the inner square S (c) by finding the farthest point from center c lying inside S(c). For this, we plot L ∞ distances from c to each point p in R along [a, b]. Bae et al. [6] showed that the upper envelope of the L ∞ distances has O(n) complexity and can be computed in O(n) time. From the centers of the annuli obtained by mapping the breakpoints of the upper envelope, we identify those annuli which are RBSA. Finally our algorithm outputs the one with the largest width. Considering all choices of p i and p j and the above discussion leads to the following result. We conclude the section with the following theorem. Maximum-Width Rainbow-Bisecting Empty Rectangular Annulus In this section, we discuss an algorithm to compute a maximum-width rainbow-bisecting empty rectangular annulus amidst a given point set P on R 2 . The following observation shows the existence of a maximum-width RBRA meeting certain conditions. Observation 2 There exists a maximum-width RBRA A with outer rectangle R out and inner rectangle R in satisfying the following conditions: (1) Each side of R out contains a point of P or lies at infinity; (2) Each side of R in contains a point of P . Proof. Consider any maximum-width RBRA A, and assume that the boundary of R out does not contain any input point. We enlarge each side of R out till each of them contains a point of P or they are at infinity. Similarly we shrink R in by pushing each side inwards until each side hits a point. Note that any two adjacent sides of R in can share a point at its corner. In this process the width of A is not decreased. By our construction, we create a new maximum-width RBRA from A that satisfies the conditions. From Observation 2, we can find a maximum-width RBRA by examining every rectangular annulus that satisfying the conditions in Observation 2. The outer rectangle R out of a rectangular annulus A can be determined by at most 4 points of P , and the inner rectangle R in of A becomes the minimum rectangle that contains the points P ∩ R out . This results in a trivial algorithm. Remark 2 For a given set of n points P , where each point in P is assigned a color from given k colors, a maximum-width RBRA can be reported in O(n 5 ) time. Figure 7: (left) A maximum-width RBRA of width w (right) A left-anchored maximum-width RBRA with uniform width. R out R in w R out w w w w R in R in A RBRA is called RBRA with uniform width, or simply uniform RBRA, when it has all four widths (top, bottom, left and right) equal. The following observation shows that we can construct a maximum width RBRA with uniform width from any maximum width RBRA. Observation 3 For any maximum-width RBRA, we can construct a maximum-width RBRA with uniform width satisfying the following characteristic: each side of its outer rectangle either contains a point of P or lies at infinity, and at least one side of its inner rectangle contains a point of P . Proof. For a maximum-width RBRA A, let A be a maximum-width RBRA that satisfies the conditions in Observation 2 created by transforming A according to the process in Observation 2. Let R out and R in be the outer and inner rectangles of A , and w be the width of A . Therefore every side of R out contains at least one point of P or lies at infinity. Now think that the width w comes from the horizontal distance of left sides (left-width) of A . We enlarge the remaining three sides of R in to form another rectangle R in , where R in ⊆ R out , and all the four widths (top, bottom, left and right) of the new annulus, A , thus formed are equal. The width of A is equal to w. Since R in ⊆ R in and thus A ⊆A , suggests A is also a RBRA. See Figure 7 for an illustration. As introduced in Bae et al. [6], we call a rectangular annulus A top-anchored (or, bottomanchorsed, left-anchored, right-anchored) if the following conditions are satisfied: (1) top (or, bottom, left, right, resp.) side of the outer rectangle of A contain a point of P or lies at infinity; (2) top (or, bottom, left, right, resp.) side of the inner rectangle of A contains a point of P . A maximum-width RBRA with uniform width has the following property. Observation 4 A maximum-width RBRA with uniform width can be either top-anchored, bottom-anchored, left-anchored or right-anchored (Figure 7(right)). In a nutshell our problem boils down to the problem of computing an anchored and uniform maximum-width RBRA. Here, we only discuss the case where the RBRA is "top-anchored". The other three cases can be similarly handled. A brief outline of our algorithm that computes a top-anchored and uniform RBRA of maximum-width is as follows: Consider the given set of points P = {p 1 , p 2 , . . . , p n }, sorted in decreasing order of their y-coordinates and X be an array of size k where X[i] indicates the number of points of color i. Consider any top-anchored and uniform RBRA A that satisfies the condition of Observation 3. Let p i ∈ P be the point lying on the top side of the outer rectangle of A. From the assumption on A, either the bottom side of the outer rectangle of A is at infinity or there is another point p j ∈ P for i < j ≤ n on it. Since A is top-anchored, there is a third point p k ∈ P on the top side of the inner rectangle of A. Note that the width of A is determined by the y-difference of p i and p k , that is, y(p i ) − y(p k ). Thus, the maximum width for top-anchored annuli is one among O(n 2 ) values {y(p i ) − y(p k ) | 1 ≤ i ≤ k ≤ n}. Initially we study the case where two points p i and p j on the top and bottom sides are fixed, and then discuss the situation where only a point p i on the top side is fixed. We discuss a decision algorithm when two points on the top and bottom sides of the outer rectangle are fixed, and use it as a sub-routine for solving the other case. Decision problem for fixed top and bottom sides Here we discuss a decision algorithm when two points p i and p j are fixed on the top and bottom sides of outer rectangle of the RBRA, where 2 ≤ i + 1 < j ≤ n. To represent the case when the bottom side of the outer rectangle lies at infinity, we use j = ∞. The decision problem DP ij (w) is defined as follows: Given: A positive real w > 0, two indices i and j where 2 ≤ i + 1 < j ≤ n, or j = ∞. Task : Decides the existence of a RBRA of width at least w and whose outer rectangle contains p i and p j on its top and bottom sides, respectively. The following observation holds on DP ij (w). Observation 5 If DP ij (w) is TRUE, then DP ij (w ) is TRUE for any w ≤ w. On the other hand, if DP ij (w) is FALSE, then DP ij (w ) is FALSE for any w ≥ w. To compute DP ij (w), we consider the set P ij := {p i+1 , . . . , p j−1 } containing points between y(p i ) and y(p j ) where 2 ≤ i + 1 < j ≤ n, and P i∞ := {p i+1 , . . . , p n }. The decision algorithm perform the y-range x-neighbor query operation [6] on points of P ij to evaluate DP ij (w). We present Algorithm 1 to answer the decision problem DP ij (w). The algorithm calls the yrange x-neighbor query twice. Also, if the algorithm decides that DP ij (w) is TRUE, then it also returns a corresponding RBRA, that is, a RBRA with uniform width w such that p i and p j on the top and bottom sides of its outer rectangle. Algorithm 1: Decision algorithm Input: A positive real w > 0, two indices i and j where 2 ≤ i + 1 < j ≤ n, or j = ∞. Output: DPij(w) and if DPij(w) is TRUE then the algorithm outputs a RBRA with uniform width w having pi and pj lying on the top and bottom sides of its outer rectangle, respectively. 1 if y(pi) − y(pj) < 2w then Next, we prove the correctness of Algorithm 1. Lemma 5 Algorithm 1 correctly computes DP ij (w) for any given w > 0 in O(n) time. Proof. The time complexity of Algorithm 1 depends on the search queries given in line numbers 3, 4, 8, 9 and 11. In line 3 and 4, the search queries can be done in O(log n) time [6]. Since the points are maintained in sorted order, the operations in lines 8, 9 and 11 can be done in O(n) time. For any horizontal gap of length at least w from L that defines the left sides of the RBRA we search for the corresponding right sides such that rainbow property is satisfied and right sides are in R. In line 11, we start searching for points p in and p out for the leftmost gap g l in set L. At the same time we maintain a count on the number of points of each color we seen so far, starting from x(l a ). At x(p in ) we have exactly k colors present. Similarly at x(p out ) we have exactly k colors present outside the region (including the boundaries) bounded by lines x(l b ), x(p out ), y(p i ) and y(p j ) from left, right, top and bottom respectively. This operation is repeated for all gaps in L and every time it advances to the next gap, the points p in and p out move forward from their previous positions. During the repetition, if Algorithm 1 finds a RBRA with uniform width w, it returns TRUE and the RBRA. Optimization Algorithm We now move to the case where we fix only a point p i ∈ P that defines the top side of the outer rectangle and compute a top-anchored and uniform RBRA A of maximum-width for the point set P . If p i lies on the top side of the outer rectangle of a top-anchored and uniform RBRA A, then the width of A should be in the set W = {y(p i ) − y(p k ) | i < k ≤ n}. Thus, we can find A by solving DP ij (w) with Algorithm 1 for every i + 1 < j and w ∈ W . There are a total of O(n 2 ) of DP ij (w) to solve, but we can find A by solving only O(n) of them. Lemma 6 For a point p i ∈ P , a top-anchored and uniform RBRA Aof maximum-width such that p i lies on the top side of the outer rectangle of A can be computed in O(n 2 ) time. Proof. For a fixed p i , there are O(n) candidate widths w of the annulus and O(n) candidate points p j which lie on the bottom side of the outer rectangle. We need to find the maximum width w * such that DP ij * (w * ) is TRUE for an index j * with i + 2 < j * . Let index j = i + 2 and width w = y(p i ) − y(p i+1 ) be the smallest possible values among the candidates, and solve DP ij (w) using Algorithm 1. If DP ij (w) is TRUE, then we set w as the next smallest candidate width w = y(p i ) − y(p i+2 ) and solve DP ij (w) again. We repeat this process until DP ij (w) becomes FALSE. From Observation 5, DP ij (w ) is FALSE for all w > w, so we don't need to solve DP ij (w ) for w > w. Thus we increase j by one, and repeat the process to find the maximum width such that DP ij (w) is TRUE. We repeat the entire process until j = ∞ or w = y(p i ) − y(p n ), and we can find the maximum width w during the process. The suggested process only increase both w and j, so we call Algorithm 1 at most O(n) times. Therefore it takes O(n 2 ) time to report a top-anchored and uniform RBRA of maximum-width for p i . We have n possible choices for p i , so we can find a top-anchored and uniform RBRA of maximum-width in O(n 3 ) time, and the other three cases (bottom-anchored, left-anchored, or right-anchored) of the maximum-width RBRA are handled similarly. We conclude the section with the following result. Theorem 3 Given a set P of n points in the plane where each one is assigned a color from given k colors, a maximum-width RBRA with respect to P can be computed in O(n 3 ) time and O(n) space. An Improved Approach In Section 4.1, we present a O(n) time decision algorithm to compute a maximum-width RBRA, where the top and bottom sides of its outer rectangle pass through two fixed points. Here we show that Algorithm 1 can be implemented in O(k 2 log n) time. Let a be any w-gap from the set L (described in Step 8 of Algorithm 1). We are interested in the leftmost w-gap b from the set R such that the corresponding inner rectangle is rainbow and the number of points outside of the annulus is maximum. Note that the inner rectangle defined by a and any w-gap b in R to the right of b is rainbow, while the number of points is less lying outside of the outer rectangle of the annulus. On the other hand, if we find the rightmost w-gap a in L such that the corresponding inner rectangle defined by a and b is rainbow, then either a = a or a lies to the right of a. Note that in the latter case we can ignore the w-gap a, since the annulus defined by w-gaps (a , b) is better than that defined by (a, b) in the sense of the number of points lying outside the outer rectangle of the annulus. Minimal Rainbow Intervals Consider the set L of points P ij ∩ [x(p l ), min{x(p i ), x(p j )} + w] × [y(p j ), y(p i )]. Then, L denotes the set of w-gaps in x-coordinates of L. Similarly R be the set of points P ij ∩[max{x(p i ), x(p j )}− w, x(p r )] × [y(p j ), y(p i )]. For our purpose we only consider the x-coordinates of the points in P ij and assume that the points are projected down on the x-axis. Let a be any point in L. We define r(a) as the leftmost point in R such that P ij ∩ [a, r(a)] is rainbow. Similarly for any point b in R, define l(b) as the rightmost point in L such that P ij ∩[l(b), b] is rainbow. Note that as a moves to the right, r(a) either moves to the right or stays at the same position. Similarly it holds for l(b). We call an interval [a, b] as minimal rainbow interval with a ∈ L and b ∈ R, if b = r(a) = r(l(b)) and a = l(b) = l(r(a)). In the following we give some properties of minimal rainbow interval. Recall that α(p) denotes the color of a point p, and let c [Q] denotes the number of points of color c in a point set Q. We are interested in the w-gaps that are generated in between minimal rainbow intervals. In the following lemma we present a bound on the number of "relevant" w-gaps. More specifically, for a given p i and p j , a w-gap is relevant if w ∈ L or w ∈ R. Lemma 7 The number of relevant w-gaps is at most O(k). Proof. Consider any w-gap A in L. Also assume that A is between the left endpoints of minimal rainbow intervals [a 1 , b 1 ] and [a 2 , b 2 ], i.e., A lies between a 1 and a 2 . Therefore, the leftmost wgap b in R such that the inner rectangle defined by (A, B) is rainbow must lies somewhere to the right of b 2 . If there lies another w-gap A between a 1 and a 2 that is to the right of A, we can simply ignore A. In a nutshell, we consider only the rightmost w-gap to the left of the left endpoint of each minimal rainbow interval and the leftmost w-gap to the right of the right endpoint of each minimal rainbow interval. Hence, the number of these "relevant" w-gaps is at most 2k. Modified Decision Algorithm Here we discuss the new decision problem DA new that computes DP ij (w). To compute all minimal rainbow intervals, first we start with the leftmost point a ∈ L, compute b = r(a), and then compute l(b). Therefore, [l(b), b] = [a 1 , b 1 ] denotes the first minimal rainbow interval. If a 1 is the rightmost point in L, then [a 1 , b 1 ] is the only minimal rainbow interval. Otherwise, we search for the second minimal rainbow interval [a 2 , b 2 ]. Since there is no point of color α(a 1 ) in [a 1 , b 1 ] other than a 1 , therefore α(b 2 ) = α(a 1 ). So, b 2 can be easily found by finding the closest point of color α(a 1 ) in P ij to the right of b 1 . Also then, a 2 = l(b 2 ). The above process continues until we reach the rightmost point in L. C([A, B]), the number of points of each color that lie outside the annulus can be found. After the completion of the above steps if there exixts a RBRA of w-width, DA new reports it. In order to perform the above operations efficiently the following data structures are being used. • Consider the 1-dimensional range tree X ij (c), where c ∈ [k], for the x-coordinates of points in P ij for counting for color c. Also at each node v of the tree, we store size(v), the number of leaves descended from v. The data structure X ij (c) can be constructed using storage O(|P ij (c)|) [10], where |P ij (c)| represents the number of points of color c in P ij . We maintain O(k) such 1D range trees. • We preprocess the points of P ij into a 2D range tree (with fractional cascading) T ij as follows: Let x 1 < x 2 < . . ., be the x-coordinates of points in P ij in sorted order. Here we map each x i into a 2D point (x i , x i+1 − x i ). We construct T ij on these 2D points. T ij can be constructed using storage O(|P ij | log |P ij |) [10]. With the above data structures in hand we can compute all minimal rainbow intervals and perform the mentioned operations on them as follows: • For any point a ∈ L (resp. for each point b in R), we find r(a) (resp. l(b)) in O(k log n) time from k given 1D range trees. Thus in O(k log n) time we compute a minimal rainbow interval. • To compute the leftmost w-gap to the right of a minimum rainbow interval [a, b], more specifically the leftmost w-gap to the right of b, we use a 3-sided range query defined by the region Now we discuss how to report a maximum-width top-anchored RBRA when a point p i ∈ P is fixed on the top side of outer rectangle of RBRA. Assume any point p i ∈ P lies on the top side of outer rectangle of RBRA. For a fixed p i , we consider the widths that lies in the set W i := {y(p i ) − y(p k ) | i ≤ k ≤ n} and report the maximum among them. The optimization algorithm is described in section 4.2. Having p i fixed, consider all possible values of j = i+2, . . . , n and also j = ∞. As j increases, more points are included in P ij and therefore we update the structures by inserting a point in each of them. Since we use O(k) tree data structures, the updates can be done in O(k log n) time. Therefore in this case we can report a maximum width top-anchored RBRA in O(k 2 n log n) time. Considering all values of p i , along with the above discussion we have the following result. R = [x(b), min(x(r 1 ), x(r 2 )) − w] × [w, ∞], Theorem 4 For a given set of n points in the plane where each point is assigned a color from given k colors, a maximum-width top-anchored RBRA can be computed in O(k 2 n 2 log n) time and O(n log n) space. Maximum-width Rainbow-Bisecting Empty Circular Annulus In this section, we compute a maximum-width RBCA from a given point set P on R 2 . Here we assume no four input points are concyclic. Lemma 9 A maximum-width RBCA A is defined by four points from P resulting one of the following potential configurations-(i) Type 1 (resp. Type 2) -C out (resp. C in ) has three points of P and C in (resp. C out ) contains one point of P and (ii) Type 3 -Each of C in and C out are defined by two points of P . Proof. For a contradiction, we assume that a maximum-width RBCA A is defined by three points. First, we deal with the case such that C in has two points p 1 and p 2 and C out has a point p 3 of P . Since C in has two points p 1 and p 2 , the center c of the annulus A will lie on their perpendicular bisector, say L. Let s be the center of the disk having points p 1 , p 2 and p 3 on its boundary. The width of annulus A will increase if we move the center c in a direction away from s along L while A remains empty and both the regions inside and outside C in and C out are rainbow, respectively. Thus A cannot be a maximum-width RBCA. The remaining cases can be treated similarly, so we prove that a maximum-width RBCA must be defined by four input points. Adopting the technique of D.Báñez et al. [12], transform the points ∈ P from R 2 to R 3 using paraboloid lifting (Figure 8) resulting the problem of computing the largest empty circular annulus problem in R 2 being mapped to the largest empty slab problem (LESP) in R 3 . Largest Empty Slab Problem: (LESP) Given: A set S of n points in R 3 Task : Find a widest empty slab where a slab through S is the open region of R 3 that is bounded by 2 parallel planes intersecting the convex hull of S. The width of the slab is the distance between the bounding planes. A candidate annulus (in R 2 ) of Type 1 (or 2) & Type 3 is being mapped (in R 3 ) to empty slabs of Type C 31 and C 22 respectively ( Figure 9). Finally, the optimal solution is obtained by tracking the slabs of Type C 31 and C 22 which span k colors on both sides of it. Generate candidate empty slabs of Types C 31 , C 22 using topological sweep [4,14] on the arrangement H of dual planes (each plane has a color) corresponding to the points in primal. During the sweep we maintain a count on the number of planes of each color in [k] present above and below for each edge of the current planar cut of each plane. Once the initial planar cut for plane k ∈ H is determined, we initialize arrays namely U P k [1 : k], LOW k [1 : k] and CO k [1 : n − 1] following sequence of edges in N k [1 : n − 1], where N k [1 : n − 1] is a list of pairs of indices indicating the lines delimiting each edge of the current planar cut for plane k . The array CO k [i] = (a, b), indicates the number of colored planes above and below each edge of N k [i] of current planar cut for k where U P k and LOW k help to initialize CO k . We get each CO k in O(n) time. As the sweep performs an elementary step, the incoming edges are swapped at the corresponding vertex in each planar cut and CO k is updated following new N k in O(1) time. The above discussion along with lemma 4 [12] leads to the following result. Theorem 5 Given a set of n points in R 2 , each one is colored with one of the k colors, the maximum-width RBCA problem can be solved in O(n 3 ) time and O(n 2 ) space. We extend the above idea to compute a maximum-width RBCA whose center lies on a given line L in the plane. We have the following lemma. Lemma 10 A maximum-width RBCA A whose center lies on a given line L is defined by three points of P where C in (resp. C out ) contains two points of P and C out (resp. C in ) contains one point of P . Proof. Similar as described in lemma 9. We conclude this section with the following corollary. Corollary 1 A maximum-width RBCA A whose center lies on a given query line L in R 2 can be computed in O(n 2 ) time and space. Conclusion In this paper, we have dealt with the problem of computing a maximum-width rainbow-bisecting empty annulus for three type of basic geometric objects among a set of given points on the plane. First, we propose an O(n 3 )-time and O(n)-space algorithm for computing a maximum-width rainbow bisecting empty axis-parallel square annulus. Next, we compute a maximum width rainbow bisecting empty axis-parallel rectangular annulus in O(k 2 n 2 log n)-time and O(n log n)space. Finally, we propose a O(n 3 )-time and O(n 2 )-space algorithm for the maximum-width rainbow bisecting empty circular annulus problem. One of the interesting open problem in the square and rectangular setting is to study the arbitrary orientation case. Another obvious direction is to improve the running time. Further, the study of any non-trivial lower bound even for the basic problem that is computing the maximum width annulus (rectangle, square or circle) among a set of points on the plane would really be an interesting one. Figure 3 : 3Three possible configurations of a maximum-width RBSA. The dashed sides represent the sides at infinity. (a) C 1 Configuration (b) C 2 Configuration (c) C 3 Configuration Figure 4 : 4An example of transformation of inner and outer squares of a RBSA (red dashed) to get another RBSA (black filled) with larger width. Lemma 2 2A maximum-width RBES of a sorted set of points can be computed in O(n) time using O(n) space. Remark 1 1Given a set of n points in the plane, each colored with one of the given k colors, a maximum-width RBLC can be computed in O(n 3 ) time using O(n) space. Figure 5 : 5(a) The horizontal part of L should be contained in the gray region bounded by p i , p j , and p k . (b) Two corners of L should be contained in the region bounded above S t and below S b . Figure 6 : 6T i−1 by adding new point p i in O(log n) time, and the size of T i is bounded by O(n) [10]. The maximum x-gap query Centers of candidate outer squares in R lie in the horizontal segment Lemma 4 4A maximum-width RBSA corresponding to C 3 configuration can be computed in O(n 3 ) time and O(n) space. Proof. For a fixed p i and p j , the centers of all possible empty annuli can be computed in O(n 3 ) time [6] and O(n) space. In another linear scan we can find the RBSA with the maximum-width. The result follows by taking all values of i and j. Theorem 2 2Given n points in R 2 where each one is colored with one of the k colors, a maximumwidth RBSA can be computed in O(n 3 ) time using O(n) space. Proof. A maximum-width RBSA results from any of the three configurations from Lemma 1. For C 1 configuration, the maximum one is reported in O(n) time using O(n) space from Lemma 2 if the points are already sorted. For C 2 configuration, we find a maximum-width RBLC in O(n log n) time using O(n) space from Theorem 1. For C 3 configuration, we compute a maximum-width RBSA in O(n 3 ) time using O(n) space from Lemma 4. Hence the statement followed. Search the rightmost and the leftmost points from the sets Pij ∩ [−∞, x(pi)] × [y(pi) − w, y(pi)] and Pij ∩ [x(pi), +∞] × [y(pi) − w, y(pi)] respectively. Let these two points be l1 and r1 respectively.4 Search the rightmost and the leftmost points from the sets Pij ∩ [−∞, x(pj)] × [y(pj), y(pj) + w] and Pij ∩ [x(pj), +∞] × [y(pj), y(pj) + w] respectively. Let these two points be l2 and r2 respectively. 5 Let p l be the rightmost one in {l1, l2} and pr be the leftmost one in {r1, r2}.6 if min{x(pi), x(pj)} < x(p l ) < max{x(pi), x(pj)} or min{x(pi), x(pj)} < x(pr) < max{x(pi), x(pj)} then 7 Return FALSE.8 Search all the horizontal gaps of length at least w from the points lying in the range (x(p l ), min{x(pi), x(pj)} + w). Let L denote the set of such gaps. // Boundary points are included in the search range. 9 Search all the horizontal gaps of length at least w from the points lying in the range (max{x(pi), x(pj)} − w, x(pr)). Let R denote the set of such gaps. // Boundary points are included in the search range. 10 if L = ∅ and R = ∅ then 11 Start with the leftmost horizontal gap (say, g l := x(la) − x(l b )) from set L. Maintain the number of points of each color on the left side of g l . From x(la) start moving forward until a point say, pin comes such that the points from x(la) to x(pin) forms a rainbow region. Move forward from pin until a point say, pout is reached such that if moved further the region outside the region bounded by x(l b ), x(pout), y(pi) and y(pj) is not rainbow. Search a w width horizontal gap between x(pin) and x(pout) and check if it lies within (max{x(pi), x(pj)} − w, x(pr)). Repeat the procedure for all the gaps in L until such a gap is found that defines the right sides of RBRA. Return TRUE if such gap exists, and RBRA of uniform width w. Otherwise Return FALSE. 12 else 13 Return FALSE. (i) If [a, b] is a minimal rainbow interval, then either α(a)[a,b] = 1 or a is the rightmost point in L; α(b) [a,b] = 1 or b is the leftmost point in R. In other words, a is the only point of that particular color in [a, b] unless a is the rightmost point in L, and so is b. (ii) No two minimal rainbow intervals are nested, and any two minimal rainbow intervals overlap. (iii) There cannot be two minimal rainbow intervals [a 1 , b 1 ] and [a 2 , b 2 ] such that a 1 and a 2 are in the same color, or b 1 and b 2 are in the same color. Also from (ii) we can order all minimal rainbow intervals from left to right, [a 1 , b 1 ], [a 2 , b 2 ],. . . , [a m , b m ] with a 1 < a 2 < . . . < a m and b 1 < b 2 < . . . < b m . Since the colors of a 1 , a 2 . . . , a m (resp. b 1 , b 2 . . . , b m ) must be all distinct, implies that m can be at most k. DA new have exactly same lines 1 to 7 as stated in Algorithm 1. The new algorithm then computes all possible minimal rainbow intervals in between y(p i ) and y(p j ) as follows. Consider the points in L. For each point a in L, we compute r(a) as follows. For each color c ∈ k, compute the closest point in P ij of color c to the right of a and then find the farthest point b from them. If b is in R, then r(a) = b; otherwise r(a) is the leftmost point in R by definition. Similarly, for each point b in R, we compute l(b). 13Consider any minimal rainbow interval[a, b]. We store k counters for[a, b]. Let C([a, b]) = #_1([a, b]), #_2([a, b]), . . . , #_k([a, b]) be the color counter vector, consisting of the number of points in P ij of each color. For each minimal rainbow interval, we store its color counter vector. For each [a, b] ∈ [a 1 , b 1 ], [a 2 , b 2 ],. . . , [a m , b m ], we repeat the following steps: (i) First we compute the rightmost w-gap A to the left of a, if any. Find the leftmost wgap B to the right of b, if any. (ii) Compute the color counter vector of the region between A and a and of the region between b and B. Also compute the counter C([A, B]) for points between A and B. (iii) From where r 1 is the leftmost point from the region [x(p i ), ∞] × [y(p i ) − w, y(p i )], and r 2 is the leftmost point from the region [x(p j ), ∞] × [y(p j ), y(p j ) + w]. Here we find the leftmost point from region R. From T ij we can find the leftmost point in R in O(log n) time. Any w-gap corresponds to a point in R and can be used to construct an empty annulus of width w.• For any rectangular region R, the color counter vector C(R) of R can be computed in O(k log n) time from k given 1D range trees.Similarly, the color counter vector of the region between A and a (resp. between b and B) and C([A, B]) (mentioned in step (ii)) can be computed in O(k log n) time.• Using C([A, B]) we can find the number of points outside the empty annulus in O(k) time.Lemma 8 DA new computes DP ij (w) for any given w > 0 in O(k 2 log n) time using O(n log n) space. Figure 8 :Figure 9 : 89Paraboloid transformation of the input points. Left side maps right side. (a) A C 31 empty slab defined by 2 parallel planes π 1 and π 2 . (b) A C 22 empty slab. Proof. DA new takes O(log n) time[6] to compute lines 3 and 4. After getting TRUE in line 7 of Algorithm 1, DA new computes all minimal rainbow intervals. Since there are O(k) minimal rainbow intervals in total, it takes O(k 2 log n) time. For any minimal rainbow interval DA new construct a w-width RBRA if exists in O(k log n) time and O(n log n) space using the data structures mentioned above. Considering all minimal rainbow intervals the total running time is O(k 2 log n). The farthest color Voronoi diagram and related problems. Manuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma, Belén Palop, Vera Sacristán, Proc. 17th EuroCG. 17th EuroCGManuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma, Belén Palop, and Vera Sacristán. The farthest color Voronoi diagram and related problems. In Proc. 17th EuroCG 2001, pages 113-116, 2001. Smallest color-spanning objects. Manuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma, Belén Palop, Vera Sacristán, Algorithms -ESA 2001, 9th Annual European Symposium. Aarhus, DenmarkManuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma, Belén Palop, and Vera Sacristán. Smallest color-spanning objects. In Algorithms -ESA 2001, 9th Annual European Symposium, Aarhus, Denmark, August 28-31, 2001, Proceedings, pages 278-289, 2001. Minimum width color spanning annulus. Ankush Acharyya, C Subhas, Sasanka Nandy, Roy, Theor. Comput. Sci. 725Ankush Acharyya, Subhas C. Nandy, and Sasanka Roy. Minimum width color spanning annulus. Theor. Comput. Sci., 725:16-30, 2018. Topological sweeping in three dimensions. Efthymios Anagnostou, G Vassilios, Leonidas J Polimenis, Guibas, Proceedings of the International Symposium on Algorithms. the International Symposium on Algorithms450Efthymios Anagnostou, Vassilios G. Polimenis, and Leonidas J. Guibas. Topological sweep- ing in three dimensions. In Proceedings of the International Symposium on Algorithms, SIGAL, volume 450 of Lecture Notes in Computer Science, pages 310-317, 1990. Selecting and covering colored points. Esther M Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Matthew J Katz, Joseph S B Mitchell, Marina Simakov, Discrete Applied Mathematics. 250Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Matthew J. katz, Joseph S. B. Mitchell, and Marina Simakov. Selecting and covering colored points. Discrete Applied Mathematics., 250:75-86, 2020. Maximum-width empty square and rectangular annulus. Arpita Sang Won Bae, Priya Ranjan Sinha Baral, Mahapatra, Computational Geometry: Theory and Applications. 96101747Sang Won Bae, Arpita Baral, and Priya Ranjan Sinha Mahapatra. Maximum-width empty square and rectangular annulus. Computational Geometry: Theory and Applica- tions, 96:101747, 2021. Color spanning objects: Algorithms and hardness results. Sandip Banerjee, Neeldhara Misra, Subhas C Nandy, Discrete Applied Mathematics. 280Sandip Banerjee, Neeldhara Misra, and Subhas C. Nandy. Color spanning objects: Algo- rithms and hardness results. Discrete Applied Mathematics., 280:14-22, 2020. Widest empty L-shaped corridor. Siu-Wing Cheng, Inform. Proc. Lett. 586Siu-Wing Cheng. Widest empty L-shaped corridor. Inform. Proc. Lett., 58(6):277 -283, 1996. Smallest color-spanning object revisited. Sandip Das, Partha P Goswami, Subhas C Nandy, Int. J. Comput. Geometry Appl. 195Sandip Das, Partha P. Goswami, and Subhas C. Nandy. Smallest color-spanning object revisited. Int. J. Comput. Geometry Appl., 19(5):457-478, 2009. Computational Geometry: Algorithms and Applications. Otfried Mark De Berg, Cheong, Mark Marc Van Kreveld, Overmars, Springer-Verlag TELOSSanta Clara, CA, USA3rd editionMark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, Santa Clara, CA, USA, 3rd edition, 2008. The largest empty annulus problem. José Miguel Díaz-Báñez, Ferran Hurtado, Henk Meijer, David Rappaport, Joan Antoni Sellarès, Int. J. Comput. Geom. Appl. 134José Miguel Díaz-Báñez, Ferran Hurtado, Henk Meijer, David Rappaport, and Joan Antoni Sellarès. The largest empty annulus problem. Int. J. Comput. Geom. Appl., 13(4):317-325, 2003. Locating an obnoxious plane. José Miguel Díaz-Báñez, Mario Alberto López, Joan Antoni Sellarès, Eur. J. Oper. Res. 1732José Miguel Díaz-Báñez, Mario Alberto López, and Joan Antoni Sellarès. Locating an obnoxious plane. Eur. J. Oper. Res., 173(2):556-564, 2006. On finding a widest empty 1-corner corridor. José Miguel Díaz-Báñez, Mario Alberto López, Joan Antoni Sellarès, Inform. Proc. Lett. 985José Miguel Díaz-Báñez, Mario Alberto López, and Joan Antoni Sellarès. On finding a widest empty 1-corner corridor. Inform. Proc. Lett., 98(5):199 -205, 2006. Topologically sweeping an arrangement. Herbert Edelsbrunner, Leonidas J Guibas, J. Comput. Syst. Sci. 381Herbert Edelsbrunner and Leonidas J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38(1):165-194, 1989. Computing the smallest color-spanning equilateral triangle. Javad Hasheminejad, Payam Khanteimouri, Ali Mohades, Proc. 31st EuroCG. 31st EuroCGJavad Hasheminejad, Payam Khanteimouri, and Ali Mohades. Computing the smallest color-spanning equilateral triangle. In Proc. 31st EuroCG 2015, pages 32-35, 2015. Computing the smallest color-spanning axis-parallel square. Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza Kazemi, Algorithms and Computation -24th International Symposium, ISAAC 2013, Proceedings. Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, and Mohammad Reza Kazemi. Computing the smallest color-spanning axis-parallel square. In Algorithms and Computation -24th International Symposium, ISAAC 2013, Proceedings, pages 634-643, 2013. Minimum-width rectangular annulus. Joydeep Mukherjee, Arindam Priya Ranjan Sinha Mahapatra, Sandip Karmakar, Das, Theoret. Comput. Sci. 508Joydeep Mukherjee, Priya Ranjan Sinha Mahapatra, Arindam Karmakar, and Sandip Das. Minimum-width rectangular annulus. Theoret. Comput. Sci., 508:74-80, 2013.
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[ "Striation lines in intermittent fatigue crack growth in an Al alloy", "Striation lines in intermittent fatigue crack growth in an Al alloy" ]
[ "Anniina Kinnunen \nDepartment of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland\n", "Ivan V Lomakin \nDepartment of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland\n", "Tero Mäkinen \nDepartment of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland\n\nNOMATEN Centre of Excellence\nNational Centre for Nuclear Research\nA. Soltana 705-400Otwock-ŚwierkPoland\n", "Kim Widell \nDepartment of Mechanical Engineering\nAalto University\nP.O. Box 1100000076Aalto, EspooFinland\n", "Juha Koivisto \nDepartment of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland\n", "Mikko J Alava \nDepartment of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland\n\nNOMATEN Centre of Excellence\nNational Centre for Nuclear Research\nA. Soltana 705-400Otwock-ŚwierkPoland\n" ]
[ "Department of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland", "Department of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland", "Department of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland", "NOMATEN Centre of Excellence\nNational Centre for Nuclear Research\nA. Soltana 705-400Otwock-ŚwierkPoland", "Department of Mechanical Engineering\nAalto University\nP.O. Box 1100000076Aalto, EspooFinland", "Department of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland", "Department of Applied Physics\nAalto University\nP.O. Box 1110000076Aalto, EspooFinland", "NOMATEN Centre of Excellence\nNational Centre for Nuclear Research\nA. Soltana 705-400Otwock-ŚwierkPoland" ]
[]
Fatigue failure of crystalline materials is a difficult problem in science and engineering, and recent results have shown that fatigue crack growth can occur in intermittent jumps which have fat-tailed distributions. As fatigue crack propagation is known to leave markings -called striations -on the fracture surface, the distances between these should also have fat-tailed distributions, if the crack propagation is intermittent. Here, we combine macroscale crack tip tracking in fatigue crack growth measurements of aluminum 5005 samples with post-mortem scanning electron microscopy imaging of the striation lines. We introduce two different methods for extracting the striation line spacing from the images. What we find is a similar distribution of striation spacings as jump sizes using one of our methods, but the average striation spacing does not correlate with the crack growth rate. We conclude that we observe avalanche-like crack propagation, reflected in both the macroscale crack tip tracking as well as the analysis of the fracture surfaces. Our results show that the fracture surfaces can be used to study the intermittency of fatigue crack propagation and in development of crack-resistant materials. The advantages and disadvantages of the two methods introduced are discussed.arXiv:2305.07460v1 [cond-mat.stat-mech] 12 May 2023
10.1103/physrevmaterials.7.053602
[ "https://export.arxiv.org/pdf/2305.07460v1.pdf" ]
258,586,235
2305.07460
8232bc03228b116ab30921a1240a0f87da0b8d66
Striation lines in intermittent fatigue crack growth in an Al alloy Anniina Kinnunen Department of Applied Physics Aalto University P.O. Box 1110000076Aalto, EspooFinland Ivan V Lomakin Department of Applied Physics Aalto University P.O. Box 1110000076Aalto, EspooFinland Tero Mäkinen Department of Applied Physics Aalto University P.O. Box 1110000076Aalto, EspooFinland NOMATEN Centre of Excellence National Centre for Nuclear Research A. Soltana 705-400Otwock-ŚwierkPoland Kim Widell Department of Mechanical Engineering Aalto University P.O. Box 1100000076Aalto, EspooFinland Juha Koivisto Department of Applied Physics Aalto University P.O. Box 1110000076Aalto, EspooFinland Mikko J Alava Department of Applied Physics Aalto University P.O. Box 1110000076Aalto, EspooFinland NOMATEN Centre of Excellence National Centre for Nuclear Research A. Soltana 705-400Otwock-ŚwierkPoland Striation lines in intermittent fatigue crack growth in an Al alloy (Dated: May 15, 2023) Fatigue failure of crystalline materials is a difficult problem in science and engineering, and recent results have shown that fatigue crack growth can occur in intermittent jumps which have fat-tailed distributions. As fatigue crack propagation is known to leave markings -called striations -on the fracture surface, the distances between these should also have fat-tailed distributions, if the crack propagation is intermittent. Here, we combine macroscale crack tip tracking in fatigue crack growth measurements of aluminum 5005 samples with post-mortem scanning electron microscopy imaging of the striation lines. We introduce two different methods for extracting the striation line spacing from the images. What we find is a similar distribution of striation spacings as jump sizes using one of our methods, but the average striation spacing does not correlate with the crack growth rate. We conclude that we observe avalanche-like crack propagation, reflected in both the macroscale crack tip tracking as well as the analysis of the fracture surfaces. Our results show that the fracture surfaces can be used to study the intermittency of fatigue crack propagation and in development of crack-resistant materials. The advantages and disadvantages of the two methods introduced are discussed.arXiv:2305.07460v1 [cond-mat.stat-mech] 12 May 2023 I. INTRODUCTION Fatigue is an important physics problem on multiple scales, ranging from the atomistic to the failure of engineering structures. Fatigue failure of ductile materials [1], such as metals, can in laboratory conditions be studied by tracking the macroscopic crack growth using various methods, but in engineering applications this is rarely a possibility. Instead one might have to rely on the analysis of post-mortem fracture surfaces [2] to gain useful insights into the fatigue failure processes. However the stochastic nature of some microscopic observables raises questions about the predictability of crack growth. An obvious question is then: can one connect the statistics of microscopic observables to the behavior of the macroscopic crack growth rate? The macroscopic crack growth rate can be observed with various methods and one can then fit crack growth laws, such as the Paris-Erdogan law [3] da dN = C∆K m(1) where a is the crack length, N the number of loading cycles, C a material-(and loading-) specific constant, K the stress intensity factor (SIF) range, and m the material-(and loading-) specific crack growth exponent. The SIF range can be written in terms of the peak SIF ∆K = (1 − R)K max using the stress ratio R = K min /K max . The power-law form of Eq. 1 points to the direction of apparent self-similarity [4,5] of fatigue crack growth. * Corresponding author: [email protected] The determination of the crack length and therefore the crack growth rate is in most cases difficult, and the methods used are either prone to errors or reduced to consider only one corner of the crack. Fractography provides an alternative look into the same problem, from the viewpoint that the topography of a post-mortem fracture surface is a characteristic of the sample microstructure and the test conditions [6], such as the different stages of crack propagation [7][8][9][10]. Extraction of features and characteristics of the fracture surface, so called quantitative fractography [11], enables the study of the sizes, shapes, orientations and other measurable values related to the features. For fatigue the most important surface features are the periodic traces -first time reported as a "platy patterns" on the fracture surface [12] observed at a high magnification using scanning electron microscopy (SEM). These traces we originally called slip bands by Thompson and Wadsworth [13] and later striations by Nine and Kuhlmann-Wilsdorf [14]. The striations are lines generated by the advancement of the fracture line and they are generally suggested as an experimental technique to probe the properties of the fatigue crack propagation. The general view has been that fatigue crack growth (especially in the Paris regime where Eq. 1 holds) is fairly regular and the mean distance between striations has a well-defined characteristic value corresponding to the macroscopic crack growth rate -indicating a oneto-one correspondence between striations and loading cycles, which has been clearly shown in an aluminum alloy using program loading [15] as well as in many other studies [16][17][18]. This view is also corroborated by acoustic emission studies [19] where the waveforms of acoustic events were observed to be nearly identical. Similarly to the general characteristics of fatigue crack growth -for example the Paris exponent mthe striation characteristics are also strongly influenced by many different factors. The loading conditions have been shown to have an effect, for example the stress ratio influences the height of striations [20] as well as their distance [15]. Similarly, small changes in microstructure [21] have been shown to affect the striation morphology and distances between them. The basic reference model for striation generation is crack tip plastic slip. During the loading phase the crack is opened by normal stress which at the crack tip generates plastic slip activity along two symmetrical directions predicted by fracture mechanics. In this phase the crack tip gets blunted and grows by material decohesion associated to dislocation flow into the tip or generated by stress concentration. As this occurs through plastic deformation, upon unloading the blunted crack tip is squished, but a new free surface remains ahead of the former crack with a new sharp tip. It is this step-by-step process of blunting and re-sharpening during each cycle that leaves on the crack path the kind of markings that we call striations [22]. An extension of this approach was proposed by Laird [23] who has given a rather different interpretation of striation formation based on the so called plastic relaxation of the crack tip based on the hypothesis of plastic collapse of the crack tip during the unloading and closure phase that leads to tip concavity. Later Forsyth [24] was able to classify two main types of striations: the ductile and the brittle one. Ductile striations lay on different individual planes corresponding to single grains that macroscopically form, all together, a plateau normal to the maximum tensile stress direction. They are called ductile because the material ahead of the crack tip undergoes plastic deformations that produce the typical curved arrays by which they advance on the fracture surface. Brittle striations, instead, develop always on crystallographic planes, usually (100) planes and appear as concentric circles departing from the initiation site, quite often brittle inclusions. This gives brittle striations the typical flat appearance without any apparent (macroscopic) plastic deformation. A characteristic feature of brittle striations is the uniform, flat and annual ring -like propagation surface that does not propagate in single crystals but on crystallographic planes that are cleavage planes. However there has also been recent imaging studies that have indicated intermittency [25,26] and heterogeneity [27] related to fatigue crack growth. Especially, when tracking the crack tip during the experiment [25,26], the jumps in the crack length have been observed to have fat-tailed distributions. If the accepted view of one striation corresponding to one loading cycle is true, one should then find -in materials and loading conditions which exhibit these fat-tailed crack tip jump distributions -a similar fat-tailed distribution of distances between striations. Although the striations are thought to correspond to loading cycles one-to-one, several studies have shown striation spacings higher than the corresponding crack growth rate [28,29] or the striation spacing growing much slower than the crack growth rate [29][30][31], at least for slower crack growth rates. This can naturally be just an effect of a limited striation distance detection accuracy, although intermittency in terms of local crack advancement arrests has been proposed [32]. Alternatively, rough fracture surfaces deviating from the simple model of striation formation could produce surface patterns where the ridges do not directly correspond to crack arrest locations. Similar problems of reconciling rough surfaces and periodic striations have also been encountered in geophysics in the context of ridges in surface topography [33]. In this paper we perform fatigue crack growth (FCG) experiments on an Al alloy and study in detail the jumps in the crack propagation. We then study the fracture surfaces of these samples with a focus on the distances between striation lines, and how these correlate with the crack growth rate. Two methods of extracting the distances between striations are presented -one which focuses on the areas of the fracture surface exhibiting clear striation markings, and one which takes the whole imaged fracture surface into account. II. MATERIAL AND METHODS A. Microstructure The material used in this study was 5005 aluminium alloy, which was provided by Alumeco Ltd. as a sheet of thickness 5 mm in a strain-hardened and partially annealed state corresponding to H24 temper. Samples for microstructural studies were cut by electrical discharge machining (EDM) and polished using a Struers Tegramin polishing machine with a final 0.3 µm OP-S suspension. To reveal the grain structure of the material Barker' anodizing in 1.8 % fluoboric acid water solution with 0.25 A/cm 2 and 30 V DC applied for 30 s was used. Further optical microscopy in polarized light was conducted using a Nikon Epiphot Inverted Metallurgical Microscope. It showed ( Fig. 1) an elongated grain structure in the rolling (RD) and transverse directions (TD). One can see that grain size is about 300, 100 and 50 µm in rolling, transverse and axial directions (AD) respectively. B. Fatigue testing The FCG measurements were performed using standard compact tension (CT) specimens with a thickness of 5 mm and W equal to 50 mm, cut with final notch shaping using EDM. The orientation of the sample is such that the crack propagates in the transverse direction and the loading occurs in the rolling direction, as showed in Fig. 1. The tests were performed using a MTS 858 hydraulic fatigue testing machine with the loading waveform being sinusoidal with a frequency of 10 Hz. Four different loading conditions were used and three experiments corresponding to each condition were made. First the effect of the stress ratio R was studied using a constant force amplitude with a maximum force F max = 1500 N and R values 0.1, 0.3, and 0.5. The effect of the force amplitude was studied using the fourth loading condition: R = 0.1 and F max = 1300 N. The tests were performed as described by the ASTM E647 standard [34] and the SIF K and SIF range ∆K determined accordingly. The yield stress of the material is 155 MPa so the standards requirement for predominantly elastic behavior is fulfilled up to crack lengths of a = 0.6W = 3.0 cm. The crack length measurements were performed optically using the experimental setup shown in Fig. 2a. The sample was imaged from the side using a Canon EOS R digital camera with an imaging frequency of 0.25 Hz, corresponding to 40 loading cycles between images. The size of the obtained images was 6720 px × 4480 px, corresponding to a resolution of 4.5 µm per pixel. Before the experiments, the sample surface was polished to a mirror-like condition using a 1 µm diamond Struers DP- Suspension as the final step. To illuminate the crack a ring LED lamp creating an analog of dark field imaging was mounted on Canon MP-E 65mm f/2.8 macro lens (×1-×5 magnification) attached to the camera. To track the advancement of the crack tip an image processing algorithm (used also in Ref. [26]) was developed and implemented in MATLAB software. The crack tip is defined as the edge pixel of the contrast area in the crack tip region defined based on the previous image. The contrast image was obtained as an outcome of a binarization procedure using local background intensity as the threshold value. Crack tip position was tracked relative to a stationary surface defect to exclude mea- surement distortions brought on by the specimen shift. Sequential processing of the images allows one to track the crack length a as a function of cycle number N (resulting curve shown in Fig. 2b). The accuracy of this tracking method is of the order of one pixel, and this small noise is filtered by requiring that the crack length increases monotonously. This is done by constructing a monotonic upper and lower envelope for the signal and taking their average to correspond to the crack length a. For determining the crack tip advancement per cycle da/dN a moving average of the raw crack length values is taken, to smoothen the signal, after which numerical differentiation is applied (see Fig. 2b for comparison between the raw and averaged curves). From the raw crack tip position a we also extract the crack tip jumps ∆a, which are just the difference in the crack tip position between consecutive images divided by the number of cycles between the images, which here is 40. This normalization is done just to make the jump size values comparable to the da/dN and does not reflect an actual improvement in the tracking resolution. In simple terms, ∆a denotes the jumps seen in the crack tip and da/dN denotes the same signal, but is sufficiently smoothened to enable the plotting of the Paris curves. This is illus-trated in Fig. 2c where both of these are plotted. In the beginning of the test the values of ∆a are much higher, as during many cycles no crack advancement is observed. The fitting of the jump size distributions, as well as later the striation spacing distributions, is done using maximum likelihood estimation [35]. C. Fractography Three fatigue samples were picked for the fractography studies, two with F max = 1500 N (R = 0.1 and R = 0.5) and one with F max = 1300 N (R = 0.1). We performed SEM imaging of the post-mortem fracture surfaces of the specimens. The SEM images were recorded using JEOL JSM-7500FA microscope operated at 15 kV, and with an Everhart-Thornley secondary electron detector to reveal a surface roughness contrast. To study the evolution of the striation line structures as a function of the crack length a, the following procedure was followed: primary images were taken at intervals of around 500 µm with ×1k magnification, and after a visual examination of the images, three regions of interest containing striation lines were recorded with a higher magnification of ×3k. This procedure is illustrated in Fig. 3a. Roughness We compute a roughness measure of the SEM images at length scale L as the root-mean-square deviation of the image intensity w(L) = (I − I L ) 2 L(2) where · L denotes an average over the length scale L. We do this in both the spatial directions to yield w x and w y , which are shown in Fig. 4. It is important to note that SEM images do not correspond to a height map of the fracture surface and thus the roughness measure of Eq. 2 is not an actual surface roughness. The values measured as SEM image intensity actually look more like the local slope or height variation of the fracture surface [36]. However the roughness measure still holds valuable information about the structure of the fracture surface. As the roughness for small length scales seems to scale as a power-law, we fit a relation w ∝ L ζ to the data. Additionally we see to which value the roughness saturates to on large length scales (by taking the average of w over the latter half of our available length scales) and compute the correlation length ξ by seeing at which length scale the fitted power-law would achieve this value. The process is visually illustrated in Fig. 4a. As the crack propagation direction (the direction perpendicular to the striation lines) in the SEM images doesn't exactly coincide with the y-direction -e.g. the lines drawn in Fig. 3a do not align with the y-direction -these directions are somewhat arbitrary. We can however utilize the roughness information to mitigate this problem, by rotating the image by an angle θ and recomputing w x and w y . We notice that the angle θ corresponding to the angle where the crack propagation (on average) aligns with the y-direction seems to be the one that minimizes the ratio w y /w x . This angle is computed for each length scale L and we take the average over the intermediate length scales (from 20 to 200 pixels, corresponding to from 0.6 µm to 6.3 µm). This observation and the aligned images are utilized in the automatic striation extraction method explained later. Manual striation extraction The first method for striation extraction is the manual method presented here (similar to the one used in Ref. [28]). As the crack advancement direction does not always coincide with the y-direction of the images, and there are other features present in the images in addition to the striation lines, we have manually marked the parts in the images containing striations. In practice this means drawing lines on the images (shown in Fig. 3a) and performing a peak-finding procedure (shown in Fig. 3b) to yield the positions of the striation lines, which lay perpendicular to the drawn lines. The peak finding is done for the inverted intensity, as we are tracking the depressions on the sample surface. The distance between striations is then directly computed from the distances between the peaks, yielding a set of values corresponding to each image (which in turn correspond to a crack length a). In total, 35865 striation line measurements along fracture surfaces of the specimens were made. Automatic striation extraction As the goal of the present study is to see if the distances between striation lines follow the same fat-tailed distributions as observed for the jumps in the crack tip, the manual tracking method presented might be problematic. It is feasible that in selecting the parts of the image to be considered, the human tendency is to select the parts where the striation line spacing is fairly regular, yielding a mismatch between the crack tip jump results and the striation spacing. This type of issue related to selection of areas and length scales has in other contexts been dubbed the "phenomenological fallacy" [33]. To try to get rid of this possible issue, we have also implemented an automatic striation extraction method. It involves first rotating the image so that the crack propagation direction (on average) aligns with the y-direction in the images. Then a simple Canny edge detector [37] algorithm is employed (again on the inverted image intensity, as we are trying to track the depressions) to yield the striation lines shown in Fig. 5. We then go through each column of pixels in the image and extract the striation distances as the distance between these lines generated by the edge detection. This method of striation distance determination has some obvious issues with multiple counting of the same distances, but this choice was made because the structures resulting from edge detection have a much more complicated structure than just straight lines. III. RESULTS A. Fatigue testing The fatigue testing results align with previous results reported in Ref. [26] for aluminum 1050 alloy. In the Paris plots (Fig. 6a) we see the Paris-Erdogan law holding for each of the loading conditions in the region from K max = 6.5 MPa √ m to K max = 13 MPa √ m with exponent values slightly over three. For SIF values less than this we see the typical threshold regime behavior of rapidly decreasing crack growth rate, and above this the crack growth rate increases quickly as the critical SIF is approached. The almost linear change in the exponent m as a function of the stress ratio R (see inset of Fig. 6a) is much smaller than in Al 1050 alloy and the exponent does not seem to tend to zero at the creep limit R → 1. The change in the maximum force from 1500 N to N has a negligible effect on the exponent m. When looking at the crack tip jump size distributions p(∆a) (Fig. 6b) we see fat-tailed distributions spanning around two orders of magnitude. They can be interpreted, similarly as in the Al 1050 alloy [26], as powerlaws with an exponential cutoff p(∆a) ∝ ∆a −γ exp − ∆a ∆a 0 (3) where γ is an exponent around 2 and ∆a 0 denotes the cutoff scale. The solid lines in Fig. 6b show maximum likelihood fits of the cutoff scales ∆a 0 (for a fixed γ = 2) for each of the loading conditions. One can clearly see (Fig. 6c) that the differences in the jump size cutoff are very small. B. Roughness When the images are rotated so that the crack propagation occurs (on average) in the y-direction, using the aforementioned method, we observe (see Fig. 4a) for small distances a power-law scaling w ∝ L ζ where the exponent ζ is initially around 0.5 and decreases to around 0.4 with increasing a (see Fig. 4b). This is significantly lower than the values of the actual roughness exponent observed for cracked aluminum alloys [38][39][40]. There is very little difference between the exponent values in the two directions ζ x and ζ y , although initially ζ x is slightly larger. This power-law scaling is observed up to a correlation length ξ, which is approximately constant for the whole experiment (see Fig. 4c). C. Striations The striations we observe (see Fig. 3a) are typical ductile striations [22] residing in channel-like areas separated by ridges. By comparing the striation morphology at the beginning of the experiment (the images corresponding to a = 14.69 mm in Fig. 3a) and in the end of the experiment (the images corresponding to a = 29.15 mm in Fig. 3a) one sees that the channels seem to get wider as the experiment progresses, as seen previously in aluminum alloys [41]. However, when comparing the automatically extracted striation spacings in the x and y-directions we see just a linear dependence. The striation spacings extracted manually do not follow a fat-tailed distribution, as can be seen in Fig. 7. Instead they follow fairly closely an exponential distribution of the form p( ) ∝ exp(− / 0 ), where the scale parameter 0 aligns well with the cutoff scale ∆a 0 of the crack tip jump size distributions, i.e. the manual extraction method manages to capture striations comparable in size to the exponential tail of the optically measured crack tip jumps -the largest jumps. When the automatic extraction is performed, one indeed sees a fat-tailed distribution of striation spacings (see Fig. 7), spanning around two orders of magnitude, as do the crack tip jump sizes. We have done a maximum likelihood fit to Eq. 3 with a fixed γ = 2 and this distribution fits the data reasonably well. Similarly to the crack tip jump sizes and the manually extracted striation spacings, the effect of loading conditions on the the cutoff scale is very small. In the very end of the tail of the distribution (sizes of several micrometers) the discrepancies between the distributions are larger. This is due to the statistics -in these bins there are very few datapoints. When looking at the evolution of the microscopic observables -the cutoff scale 0 from the automatically extracted striation spacings, the mean striation spacing extracted using both the manual and automatic methods, and the correlation length in the crack propagation direction ξ y -one can clearly see (Fig. 8) that they stay approximately constant for the whole duration of the ex-periment. During the same experiment, the macroscopic crack growth rate varies around four orders of magnitude. If the one-to-one correlation between striation spacings and the macroscopic crack growth rate held, one would expect the curves to have the same slope in this log-logplot. Three of these microscopic observables ( 0 from automatic extraction, from manual extraction, and ξ y ) have roughly the same value -corresponding roughly to the macroscopic crack growth rate at the end of the experiment -and the from the automatic extraction has a slightly smaller value due to the automatically extracted striation spacings spanning a larger range of values. IV. CONCLUSIONS Starting with crack propagation, comparing the FCG results with similar ones for Al 1050 alloy [26], we find that the change in the Paris law exponent is much smaller with increasing R. We suggest that this difference can be attributed to a decrease in the sample plasticity, as Al 5005 is as an alloy significantly more brittle than Al 1050. Additionally we note that a change in the Paris exponent makes the usual models [42][43][44] for the effect of the stress ratio R not effective here, as they only apply a R dependent prefactor to the SIF value, therefore not changing the slope of the curves whereas here the slopes change. The statistical analysis of the crack tip advancement on a microscopic scale (down to a few microns) shows a fat-tailed distribution of crack tip jump sizes. The distribution can be modelled as a power-law with an exponential cutoff where the exponent is around 2 for all the loading conditions. The changes in the cutoff scale between different loading conditions are too small to make meaningful distinctions but this seems logical with the changes in the Paris curves also being small. This exponent 2 also agrees with the Al-1050 results, hinting at a degree of universality, but much more statistics would be needed for definitive conclusions. We can not for example exclude some other fat-tailed distributions, such as the the streched exponential. However, a plausible explanation for this exponent can be found from previous analysis of aluminum fracture surfaces [36] where the distribution of microcrack sizes was found the be power-law distributed with an exponent around 2. If one assumes the coalescence of these microcracks as the primary crack growth mechanism [45], this would then lead to a crack tip jump distribution with the same exponent. On the fractography side we study the roughness extracted from the images, which on small scales scales as a power-law of the length scale with an exponent slightly below 0.5. This exponent value also decreases during the experiment. When interpreting these results, one should however take into account that we are not measuring the surface height, only the SEM image intensity. We then introduce two methods of striation spacing extraction: the manual and the automatic one. The manual extraction method focuses on the features that can clearly be identified as striations and the automatic one considers all depressions on the fracture surface. The two extraction methods agree roughly statistically on the striation spacings when the spacing is above 1 µm. The manual one neglects the smaller distances between striations, that seem to be at least statistically captured by the automatic method. The automatic method also produces a striation spacing distribution which roughly matches the one seen for the crack tip jump sizes. However none of the microscopic quantities measured (the striation spacings or the correlation length associated with roughness) capture features that would correspond to the macroscopic crack growth rate spanning multiple orders of magnitude. The only microscopic quantity where any significant change during the experiment can be seen is the small length scale power-law exponent ζ of the roughness. If one accepts that the spacing between striation lines does not evolve during the experiment, one could envisage a mechanism of striation channel widening to account for the changes in the crack growth rate. The motivation for this approach comes from the clearly different striation morphology when comparing the beginning and end of the experiment. However, the results of our automatic tracking do not support this mechanism but the limited field of view of our SEM imaging also does not allow for a full characterization of this phenomenology. One should also note that this type of behavior would deviate from the constant shape or statistics of avalanche fronts observed for other crack avalanching systems in non-fatigue loading [46][47][48]. We have shown that the striation spacings in intermittent crack growth have fat-tailed distributions, in accordance with the one-to-one correspondence between striations and loading cycles. The manual extraction method only captures the striations with a well-defined mean value -corresponding to the largest jumps -but the automatic one also shows a plethora of features with narrower spacings. These statistically match the crack tip jump sizes, but it is a matter of terminology if these features should actually be called striations or apparent striations [32,49]. One should note that the images captured represent only a part of the fracture surface and clear striation-like markings are seen only on parts of the images. Generally crack propagation seems to be a much more complex phenomenon than just crack advancement lines with a well-defined spacing, as illustrated by the inconsistency between the striations and the macroscopic crack growth rate. On this rough and complicated fracture surface, the observed ridges might not correspond to crack arrest locations implied by the simplified model for striation formation. Further work should be done to see if the observed universal features of intermittent crack growth in fatigue extend to a wider variety of materials and to explore the effects with much better statistics. It would be interesting to see if the area fraction of the fracture surface exhibiting striation-like markings is correlated with the intermittency of crack propagation. One should note that in Al alloys increasing the Mg content further is known to introduce dynamic strain aging, which might significantly complicate the crack propagation dynamics. The validity of the striation line extraction methods introduced here should also be verified with other materials. By performing direct measurements of the fracture surface topography the possible connection to the microcrack size distribution should be explored. FIG. 1 . 1The grain structure of the 5005 alloy used in this study, showing elongated grains in the rolling and transverse directions. The plane of crack propagation is also indicated. FIG. 2 . 2a) A representation of the imaging setup used in the fatigue tests, showing the position of the camera in relation to the sample, and a schematic of the crack tip tracking procedure. b) The resulting raw crack tip positions as a function of the number of cycles, and the result of averaging used to yield the crack tip advancement per cycle da/dN . The inset includes a zoomed-in view, showing the jumps in the crack position. c) The jumps in the crack tip position (blue) and the crack tip advancement per cycle (orange) as a function of the number of cycles. FIG . 3. a) SEM images obtained at a = 15.69 mm and a = 29.15 mm showing the primary image and the three secondary images. The lines on the images correspond to the lines along which the striations are tracked in the manual extraction method. b) The profiles «a» and «b» corresponding to the respective lines in panel a, and the results of the peak finding (triangles) used in the manual extraction method to extract the striation lines. FIG . 4. a) An example of the surface roughness extracted from one image in the two perpendicular directions wy and wy as a function of the length scale L, when the image is rotated so that the crack propagation occurs in the y-direction. The solid black line corresponds to the fitted power-law (for wy) and the dashed line to the saturation value. b) The power-law exponent ζ extracted from the roughness curves as a function of the crack length a. c) The correlation length ξ extracted from the roughness curves as a function of the crack length a. FIG. 5 . 5An example of the automatic striation extraction. The top image is the raw (rotated) SEM image and the bottom one the result after striation extraction. The distance between striations is then the distance between lines in the bottom image in the y-direction, computed for each pixel column of the image. FIG. 6 . 6a) The Paris curves for each of the loading conditions and the exponent fits (black lines) according to Eq. 1 to each of them in the Paris regime. b) The probability distributions of the crack advancement jump sizes ∆a for each of the loading conditions. The lines (with corresponding colors) are maximum likelihood fits to Eq. 3 with a fixed exponent γ = 2. c) The fitted values of the exponent m (Eq. 1) as a function of the stress ratio R. d) The fitted cutoff sizes of the crack advancement jumps ∆a0 (Eq. 3) as a function of the stress ratio R. FIG. 7 . 7The distributions of three quantities: crack advancement jump size ∆a and the distance between striations measured with manual and automatic tracking. The three plots correspond to different loading conditions, a) F = 1500 N and R = 0.1, b) F = 1500 N and R = 0.5, and c) F = 1300 N and R = 0.1. FIG. 8 . 8The Paris curve of one experiment (R = 0.1, Fmax = 1500 N) with the the evolution of the microscopic quantities as a function of the SIF range superimposed. The quantities are the average striation spacing determined by the manual and automatic extraction methods, the cutoff scale fitted to the automatically extracted striation spacings 0 and the correlation length ξy in the crack propagation direction determined by the roughness analysis. ACKNOWLEDGMENTS M.J.A. and T.M. acknowledge support from the European Union Horizon 2020 research and innovation programme under grant agreement No 857470 and from European Regional Development Fund via Foundation for Polish Science International Research Agenda PLUS programme grant No MAB PLUS/2018/8. 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[]
[ "Information Spectrum Converse for Minimum Entropy Couplings and Functional Representations", "Information Spectrum Converse for Minimum Entropy Couplings and Functional Representations" ]
[ "Yanina Y Shkel [email protected] \nSchool of Computer & Communication Sciences\nEcole Polytechnique Fédérale de Lausanne (EPFL\nSwitzerland\n", "Anuj Kumar \nSchool of Computer & Communication Sciences\nEcole Polytechnique Fédérale de Lausanne (EPFL\nSwitzerland\n", "Yadav \nSchool of Computer & Communication Sciences\nEcole Polytechnique Fédérale de Lausanne (EPFL\nSwitzerland\n" ]
[ "School of Computer & Communication Sciences\nEcole Polytechnique Fédérale de Lausanne (EPFL\nSwitzerland", "School of Computer & Communication Sciences\nEcole Polytechnique Fédérale de Lausanne (EPFL\nSwitzerland", "School of Computer & Communication Sciences\nEcole Polytechnique Fédérale de Lausanne (EPFL\nSwitzerland" ]
[]
Given two jointly distributed random variables (X, Y ), a functional representation of X is a random variable Z independent of Y , and a deterministic function g(·, ·) such that X = g(Y, Z). The problem of finding a minimum entropy functional representation is known to be equivalent to the problem of finding a minimum entropy coupling where, given a collection of probability distributions P1, . . . , Pm, the goal is to find a coupling X1, . . . , Xm (Xi ∼ Pi) with the smallest entropy Hα(X1, . . . , Xm). This paper presents a new information spectrum converse, and applies it to obtain direct lower bounds on minimum entropy in both problems. The new results improve on all known lower bounds, including previous lower bounds based on the concept of majorization. In particular, the presented proofs leverage both -the information spectrum and the majorization -perspectives on minimum entropy couplings and functional representations.
10.48550/arxiv.2305.05745
[ "https://export.arxiv.org/pdf/2305.05745v1.pdf" ]
258,587,904
2305.05745
373750942e2493a1d75257a084e62cca759f7fd4
Information Spectrum Converse for Minimum Entropy Couplings and Functional Representations Yanina Y Shkel [email protected] School of Computer & Communication Sciences Ecole Polytechnique Fédérale de Lausanne (EPFL Switzerland Anuj Kumar School of Computer & Communication Sciences Ecole Polytechnique Fédérale de Lausanne (EPFL Switzerland Yadav School of Computer & Communication Sciences Ecole Polytechnique Fédérale de Lausanne (EPFL Switzerland Information Spectrum Converse for Minimum Entropy Couplings and Functional Representations Given two jointly distributed random variables (X, Y ), a functional representation of X is a random variable Z independent of Y , and a deterministic function g(·, ·) such that X = g(Y, Z). The problem of finding a minimum entropy functional representation is known to be equivalent to the problem of finding a minimum entropy coupling where, given a collection of probability distributions P1, . . . , Pm, the goal is to find a coupling X1, . . . , Xm (Xi ∼ Pi) with the smallest entropy Hα(X1, . . . , Xm). This paper presents a new information spectrum converse, and applies it to obtain direct lower bounds on minimum entropy in both problems. The new results improve on all known lower bounds, including previous lower bounds based on the concept of majorization. In particular, the presented proofs leverage both -the information spectrum and the majorization -perspectives on minimum entropy couplings and functional representations. The Functional Representation Lemma (FRL) [1] states that it is possible to decompose a random variable X into an arbitrarily correlated random variable Y and a residual random variable Z, independent of Y . More precisely, given any pair of random variables (X, Y ) ∼ P XY , there exists a random variable Z, independent of Y such that X is a deterministic function of (Y, Z). FRL has been widely used in proving various results in multi-user information theory [2]- [4], privacy and secrecy [5], and entropic causal inference [6], [7]. Several properties of functional representations have been studied in the literature. One such property of functional representations is the mutual information I(X; Z). Bounds on I(X; Z) have been studied in [4] in the form of a strong functional representation lemma. This problem has also been studied in [8] and [9] with applications to one-shot channel simulation with unlimited common randomness. It has been applied to prove the result on constrained min-max remote prediction [10] and to prove the asymptotic achievability in Gelfand-Pinsker's Theorem [11]. A closely related line of work is the setting of privacy funnel function [12]- [16]. Here, the goal is to maximize I(X; Z) while enforcing the Markov chain Y ↔ X ↔ Z. The motivation is to have perfect privacy for the information Y while realizing maximum information about X, see [5], [15], [16] for connections with FRL. Another important property, which is the focus of this paper, is the entropy of the residual random variable Z. The minimization of H(Z) has been applied in identifying the direction of causality [6], [7], and the compression length in a variable and fixed length secure compression [5], [17]. It could also be applied to studying a noisy communication channel [18], if the goal is to minimize the entropy of the noise source. Lower entropy of Z also means that less auxiliary randomness is needed to construct it from (X, Y ). This is of prime importance as randomness does not come for free and various expensive methods have been designed to generate it in practical systems [19], [20]. B. Minimum Entropy Coupling The minimum entropy coupling problem aims to find the joint distribution (P X1X2···Xm ) with minimum joint Rényi entropy H α (X 1 , X 2 , . . . , X m ), given the m marginal distributions X 1 ∼ P 1 , X 2 ∼ P 2 , . . . , X m ∼ P m (see Definition 2). This problem has been widely studied in [21]- [24] and is closely related to the functional representation lemma. It has been shown in [7], [24] that finding the minimum entropy of Z is the same as solving the minimum entropy coupling problem for the marginal distributions {P X|Y =y } y∈Y (see Lemma 1 and Appendix for details). Thus, the lower bounds on H(Z) provided in this paper are also lower bounds on the joint entropy in the minimum entropy coupling problem. Additional scenarios where the problem of minimum entropy coupling arises include the entropic causal inference [6], [7], dimension reduction [25], contingency tables and transportation polytopes in [26], [27] and randomness generation [19], [20]. The computation of the minimum entropy coupling was shown to be NP-hard in [22] and [25]. In [21], it is shown that any coupling has the joint entropy at least H(∧ m i=1 P i ), where ∧ denotes the greatest lower bound with respect to the majorization of probability distributions. A polynomial-time approximation algorithm for the construction of coupling within log m bits from H(∧ m i=1 P i ) is also provided in [21]. Li [24] improved upon the result of [21] by providing construction of a coupling with joint entropy being within 2 − 2 2−m bits from the H(∧ m i=1 P i ). Recently, Compton [28] showed that the greedy coupling algorithm provided in [6] is always within log 2 (e) bits from H(∧ m i=1 P i ) which further improves upon the result in [24] for m > 2. In this paper, we focus on lower bounds on the Rényi entropy H α (Z) (or equivalently H α (X 1 , X 2 , . . . , X m )) for every α ∈ [0, ∞), and prove a new converse in terms of the information spectrum of Z. This converse improves upon the information spectrum converse in [5] and the majorization converse in [21] for Rényi entropy of any order α. Similar results have also been recently studied in a parallel work by Compton et. al in [29] (see Remark 1 for details). The rest of the paper is organized as follows. In Section II, we state our notation, formulate the problem, and review the known lower bounds. We present our main results in Section III. In Section IV, we compare all the lower bounds and make concluding remarks. Finally, we postpone some of the proofs to Section V and the Appendix. II. PRELIMINARIES A. Notations We denote the probability mass function (PMF) of a random variable using a capital letter, say P , while the probability of an event is denoted using the bold-face letter P and the expectation of a random variable is denoted with E. Given a random variable X, its support (and sets in general) is denoted by X , while a realization is denoted by lower case letter, for example, x ∈ X . The information of a random variable X is ı X (x) := log 1 P X (x) ∀x ∈ X ,(1) where all logarithms have base two unless specified otherwise. The bold-face F X denotes the cumulative distribution function (CDF) of X. With some abuse of notation, we use F ı X to denote the CDF of the information ı X (X) of X; F ı X is also known as the information spectrum of X. Recall that the Shannon entropy of a random variable X can be written as H(X) = E [ı X (X)] .(2) Similarly, the Rényi entropy of X can be written as H α (X) = 1 1 − α log E 2 (1−α)ı X (X)(3) for α ∈ [0, 1) ∪ (1, ∞) and H α (X) := H(X) for α = 1. Given a random variable X with PMF P X we also use the notation H(P X ) and H α (P X ) to denote H(X) and H α (X). Given two PMFs P = (p 1 , p 2 , · · · ) and Q = (q 1 , q 2 , · · · ) with countably infinite supports, let the probability masses for both be given in non-increasing order, that is p 1 ≥ p 2 . . . and q 1 ≥ q 2 . . . . Then, we say that the distribution P majorizes Q, or Q m P , if k i=1 q i ≤ k i=1 p i(4) for all k ∈ {1, 2, . . . }. In the case that P and Q have finite supports with different cardinalities, we apply the same definition by padding both the PMFs with zeros. B. Problem Formulation Consider two jointly distributed random variables (X, Y ) ∼ P XY taking values in a countable set X and a finite set Y, respectively. Recall that the Functional Representation Lemma [1], [4], [5] says that there exists a random variable Z taking values in some set Z, such that 1) X is a deterministic function of Y and Z i.e., H(X|Y, Z) = 0(5) 2) Y and Z are independent of each other i.e., I(Y ; Z) = 0.(6) In other words, X can be represented as a deterministic function of two independent random variables Y and Z. Definition 1 (Minimum Entropy Functional Representation). The minimum Rényi entropy of functional representation of (X, Y ) ∼ P XY is H α (P XY ) := inf P Z|XY : H(X|Y,Z)=0, I(Y ;Z)=0 H α (Z) (7) for α ∈ [0, ∞). The coupling of m PMFs P 1 , P 2 , . . . , P m refers to a joint distribution of P X1X2···Xm subject to the constraints X i ∼ P i for all i ∈ {1, 2, . . . , m}. Definition 2 (Minimum Entropy Coupling). The minimum Rényi entropy of a coupling of a set of PMFs is H * α (P 1 , . . ., P m ) := inf P X 1 X 2 ···Xm : Xi∼Pi H α (X 1 , . . . , X m ) (8) for α ∈ [0, ∞). Lemma 1. Let (X, Y ) ∼ P XY be jointly distributed random variables with Y = {y 1 , . . . , y m }. Then H α (P XY ) = H * α (P X|Y =y1 , . . . , P X|Y =ym ).(9) That is, the problem of finding the functional representation with the minimum entropy is a minimum entropy coupling problem. Proof. The lemma is proved in [6], [24]. For the sake of completeness, the proof is presented in the Appendix. We state all results in terms of the functional representation conditions for the remainder of the paper. However, in light of Lemma 1, we see that the same results also apply to the minimum entropy coupling problem. C. Prior Work It is shown in [21] that H α (P XY ) ≥ H α (∧ y∈Y P X|Y =y )(10) where ∧ represents the greatest lower bound with respect to majorization, m , of the set {P X|Y =y } y∈Y . For details, refer to [21], [30]. Likewise, [5], [21] independently presented the following lower bound for α = 1, H α (P XY ) ≥ sup y∈Y H α (P X|Y =y ).(11) Although the general case is not addressed in [5], [21], this lower bound could readily be extended to Rényi Entropy of order α ∈ [0, ∞), see Section III and Appendix for details. The following information spectrum converse is essentially shown in [5]. Let Z be any random variable satisfying (5) and (6). Then, for any real valued t ∈ [0, ∞) and τ > 0, P[ı Z (Z) > t] ≥ sup y∈Y P[ı X|Y (X|Y ) > t+τ |Y = y]−exp(−τ ).(12) Our main contribution in this work is to strengthen (12) by removing the dependence on τ , as well as to leverage this new information spectrum converse in order to get direct lower bounds on H α (P XY ). III. MAIN RESULTS A. Information Spectrum Converse Theorem 1. Let (X, Y ) be jointly distributed random variables supported on countable X and countable Y. Consider any random variable Z that satisfies (5) and (6). Then, for any real-valued t ∈ [0, ∞) we have, P[ı Z (Z) > t] ≥ sup y∈Y P[ı X|Y (X|Y ) > t|Y = y].(13) The proof of Theorem 1 is given in Section V. In this paper, we focus on countable X and finite Y, though the results of Theorem 1 could be extended to a more general setting. In order to connect Theorem 1 to the majorization lower bound (10), we propose a similar definition with the ordering based on the information spectrum. Definition 3 ( ı ). Consider two PMFs P and Q with countably infinite supports. We write Q ı P , if the information spectrum of Q lower bounds the information spectrum of P . In other words, let U ∼ Q and V ∼ P . Then P [ı U (U ) ≤ t] ≤ P [ı V (V ) ≤ t](14) or equivalently, P [ı U (U ) > t] ≥ P [ı V (V ) > t](15) for all t ∈ [0, ∞). The following lemmas connect the above definition to the definition of majorization. Lemma 2. Given two PMFs Q and P we have that Q ı P ⇒ Q m P(16) where m denotes majorization. The proof of Lemma 2 is based on standard induction arguments, see Appendix for details. (See also [31], where this result was independently proven.) Note that in general, the reverse is not true: it is possible to have Q m P but not Q ı P . There exists Q * ∈ F such that Q m Q * for all Q ∈ F. In addition, Q * could be computed using a greedy algorithm with complexity linear in m i=1 |S i |. This result could also be extended for a countable set {S 1 , S 2 , . . . }. We leave the complete details for future work. The idea behind the greedy construction of Q * is as follows. We construct Q * = (q * 1 , q * 2 , . . . ) with q * 1 ≥ q * 2 . . . in a greedy way. First, assign q * 1 = min i∈{1,...,m} max s∈Si P i (s) .(18) This is the maximum value we can assign to q * 1 while satisfying the information spectrum constraint defined by F. We continue with this strategy: at each step k we assign the maximum possible value to q * k until the total probability of all q * k reaches one. The fact that this distribution majorizes all other Q ∈ F is proven using a standard induction argument. Likewise, the number of steps the algorithm needs to take is linear in m i=1 |S i |, since the constraints defined by F is a step function with at most steps m i=1 |S i |, see the Appendix. B. Lower bounds on Shannon and Rényi Entropy While Theorem 1 does not provide direct lower bounds on H α (P XY ), the next two corollaries could be used to compute such lower bounds. Corollary 1. Let (X, Y ) ∼ P XY be jointly distributed random variables supported on countable X and countable Y. Then H α (P XY ) ≥ K α (P XY )(19) for any α ∈ [0, ∞), where K α (P XY ) :=        1 1−α log 1 + ∞ 0 J α (t)dt , if α ∈ [0, 1) or α ∈ (1, ∞) ∞ 0 G(t)dt, α = 1 (20) with G(t) := sup y∈Y P[ı X|Y (X|Y ) > t|Y = y](21) and J α (t) := ln 2(1 − α)G(t)2 (1−α)t . Proof. Theorem 1 could be restated as 1 − F ı Z (t) ≥ G(t).(23) Recall that for any non-negative random variable V , its expectation could be related to its CDF via E[V ] = ∞ 0 (1 − F V (t))dt,(24) see, for example [32]. Since ı Z (Z) ≥ 0, we have that H(Z) = E[ı Z (Z)] = ∞ 0 1 − F ı Z (t) ≥ ∞ 0 G(t)dt,(25) where (25) follows from (23). Equation (19) is proved similarly for α ∈ [0, 1) ∪ (1, ∞), see Appendix for details. We remark that Corollary 1 is different from all other lower bounds in this paper in that it does not involve computing the entropy of a random variable. For example, we compute the expectation of a random variable with CDF 1 − G(t) in order to obtain a lower bound on Shannon entropy (α = 1). In general, there may not be an information random variable with this CDF. The next corollary incorporates this additional constraint in its application of Theorem 1. Corollary 2. Let (X, Y ) be jointly distributed random variables supported on finite X and finite Y. Further, let the parameter α ∈ [0, ∞). Then, H α (P XY ) ≥ inf Q∈F H α (Q) = H α (Q * )(26) where F = {Q : Q ı P X|Y =y ∀y ∈ Y} and Q * ∈ F is as given in Lemma 3. Proof. The first inequality is a restatement of Theorem 1 using the relation ı . According to Lemma 3, Q * majorizes every distribution in F. Rényi Entropy is Schur concave [33] with respect to majorization: that is, if Q m P then H α (Q) ≥ H α (P ). Thus, Q * achieves the infimum over F. Remark 1. Additionally, Note that for α = 1, these result have also been shown in a parallel work [29] using 'profile method'. Our Corollary 1 and Corollary 2 are equivalent to Theorem 3.3 and Theorem 5.5 in [29], respectively. The authors also showed that these results also hold for a concave function F : R d → R, such that F (p 1 , p 2 , p 3 , ..., p n ) := n i=1 f (p i ), where f : R → R is IV. COMPARISON OF LOWER BOUNDS A. Analytical Comparison The following theorem compares the lower bounds in Corollary 1 and 2 to (10) and (11). with Q * ∈ F as in Lemma 3. Let K α (P XY ) be as in Corollary 1. We make the following statements for all α ∈ [0, ∞): 1) In general, we have that H α (Q * ) ≥ K α (P XY ) ≥ sup y∈Y H α (P X|Y =y )(28) and, H α (Q * ) ≥ H α (∧ y∈Y P X|Y =y ) (29) ≥ sup y∈Y H α (P X|Y =y ).(30) 2) If there exists y ∈ Y such that P X|Y =y ı P X|Y =ŷ for allŷ ∈ Y, then H α (Q * ) = K α (P XY ) (31) = H α (∧ y∈Y P X|Y =y ) (32) = sup y∈Y H α (P X|Y =y ).(33) 3) If there exists y ∈ Y such that P X|Y =y m P X|Y =ŷ for allŷ ∈ Y, then H α (∧ y∈Y P X|Y =y ) = sup y∈Y H α (P X|Y =y ).(34) In particular, in light of (28), this implies that Proof. To see (29), note that for all y ∈ Y we have the following relation H α (Q * ) ≥ K α (P XY ) ≥ H α (∧ y∈Y P X|Y =y ).(35Q * m ∧ y∈Y P X|Y =y m P X|Y =y .(36) The relations holds by Lemma 2 together with Q * ∈ F, and the fact that ∧ y∈Y P X|Y =y is the greatest lower bound with respect to majorization. Therefore, (29) holds by Schur concavity of Rényi entropy. As already observed in [21], (30) and (34) hold by the same application of majorization and Schur concavity. Let U ∼ Q * . We state the rest of the proof for the α = 1 case. The other cases follow analogously. For (28), we have that H(Q * ) = ∞ 0 P[ı U (U ) > t]dt (37) ≥ ∞ 0 sup y∈Y P[ı X|Y (X|Y ) > t|Y = y]dt (38) ≥ sup y∈Y ∞ 0 P[ı X|Y (X|Y ) > t|Y = y]dt (39) = sup y∈Y H(P X|Y =y )(40) where (38) follows since Q * ∈ F, and (37) and (40) follow from (24). If there exists y ∈ Y such that P X|Y =y ı P X|Y =ŷ for allŷ ∈ Y then P X|Y =y will have the same information spectrum as Q * , and (38) and (39) become equalities. This shows that H α (Q * ) = K α (P XY ) = sup y∈Y H α (P X|Y =y ). Finally, (32) holds by (34) and Lemma 2. B. Numerical Comparison As we see in Theorem 2, the lower bound in Corollary 2 outperforms all other lower bounds. It is not clear how Corollary 1 compares to (10), in general. Numerical evaluations show that Corollary 1 outperforms (10) for low values of α, but is outperformed by (10) for high values of α. This is demonstrated by the following numerical example which has been carefully chosen to highlight the nuances between the different lower bounds. The comparison of lower bounds on Rényi entropy for this example is provided in Figure 1. Observe that the lower bounds (10) and (11) cannot exceed log |X | because they are limited to distribution with support size of X . Corollaries 1 and 2, however, take into account possible support size enlargement for the functional representation variable Z. This can be seen in Figure 1 for α = 0 (that is, log of the support size). This advantage of Corollaries 1 and 2 is even more apparent for binary supported distribution in the following example, see also Figure 2. The comparison of lower bounds on Shannon entropy for this example is provided in Figure 2. C. Concluding Remarks In order to provide context for the lower bounds in Figures 1 and 2 we include a plot of an upper bound. A number of theoretical upper bounds are available, see for example, [5], [6], [21], [24]. We implement the construction from [6] for our examples. We leave the question of improved theoretical upper bounds to future work and believe that combining information spectrum and majorization techniques could also be a fruitful strategy for this problem. V. PROOF OF THEOREM 1 Theorem 1. Let's begin by fixing a y ∈ Y and a real-valued t ∈ [0, ∞). Now, let us define X y := {x ∈ X : P X|Y (x|y) > 0}(47) and X y,t := {x ∈ X y : P X|Y (x|y) < 2 −t } (48) = {x ∈ X y : t < ı X|Y (x|y) < ∞}. Define also Z t := {z ∈ Z : P Z (z) < 2 −t } = {z ∈ Z : ı Z (z) > t}.(50) Recall that X is a deterministic function of Y and Z i.e., for some g : Y × Z → X , we have X = g(Y, Z). Further, define Z y (x) := {z ∈ Z : g(y, z) = x} and observe that P X|Y Z (x|y, z) = 1 , if z ∈ Z y (x) 0 , if z ∈ Z y (x)(51) i.e., for y ∈ Y fixed as above, every element of X y maps to a disjoint subset of Z. Therefore, P X|Y (x|y) = z∈Z P X|Y Z (x|y, z)P Z|Y (z|y) (52) = z∈Zy(x) P X|Y Z (x|y, z)P Z|Y (z|y) (53) (a) = z∈Zy(x) P Z|Y (z|y) (b) = z∈Zy(x) P Z (z).(54) Eq. (52) follows by the law of total probability. Eq. (53) and (54)(a) follow from 51, while (54)(b) follows from the fact that Y and Z are independent. Putting everything together, P[ı X|Y (X|Y ) > t|Y = y] = x∈Xy,t P X|Y (x|y) (55) (a) = x∈Xy,t z∈Zy(x) P Z (z) (b) = z∈ x∈X y,t Zy(x) P Z (z) (56) ≤ z∈Zt P Z (z) = P[ı Z (Z) > t].(57) Eq. (56)(a) follows from (54), Eq (56)(b) follows since every element of X y maps to a disjoint subset of Z, Eq. (57) follows from noting that ∀ z ∈ x∈Xy,t Z y (x), we have P Z (z) < 2 −t and therefore x∈Xy,t Z y (x) ⊆ Z t . Thus, we have shown that P [ı Z (Z) > t] ≥ P ı X|Y (X|Y ) > t|Y = y .(58) Since (58) holds for ∀ y ∈ Y, it holds for the supremum over Y. APPENDIX A. Proof of Lemma 1: Lemma 1. Let the set Y := {y 1 , y 2 , y 3 , · · · , y |Y| }. Recall that X is a deterministic function of Y and Z. Thus, X = g(Y, Z). For a fixed y ∈ Y, we define the random variable X y as X y := g(y, Z) = g y (Z) for some function g y : Z → X . Further, note that the random variable X y is distributed according to P X|Y =y . Now, H(Z, g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)) = H(g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)) + H(Z|g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)). Also, H(Z, g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)) = H(Z) + H(g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)|Z) (60) = H(Z).(61) From (59) and (61), we have H(Z) ≥ H(g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)) = H(X y1 , X y2 , · · · , X y |Y| ) (62) ≥ inf P X 1 X 2 ···Xm : Xi∼Pi H(X y1 , X y2 , · · · , X y |Y| ).(63) Note that Eq. (62) holds for every random variable Z satisfying (5) and (6), and for every joint distribution of the random variables (X y1 , X y2 , · · · , X y |Y| ). Therefore, the best possible lower bound on H(Z) is obtained by minimizing the joint entropy H(X y1 , X y2 , · · · , X y |Y| ), which is equivalent to solving the minimum entropy coupling problem given the random variables X y1 , X y2 , · · · , X y |Y| distributed according to P X|Y =y1 , P X|Y =y2 , · · · , P X|Y =y |Y| , respectively. Remark 2. The above proof can be easily extended for the Rényi entropy of order α ∈ [0, 1) ∪ (1, ∞). Let Z, g y1 (Z), g y2 (Z), · · · , g y |Y| be random variables as above, then H α (Z, g y1 (Z), g y2 (Z), · · · , g y |Y| (Z)) = H α (Z) ≥ H α (g y1 (Z), g y2 (Z), · · · , g y |Y| ). Note that Equation 64 holds from noting that for any two random variables X and Y , the following holds H α (X, Y ) ≥ H α (X) and H α (X, Y ) ≥ H α (Y ). Further, if Y = f (X) then H α (X, Y ) = H α (X) ≥ H α (Y ). Thus, we have the following result from (64): H α (Z) ≥ inf P X 1 X 2 ···Xm : Xi∼Pi H α (X y1 , X y2 , · · · , X y |Y| ).(65) B. Proof of Lemma 2 and 3 Lemma 2. Define t i = log 1 q i and s i = log 1 p i ,(66) and note that t 1 , t 2 , . . . and s 1 , s 2 , . . . are both increasing sequences. We prove the claim by induction. For the base case, suppose p 1 < q 1 and note that this implies that s 1 > t 1 . Then q 1 ≤ P [ı U (U ) ≤ t 1 ] ≤ P [ı V (V ) ≤ t 1 ] = 0(67) which is a contradiction. Therefore p 1 ≥ q 1 . For the inductive step, assume that for some k k i=1 q i ≤ k i=1 p i .(68) If p k+1 ≥ q k+1 , then we are done. Else, for the case p k+1 < q k+1 , we note that s k+1 > t k+1 . Then k+1 i=1 q i ≤ P [ı U (U ) ≤ t k+1 ] ≤ P [ı V (V ) ≤ t k+1 ] (69) = k i=1 1{s i ≤ t k }p i < k+1 i=1 p i .(70) This completes the proof. where S i ∼ P i . Note G(t) is an increasing step function that goes from zero to one in at most m i=1 |S i | steps. Also note that every Q ∈ F satisfies k i=1 q i ≤ G log 1 q k(72) where, again, we assume q 1 ≥ q 2 ≥ q 3 . . . . We construct Q * = (q * 1 , q * 2 , q * 3 , . . . ) with q * 1 ≥ q * 2 ≥ q * 3 . . . in a greedy way. First, assign q * 1 = min i∈{1,...,m} max s∈Si P i (s) . This is the maximum value we can assign to q * 1 while satisfying the information spectrum constraint (72). We continue with this strategy: at each step k we assign the maximum possible value to q * k until the total probability of all q * k reaches one. We claim that the algorithm will terminate in at most 2 m i=1 |S i | iterations. Indeed, for every 'step' in G(t), the algorithm can assign at most two probability values to Q * before it needs to start assigning smaller values and move on to the next 'step'. We also claim that the resulting distribution Q * will majorize all other distributions that satisfy the information spectrum constraint (72). To this end, let Q ∈ F be arbitrary such distribution. We show that Q m Q * inductively. For the base case, we have that q * 1 ≥ q 1 since by construction we assigned the maximum possible value to q * 1 . For the inductive step, assume that for some integer k we have that k i=1 q k ≤ k i=1 q * k . If q * k+1 ≥ q k+1 , we are done. Otherwise, define t k+1 = log 1 q * k+1 and s k+1 = log 1 q k+1 , and note that s k+1 < t k+1 and G(s k+1 ) ≤ G(t k+1 ). Let g k+1 = G(s k+1 ) − k i=1 q * i(75) and note that g k+1 < q k+1 . This is a consequence of the greedy construction; if this was not the case, we would assign q k+1 to q * k+1 . Also, q * k+1 ≥ g k+1 , again by the property of the greedy construction. This is because we could always assign the value of g k+1 to q * k+1 without violating the information spectrum constraint. Thus we have that k i=1 q * i + q * k+1 ≥ G(s k+1 ) ≥ k+1 i=1 q i(76) and this completes the proof. C. Proof of Corollary 1: Corollary 1. Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-B Minimum Entropy Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 II-B Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 II-C Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 III-B Lower bounds on Shannon and Rényi Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 IV-B Numerical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 IV-C Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lemma 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 B Proof of Lemma 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 C Proof of Corollary 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 I. INTRODUCTION A. Functional Representation Lemma Lemma 3 . 3Let {P 1 , . . . , P m } be a finite collection of PMFs supported on finite {S 1 , . . . , S m } and let F = {Q : Q ı P i ∀i ∈ {1, . . . , m}}. a non-negative and concave function satisfying f (0) = 0. Theorem 2 . 2Let (X, Y ) be jointly distributed random variables supported on finite X and finite Y. Let F = {Q : Q ı P X|Y =y ∀y ∈ Y}, Example 1 ( 1Rényi entropy). Let (X, Y ) be jointly distributed random variables with X = {1, . . . , 6}, Y = {y 1 , y 2 , y 3 } and the conditional PMF, P X|Y (·|y 1 ) = P X|Y (·|y 3 ) = [0.35, 0.35, 0.25, 0.04, 0.005, 0.005]. (43) Then, ∧ y∈Y P X|Y =y = [0.35, 0.275, 0.125, 0.125, 0.12, 0.005], Q * = [0.35, 0.15, 0.125, 0.125, 0.1, 0.1, 0.04, 0.005, 0.005]. Fig. 2 . 2Lower bound on Shannon entropy for collections of distributions (indexed by p) in Example 2. Note that Corollary 2 matches the upper bound and is therefore exactly H 1 (P XY ). Lower bounds(10)and(11)match for this example. Example 2 ( 2Binary Support Size). Let (X, Y ) be jointly distributed random variables with |X | = |Y| = 2 and the conditional PMF, P X|Y (·|y 1 ) = [0.9, 0.1] and P X|Y (·|y 2 ) = [1 − p, p] (44) where p ∈ [0, 0.5]. Then, ∧ y∈Y P X|Y =y = Si (S i ) > t] Remark 3. Note that the slightly modified version of Equation (58) i.e.,is also true. The proof follows via the same construction except the fact that the definitions of the sets X s,t and Z t does not involves the strict inequality. From equation(3), we havewhere,From Theorem (1), we have:where t :Here, (80) follows from noting that ı α Z (Z) > 0 and(24).This completes the proof for the case of 0 ≤ α < 1.Case 2: α ∈ (1, ∞):Here, we will use the modified version (cf. Remark(3)) of Theorem(1)i.e.,where t := 2 (1−α)t . Now,Here, (86) follows from noting that ı Z (Z) ≥ 0 and(24).(87) follows from noting that ı α Z (Z) ∈ (0, 1) for α ∈ (1, ∞).Remark 4. 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Available: https://proceedings.mlr.press/v206/compton23a.html Inequalities: Theory of majorization and its applications. A W Marshall, I Olkin, B C Arnold, A. W. Marshall, I. Olkin, and B. C. Arnold, "Inequalities: Theory of majorization and its applications," 1980. Infinite divisibility of information. C T Li, IEEE Transactions on Information Theory. 687C. T. Li, "Infinite divisibility of information," IEEE Transactions on Information Theory, vol. 68, no. 7, pp. 4257-4271, 2022. B Hajek, Random Processes for Engineers. Cambridge University PressB. Hajek, Random Processes for Engineers. Cambridge University Press, 2015. Convex functions, partial orderings, and statistical applications. J E Peari&apos;c, F Proschan, Y L Tong, J. E. Peari'c, F. Proschan, and Y. L. Tong, "Convex functions, partial orderings, and statistical applications," 1992.
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[ "Kelson Kaj \nUniversity of California\nSan DiegoCA\n", "Kevin A Cremin \nUniversity of California\nSan DiegoCA\n", "Ian Hammock \nUniversity of California\nSan DiegoCA\n", "Jacob Schalch \nUniversity of California\nSan DiegoCA\n", "D N Basov \nDepartment of Physics\nColumbia University\nNew YorkNY\n", "R D Averitt [email protected] \nUniversity of California\nSan DiegoCA\n" ]
[ "University of California\nSan DiegoCA", "University of California\nSan DiegoCA", "University of California\nSan DiegoCA", "University of California\nSan DiegoCA", "Department of Physics\nColumbia University\nNew YorkNY", "University of California\nSan DiegoCA" ]
[]
Terahertz nonlinear optics is a viable method to interrogate collective phenomena in quantum materials spanning ferroelectrics, charge-density waves, and superconductivity. In superconductors this includes the Higgs amplitude and Josephson phase modes. We have investigated the nonlinear c-axis response of optimally doped La1.85Sr0.15CuO4 using high-field THz time domain spectroscopy (THz-TDS) at field strengths up to ∼80 kV/cm. With increasing field, we observe a distinct redshift of the Josephson plasma edge and enhanced reflectivity (above the plasma edge) arising from third harmonic generation. The non-monotonic temperature dependent response is consistent with nonlinear drive of the Josephson Plasma Mode (JPM) as verified with comparison to theoretical expectations. Our results add to the understanding that, using THz light, the JPM (in addition to the Higgs mode) provides a route to interrogate and control superconducting properties.
10.1103/physrevb.107.l140504
[ "https://export.arxiv.org/pdf/2211.17184v2.pdf" ]
254,096,470
2211.17184
02140eb000ed5bdef36da0061af2883e110bfe14
Kelson Kaj University of California San DiegoCA Kevin A Cremin University of California San DiegoCA Ian Hammock University of California San DiegoCA Jacob Schalch University of California San DiegoCA D N Basov Department of Physics Columbia University New YorkNY R D Averitt [email protected] University of California San DiegoCA Mar 2023 Terahertz third harmonic generation in c-axis La 1.85 Sr 0.15 CuO 4* These two authors contributed equally † Corresponding author Terahertz nonlinear optics is a viable method to interrogate collective phenomena in quantum materials spanning ferroelectrics, charge-density waves, and superconductivity. In superconductors this includes the Higgs amplitude and Josephson phase modes. We have investigated the nonlinear c-axis response of optimally doped La1.85Sr0.15CuO4 using high-field THz time domain spectroscopy (THz-TDS) at field strengths up to ∼80 kV/cm. With increasing field, we observe a distinct redshift of the Josephson plasma edge and enhanced reflectivity (above the plasma edge) arising from third harmonic generation. The non-monotonic temperature dependent response is consistent with nonlinear drive of the Josephson Plasma Mode (JPM) as verified with comparison to theoretical expectations. Our results add to the understanding that, using THz light, the JPM (in addition to the Higgs mode) provides a route to interrogate and control superconducting properties. I. INTRODUCTION Terahertz spectroscopy has emerged as a powerful probe of non-equilibrium dynamics in quantum materials [1,2]. More recently, the generation of intense terahertz (THz) high-field pulses has enabled nonlinear drive of the low energy electrodynamics, offering new insights into the many-body response while also providing a route for on-demand control of emergent properties [3][4][5][6]. Superconductors are particularly amenable to high-field interrogation and manipulation of the condensate with terahertz pulses. This includes the Higgs amplitude mode in conventional superconductors where light couples nonlinearly to the condensate, driving oscillations of the order parameter amplitude at twice the pump frequency, leading to harmonic generation [7][8][9][10][11][12][13]. Higgs mode spectroscopy has been extended to cuprate superconductors with novel mode dynamics associated with the d-wave gap symmetry and to iron pnictides where multiband effects have been observed [14,15]. In the high-T c cuprates the phase mode response (Josephson plasma mode -JPM) manifests at THz frequencies [16,17]. Briefly, the copper-oxygen planes are weakly interacting and Josephson coupling dictates c-axis Cooper pair tunneling based on the interlayer phase difference of the superconducting order parameter between adjacent planes. This results in a plasma edge in the caxis reflectivity at the Josephson plasma frequency ω p . For single layer cuprates, ω p typically manifests at THz frequencies with ω 2 p proportional to the condensate density n s (e.g., see Fig.1, discussed in more detail below). As such, the JPM serves as a reporter of the condensate response which includes nonlinearities such as fieldinduced renormalization of ω p and harmonic generation [18][19][20][21]. We investigate the nonlinear spectral response of c-axis La 1.85 Sr 0.15 CuO 4 (LSCO) using high field THz-TDS as a function of field strength and temperature. A redshift of ω p with increasing field (2.4 kV/cm up to 80 kV/cm) arises from the Josephson effect. With increasing temperature the maximum redshift increases from ∼110 GHz at 10 K ∼220 GHz at 32 K. This temperature dependent frequency shift cannot be explained solely using the Josephson equations which predict a high-field shift of the JPM is a constant fraction of the equilibrium JPM frequency at each temperature. This is the opposite of the observed behavior, and could be related to increased quasiparticle damping at higher temperatures. Commensurate with this is broadband third harmonic generation above the plasma edge which exhibits a slight increase with decreasing temperature below T c , dropping off in the normal state. The temperature dependence is compared with calculations based on the theory in Reference [21]. The qualitative agreement between calculations and experiment suggests that the temperature dependence is related to the competing factors of Josephson coupling strength, the resonance of the pump with the JPM, thermal population of excited plasmon states, and quasiparticle damping. For our broadband drive, we estimate a power conversion efficiency of ∼ 6×10 −5 . II. METHODS THz radiation is generated via optical rectification using a Ti:sapphire regenerative amplifier (1 KHz, 800 nm, 100 fs, 3 mJ) using tilted pulse front generation in a Mg-LiNbO 3 (LNO) crystal [22,23]. The THz output from the LNO crystal surface is collimated with a lens (f = 120 mm) and focused onto the sample with an angle of incidence of 15 o and a beam diameter of ∼2.3 mm FWHM (pulse energy ∼2 µJ). Before reaching the sample, the THz light passes through a pair of wire grid polarizers which are used to attenuate the THz pulse, covering the range from 2 -80 kV/cm. The reflected beam from the sample surface is collected and focused onto a 300 µm thick (110) GaP crystal for EO sampling. The GaP crys- tal is mounted on a 2 mm thick (100) GaP crystal to delay the internal etalon inside the GaP to times beyond our temporal measurement window. Roughly 1% of the pump beam is split off and used for gating the THz pulse in the GaP crystal. The entire THz beam path is in a vacuum chamber (schematic of the experimental setup is shown in Fig. 2a). The bandwidth of the pulses extends from 0.2 -3 THz (see Fig. 3a) with a maximum at ∼0.65 THz. The full spectral content is used for the nonlinear studies and is not spectrally filtered. The LSCO crystal was cut and polished to expose the a-c plane and was grown via a traveling-solvent floatingzone method [24] with a surface size of approximately 3mm × 3mm. Reflectivity measurements were performed by taking time domain scans of the electric field reflected off the sample and from a gold reference mirror at the sample position in both nonlinear and linear regimes. The reflectivity was then obtained by calculating the Fourier transforms of the time domain scans and taking their ratios, R N L = |E N L (ω)/E Au N L (ω)| 2 . Measurements were performed above and below T c = 38K as described below. III. RESULTS We first measured the temperature dependent c-axis response at the lowest available electric field to characterize the linear response. The c-axis reflectivity is plotted in Fig. 1(a) for several temperatures above and below T c = 38 K, and shows a clear plasma edge emerge and sharpen with decreasing temperature from 32 K to 10 K, coinciding with an increase in superconducting condensate. The low temperature (10K) plasma edge ω p is at ∼1.7 THz is in agreement with previous caxis measurements for x = 0.15 doping [16,25]. Figure 1(b) shows the plasma frequency ω p (T ) normalized by the low temperature measurement ω p (10 K) and scales with BCS-like order parameter temperature dependence, as ω 2 p ∝ tanh(2.26 T c /T − 1)) [26,27]. The inset of Fig.1(b) displays the normalized loss function -Im(1/ǫ) at each temperature. For frequencies below ω p the reflectivity is ∼90%, whereas near-unity reflection is expected as ω → 0. This deviation is attributed to imperfect referencing since the sample size and beam diameter are comparable. This effect is more pronounced at lower frequencies as the focused THz beam diameter is frequency dependent. However, the low temperature ω p (T ) (and associated temperature dependence) is consistent with previous studies for x = 0.15 doping, indicative of a high-quality crystal. The reflectivity presented in Fig. 1 and 2 are normalized to match reflectivity measurements using Fourier transform infrared spectroscopy (FTIR) on the same LSCO crystal at 1 THz [28]. The nonlinear terahertz reflectivity results are shown in Fig. 2b-f. As shown in Fig. 2b (base temperature 10 K), two pronounced effects occur: There is a redshift of the plasma edge with increasing field and, above the plasma edge, there is an increase in the reflectivity corresponding to third harmonic generation (as described in greater detail below). With increasing temperature, there is a decrease in the condensate density, but there is still a clear redshift in the plasma edge and enhanced reflectivity. At 32 K (Fig. 2e), the plasma edge shift as a function of field is ∼220GHz, larger than the ∼110GHz shift at 10 K, but the reflectivity increase is relatively small. Above T C (Fig. 2f), there is no longer a superconducting response and the nonlinear increase in reflectivity is minimal to nonexistent. The nonlinear reflectivity data in Figure 2 is informative as it reveals both the JPM redshift and the increase in reflectivity above ω p . However, the increase in reflectivity requires a more careful analysis as we now discuss. Figure 3a plots the spectral amplitude of the electric fields reflected from the LSCO at 10K, comparing the reflectivity of the nonlinear (NL) and linear (L) response. The red curve is E N L (ω) at the highest field (80kV/cm). The black curve is E * L (ω) at the lowest field, appropriately normalized to enable quantitative comparison with the spectral changes that arise in the nonlinear regime [29]. As with the reflectivity data in Figure 2, the redshift of the plasma edge is evident when comparing E * L (ω) and E N L (ω) in Fig. 3a. Moreover, the increase in the spectral amplitude above the plasma edge is also evident (blue shaded region). The peak spectral amplitude of the terahertz pulses is at ∼0.65 THz. The second peak in E N L (ω) is at ∼1.9 THz, consistent with third harmonic generation. The data in Fig. 3a reveals that the nonlinear signal is relatively small in comparison to the peak amplitude of the incident pulse (i.e.,E * L (ω) at 0.65 THz), but considerably larger than the spectral spectral amplitude of E * L (ω) between 2 -3 THz. Nonetheless, the dynamic range is sufficient to enable a determination of the field dependence of the integrated spectral amplitude above the plasma edge (Fig. 3b), which further verifies that the enhanced signal arises from third harmonic generation (THG). The THG response in Fig. 3 (b) is quantified by integrating E 3ω ≡ E N L (ω) − E * L (ω) dω(1) where E N L (ω) and E * L (ω) are the reflected signals from the sample as defined above. To avoid integrating over the Josephson plasma edge, the integration is taken from 1.8 THz to 2.4 THz. Figure 3(b) displays the magnitude of the third harmonic E 3ω as a function of field strength at 10 K, along with a cubic polynomial fit (red line, R-squared value of 0.99). This further confirms the reflectivity/spectral amplitude increase arises from third harmonic generation in the superconducting phase. The lowest field strengths have more third harmonic intensity than the cubic fit, which may be attributed to the fit over-favoring the highest fields. IV. DISCUSSION The experimental observations can be further understood by examining the higher order terms in the phase dynamics of layered superconductors described by the Josephson equations [18,30]. In a layered superconductor, the interlayer phase difference θ(t) changes with time according to the second Josephson equation ∂θ(t) ∂t = 2edE(t)(2) where 2e is the cooper pair charge, d is the interlayer spacing (∼1 nm), is Planck's constant divided by 2π, and E(t) = E 0 sin(ω pump t) is an electric field along the c-axis with E 0 the field strength which oscillates at frequency ω pump . Solving equation 2 yields the relation θ(t) = (2edE 0 / )cos(ω pump t). Since the c-axis superfluid density ρ c scales as the order parameter phase difference, ρ c ∝ cosθ and ρ c ∝ ω 2 p as shown in Fig. 1(b), the plasma frequency renormalizes according to ω 2 N L = ω 2 p cosθ(t) where ω N L is the new Josephson plasma frequency under intense field strengths. By inserting θ(t) and expanding the ω N L we arrive at ω 2 JP M = ω 2 JP M0 cos(θ) = ω 2 JP M0 cos θ 0 cos(ω pump t) ≈ ω 2 JP M0 1 − θ 2 0 4 − θ 2 0 cos(2ω pump t) 4 + . . .(3) where θ 0 = 2edE 0 / . From this expansion we can see that the next leading order term is subtracting, resulting in a redshift in the plasma frequency. This is what is observed below T c for high fields as shown in Fig. 2. The tunneling interlayer current depends on the phase difference as I(t) = I 0 sin[θ(t)], and solving for I(t) gives I(t) = I c sin θ 0 cos(ω pump t) ≈ I c θ 0 cos(ω pump t) − θ 3 0 6 cos 3 (ω pump t) + . . .(4) where the leading higher order in the expansion is cubic. This expanded term leads to driving the current at the third harmonic and is observed as THz emission at 3ω, which manifests as a reflectivity increase above the plasma edge as shown in Fig. 2(a)-(d). The above equations predict the third harmonic signal to scale with the superfluid density, as observed in previous work on LBCO [20]. The temperature dependence of the third harmonic emission for our LSCO studies is shown in Fig. 4 (black dots). Clearly, the signal does not solely scale with the superfluid density (which is directly proportional to the square of the JPM frequency), but instead decreases slightly for temperatures lower than 27K. Recent work has shown that detailed calculations are necessary to understand the temperature dependence of the third harmonic signal. Calculations based on this theory are plotted as red dots in Fig. 4 [31]. The third harmonic signal is calculated from the nonlinear optical kernel of the Josephson phase, and its overlap with the spectral amplitude of the pump pulse, as shown in equation 5. I N L (ω) = d(ω ′ )A(ω − ω ′ )K(ω ′ )A 2 (ω ′ ) K ∝ ω 3 J coth(βω J /2) 4ω 2 J − (ω + iγ(T )) 2(5) A is the spectral amplitude of the electric field profile and K is the nonlinear optical Kernel. This kernel does have an overall scaling with the superfluid density as ω 3 J . However, there is also a factor in the kernel, coth(βω J /2), that comes from the thermal excitation of plasmon modes, which causes third harmonic emission to increase with temperature [21]. The kernel also has a resonance at the Josephson plasma frequency. For our experiment, the pump pulse is centered at a frequency lower than the Josephson plasma frequency for all measured temperatures. However, with increasing temperature this process is closer to being on resonance since the I P Q R S T U V W X Y ` a b c d f g h i q r s t u w y FIG. 3. Third harmonic generation from c-axis LSCO at 10K. (a) Linear (black) and nonlinear (red) spectral amplitude of the THz pulses reflected from the sample. The nonlinear spectrum is at the maximum field strength of ∼80 kV/cm, and the linear spectrum is at the minimum field strength (renormalized as explained in the text). The shaded region in blue is the spectral content attributed to third harmonic generation. The inset contains a plot of the spectral amplitudes with a linear y-axis. (b) Third harmonic generation as a function of incident THz field strength at 10K. Each data point is from integrating Eq. 1 over the third harmonic region. The red line is a cubic polynomial fit, and the shaded area is bounded by cubic fits of the data including the upper and lower bounds of the data including the error bars. condensate density (and hence ω p ) is reduced. Finally, quasiparticle damping increases with temperature, causing third harmonic emission to decrease. Thus, there are four competing factors that determine the overall temperature dependence of third harmonic emission due to Josephson plasma waves. These calculations capture the qualitative features of the data including a maximum in third harmonic emission at 27K, a decrease as the temperature is decreased, and a sharp decline in third harmonic emission in the vicinity of T C . The work in [21] and the results shown here motivate looking at the temperature dependence of third harmonic emission from other c-axis cuprates since the non-monotonic temperature dependence goes beyond basic Josephson-equations predictions. To our knowledge, third harmonic emission from c-axis cuprates has only been previously reported in La 2−x Ba x CuO 4 [20] and shares some similarities with our results. More detailed studies of third harmonic generation and THz nonlinearities have been performed in many superconductors, including cuprates, focusing on light polarized in the ab-plane. The interpretation of the data has been in terms of the Higgs mode and quasiparticle contributions [7,8,14,[32][33][34]. We note that there is a prediction of contribution of plasma waves to the third harmonic generation for light polarized in the ab-plane [21]. The nonlinear THz response of cuprates can also be studied with pump-probe protocols, with [21] giving predictions for the response using the nonlinear optical kernel formalism. Experimentally, this pump-probe protocol has been used to study LBCO, where both amplification of Josephson Plasma Waves [19] and long-lived oscillations in the THz pump-probe signal [35] were observed. V. CONCLUSION We have explored the nonlinear c-axis response of LSCO and have observed THz third-harmonic generation arising from the Josephson plasma mode. The emission from the sample under intense THz radiation displays cubic behaviour indicative of third harmonic generation, which has been shown to be consistent with phase dynamics between the copper-oxygen planes in LSCO. An interesting future direction on the cuprates would be to investigate the relationship between the phase mode and amplitude mode and their potentially coupled contributions to the nonlinear terahertz response. FIG. 1 . 1(a) Low field c-axis reflectivity of La1.85Sr0.15CuO4 at several temperatures. At 10K, ωp is ∼1.7 THz. (b) Temperature dependence of ωp below Tc normalized to the low temperature measurement ωp(T = 10 K). The dashed line is a fit to a BCS-like order parameter temperature dependence, as described further in the text. The inset shows the (normalized) loss function, -Im(1/ǫ), using the same legend in panel (a). FIG. 2 . 2La1.85Sr0.15CuO4 c-axis THz reflectivity. (a) Schematic of experimental setup. Panels (b)-(f) show the reflectivity taken at temperatures 10 K, 20 K, 27 K, 32 K, and 45 K respectively. At each temperature, the reflectivity was measured for fields ranging from 2.4 -80 kV/cm as indicated in the legend. FIG. 4 . 4Magnitude of c-axis THG from LSCO versus temperature. All data points were taken with maximum field strengths of 80 kV/cm. The black circles are experimental data and the red circles are calculations as described in the text. The experimental data and calculations are normalized to the maximum value, which occurs at 27K in both experiment and simulation. . the Vannevar Bush Faculty Fellow ONR-VB. the Vannevar Bush Faculty Fellow ONR-VB: N00014-19- 1-2630. . J Zhang, R Averitt, 10.1146/annurev-matsci-070813-113258Annual Review of Materials Research. 4419J. Zhang and R. 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[]
[ "Network embedding unveils the hidden interactions in the mammalian virome", "Network embedding unveils the hidden interactions in the mammalian virome" ]
[ "Timothée Poisot \nDépartement de Sciences Biologiques\nUniversité de Montréal\n\n", "Marie-Andrée Ouellet \nDépartement de Sciences Biologiques\nUniversité de Montréal\n\n", "Nardus Mollentze \nInstitute of Biodiversity, Animal Health & Comparative Medicine\nUniversity of Glasgow\nGlasgowUK\n\nMRC -University of Glasgow Centre for Virus Research\nGlasgowUK\n", "Maxwell J Farrell \nDepartment of Ecology and Evolutionary Biology\nUniversity of Toronto\nTorontoONCanada\n", "Daniel J Becker \nDepartment of Biology\nUniversity of Oklahoma\nNormanOKU.S.A\n", "Liam Brierley \nDepartment of Health Data Science\nUniversity of Liverpool\nLiverpoolUK\n", "Gregory F Albery \nDepartment of Biology\nGeorgetown University\nWashingtonD.CU.S.A\n", "Rory J Gibb \nCentre on Climate Change and Planetary Health\nSchool of Hygiene and Tropical Medicine\nLondon, LondonUK\n\nCentre for Mathematical Modelling of Infectious Diseases\nSchool of Hygiene and Tropical Medicine\nLondon, LondonUK\n", "Stephanie N Seifert \nPaul G. Allen School for Global Health, WSU\nPullmanWAU.S.A\n", "Colin J Carlson \nCenter for Global Health Science and Security\nGeorgetown University Medical Center\nWashingtonD.CU.S.A\n\nDepartment of Microbiology and Immunology\nGeorgetown University Medical Center\nWashingtonD.CU.S.A\n" ]
[ "Département de Sciences Biologiques\nUniversité de Montréal\n", "Département de Sciences Biologiques\nUniversité de Montréal\n", "Institute of Biodiversity, Animal Health & Comparative Medicine\nUniversity of Glasgow\nGlasgowUK", "MRC -University of Glasgow Centre for Virus Research\nGlasgowUK", "Department of Ecology and Evolutionary Biology\nUniversity of Toronto\nTorontoONCanada", "Department of Biology\nUniversity of Oklahoma\nNormanOKU.S.A", "Department of Health Data Science\nUniversity of Liverpool\nLiverpoolUK", "Department of Biology\nGeorgetown University\nWashingtonD.CU.S.A", "Centre on Climate Change and Planetary Health\nSchool of Hygiene and Tropical Medicine\nLondon, LondonUK", "Centre for Mathematical Modelling of Infectious Diseases\nSchool of Hygiene and Tropical Medicine\nLondon, LondonUK", "Paul G. Allen School for Global Health, WSU\nPullmanWAU.S.A", "Center for Global Health Science and Security\nGeorgetown University Medical Center\nWashingtonD.CU.S.A", "Department of Microbiology and Immunology\nGeorgetown University Medical Center\nWashingtonD.CU.S.A" ]
[]
At most 1-2% of the global virome has been sampled to date. Recent work has shown that predicting which host-virus interactions are possible but undiscovered or unrealized is, fundamentally, a network science problem. Here, we develop a novel method that combines a coarse recommender system (Linear Filtering; LF) with an imputation algorithm based on low-rank graph embedding (Singular Value Decomposition; SVD) to infer host-virus associations. This combination of techniques results in informed initial guesses based on directly measurable network properties (density, degree distribution) that are refined through SVD (which is able to leverage emerging features). Using this method, we recovered highly plausible undiscovered interactions with a strong signal of viral coevolutionary history, and revealed a global hotspot of unusually unique but unsampled (or unrealized) host-virus interactions in the Amazon rainforest. We develop several tests for quantifying the bias and realism of these predictions, and show that the LF-SVD method is robust in each aspect. We finally show that graph embedding of the imputed network can be used to improve predictions of human infection from viral genome features, showing that the global structure of the mammal-virus network provides additional insights into human disease emergence.2The novel coronavirus SARS-CoV-2 is only the newest of the thousands of mammalian viruses that might have the capacity to infect human hosts. Despite growing interest in viral ecology, data remains limiting, as most of the global virome remain undocumented. Computational methods that can infer undiscovered associations in a partially-observed host-virus network can fill-in some of these gaps[1]. At least 20% to 40% of host-parasite associations are estimated to be unrecorded in locally-collected, highly-complete datasets [2]; a much higher proportion are likely unrecorded in the high-sparsity datasets cataloguing the global virome. An even greater proportion of host-virus interactions may be biologically plausible (i.e., a virus might have the capacity to infect a host) but as-yet unrealized for lack of ecological opportunities. These are often the links with the greatest relevance to actionable science: at least 10,000 mammalian viruses likely have the unrealized capacity to infect human hosts [3], while an even greater number could be shared thousands of times between mammals as they track shifting habitats in a changing climate[4].Here, we propose a novel method for predicting unknown links in partially-sampled networks, and apply it to the largest database of host-virus associations currently available. The method is based on a combination of linear filtering, which uses high-level network information to generate an initial guess as to the probability of an interaction, and singular value decomposition, which uses the structure of a low-rank approximation (which has a better signal-to-noise ratio [5]) of the entire network to impute interactions that were presumed negatives. In combination, this method uses existing knowledge on the entire network, but also can be tuned in such a way that its adjacency matrix is approximated at a rank that maximizes the amount of information used for imputation. Importantly, this method relies only on network structure, and does not consider (or require) external information specific to the hosts and viruses involved. We used this method to predict host-virus associations that are either undetected, or are biologically plausible but possibly unrealized in the real world. We found that the imputed network carries a weaker signal of sampling biases, while preserving the real signal of coevolutionary history. Applying methods from community ecology, we found that the Amazon may be more of a global hotspot of undescribed host-virus associations than currently assumed. Finally, we applied graph embedding to the observed and imputed networks, and used these as predictive features to augment a previously-published model that predicts which viruses can infect humans based on summaries of viral genome composition[6], testing whether knowledge about the global dynamics of cross-species transmission are informative for the limited case of predicting human disease emergence.3
10.1016/j.patter.2023.100738
[ "https://arxiv.org/pdf/2105.14973v2.pdf" ]
247,749,113
2105.14973
b8eb7d344c07b98197a89b4c00067a2d8d6311c9
Network embedding unveils the hidden interactions in the mammalian virome March 28, 2022 24 Mar 2022 Timothée Poisot Département de Sciences Biologiques Université de Montréal Marie-Andrée Ouellet Département de Sciences Biologiques Université de Montréal Nardus Mollentze Institute of Biodiversity, Animal Health & Comparative Medicine University of Glasgow GlasgowUK MRC -University of Glasgow Centre for Virus Research GlasgowUK Maxwell J Farrell Department of Ecology and Evolutionary Biology University of Toronto TorontoONCanada Daniel J Becker Department of Biology University of Oklahoma NormanOKU.S.A Liam Brierley Department of Health Data Science University of Liverpool LiverpoolUK Gregory F Albery Department of Biology Georgetown University WashingtonD.CU.S.A Rory J Gibb Centre on Climate Change and Planetary Health School of Hygiene and Tropical Medicine London, LondonUK Centre for Mathematical Modelling of Infectious Diseases School of Hygiene and Tropical Medicine London, LondonUK Stephanie N Seifert Paul G. Allen School for Global Health, WSU PullmanWAU.S.A Colin J Carlson Center for Global Health Science and Security Georgetown University Medical Center WashingtonD.CU.S.A Department of Microbiology and Immunology Georgetown University Medical Center WashingtonD.CU.S.A Network embedding unveils the hidden interactions in the mammalian virome March 28, 2022 24 Mar 2022† Correspondence should be directed to '[email protected]' * These authors share lead author status. 1 At most 1-2% of the global virome has been sampled to date. Recent work has shown that predicting which host-virus interactions are possible but undiscovered or unrealized is, fundamentally, a network science problem. Here, we develop a novel method that combines a coarse recommender system (Linear Filtering; LF) with an imputation algorithm based on low-rank graph embedding (Singular Value Decomposition; SVD) to infer host-virus associations. This combination of techniques results in informed initial guesses based on directly measurable network properties (density, degree distribution) that are refined through SVD (which is able to leverage emerging features). Using this method, we recovered highly plausible undiscovered interactions with a strong signal of viral coevolutionary history, and revealed a global hotspot of unusually unique but unsampled (or unrealized) host-virus interactions in the Amazon rainforest. We develop several tests for quantifying the bias and realism of these predictions, and show that the LF-SVD method is robust in each aspect. We finally show that graph embedding of the imputed network can be used to improve predictions of human infection from viral genome features, showing that the global structure of the mammal-virus network provides additional insights into human disease emergence.2The novel coronavirus SARS-CoV-2 is only the newest of the thousands of mammalian viruses that might have the capacity to infect human hosts. Despite growing interest in viral ecology, data remains limiting, as most of the global virome remain undocumented. Computational methods that can infer undiscovered associations in a partially-observed host-virus network can fill-in some of these gaps[1]. At least 20% to 40% of host-parasite associations are estimated to be unrecorded in locally-collected, highly-complete datasets [2]; a much higher proportion are likely unrecorded in the high-sparsity datasets cataloguing the global virome. An even greater proportion of host-virus interactions may be biologically plausible (i.e., a virus might have the capacity to infect a host) but as-yet unrealized for lack of ecological opportunities. These are often the links with the greatest relevance to actionable science: at least 10,000 mammalian viruses likely have the unrealized capacity to infect human hosts [3], while an even greater number could be shared thousands of times between mammals as they track shifting habitats in a changing climate[4].Here, we propose a novel method for predicting unknown links in partially-sampled networks, and apply it to the largest database of host-virus associations currently available. The method is based on a combination of linear filtering, which uses high-level network information to generate an initial guess as to the probability of an interaction, and singular value decomposition, which uses the structure of a low-rank approximation (which has a better signal-to-noise ratio [5]) of the entire network to impute interactions that were presumed negatives. In combination, this method uses existing knowledge on the entire network, but also can be tuned in such a way that its adjacency matrix is approximated at a rank that maximizes the amount of information used for imputation. Importantly, this method relies only on network structure, and does not consider (or require) external information specific to the hosts and viruses involved. We used this method to predict host-virus associations that are either undetected, or are biologically plausible but possibly unrealized in the real world. We found that the imputed network carries a weaker signal of sampling biases, while preserving the real signal of coevolutionary history. Applying methods from community ecology, we found that the Amazon may be more of a global hotspot of undescribed host-virus associations than currently assumed. Finally, we applied graph embedding to the observed and imputed networks, and used these as predictive features to augment a previously-published model that predicts which viruses can infect humans based on summaries of viral genome composition[6], testing whether knowledge about the global dynamics of cross-species transmission are informative for the limited case of predicting human disease emergence.3 Predicting the host-virus network The combined linear filtering and singular value decomposition (LF-SVD) model relies on four hyper-parameters describing the relative importance of network structure (LF: indegree, out-degree, and connectance) and matrix rank used for approximation (SVD). After tuning of the hyper-parameters, the best model (which used initial values emphasising network connectance, and performed SVD at rank 12) achieved a ROC-AUC of 0.84 (ED Table 1, ED Figure 1). Although analyses of ecological networks usually gravitate towards using degree-based (over connectance-based) models, this choice of best model is unsurprising. Assuming that the overwhelming majority of interactions are unsampled (which is supported by the observation that imputation increased the number of interactions by a factor of about 15), degree holds very little information besides sampling effort. We applied four tests of whether model performance was undermined by biases in the partially-observed network, a common problem in predicting host-pathogen interactions. First, we tested the effect of passive sampling bias with a regression of host species' viral diversity against citation counts, a commonly-used proxy for scientific research effort; we found that consistently, citations had a weaker effect predicting viral richness after imputation (ED Table 2). Second, we examined the top ten hosts that shared viruses with humans, to assess the influence of impact bias, a specific form of active sampling bias driven by relevance to human health. We found that while domesticated or lab animals dominated this list in the observed network, the imputed network removed most of these species (ED Figure 2). Third, we assessed phylogenetic patterns in the number of 'missing' viruses linked to each host after imputation. Missing viruses displayed only moderate phylogenetic signal (Pagel's = 0.35), suggesting they are distributed fairly equally across the mammalian tree of life-a finding that matches other recently-published observations [7]. An additional taxonomic analysis identified four clades-the cetaceans, a subclade of mostly insectivorous bats, and two subclades of New World rodents-with fewer missing viruses than other mammals (ED Figure 3, ED Table 3). This suggests sampling efforts in these taxa may be more complete than viral sampling more generally [8]. Finally, we mapped the global geographic hotspots of interactions based on mammal ranges, and found significant under-representation in the Amazon and Congo basins; these coldspots are significantly reduced (though not fully eliminated) in the imputed network, suggesting that our model is able to successfully anticipate the thousands of undiscovered ecological interactions that occur in hyperdiverse but undersampled tropical rainforests (ED Figure 4). Emergent properties of the imputed host-virus network Compared to the 5,494 interactions recorded in our original mammal-virus dataset, our model predicted a total of 75,901 new interactions (ED Figure 5). With a total of 81,395 interactions, the imputed network has a connectance of 0.09, which is well within the range of connectances for antagonistic bipartite networks [9]. The best scoring model has a false discovery rate of 9.3%, meaning that it is potentially over-predicting about 7,060 interactions. The same model has a false omission rate of 23%, which would suggest a number of undiscovered interactions of the order of 10 5 for this dataset. This being said, these numbers should be interpreted within the context of data constraints: the initial dataset is biased towards extreme sparsity, and for this reason it is likely that the imputed network is less severely incomplete than the false omission rate would suggest. We next examined the post-imputation network for meaningful biological signals. The "evolutionary distance effect" is often the best-supported signal in host-virus networks: closely-related hosts share both viruses (through coevolution) and microbiologically relevant traits (through identity by descent) which facilitate cross-species transmission, leading to a correlation between evolutionary distance and virome similarity [10]. We tested this property in both the pre-and post-imputation networks by examining viral sharing, both pairwise among all hosts and between humans and other mammals. We found a strong and consistent phylogenetic distance effect in both viral sharing (whether two hosts share any viruses at all) and the total number of viruses shared, both pairwise among mammals and specifically with humans (ED Figure 6); although imputation reduced the signal of these effects, all but one (binary viral sharing with humans) remained significant even after imputation (ED Tables 4, 5). These results suggest that the missing interactions identified by our model have a high biological plausibility, and that-even though we do not incorporate phylogeny or any other host traits into our analyses-the latent factors that structure the network are identified and successfully recapitulated by the model. Finally, we evaluated the effect of imputation on the spatial distribution of viral biodiversity. The local contribution to beta-diversity approach [11] -essentially a partition of the variance in the community matrix -measures the extent to which a single location differs from the expectation based on the entire range considered. When applied to interactions [12], it reveals areas where, although the network might not be structurally different, it is composed of interactions that do not usually occur together. In biological terms, this means that novel host-jumps are possible through different host-virus pairs being in contact. Comparing the uniqueness of the viral community composition based on host spatial distribution before and after imputation reveals an undocumented hotspot of unique host-virus associations in the Amazon (Figure 1, ED Figure 7). Predicting viruses with zoonotic potential We finally explored whether network-wide prediction offered useful insights into zoonotic potential, the ability of a virus to infect humans (a subset of links with one focal node in the network). Surprisingly, we found that the imputation method did not predict known human-associated viruses any better than random (AUC = 0.51; ED Table 6). This finding does reassuringly imply that zoonotic viruses are not contributing a particularly strong structural bias to the predictions, but indicates that the model performs poorly when predictions are restricted to one fairly-atypical node out of over 1,000. Indeed, while the ability of the model to predict the viruses associated with a given host generally increased as hosts are linked to more viruses, performance was poor for hosts linked to unusually-high numbers of viruses relative to the rest of the dataset (of which humans were the most extreme; ED Figure 8). A similar, but less extreme, pattern was observed among viruses linked to above-average numbers of hosts. Thus, although our best model focusing exclusively on connectance performed well in general, models incorporating in-or out-degree or specialized to a particular node may be needed for better-sampled nodes. We next investigated whether the imputed host-virus network could be applied in specialised models aimed at identifying human-infecting viruses. Viral host breadth is a widely-used predictor of zoonotic ability, but is generally unavailable for poorly-studied viruses [13,14,6]. To test whether the structural information on host range from our imputed network can be made accessible for prediction, we revisited a recently-developed model that applies boosted regression tree models to predict zoonotic potential based on the genome composition of animal viruses [6]. We extracted the position of viruses in the pre-and post-imputation networks by removing humans (as well as viruses linked only to humans in the observed data) and applying random dot product graph embedding, which generated a total of 12 latent features that describe each virus's relationship to other viruses and animal hosts in the network. We then added these features to the genome compositionbased model, and compared performance on the same set of viruses. Models incorporating the embeddings performed significantly better than a genome composition-only model, despite the fact that humans were removed from the network. Using embeddings derived from the post-imputation network consistently produced better predictions (mean test-set AUC = 0.875, SD = 0.04; Figure 2). Averaging predictions across the top 10% of repeated training iterations [15] further improved performance (AUC = 0.898). Moreover, of the top 20 viruses predicted by the algorithm, eleven already have serological or otherwisecircumstantial evidence of human infection (ED Table 7), as do many of the other highly ranked viruses (ED Figure 9). The performance gains from using the imputed network suggest that LF-SVD is a viable option for improving applications reliant on knowledge of the global virome, including the identification of potential zoonotic viruses. However, more work is needed to establish the exact operating conditions under which such an approach can be safely applied; in particular, the number of animal hosts which need to have been found before reliable inferences on zoonotic risk can be made for novel viruses (cf. ED Figure 8) is difficult to assess without detailed data on the order in which hosts are linked 6 to viruses (expected to be nonrandom given sampling biases). Conclusions These results indicate that the structure of the observed host-virus network contains meaningful information about the rules of cross-species transmission. The imputation process recovers more of this information, even without the use of mechanistic predictors like host phylogeny, retaining biologically relevant signals while reducing key biases in current observational data. Thus, future efforts to predict viral emergence may be able to leverage the use of recommender systems as a data-inflation step to make better predictions. However, these approaches (and notably their validation) remain limited by how poorly characterized the host range of most viruses is; the majority of viruses are either undiscovered or known from a single host. As the global virome becomes better sampled, these approaches will be increasingly reliable not just for biological inference but for actionable efforts to prevent zoonotic emergence. Acknowledgements: This work was supported by funding to the Viral Emergence Research Initiative (VERENA) consortium including NSF BII 2021909 and a grant from Institut de Valorisation des Données (IVADO). NM was funded by the Wellcome Trust (217221/Z/19/Z). TP and MAO were funded by the Fondation Courtois. This research was enabled in part by support provided by Calcul Québec (www.calculquebec.ca) and Compute Canada (www.computecanada.ca). We acknowledge that this study was conducted on land within the traditional unceded territory of the Saint Lawrence Iroquoian, Anishinabewaki, Mohawk, Huron-Wendat, and Omàmiwininiwak nations. Code availability statement: The code for LF-SVD tuning, imputation, and analysis, has been archived at 10.5281/zenodo.4850581. The code for prediction of zoonotic potential has been archived at 10.5281/zenodo.4850643. Methods Model design and implementation Host-virus association data We used a recently published dataset called CLOVER [16], which is the largest open dataset describing the mammal-virus network currently available, and combines data from four sources that each cover overlapping-but-distinct portions: the Host-Pathogen Phylogeny Project (HP3) dataset [13], the ENHanCEd Infectious Diseases Database (EID2) [17], the Global Mammal Parasite Database version 2.0 (GMPD2) [18], and an unnamed dataset recently published by Shaw et al. [19]. By reconciling these datasets and their underlying taxonomy, the CLOVER dataset is able to achieve a 30% reduction in matrix sparsity over the next most detailed dataset. The CLOVER dataset describes 5,494 interactions between 829 viruses and 1,081 mammalian hosts. The majority of these interactions have been recorded in wild animals, using a combination of detection methods (usually serology, PCR, or viral isolation). A small portion of records assimilated from NCBI's GenBank into these other datasets may also record experimental infections, which provide insight into biological compatibility but not necessarily opportunity for infection in nature. Each of the component datasets, and the CLOVER dataset, are presence-only (i.e., they only report an edgelist of known interactions, and do not include true negatives). Imputation model description The imputation model uses two steps to chain linear filtering (which can recommend potentially false-negative interactions [20]) to recommendation based on singular value decomposition (which adequately captures the low-rank structure of ecological association networks [21]). This imputation model is hereafter termed LF-SVD. The LF step relies on four hyper-parameters expressed as an array of weights = [ 1 , 2 , 3 , 4 ] , which are respectively the relative importance of the original (i.e., observed) value of the interaction, in-and out-degree, and of connectance (the constraint ∑ = 1 is always enforced). LF creates a potential matrix from an observed matrix of size ( , ) by assigning every interaction between species and an initial score given by the dot product of weights and properties of , = [ , 1 ∑ , 1 ∑ , 1 ∑ ] ⋅ . This corresponds to a weighted average of averages, wherein ∀( , ), 0 ≤ ≤ 1. We compared three parameterizations of the : connectance only ([0, 0, 0, 1] ), degree only ([0, 1, 1, 0] ), and hybrid ([0, 1, 1, 1] ). While technically there is an infinite number of possible configurations for the LF weight vector, the computational cost of a grid search is prohibitive, and these parameterizations have the added benefit of corresponding to phenomenological assumptions about what drives network structure that have been well laid out in the literature [9]. In this application, we set 1 = 0, as the initial value of the interaction is ignored, reducing th number of hyper-parameters to tune from four to three. We updated the initial values produced by LF using (truncated) singular value decomposition (SVD) imputation. Like Principal Component Analysis (PCA), SVD is an embedding of a starting matrix into latent subspaces; compared to PCA, SVD is a more general solution that also handles numerical instability due to very small entries well [22], which is a likely scenario as some interaction probabilities are expected to be small. As all entries of both and are in ℝ, we can decompose either of these matrices as , where and are unitary matrices known as the left and right subspaces, and is a diagonal matrix containing the singular vales of the decomposed matrix. To impute the interaction ( , ), we create a matrix = , wherein = (according to the LF model). To decompose this matrix at low-rank , we set the values of larger than to 0, and calculate the approximate version of as̄ = The overall SVD step was conducted as follows: for every interaction ( , ), we first set its value according to the LF model, and perform the truncated SVD step as outlined above. We then update so that =̄ . The SVD step is repeated 20 times (after preliminary assays revealed that the absolute change after 10 iterations was consistently smaller than 10 −3 ), and the final value after 20 iterations is the score for the imputed interaction. Note that due to the nature of SVD, the score is not bound to [0, 1]. The tuning and imputation of the LF-SVD model were performed in the software Julia 1.6 [23], using the EcologicalNetworks.jl package [24]. Hyper-parameters tuning, thresholding, and evidence scoring In order to tune the hyper-parameters (LF weight vector, SVD rank), we picked a calibration set of 800 positives and 800 assumed negative interactions, and imputed them using each possible model ( = 60). This makes the strong assumption that the 800 negative interactions we picked in the calibration set were indeed true negatives; although the model ended up recommending a large number of interactions, ecological networks are known for their sparsity, and we judged this assumption acceptable based on an overall examination of model performance. For each set of 1600 predictions returned by the models, we derived confusion tables at thresholds ranging from the lowest to the highest score, using a stepsize of 1000. From this confusion table, we calculated the ROC-AUC, true/false positive/negative rates, positive/negative predictive values, false discovery/omission rates, critical success index, accuracy, and informedness (a.k.a Youden's J). The model with the highest ROC-AUC was picked as the best model, and used for the rest of this manuscript. The exact cutoff to use to transform the continuous output of LF-SVD into a binary classifier (i.e. the interaction is recommended or not) was determined by picking the threshold value maximizing Youden's J statistic. Each interaction is given an evidence score, which is obtained by dividing the values post-imputation (LF-SVD) by the values pre-imputation (LF), minus one. An evidence of 0 means that the imputation did not change the value, and increasingly positive values meant that the change due to imputation was stronger. This interaction evidence was used to rank interactions when required for the analyses. Analysis of the imputed network Additional data sources For phylogenetic analyses, we used a recently published mammalian supertree published by Upham et al. [25], which has been taxonomically harmonized to the CLOVER dataset for ease of analysis. For geographic analyses, we used the IUCN Red List (iucnredlist.org) species distribution maps for mammals, downloaded on June 6, 2019. For citation counts, we extracted total virus-related publications for each species (by searching for host species binomial plus all known synonyms and "virus" or "viral") from the PubMed database using the R package rentrez [26]. Testing effects of biased data collection Observed host-pathogen association networks compiled from published records are influenced by a passive sampling bias resulting from differential research across host and pathogen species. In comparative analyses of viral richness per host species, the number of publications per host species is often included as a covariate in an attempt to control for variable sampling effort across hosts [27]. This estimate of sampling bias is consistently positively related to viral richness, and typically is the strongest predictor, explaining more variation than other biological covariates [13,28,29,30,31]. To explore whether network imputation via LF-SVD is extrapolating sampling biases across host species, we conducted a set of phylogenetic regressions of the relationship between viral richness and the number of publications per host species (both in total and limited to those including keywords about viruses). Models were fit using the formulation of phylogenetic least squares regression provided via the pgls function (Pagel's lambda estimated via maximum likelihood) in the R package caper [32,33]. By comparing models of observed viral richness to estimates after imputation with LF-SVD, we investigate the slope of the relationship and the explained variance in viral richness to assess how strongly passive sampling biases are retained in the LF-SVD imputed network. In addition to passive sampling bias, host-virus association data are frequently shaped by active or impact bias, where surveillance is targeted based on relevance to human health or economics. This is easily detected in records of viral sharing with humans. In principle, the species with the highest similarity to the human virome should be species that are closely related to humans (primates) or frequently live alongside humans (domesticated animals or synanthropic wildlife, particularly rodents that can live in human settlements), but domesticated animals and laboratory model systems will also score disproportionately in this metric, because of sampling effort. As a new test of model bias, we propose that imputation should reduce the signal of the latter group in viral sharing with Homo sapiens, leaving mostly the former. To test the effect of active sampling bias, we examined the top 10 hosts based on similarity to Homo sapiens, pre-and post-imputation. Prior to imputation, the top 10 list (based on Jaccard similarity of host and human viral community) includes six livestock or companion animals (Bos taurus, Equus cabalus, Sus scrofa, Ovis aries, Capra hircus, and Canis lupus familiaris), three primates (Pan troglodytes, Macaca mulatta, and Macaca fascicularis), and one synanthropic and commonly-studied laboratory animal (Mus musculus). After imputation, four of the domesticated or primate species remained (Canis lupus familiaris, Equus caballus, Sus scrofa, and and Pan troglodytes). The updated list includes two more primates (Gorilla beringei, Gorilla gorilla) and four more mice or rats (Hylaeamys megacephalus, Permoyscus maniculatus, Proechimys guyannensis, and Zygodontomys brevicauda). This mostly reflects changes in the network connectivity; all but one of these are in the top 10 species to gain links (with Z. brevicauda replaced by Rattus rattus). Phylogeographic signals of missing interactions The distribution of missing viruses (each host species' total number of predicted but unknown host-virus links) across space, and across the evolutionary tree, are interlinked patterns that are of significant interest to viral ecologists [13]. These patterns inform scientists' understanding of where undiscovered zoonotic threats might emerge and can be used to target sampling to locations and taxa with the greatest number of undiscovered viruses. However, these predictions are also difficult to disentangle from sampling bias, which can create spurious patterns that are undermined on closer analysis [7]. To assess phylogenetic patterns in the number of missing viruses, we used the previouslyspecified supertree [25]. To match virus data against the phylogeny, we averaged missing virus counts for 30 species (n = 14 tips in the supertree). We used the caper R package to first broadly estimate phylogenetic signal as Pagel's [34]. We next applied a graph partitioning algorithm, phylogenetic factorization, to more flexibly identify mammal clades that differ in missing virus counts. We used the phylofactor R package to partition counts of missing viruses in a series of generalized linear models with a negative binomial distribution [35]. We determined the number of significant clades using Holm's sequentially rejective test with a 5% family-wise error rate. We identified weak-to-moderate overall phylogenetic signal in the number of missing viruses ( = 0.35), although this estimate was distinct from both phylogenetically indepent models and Brownian motion models of evolution (both p < 0.01). Phylogenetic factorization in turn identified only four small clades with significantly different counts of missing viruses, all of which had fewer missing viruses than the remaining mammal phylogeny (ED Table 3). These clades included ceteceans (̄ = 18, n = 30) and a subclade of primarily insectivorous Yangochiroptera (̄ = 43, n = 109) as well as two subclades of the New World rodent subfamily Sigmodontinae (̄ = 11, n = 11;̄ = 16, n = 15). Overall, these results indicate that, with the exception of some coldspots likely driven by oversampling (or, in the case of cetaceans, a peripheral role in the host-virus network), missing viruses are distributed fairly equally across the mammalian tree of life-a finding that matches other recently-published work [7]. To assess geographic patterns in the number of missing viruses, we evaluated the number of known and missing viruses at the level of each host species and joined these to each host's IUCN range map. We mapped the total number of hosts with recorded interactions, the total number of known and predicted missing interactions, and the normalized difference between missing interactions and host diversity. Known interactions are recorded disproportionately in Europe and Asia, and to a lesser degree North America, a pattern that reveals strong sampling bias in viral inventories (ED Figure 4). This pattern is substantially reduced in the missing interactions, which globally track the true distribution of mammal diversity fairly well (better, in some places, than the hosts with viral interactions recorded in CLOVER). However, the normalized difference map still revealed a bias towards interactions predicted in North America and Eurasia, with coldspots in South America and Africa (ED Figure 4). Coevolutionary signal in viral sharing To test for the signal of evolutionary history in the viral sharing network, we analyzed two outcome variables (viral sharing as a binary state, and as the total number of viruses shared) for two data structures (the entire pairwise host-host viral sharing matrix, or each hosts' sharing with Homo sapiens, i.e., its role in zoonotic disease) in both the pre-and postimputation network. We analyzed these variables as a function of phylogenetic distance using generalized linear models (GLMs), with viral sharing coded as a binomial outcome (logit link) and the count data modeled using a Poisson distribution. GLMs were fit using the stats package in R, and adjusted R-squared values were derived using the rsq package. Model coefficients and significance are given in ED Tables 4 and 5. Response curves were finally plotted using the automated smoothing in the ggplot package with the same specifications. We found that viral sharing, as a binary outcome, decoupled substantially from phylogeny after imputation. In large part, this can be explained by the fact that, with a 16-fold increase in connectance, binary sharing should become substantially less informative after imputation. (This also makes biological sense: for example, nearly all mammal species should share the capacity to be infected with true generalist viruses like rabies and influenza A.) In particular, the phylogenetic signal of viral sharing with humans became insignificant (p = 0.52) after imputation, the only insignificant relationship among those we tested. While the count data also recovered a reduction in effect size after imputation, we found that this reduction was much smaller, and that the phylogenetic signal of sharing with Homo sapiens was slightly more explanatory in the post-imputation network. Community uniqueness analysis We performed a measure of community compositional uniqueness using the local contribution to beta-diversity approach [11], and specifically its extension to interaction data following [12]. LCBD identifies locations (here, pixels) in which the community composition contributes more to the overall dissimilarity. For this section, we will note the sites-by-items matrix, often referred to as a "community data matrix," in which locations are rows, and items (host,viruses, interactions) are columns. The total beta-diversity is measured as = Var( ), after rows and columns with a marginal sum of 0 have been removed. The matrix is then transformed by centering and squaring the values, so that = [ ] = [( −̄ ) 2 ]. The sum of squares in is then simply given by SS total = ∑ ∑ . From there, measuring the LCBD (i.e. the actual contribution of each location to ) is done by summing the matrix row-wise, and dividing by the total sum of squares: LCBD = ∑ SS total . Within every location, this value indicates the degree of uniqueness of this location (sampling unit) compared to all other sampling units in the data. LCBD values are typically, but not necessarilly, measured after has been transformed using Chord's or Hellinger's distance. This, however, assumes that sampling is close to complete, which is an unreasonable assumption in our observed dataset; as applying an Hellinger transformation post-but not pre-imputation would prevent a comparison of the results, we work on the raw matrices. Prediction of zoonotic potential We next tested whether the expanded host-range information available in the imputed network could improve zoonotic risk prediction in cases where information on individual viruses is limited. We expanded a recently developed model which combines summary statistics of viral genome composition and compositional similarity to human genes to predict zoonotic risk [6]. We extracted pseudo-traits from the host-virus network using latent variables, by extracting the left latent subspace of a random dot product graph decomposition [36]. We used the same number of dimensions (12) as for the low-rank approximation based on the imputation method. The feature matrix for viruses is given bȳ = 12 √ 12 where and are the truncated left-subspace and singular values matrices of the decomposition of the network. This method was selected because the latent traits extracted this way can reproduce the original network within an arbitrary precision threshold, and have been shown to capture evolutionary signal on network structure [36]. To avoid leaking data on observed human infection into subsequent model training and evaluation steps, these network embeddings were generated while excluding humans. We also removed all viruses which had thus far only been linked to humans, since -after the removal of humans from the network -these viruses were uniquely identifiable as some of the only included viruses with no links in the network (another potential data leak; a small number of viruses with no known mammalian hosts were similarly unlinked, but these were rare enough that a model which predicted all unlinked viruses as human-infecting would have had reasonably high performance). Full genomes were available for 612 of the remaining viruses. We used the reference sequence for each virus whenever available, or the longest complete genome otherwise. These genomes were used to calculate the relevant genome composition measures described in [6]. These were combined with the embeddings to train a series of gradient boosted classification and regression tree models to distinguish between viruses known to infect humans and other viruses. Viruses were randomly split into three datasets, using 70% for training, 15% for model calibration, and the remaining 15% for evaluating model performance [6]. This training/calibration/test procedure was repeated 1000 times to assess variability in performance arising from current limited knowledge of the human-infecting virome. We compared models trained on either the original viral genome composition descriptors from [6], or a combination of viral genome composition and embeddings derived from either the observed network or from the imputed network. Finally, the best model by ROC-AUC (the model using viral genome features and embedding-features describing the imputed network) was used to predict the probability of human infection for all 612 viruses. For this purpose, predictions were averaged across the best-performing 10% of models in which each virus occurred in the test data, a process akin to bagging [15]. Model performance was re-evaluated while excluding the virus being predicted, to avoid selecting models based on their performance on the virus being predicted. Figure 1: Imputed network reveals undiscovered hotspots of unique host-virus associations in the Amazon. Top: difference between (ranged) compositional uniqueness of the viral community based on host presence (see ED Fig. 7). Dark green areas indicate that the imputed network suggests a higher originality of the viral community than available data would. Bottom: comparison between the number of hosts and the uniqueness of the viral community. Assuming random discovery of viruses through host sampling, this relationship would be overall linear and positive, as is the case pre-imputation. Adding imputed interactions removes some of the sampling biases, and shows how areas with lower host richness have more unique contributions to viral uniqueness, which suggests that they harbor viruses not shared by more speciose locations. Extended Data Figure 1: Receiver operating characteristic (ROC) curve for the best model. All models (from SVD rank 1 to 20, and using three linear filtering parameterization) have been compared on the same training/validation dataset by measuring the area under the ROC curve. The best model has an area under the ROC curve (AUC) of 0.85. The threshold turning the continuous prediction of LF-SVD into a binary classification was picked to simultaneously maximize the true positive rate and minimize the false positive rate, in practice by maximizing Youden's informedness. By coincidence, this threshold is also approximately equal to 0.85 in the best model. Extended Data Figure 2: Pairwise host-similarity to Homo sapiens changes post imputation. The ten hosts with the most viral overlap to Homo sapiens (as measured by Jaccard similarity) tend to be livestock. By contrast, the ten most similar hosts after imputation are mostly composed of primates and rodents, which suggests that LF-SVD is able to overcome taxonomic bias in the original dataset. Extended Data Figure 6: Evolutionary signals dominate viral sharing. Phylogenetic distance among all mammals (top) or from humans (bottom) structure viral sharing measured as a binary trait (left) or based on the total number of shared viruses (right). In the imputed network, mosts hosts have many more ; as a result, phylogenetic distance is less informative of whether hosts share viruses, because most hosts share at least one virus, but the phylogenetic signal of the count data is much stronger. Curves are given as generalized linear model smooths, with a Poisson distribution for count data and a binomial distribution with a logit link function for viral sharing. Figures and Tables Extended Data Figure 7: Imputation changes the spatial location of viral uniqueness hostpots. The top and bottom panel show, respectively, the LCBD calculated on viral community composition before and after imputation. Although sampling biases in the original dataset (notably an over-sampling of livestock viruses) puts a lot of emphasis on Europe, the main hostpot post-imputation is in the Amazon. Linear regression between the two layers reveals that the bias reduction only has a moderate effect on the overall relative patterns (constrained to have a 0 intercept; = 1080; 2 = 0.92. Extended Data Figure 9: Ranking viruses by their predicted probability of human infection accurately predicts known infections. Viruses are arranged by the mean prediction produced by a bagged version of the model trained on both genome composition features and an embedding representing the imputed network (black line). Error bars show the region containing 95% of the predictions used for bagging. Dashed lines highlight the cut-off which maximizes informedness (Youden's J) when converting mean predicted probabilities to binary predictions. A second panel shows the most reliable detection method providing evidence of human infection for each virus in the CLOVER database. For the purposes of model training, viruses linked to humans through serological detections only or where the detection method was unspecified were labelled as negative; the model nevertheless identifies the majority of these as human-infecting. Extended Data Table 1: Model performance for the top 10 models by AUC. Metrics include the AUC and cutoff (expressed as a pseudo-probability), the true positive and true negative rates (TPR, TNR), the positive and negative predictive values (PPV, NPV), the false negative and positive rates (FNR, FPR), the false discovery and false omission rates (FDR, FOR), the critical success index (CSI), accuracy (ACC), and Youden's J. Extended Data Table 2: Imputation reduces the effect of sampling bias. To explore whether network imputation via LF-SVD is extrapolating existing research biases, we conducted a set of comparative analyses investigating the how the explanatory power of sampling effort on viral species richness changes after network imputation. We find that after imputation, the slope of the relationship ( ) decreases, and sampling effort explains less of the variance in viral richness ( 2 ), suggesting that imputation via LF-SVD is not merely recapitulating the observed sampling effort per host. Statistics are given for a phylognetic generalized linear model fit with the maximum likelihood estimate of Pagel's . Predictors and responses were log-10 transformed prior to analyses. Figure 2 : 2Imputing the viral sharing network improved prediction of human infection ability. (A) An existing model of human infection-risk using viral genomic features is improved when network embeddings are added as viral traits; models that use embeddings from the imputed network perform better than those using the observed network. Violins and boxplots show the ROC-AUC for test-set predictions across 1000 replicate 70%:15%:15% train:calibrate:test splits ( = 612). P-values from pairwise Kruskall-Wallis rank sum tests are shown for all comparisons. Diamonds indicate the performance of a bagged model which averages predictions from the 100 best-performing models, based on test-set AUC iteratively re-calculated while excluding the virus being predicted. (Mean AUC: genome composition model = 0.723; genome composition + observed network = 0.830; genome composition + imputed network = 0.875.) (B) Predictive feature importance in the combined (genome composition + imputed network) model; network embeddings are consistently the top predictive features, compared to biologically-informative measures of genome composition. Extended Data Figure 3 :Figure 4 : 34Phylogenetic bias in missing viruses. Phylogenetic factorization determined that the majority of species have no phylogenetic signal in the number of missing viruses estimated by the LF-SVD model, with the exception of a handful of small clades that included cetaceans (clade 1), a mostly insectivorous subclade of the Yangochiroptera (clade 2), and two small rodent clades (clades 3 and 4), all of which have significantly fewer than Geographic biases in sparsity are reduced, but not entirely eliminated, by the imputation model. Host diversity (number of species) in the CLOVER dataset closely tracks true patterns of global biodiversity (A), but the total number of interactions recorded does not (B), due to a high degree of sampling bias. Model-based predictions of undiscovered interactions (C) much more closely track true biodiversity gradients, but likely underestimates in South America and Africa due to sparsity (D). Numbers of interactions are given in thousands (B,C). Hotspots in (D) are given as the difference between the number of undiscovered interactions and underlying host diversity, both rescaled between 0 and 1.Extended DataFigure 5: The global virome, pre-and post-imputation. Network layouts reflect the first two dimensions of a tSNE embedding on four dimensions, wherein the positions of nodes where initially picked based on a principal components analysis. Hosts are shown as circles and viruses as downwards-pointing triangles, and the relative size of each point scales linearly with degree (using the same scale for both figures, i.e. two nodes with the same degree will have the same size in the left and right panels). : 3 3Phylofactorization of missing viruses. Significant clades identified from a phylogenetic factorization of missing virus counts. Included taxa are listed alongside the number of species and the mean number of missing viruses for each clade in comparison to the paraphyletic remainder. Clade codes match ED Figure Extended DataFigure 8: Predictive performance of LF-SVD generally increases with increased connectivity. Points represent individual host species, and show the probability that a randomly sampled virus known to infect that host will be ranked above a randomly sampled virus which has not been observed to do so (measured as the area under the receiver operating characteristic curve [AUC]). While hosts subject to extreme study bias such as humans cannot be predicted, this does not appear to degrade performance on other species.Bos taurus Capra hircus Equus caballus Homo sapiens Macaca mulatta Mus musculus Ovis aries Pan troglodytes Sus scrofa 0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 1.5 2.0 2.5 log 10   Number of known viruses   AUC A Bluetongue virus Rabies lyssavirus West Nile virus 0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 1.5 2.0 2.5 log 10   Number of known hosts   AUC B 0.00 0.25 0.50 0.75 1.00 Predicted probability Strong evidence of human infection True False No detection Not specified Serology Genetic Isolation Virus species Detection method Extended DataTable 4: Phylogenetic signal in viral sharing, pre-and post-imputation. Statistics are given for a generalized linear model fit with a binomial distribution for the outcome variable (whether any viruses at all are shared between two hosts). Significance given as *** indicates p < 0.001.1 Ziphiidae, Physeteridae, Phocoenidae, Monodontidae, Delphinidae, Eschrichtiidae, Balaenopteridae, Balaenidae 30 18 71 2 Nycteridae, Emballonuridae, Natalidae, Molossidae, Vespertilionidae 109 43 73 3 Calomys, Graomys, Phyllotis, Loxodontomys, Abrothrix 11 11 70 4 Bibimys, Oxymycterus, Necromys, Akodon, Thaptomys 15 16 70 32 Sharing Data source S.E. p 2 (adj.) Pairwise (all hosts) Pre-imputation 2.23 e-2 8.44 e-05 *** 9.8% Pairwise (all hosts) Post-imputation 3.50 e-03 1.30 e-4 *** 0.07% With Homo sapiens Pre-imputation 3.32 e-2 1.07 e-2 *** With Homo sapiens Post-imputation 1.46 e-2 2.30 e-2 0.524 -0.09% Extended DataTable 5: Phylogenetic signal in number of viruses shared, pre-and postimputation. Statistics are given for a generalized linear model fit with a Poisson distribution for the outcome variable. Significance given as *** indicates p < 0.001.SharingData source S.E. p 2 (adj.) Pairwise (all hosts) Pre-imputation -2.04 e-02 4.97 e-05 *** 8.5% Pairwise (all hosts) Post-imputation -6.12 e-03 7.20 e-06 *** 3.0% With Homo sapiens Pre-imputation 6.53 e-03 4.50 e-04 *** 2.2% With Homo sapiens Post-imputation 4.76 e-0.3 1.08 e-04 *** 3.8%Extended DataTable 6: The top 10 predicted (novel) zoonotic links in the postimputation network. Evidence of interaction generated by the imputation model is contrasted against prior predictions by[6], who implemented a model that successfully predicts zoonotic potential from viral genome composition bias. Extended DataTable 7: The top 20 predicted (novel) zoonotic viruses in the extended model. All are classified as "very high" risk by the combined model, which uses both viral genome compositions and imputed network embeddings. Prior risk assignments from[6]are also given where possible. (*: Indicates that a virus has serological evidence of human infection in CLOVER, which was not included as a positive in the genomic model, but was considered evidence of association in the mammal-virus network; however, note that Homo sapiens and its associations were dropped before generating embeddings. †: Indicates that a virus has recorded evidence of human infection in CDC's ArboCat, though original source literature is not traceable. ‡: Indicates that a virus is accepted as a human virus by[37]. 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[ "Improving Visual-textual Sentiment Analysis by Fusing Expert Features", "Improving Visual-textual Sentiment Analysis by Fusing Expert Features" ]
[ "Junyu Chen \nUniversity of Rochester\n\n", "Jie An \nUniversity of Rochester\n\n", "Hanjia Lyu \nUniversity of Rochester\n\n", "Jiebo Luo [email protected] \nUniversity of Rochester\n\n" ]
[ "University of Rochester\n", "University of Rochester\n", "University of Rochester\n", "University of Rochester\n" ]
[]
Visual-textual sentiment analysis aims to predict sentiment with the input of a pair of image and text. The main challenge of visual-textual sentiment analysis is how to learn effective visual features for sentiment prediction since input images are often very diverse. To address this challenge, we propose a new method that improves visual-textual sentiment analysis by introducing powerful expert visual features. The proposed method consists of four parts: (1) a visual-textual branch to learn features directly from data for sentiment analysis, (2) a visual expert branch with a set of pre-trained "expert" encoders to extract effective visual features, (3) a CLIP branch to implicitly model visual-textual correspondence, and (4) a multimodal feature fusion network based on either BERT or MLP to fuse multimodal features and make sentiment prediction. Extensive experiments on three datasets show that our method produces better visual-textual sentiment analysis performance than existing methods.
10.48550/arxiv.2211.12981
[ "https://export.arxiv.org/pdf/2211.12981v1.pdf" ]
253,801,570
2211.12981
886f664ee3cb87bb3f8f299e623e65ee0e7f9397
Improving Visual-textual Sentiment Analysis by Fusing Expert Features Junyu Chen University of Rochester Jie An University of Rochester Hanjia Lyu University of Rochester Jiebo Luo [email protected] University of Rochester Improving Visual-textual Sentiment Analysis by Fusing Expert Features Visual-textual sentiment analysis aims to predict sentiment with the input of a pair of image and text. The main challenge of visual-textual sentiment analysis is how to learn effective visual features for sentiment prediction since input images are often very diverse. To address this challenge, we propose a new method that improves visual-textual sentiment analysis by introducing powerful expert visual features. The proposed method consists of four parts: (1) a visual-textual branch to learn features directly from data for sentiment analysis, (2) a visual expert branch with a set of pre-trained "expert" encoders to extract effective visual features, (3) a CLIP branch to implicitly model visual-textual correspondence, and (4) a multimodal feature fusion network based on either BERT or MLP to fuse multimodal features and make sentiment prediction. Extensive experiments on three datasets show that our method produces better visual-textual sentiment analysis performance than existing methods. Introduction Visual-textual sentiment analysis (You et al. 2015) is an interesting task at the intersection of computer vision and natural language processing. Given a pair of image and text as the input, the visual-textual sentiment analysis task aims at recognizing the polarized sentiments (i.e., positive, negative, and neutral) or fine-grained emotions (e.g., happy, angry, calm, etc.) from the input. A typical application of visualtextual sentiment analysis is to predict the polarized sentiment of the user posts in social networks, where users post texts and images to jointly express their feelings. The key to visual-textual sentiment analysis is effective visual/textual features, from which the sentiment can be inferred by fusing the visual and textual information. Most existing works (You et al. 2016a;Zhu et al. 2019) follow the framework of training two encoders to extract visual and textual information, respectively, where the visual encoder is usually based on CNN and text encoder is based on either RNN or Transformer. Next, a feature fusion module is used to infer the sentiment from extracted visual and textual features. Although significant advances have been made by this framework, a main challenge still exists in visual-textual sentiment analysis -it is difficult to extract effective visual features and recognize the abstract sentiment (unlike objects or scenes) of images. The reason behind this is that sentiment is a comparatively higher level information, which is more difficult to learn with a network. On the other hand, images in visual-textual sentiment datasets come from multiple different domains, which are highly diverse. For example, as shown in Figure 1, some input images are screenshots of texts. Therefore, a powerful visual feature extractor that can handle the heterogeneity of various domains of images is desired for visual-textual sentiment analysis. In this paper, we focus on the visual-textual sentiment analysis task and aim to improve the performance of existing methods with a more powerful visual feature extractor. The proposed algorithm ( Figure 2) consists of four parts. First, a visual-textual branch based on the Swin Transformer (Liu et al. 2021) and RoBERTa ) is used to learn visual and textual features for sentiment prediction, which captures the important information for sentiment prediction directly from the dataset. Second, a visual expert branch with a set of pre-trained "expert" visual encoders is used to equip the proposed method with a more powerful visual feature extractor. It is hard to extract effective visual features only based on the learnable image encoder. Therefore, we adopt a few pre-trained visual features to extract more diverse visual features, including face , object (Jocher et al. 2022), scene (Zhou et al. 2017), and OCR, which are intuitively useful for sentiment analysis (Yuan et al. 2013;Xu and Mao 2017;Cheema et al. 2021) but hard to learn directly from the dataset. It is worth noting that our experiments demonstrate that the above-mentioned visual features can indeed improve the accuracy of the visual-textual analysis. Third, a CLIP branch to implicitly model the visual-textual correspondence, which previous methods usually learn via the attention mechanism (Xu et al. 2020;Huang et al. 2020;Yang et al. 2021;Hu and Yamamura 2022). Since CLIP is trained with a large number of image-text pairs, on the one hand, it improves the generalization ability of the proposed method to handle unseen or weakly sentiment images. On the other hand, because CLIP is trained to minimize the cosine distance between paired images and textual captions, imagetext correspondence can be implicitly induced with the extracted visual and textual features from the CLIP encoder. Finally, a multimodal feature fusion network based on either BERT (Devlin et al. 2018) Figure 1: Examples from a visual-textual sentiment dataset, where image and text jointly deliver the sentiment and it is challenging to predict the sentiment only from the image in some cases. well in different datasets and experimental settings. Our contributions are three folds: • We use the pre-trained CLIP model to extract aligned visual and textual features, which improves the performance of sentiment prediction by allowing the proposed method to implicitly model the visual-textual correspondence. • We adopt a set of "expert" visual encoders which are pre-trained for different visual tasks. These expert features provide a strong visual prior and enable the proposed method to capture complicated and subtle visual information for sentiment analysis. • Our whole framework, which consists of the CLIP, visualexpert, and trained visual-textual branches, achieves better results than the state-of-the-art methods on most existing datasets. Related Work Textual Sentiment Analysis. Textual sentiment analysis is a well-established task that aims to extract the sentiment embedded in the text. Traditional methods mainly focus on hand-crafted and lexicon-based features. Two widely used tools are SentiWordNet (Esuli and Sebastiani 2006) and VADER (Hutto and Gilbert 2014): the former assigns scores to individual words to predict the sentiment, and the latter further takes into account heuristic rules including capitalization, punctuation and degree modifiers for a given sentence. Since they are easily interpretable and computationally efficient, they are widely adopted as a sentiment classifier for downstream tasks such as consumer reviews (Hamouda and Rohaim 2011) and Twitter viewpoint analysis (Bose et al. 2021;Lyu et al. 2022). Recently, with the successes of machine learning techniques, convolutional and sequential models like CNN and LSTM have been widely adopted. Wang et al. (2016) used a CNN to extract the affective information in local regions and incorporated it into an LSTM, which considers both local information and the long-distance dependency of sentences. More recently, BERT-based models (Devlin et al. 2018) are dominating various NLP tasks. Some studies further investigate the effectiveness of aligning the genre of text between pre-training and testing. BERTweet (Nguyen, Vu, and Nguyen 2020), for example, has set a new benchmark for many semantic evaluation tasks of tweets (Barbieri et al. 2020), and shows a good generalization ability to other social media text (Guo et al. 2020). Image Sentiment Analysis. Image sentiment analysis intends to explore the sentiment evoked or expressed by images, which has become popular since people start to share images on social media platforms. Traditional methods are based on low-to mid-level feature extraction. Siersdorfer et al. (2010) used color histogram and SIFT bag-of-visualterm as visual features, along with sentiment scores extract from textual metadata of images like title and description. Borth et al. (2013) proposed to analyze visual sentiment by detecting visual concepts in 1,200 adjective-noun pairs which are considered sentiment-related. The classifier was based on image features including color histogram, local binary pattern, and GIST descriptor. With the development of deep learning, many later works are based on CNN to take advantage of the expressiveness of pre-trained models. Xu et al. (2014) proposed to adopt the CNN that is pre-trained on ImageNet for sentiment analysis tasks. You et al. (2015) designed a progressive fine-tuning procedure and explored the performance boost under the transfer learning setting for polarized sentiment classification on Flicker and Twitter images. Song et al. (2018) and You, Jin, and Luo (2017) both adopted attention modules with CNN, which force the model to focus on salient regions that mostly reveal the sentiment. Compared with textual data, one major challenge for image sentiment analysis lies in a higher level of abstraction and subjectivity. Some work tried to address this issue by explicitly adding additional modules for detecting extra features. Yuan et al. (2013) employed scene-based attributes to define mid-level features, and added a facial expression recognition step, which helped the sentiment prediction task when applied to images with faces. Ortis et al. (2018) proposed to use a pre-trained image caption module to extract objective image description, which improved the classifier performance. Visual-textual Sentiment Analysis. Visual-textual sentiment analysis has become popular nowadays, where a more comprehensive sentiment prediction can be achieved by considering various aspects. The main idea is to effectively combine the information from two modalities that could affect Figure 2: Framework of the proposed method. Our method consists of four parts: (1) a visual-textual branch based on Swin Transformer and RoBERTa to learn visual and textual features for sentiment prediction, (2) a visual expert branch to equip the method with a strong visual prior, (3) a CLIP branch to implicitly model the visual-textual correspondence with aligned embeddings, and (4) a multimodal feature fusion module to integrate all information and make the sentiment prediction. the sentiment. To this end, different fusing methods have been explored. You et al. (2016b) proposed a cross-modality consistent regression model to force the representation extracted from text and image to be consistent. Xu et al. (2020) introduced a dual attention module to capture the correlations between image and text. Du et al. (2022) proposed a gated attention mechanism to encourage the model to focus on the text that is more representative of sentiment, based on visual information. Recently, some studies have explored the benefits of pretext tasks. Ling, Xia et al. (2022) adopted multiple pre-training tasks including masked language/region modeling and textual/visual opinion generation, to facilitate the extraction of fine-grained aspect-based sentiment and the alignment across modalities. Li et al. (2022) utilized sentiment-label-based and dataaugmentation-based contrastive learning to help the model capture the sentiment-related features in multimodal data. Concerned with the noisy nature of user-generated content on social media, researchers normalize and pre-process informal text data into a more informative version, while less attention has been paid to the visual modality. Most existing datasets in this field are collected from social media (Niu et al. 2016;Yang et al. 2020;You et al. 2016b). Compare with other image classification tasks like the Ima-geNet challenge (Deng et al. 2009), images from social media could belong to arbitrary and diverse domains. Instead of using a single encoder for the image to extract a taskspecific sentiment-related feature, Xu and Mao (2017) and Yang et al. (2020) explicitly identified object and scene as semantic features of images, which help to alleviate this issue by considering more aspects. Method To address the challenge of extracting effective visual features for visual-textual sentiment analysis, we propose a new method that takes advantage of the power of pre-trained expert visual encoders and large visual-language pre-trained models for sentiment analysis. As Figure 2 shows, our framework consists of four main parts: (1) a trainable visualtextual branch for extracting multimodal features directly from the training dataset for sentiment analysis, (2) a visual expert branch for input images, which includes a few effective and diverse visual feature extractors to handle noisy and open-domain images, (3) a CLIP branch for both image and text input that implicitly model the cross-modal correspondence via semantic-level features, and (4) a multimodal feature fusion network to integrate all information and make the sentiment prediction. In this section, we will introduce the architecture of each branch and the pre-training procedure. Visual-Textual Branch The visual-textual branch aims at extracting sentimentrelated features directly from the dataset, which comprises two encoders for the input text and image, respectively. Text Encoder. We adopt the BERTweet-large (Nguyen, Vu, and Nguyen 2020) eri et al. 2020), which aims at predicting the class label of tweet posts. Because text sentiment analysis is also a text classification task, we use BERTweet to take advantage of its good performance in extracting sentiment-related features. To adapt the BERTweet model for sentiment analysis, we first train the BERTweet model with only the text in the visual-textual sentiment datasets. Then, we remove the classification head of the BERTweet model and use the tanh activation before the classification head as the text feature. Image Branch. We use Swin Transformer (Liu et al. 2021) as the image encoder. Swin Transformer is a powerful architecture that combines the benefit of both CNN and attentionbased transformers. In many vision tasks including image classification, object detection, and semantic segmentation, Swin Transformer has better performance due to its better scalability and less inductive bias (Khan et al. 2021). To balance the analysis performance and the computation burden, we use the Swin-base architecture. We first load the pre-train parameters of the Swin-base on ImageNet-21k at a resolution of 384x384, which can enhance its ability to handle noisy open-domain images in visual-textual sentiment datasets. Next, similar to the text encoder, we train the Swin-base model with only the images in datasets and use the feature of the last layer before the classification head as the image feature for sentiment analysis. Visual Expert Branch Visual information is important to sentiment analysis. Most existing methods train a CNN to extract visual features from the input image. For data such as a Twitter post with images, the input image might belong to arbitrary categories, such as portrait, animal, scene, and even poster or screenshot of cellphones. It is challenging to train a single network to extract effective features given such a diverse input. To this end, we adopt several pre-trained neural networks to extract face, scene, object, and optical character recognition (OCR) features in images. The pre-trained networks can provide direct and diverse visual features, which provide visual information from various perspectives by concentrating on different aspects of images. Details are as follows. Face Features. Human faces in images are an important source of information for accurate sentiment recognition since facial expression is one of the most straightforward ways for a human to express emotions. For visual-textual sentiment analysis, as shown in Table 1, portrait and selfie photos with faces occupy more than 40% of images in datasets. Therefore, it is necessary to extract face features and make sentiment analyses based on them. To extract effective face features, we first use the MTCNN face detector ) to filter out images without faces. Then we adopt the Facenet (Schroff, Kalenichenko, and Philbin 2015) pre-trained on VGGFace2 (Cao et al. 2018) as the face feature extractor, where the last layer features of the largest face in images are used as the face features. Scene and Object Features. Scenes and objects have been proven by existing methods (Borth et al. 2013;Yang et al. 2021) that are important visual concepts for sentiment analysis. For example, a photo of a delicious meal in a fancy restaurant certainly reveals a sense of enjoyment, while a bold tree standing alone in the graveyard possibly shows the feeling of depression and dreariness. Based on these observations and following Yang et al. (2021), we equip the proposed method with a scene and object feature extractor to give the knowledge of which scenes an image could belong to, and which objects appear in an image. More specifically, we use YOLOv5x6 (Jocher et al. 2022) pre-trained for object detection on the COCO dataset (Lin et al. 2014) as the object feature extractor. The output of the last layer, i.e., the predicted logits for each category are summed up into a vector, which represents the number and type of objects contained in an image and is used as the object feature. Scene features are extracted by a DenseNet161 (Huang et al. 2017) pre-trained on the Place365 (Zhou et al. 2017) dataset for scene recognition, where the pre-trained model has the ability to recognize 365 common scenes such as "airfield ", "campus", and "highway", etc. We directly use the output logit of each image as the scene feature. OCR Features. Images containing text are very popular in user posts on social media platforms like Twitter, which account for a large portion of the visual-textual sentiment datasets. For example, images of Internet memes, social event posters, and screenshots of text messages all have texts within images, where the text and image itself jointly express the emotion that the user wants to deliver. However, such important text information within images usually cannot be implicitly learned by a visual encoder for sentiment analysis. In fact, most image sentiment analysis methods do not handle such images despite their popularity. To improve the sentiment analysis performance on images with texts, we employ an OCR encoder to explicitly extract text within images and then use the feature of the extracted image text for sentiment analysis. More specifically, we use the pre-trained MPNet (Song et al. 2020) as the OCR encoder. To obtain OCR features, we first extract the English words in images with Google's Tesseract-OCR Engine. 1 Next, we concatenate the extracted words into a sequence and feed it into the MPNet, where the output sentence feature of MPNet is used as the OCR feature. To reduce the influence of meaningless words in images, we only extract OCR features for images with 5 or more extracted words. CLIP Branch CLIP (Radford et al. 2021) is a powerful visual-language pre-trained model, which consists of an image encoder and a text encoder pre-trained on a huge dataset of 400 million image-text pairs to learn the correspondence of image and text inputs. Because CLIP is trained with contrastive loss to minimize the cosine similarity between the paired text and image, its encoders can capture the high-level correlation between image and text, which cannot be learned by other feature extractors based on a single modality. In addition, since CLIP is pre-trained with a huge dataset, it has a better generalization ability to handle noisy inputs for sentiment analysis. In our CLIP branch, we use the image and text encoder of CLIP to extract visual and textual features, respectively, where the image encoder is a 24-layer 16-head vision transformer while the text encoder is a 12-layer 12head transformer. Multimodal Feature Fusion Given the extracted visual and text features, we introduce a multimodal feature fusion module that makes sentiment predictions by fusing all visual and textual information. Regarding the architecture of the feature fusion module, we either use a three-layer BERT model or a lightweight three-layer MLP. When the BERT model is used, as Figure 3 shows, we first zero pad all the extracted features to the same size, which is 1024 in our experiments. Next, all features are put together to form a sequence. Finally, the sequential features are fed into the BERT model, which fuses the information from the visual-textual, visual expert, and CLIP branches to make the final sentiment prediction. For the MLP, the visual-textual features are directly concatenated into a onedimensional feature vector and then fed into the MLP to obtain the sentiment prediction. Experiments To demonstrate the performance of the proposed method, we conduct experiments on three publicly available datasets. In this section, we first introduce the datasets and our preprocessing approach. Then we show the compared baselines. Finally, we show the experiment results and our key findings. Datasets and Preprocessing We conduct experiments on three publicly available visualtextual datasets. MVSA-Single and MVSA-Multiple. The MVSA dataset (Niu et al. 2016) was collected from Twitter posts with both text and image. MVSA-Single contains 4,869 image-text pairs and each image-text pair contains a human-annotated sentiment label, i.e., positive, negative, or neutral. The MVSA-Multiple dataset contains 19,600 image-text pairs, and each sample is labeled by three annotators. Following the preprocessing procedure of Xu and Mao (2017), we collect 4,511 and 17,025 image-text pairs from MVSA-Single and MVSA-Multiple, respectively. TumEmo. TumEmo (Yang et al. 2020) is a large visualtextual emotion dataset consists of 195,265 image-text pairs collected from Tumblr. The ground truth emotional annotation of each pair is labeled by a set of rules pre-defined on text, where an image-text pair is categorized into one of seven emotional classes (i.e., angry, bored, calm, fearful, happy, loving, and sad) if the text contains certain keywords related to that emotion class. All hashtags that could reveal the ground truth label are removed. For the pre-processing of the text data, we follow BERTweet's normalization procedure, where emojis, user names, and URLs are replaced with special placeholders. Besides, all the usages of punctuations and abbreviations are made consistent among sentences. Table 1 shows the detailed statistics of the datasets after pre-processing. Since all three visual-textual sentiment datasets do not provide the official train-test lists, most existing methods split the train and test set randomly. For a fair comparison, we follow the same split of Li et al. (2022) 2 for the two MVSA datasets, and follow Yang et al. (2021) 3 for the TumEmo dataset, where only a subset of TumEmo is used. To better evaluate the effectiveness of our framework, we additionally conduct a 10-fold cross-validation for the two MVSA datasets and train the model with the entire TumEmo dataset in our ablation study. Model Training In model training, the CLIP and visual expert branches are fixed. Our training procedure consists of two steps. Single-Modal Training. We first train the visual-textual branch with either texts or images in visual-textual sentiment datasets to let the visual and textual encoders learn enough knowledge for sentiment analysis from single-modal data. For the visual encoder, we train the Swin Transformer with only images in datasets for 20 epochs with a learning rate 1e−4 and batch size 32. For the textual encoder, BERTweet is trained using only texts for 20 epochs with a learning rate of 5e−5 and batch size of 64. Multi-Modal Training. We then train the whole model with both images and texts in datasets. We first initialize the visual-textual branch (i.e., Swin Transformer and BERTweet) with the best parameter Method MVSA-Single MVSA-Multiple TumEmo Acc↑ F1↑ Acc↑ F1↑ Acc↑ F1↑ MultiSentiNet (Xu and Mao 2017) 0.6819* 0.6771* 0.6815* 0.6639* 0.6418 0.5962 Co-Mem (Xu, Mao, and Chen 2018) 0.7051* 0.7001* 0.6992* 0.6983* 0.6426 0.5909 MVAN (Yang et al. 2020) 0.7298* 0.7139* 0.7183* 0.7038* 0.6553 0.6543 MGNNS (Yang et al. 2021) 0.7377* 0.7270* 0.7249* 0.6934* 0.6672 0.6669 CLMLF (Li et al. 2022) 0 Table 2: Quantitative results on three datasets. All the results marked with * use an unknown train-test split list, which cannot be compared with other methods directly and should be treated as references. The best performance is highlighted in bold and the second best is underlined. on the validation set in single-modal training. Then we train the visual-textual branch and the final multimodal feature fusion module (BERT or MLP) together in an end-to-end manner. We use the AdamW optimizer (Loshchilov and Hutter 2017) with learning rate 5e−6 on the two MVSA datasets and 1e−5 on TumEmo dataset. The batch size is set to 16 for all datasets. The dropout rate is set to 0.5 to prevent overfitting. The training procedure stops when it reaches a maximum of 30 epochs, or the losses on the validation set no longer decrease over 3 epochs. Since all three datasets are imbalanced, we save the model parameters when the model reaches the highest F1 score on the validation set and use them for testing. Baselines We compare the proposed method with the following baselines for visual-textual sentiment analysis: Co-Memory (Xu, Mao, and Chen 2018) utilizes the attention mechanism to model the interaction of visual contents and textual words for sentiment analysis. MultiSentiNet (Xu and Mao 2017) explicitly identifies object and scene as semantic features of images, which guides the learning of text representations through attention and aggregates with the textual feature for sentiment analysis. MVAN (Yang et al. 2020) incorporates Co-Memory and MultiSentiNet, where scene-guided and object-guided text features and text-guided scene/object features are learned for better modeling the correspondence between two modalities. MGNNS (Yang et al. 2021) introduces a multi-channel graph neural network to model the object, scene and text representations based on the global co-occurrence characteristics of the whole dataset. CLMLF (Li et al. 2022) designs a contrastive learning framework to help the model learn general representations of both images and texts for visual-textual sentiment analysis. Table 2 shows the quantitative comparison between the proposed method and other state-of-the-art approaches. On the MVSA-Single and TumEmo datasets, the proposed method outperforms the state-of-the-art methods, which demon-strates the effectiveness of the proposed improvement strategies for visual-textual analysis. As discussed previously, the proposed visual-textual branch is based on the pre-trained Swin Transformer and BERTweet, which allows the proposed method to employ the power of state-of-the-art neural network architectures. Regarding the visual expert branch, it has four different visual features that cover more comprehensive visual information than existing methods. For example, MultiSentiNet (Xu and Mao 2017), MVAN (Yang et al. 2020), and MGNNS (Yang et al. 2021) only use pretrained scene and object features while OCR and face features are not used. It is hard for a trainable visual encoder to learn these missing visual features. The CLIP branch enables the implicit modeling of visual-textual correspondence. We think all the above-mentioned benefits of the proposed method lead to its superior performance. Regarding the architecture of the multimodal feature fusion module, it is worth mentioning that a large model does not always lead to better performance because our experiments show that BERT can achieve better results on MVSA-Single while MLP outperforms BERT on TumEmo. We think different architectures may suit different datasets. Therefore, we leave both BERT and MLP as options for a multimodal feature fusion module. Quantitative Comparison Ablation Study To reveal the importance of each branch, we conduct an ablation study by removing each encoder in the proposed method and see the performance change of sentiment analysis. Table 3 shows the results of the ablation study. For the two MVSA datasets, we report averaged metrics of 10-fold cross-validation. For TumEmo, we train the model with all data and report the metrics on the test set. As Table 3 shows, the largest performance drop happens when BERTweet is removed. This shows that text information is of pivotal importance for visual-textual sentiment analysis. We think the reason is that learning sentiment from texts is easier than images since adjectives can deliver sentiment information more straightforwardly than images. For the MVSA-Multiple and TumEmo datasets, removing the CLIP image encoder causes a big performance drop while removing Swin Transformer does not, which shows In the ablation study on the TumEmo dataset, we find that the OCR and scene features do not help in sentiment analysis. The potential reasons may be: (1) the pre-trained OCR and scene encoders are not strong enough to extract features for more complicated seven-class emotion classification, (2) for the OCR encoder, the text information in images may not be consistent with the input texts, and (3) for the scene encoder, the emotion label may be more subjective than the sentiment, which is more difficult to infer from the scene information. We leave a more comprehensive analysis and improvement to future work. Figure 3 : 3Multimodal feature fusion modules. (a) When BERT is used, we first pad all features to the same size and feed them into BERT to perform sentiment prediction. (b) If MLP is used, we connect all features to a single vector and make the prediction via the MLP. or MLP is used to fuse multidomain multimodal features and make sentiment prediction. Our experiment results show that BERT and MLP perform You only allow imprisonment...... Because you are #weak.... and full of #fear..... Control can only work if you #Believe Text: these came in the post for my brother now he wo n't stop trying to make me jealousWe've posted plenty of smiling #dogs before, but this one is the face of true #happiness Good coffee, good scones, good friend. That's all about it. ) #scones #TorontoLife Pamela Wallin case prompts more unwelcome questions for Stephen Harper #cdnpoli httpt.coPt7tAgt72D Positive Positive Negative Negative Without the context provided by text Positive or Negative? Input Image : ImageThe fishing is a little slow but the flowers are vibrant and beautiful.Multi-modal Feature Mixer Positive Feature Sequence Text Branch Image Branch Trained Branches CLIP Text Encoder CLIP Image Encoder CLIP Branches (Fixed) Input Text: Face Recognition Scene Recognition Object Detection OCR & Text Encoding Visual Expert Branches (Fixed) Tokenize Padding Padding Padding Concat model as the text encoder, which is pretrained on 850M English Tweets corpus inspired by the state-of-the-art RoBERTa) model for various NLP tasks. BERTweet is a variant of the powerful BERT(Devlin et al. 2018) architecture. BERTweet achieves the best performance on the TweetEval benchmark (Barbi Table 3 : 3Ablation study results. Here all results are based on using the BERT architecture as the multimodal feature fusion module.that the CLIP image encoder can extract unique image features different from the learnable Swin Transformer. 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Cross-modality consistent regression for joint visual-textual sentiment anal- ysis of social multimedia. In ACM International Conference on Web Search and Data Mining, 13-22. Sentribute: image sentiment analysis from a mid-level perspective. J Yuan, S Mcdonough, Q You, J Luo, International Workshop on Issues of Sentiment Discovery and Opinion Mining. Yuan, J.; Mcdonough, S.; You, Q.; and Luo, J. 2013. Sen- tribute: image sentiment analysis from a mid-level perspec- tive. In International Workshop on Issues of Sentiment Dis- covery and Opinion Mining, 1-8. Joint face detection and alignment using multitask cascaded convolutional networks. K Zhang, Z Zhang, Z Li, Y Qiao, IEEE Signal Processing Letters. 2310Zhang, K.; Zhang, Z.; Li, Z.; and Qiao, Y. 2016. Joint face detection and alignment using multitask cascaded convolu- tional networks. IEEE Signal Processing Letters 23(10): 1499-1503. Places: A 10 million Image Database for Scene Recognition. 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[ "https://github.com/Link-Li/CLMLF/tree/main/dataset/data", "https://github.com/YangXiaocui1215/MGNNS/tree/master/" ]
[ "Reversal of quantised Hall drifts at non-interacting and interacting topological boundaries", "Reversal of quantised Hall drifts at non-interacting and interacting topological boundaries" ]
[ "Zijie Zhu \nInstitute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland\n", "Marius Gächter \nInstitute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland\n", "Anne-Sophie Walter \nInstitute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland\n", "Konrad Viebahn \nInstitute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland\n", "Tilman Esslinger \nInstitute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland\n" ]
[ "Institute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland", "Institute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland", "Institute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland", "Institute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland", "Institute for Quantum Electronics & Quantum Center\nETH Zurich\n8093ZurichSwitzerland" ]
[]
The transport properties of gapless edge modes at boundaries between topologically distinct domains are of fundamental and technological importance. Therefore, it is crucial to gain a better understanding of topological edge states and their response to interparticle interactions. Here, we experimentally study long-distance quantised Hall drifts in a harmonically confined topological pump of non-interacting and interacting ultracold fermionic atoms. We find that quantised drifts halt and reverse their direction when the atoms reach a critical slope of the confining potential, revealing the presence of a topological boundary. The drift reversal corresponds to a band transfer between a band with Chern number C = +1 and a band with C = −1 via a gapless edge mode, in agreement with the bulk-edge correspondence for non-interacting particles. We establish that a non-zero repulsive Hubbard interaction leads to the emergence of an additional edge in the system, relying on a purely interaction-induced mechanism, in which pairs of fermions are split. arXiv:2301.03583v1 [cond-mat.quant-gas] 9 Jan 2023
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[ "https://export.arxiv.org/pdf/2301.03583v1.pdf" ]
255,545,987
2301.03583
d8ca8230852cd710b6411b2e5167e789365577cb
Reversal of quantised Hall drifts at non-interacting and interacting topological boundaries Zijie Zhu Institute for Quantum Electronics & Quantum Center ETH Zurich 8093ZurichSwitzerland Marius Gächter Institute for Quantum Electronics & Quantum Center ETH Zurich 8093ZurichSwitzerland Anne-Sophie Walter Institute for Quantum Electronics & Quantum Center ETH Zurich 8093ZurichSwitzerland Konrad Viebahn Institute for Quantum Electronics & Quantum Center ETH Zurich 8093ZurichSwitzerland Tilman Esslinger Institute for Quantum Electronics & Quantum Center ETH Zurich 8093ZurichSwitzerland Reversal of quantised Hall drifts at non-interacting and interacting topological boundaries The transport properties of gapless edge modes at boundaries between topologically distinct domains are of fundamental and technological importance. Therefore, it is crucial to gain a better understanding of topological edge states and their response to interparticle interactions. Here, we experimentally study long-distance quantised Hall drifts in a harmonically confined topological pump of non-interacting and interacting ultracold fermionic atoms. We find that quantised drifts halt and reverse their direction when the atoms reach a critical slope of the confining potential, revealing the presence of a topological boundary. The drift reversal corresponds to a band transfer between a band with Chern number C = +1 and a band with C = −1 via a gapless edge mode, in agreement with the bulk-edge correspondence for non-interacting particles. We establish that a non-zero repulsive Hubbard interaction leads to the emergence of an additional edge in the system, relying on a purely interaction-induced mechanism, in which pairs of fermions are split. arXiv:2301.03583v1 [cond-mat.quant-gas] 9 Jan 2023 The existence of individual edge modes at topological boundaries plays a crucial role in quantum Hall physics. More specifically, a non-trivial topology in the bulk of a material ensures that its edge modes are gapless and chiral. Gaplessness is related to the bulk-edge correspondence, stating that the number of topological edge modes is equal to the difference in Chern number across an interface [1]. Consequently, a gapless mode should allow an adiabatic transfer from one band to another, resulting in a reflection of transverse bulk currents in the opposite direction if the two bands feature opposite Chern numbers. However, the coherence time in most electronic materials is not sufficient to observe this effect, and edges are generally probed spectroscopically [1][2][3][4]. Moreover, studies of edge physics in engineered quantum systems, such as ultracold atoms and photonics, have so far been focussed on chirality [5][6][7][8][9] and localisation [10][11][12][13]. A boundary reflection has not been detected [14,15], or it was disregarded [16,17], and to our knowledge it has never been studied for variable interaction strength. Here, we observe the reversal of quantised bulk drifts due to harmonic trapping in a topological Thouless pump, the temporal analogue of the quantum Hall effect [18][19][20]. The reflection is a fundamental manifestation of confined topological matter and directly shows the gapless nature of topological edge modes. Going beyond the non-interacting regime, we discover the emergence of a second edge for repulsive Hubbard U . The experiments are performed with ultracold fermionic potassium-40 atoms, which are loaded into the potential of a generalised optical lattice formed by a combination of standing and running waves of wavelength λ = 1064 nm [22]. This creates an array of decoupled, one-dimensional tubes. Along the tube direction, the periodically modulated Rice-Mele-Hubbard Hamiltonian with harmonic confinement is realised, H(τ ) = − j,σ t + (−1) j δ(τ ) ĉ † jσĉ j+1σ + h.c.(1)+ ∆(τ ) j,σ (−1) jĉ † jσĉ jσ + U jĉ † j↑ĉ j↑ĉ † j↓ĉ j↓ + j,σ V jĉ † jσĉ jσ , whereĉ jσ is the fermionic annihilation operator for spin σ ∈ {↑, ↓} on site j, and t denotes the average tunnelling. An adiabatic modulation of bond dimerisation δ(τ ) = δ 0 cos(2πτ /T ) and sublattice offset ∆(τ ) = ∆ 0 sin(2πτ /T ) traces a closed trajectory in the δ-∆ plane around the origin, referred to as critical point. Therefore, an insulator or homogeneously filled band at U = 0 describes a topological pump [18,20] with T being the pump period. Experimentally, the topological pump manifests itself as a quantised drift of the atom position by one unit cell per pump cycle [23][24][25][26][27]. The harmonic confinement is characterised by the trap frequency ν, entering Eq. 1 as V j = 1 2 m(2πνaj) 2 ≡ V 0 j 2 (a = λ/2, lattice spacing; m, atomic mass). Due to the confinement, the atoms are initially localised at the centre of the trap. Topological pumping then leads to a quantised drift of atoms against the confining potential (Fig. 1A). Our measurements show that the quantised drift changes its direction at a certain distance from the trap centre. We will demonstrate that this happens when the gradient of the confinement overcomes the band gap and a boundary between topological and trivial regions emerges. For repulsive interactions, we observe another reflection, closer to the trap centre, while a part of the atoms keeps drifting in the original direction (Fig. 1A). In the following, we develop a description of the reflection in terms of gapless edge modes and the bulkedge correspondence within the framework of the Harper-Hofstadter-Hatsugai (HHH) model with one real (x) and one synthetic (n) dimension. The model features bulk Using Floquet theory, the 1D Rice-Mele pump can be mapped to a 2D Harper-Hofstadter-Hatsugai model with a linear gradient along the synthetic dimension n which represents the photon number. The magnetic flux per plaquette is Φ = 1/2 in units of the magnetic flux quantum [21]. The gradient along n leads to a transverse Hall drift along x (red arrows) due to the nontrivial topology of the bands. (C) Schematic spectrum of the mapped 2D Hofstadter model in a semi-infinite geometry. The lowest two bands have C = ±1, respectively. The linear gradient induces Bloch oscillations in the synthetic reciprocal space (dashed arrows). A gapless edge mode (solid arrow) appears at the topological interface. The reflection of the Hall drift can be understood as atoms being transported from the lower band (C = 1) to the higher band (C = −1) via the topological edge mode. Chern bands with C = +1 and C = −1. An exact mapping between the non-interacting 1D Rice-Mele Hamiltonian (Eq. 1) and the two-dimensional (2D) HHH model can be obtained using Floquet theory, illustrated in Fig. 1B (for derivation see, e.g., refs. [19] and [21]). A linear gradient along the synthetic dimension n appears in the mapping since the state with n photons acquires an energy of −n ω, where ω = 2π/T is the pump frequency. The gradient along n or, equivalently, an external force causes Bloch oscillations along the synthetic reciprocal dimension k n which, in turn, lead to a Hall drift or 'anomalous velocity' along the transverse real direction x [14,15]. The bulk Hall drift along x corresponds exactly to the quantised displacement measured in the topological pump. The trap induces a boundary between topological (C centre = 1) and trivial (C right = 0) regions and a single gapless edge mode emerges, according to the bulk-edge correspondence: C centre − C right = 1. The edge modes connects two bands of opposite Chern invariant, as shown in Fig. 1C. Thus, a Bloch oscillation transfers the atoms from the ground to the first excited band via that edge mode. Since the first excited band has Chern number −1 the atoms are now moving 'backwards', resulting in a reversal of the quantised Hall drift. Fig. 2 shows experimental in-situ images of the atomic cloud as a function of time τ at U = 0. The data shows a quantised drift of 1.00(1) × 2a/T up to about 60 T , which confirms the long coherence time of Bloch oscillations which induce the transverse drift. At τ 75 T the atoms change their drift direction, which is a key observation of this work. The expected topological boundary (red dashed line) represents the position at which the local tilt from the external harmonic (7). Beyond this position the total sublattice offset ceases to change sign, rendering the region outside x edge topologically trivial. The boundary caused by the harmonic confinement is not infinitely sharp, but smoothened over several lattice sites. This leads to a small T -dependence of the reflection position (Fig. S1), compared to its absolute value, and the calculation above should be understood as the outermost point of the reflective region. The reflected atoms exhibit a quantised drift of −0.99(3) × 2a/T in the opposite direction, in agreement with a transfer to the first excited band with C = −1. The linear relation between the position of topological boundary x edge and the maximum sublattice offset ∆ 0 is further confirmed by measuring the reflection in different lattices (Fig. S1). The reflection is observed under all parameter settings tested in this work, highlighting that the existence of the topological boundary is robust. potential ∆ ext (j) ≡ 1 2 |V j − V j−1 | = V 0 j − 1 2 equals the maximum sublattice offset ∆ 0 , thus, x edge / (2a) 1 2 ∆ 0 /V 0 = 92 In addition to the reflection, we also observe a cloud of atoms temporarily remaining at the boundary before gradually dissolving. This process can be understood via the presence of topologically trivial edge states, which hybridise with the gapless edge modes. To simplify the picture, let us consider a sharp domain wall between C = 1 and C = 0 (Fig. 2C). According to the bulkedge correspondence, the topologically nontrivial region contributes exactly one gapless mode whereas the trivial region can contribute gapped edge modes. Due to tunnel coupling at the interface, hybridisation takes place [34] and gaps on the order of the pump frequency 2π/T emerge. Bloch oscillations along k n can now lead to nonadiabatic 'Landau-Zener' transfers between topological and trivial edge modes, causing an incomplete transfer to the higher band, and atoms remaining at the boundary. Subsequent Bloch oscillations will transfer atoms back into the topological domain, leading to the dissolu- [21]. In a local density approximation picture, the local tilt ∆ext shifts the δ-∆ pump trajectory upwards. Depending on whether or not the trajectory encloses the critical point, the pump is rendered topological or trivial. (B) Measured band populations as a function of time τ . Each density image is averaged over six individual measurements with the parameters V0 = 0.0191(6)t, ∆0 = 3.2(2)t and T = 3 ms = 10.7(2) /t. The total atom number remains constant, within error bars, throughout the experiment. Due to the underlying honeycomb lattice geometry in the x-z plane, the first Brillouin zone has a diamond shape [21]. The band population is inverted when the bulk current is reflected off the topological interface, manifesting the gapless nature of the topological edge mode. (C) Hybridisation of the edge modes at the topological interface due to tunnelling. Bloch oscillations along kn can lead to Landau-Zener transfers between topologically trivial and nontrivial edge modes. The population of trivial edge modes explains the atoms being left at the boundary. Hybridisation never changes the total number of gapless edge modes at the boundary. tion of the cloud at the boundary. While the harmonic confinement leads to a more complex level structure [21], the underlying process remains qualitatively the same. We support the in-situ images with measurements of band population before, during, and after the reflection, as shown in Fig. 2B [21]. Before the reflection, we find a filled ground band, which is consistent with the observation of a quantised Hall drift. At the reflection (τ 72 T ), we observe an inversion of the population to the first excited band. After the reflection, the inverted population remains almost unchanged while the atoms are travelling back, highlighting the absence of incoherent relaxation to the ground band, even after more than a hundred Bloch oscillations. We further explore the effect of attractive and repulsive interactions on the topological boundary. For attractive Hubbard U = −3.48(7)t = 1.27(7)∆ 0 (Fig. 3A), the quantised Hall drift is reversed at the same position as in the non-interacting situation. This can be explained in terms of the Rice-Mele model in which fermions in the strongly attractive regime approach the limit of hard-core bosons [22], and the condition for the emergence of a topological boundary ∆ ext (j) = ∆ 0 remains unchanged. For repulsive Hubbard U = 3.48(7)t (Fig. 3B), we observe a second reflection in addition to the original one. Compared to the non-interacting case, this reflection appears much closer to the trap centre. The zoomed-in image (Fig. 3C) shows that a proportion of the atoms start to move backwards after about 12 cycles. In con- (7)t leads to the same reflection behaviour as observed for U = 0 (measurement parameters otherwise identical to Fig. 2A). (B) Repulsive Hubbard interactions of U = 3.48(7)t lead to the emergence of a second reflection, closer to the trap centre, which we attribute to an interacting topological boundary. A zoom-in (C) shows that the early reflection happens after about twelve cycles. The white dashed lines are guides to the eye, calculated as linear fits to the cloud position, extracted as the sum of a skewed and a regular Gaussian. (D) Microscopic description of the interaction-induced reflection for repulsive Hubbard U . When the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) becomes smaller than the Hubbard interaction, formation of double occupancies is prohibited and one atom is left in the higher-energy site of a unit cell, which then drifts in the opposite direction. (E) The critical point in the δ-∆ plane is split into two in the presence of repulsive Hubbard U [35]. When the pump trajectory encloses both critical points, a quantised drift is expected, as in the non-interacting system. The local tilt given by the external potential ∆ext shifts the trajectory along the ∆-axis, eventually enclosing just one critical point. Since a single split critical point features half the topological charge of the orignal one, the material's topology changes and a boundary emerges. trast to the drift reversal in the non-interacting system, a large fraction of the atom cloud still undergoes quantised drifting in the original direction. In the following, we develop a microscopic picture of the interaction-induced partial reflection in the limiting case of two isolated spins (↑, ↓), which approximates our initial state in a unit cell (Fig. 3D). As long as the maximum energy offset between two neighbouring sites 2 (∆ 0 − ∆ ext ) is larger than the Hubbard U , the formation of a double occupancy is energetically allowed and the quantised drift persists [22], even in the presence of an external potential. However, when ∆ ext becomes larger, the energy offset between two neighbouring sites remains always smaller than U , double occupancy formation becomes prohibited. In the latter case, one atom of a singlet pair is transferred to the energetically excited site of a unit cell, which will subsequently drift in the opposite direction. The other atom, in the lower-energy site, will move onwards because onsite interactions become irrelevant if there is only one atom per unit cell. Since the underlying Hamiltonian (Eq. 1) is SU(2)-symmetric, spin-↑ and spin-↓ have equal probability of being reflected and they should remain correlated after the splitting process. The full many-body description of the interactioninduced reversal requires the development of suitable topological invariants for smooth confinements and strong interactions, which goes beyond the scope of this work. Nevertheless, we obtain an intuition of the boundary's topological origins using again the idea of shifted pump trajectories in the δ-∆ plane with a staggered potential (c.f. Fig. 2A). Numerical simulations have shown that a repulsive Hubbard U can split the critical point at the origin into two separate ones [35], each retaining half the original topological charge. The distance between the new critical points is approximately U up to a correction on the order of the tunnelling t [36]. When the trajectory encloses both critical points, quantised drift of two spins (↑, ↓) is expected, as in the non-interacting system. As the position-dependent local tilt ∆ ext (j) shifts the trajectory upwards, it will enclose only one of the critical points beyond certain position along x (Fig. 3E). This indicates a transition of topological properties and the emergence of an interacting topological edge in real space. The estimation of the interacting boundary at ∆ ext (j) ∆ 0 − U/2 lies close to the centre and agrees with the microscopic picture discussed above. Similar to the non-interacting case, this boundary should be considered as the outermost position of the reflective region. In conclusion, we have experimentally observed a reversal of quantised Hall drifts at a topological boundary in a harmonic potential. The reflection is a direct manifestation of the gapless nature of topological edge modes between Chern bands of opposite sign. We explore the effect of Hubbard interactions, both attractive and re-pulsive, and find an asymmetric behavior with respect to U = 0. While on the attractive side the topological boundary is unaffected, repulsive interactions lead to the emergence of a second interface, featuring a splitting of quantised drifts. As a result, our experiments could enable the realisation of circular current patterns for constructing novel many-body phases [37]. More broadly, our work allows the exploration of the bulk-edge correspondence in the presence of interactions [38], as well as the investigation of edge reconstruction [39] in the quantum Hall effect and in interacting topological insulators. SUPPLEMENTAL MATERIALS Dependence of the drift reversal on experimental parameters The expected drift reversal happens when the maximum local site offset over one pump-cycle ∆ 0 is equal to the local tilt given by the harmonic trap. This position given by x edge ∆ 0 a/V 0 with a = λ/2 and V 0 = 1 2 m(2πνa) 2 . By measuring the reflection point in lattices with different ∆ 0 , we verify the relevant scaling x edge ∝ ∆ 0 , as shown in Fig. S1. The blue line in Fig. S1A marks the theoretically expected x edge with the uncertainty propagated from the uncertainty of the trap frequency ν. The disagreement between theory and experiment for larger values of ∆ 0 can be explained by the finite waist of the lattice beams. In order to explore the edge in our system, the atoms are pumped by almost 100 unit cells (∼ 100 µm). Due to the Gaussian envelope of the transverse beams, which are essential to realise the pump, the lattice is effectively shallower far away from the centre. Thus, ∆ 0 decreases towards the edge and atoms are reflected sooner. We also find a small dependence of the reflection point on the pump period, compared to its absolute value, which spans roughly 10 unit cells when changing T from 2 ms to 10 ms (Fig. S1B). This can be understood by considering the energy spectrum of the Harper-Hofstadter-Hatsugai (HHH) model in a harmonic potential, which will be discussed below. Experimental sequence We start by preparing a degenerate cloud of fermionic 40 K in a crossed dipole trap. We have a spin mixture of m F = {−9/2, −7/2} except for the measurements in Fig. 2B and Fig. S1, where a spin-polarised cloud in the magnetic state F = 9/2, m F = −9/2 is used. The spinpolarised cloud is directly loaded into the pumping lattice, while the spin mixture is first loaded into an intermediate chequerboard lattice with strongly attractive interactions. The two-step loading precludes the presence of atoms in the higher band and gives a larger fraction of atoms in doubly occupied unit cells [22]. After pumping the system for varying times, we either take a in-situ absorption image to measure the density or detect the band population with band-mapping. The latter is implemented with an exponential ramp to switch off the lattice beam in 500 µs, followed by a time-of-flight expansion of 25 ms before absorption imaging. Realisation of a Thouless pump in the Rice-Mele model The lattice setup is comprised of non-interfering standing waves in x, y, and z directions, together with interfering laser beams in the x-z plane. All the lattice beams come from a single laser source at wavelength λ = 1064 nm. These potential combine to form a honeycomb lattice in the x-z plane, which can be considered as isolated tubes of one-dimensional superlattices along x in the limit of deep transverse lattices. In each 1D tube the potential can be modeled by a one-dimensional superlattice with two sites per unit cell. With this setup, we realise the Rice-Mele model [22]. In the tight-binding limit, the Rice-Mele model can be described with three numbers: the offset energy ∆ between the two sites of a unit cell, the averaged nearest-neighbour tunneling t and the bond dimerisation δ which gives half the difference between the inter-and intra-dimer tunnellings. By having a dynamical control of the relative phase ϕ between the laser beams generating the interfering and the non-interfering lattice, we manage to shift the two with respect to each other. This shift modulates ∆ and δ periodically, which can be depicted as a closed trajectory in the ∆-δ coordinate (Fig. S2). In the adiabatic limit, this realises a Thouless pump with its hallmark quan-tised transport. In this case, the atomic displacement is given by the number of revolutions around the origin of the ∆-δ plane. φ = 0 φ = ∏/2 φ = ∏ φ = 3∏/2 x E x E x E x E FIG. S2. Realisation of a Thouless pump in the Rice-Mele model. In our system the 1D lattice potential can be modelled by a superlattice with two sites per unit cell, which is depicted in the bottom part of this figure as a function of relative phase ϕ. The local site offset ∆ as well as the bond dimerisation δ is depicted in the ∆-δ plots corresponding to the respective potentials. The resulting pump displacement corresponds to the number of revolutions around the origin in the ∆-δ plane. Mapping a 1D Thouless pump to a 2D Hofstadter Model with quantum Hall response A 1D Thouless pump with a period of T , as realised in our experiment, can be mapped to a 2D topological tight-binding (HHH) model with an applied electric field E = 2π qT where q can be thought of as a fictitious charge of netural atoms. Due to the topological bandstructure, this electric field leads to a transverse current I trans = q T of one atom per period, when considering a fully occupied band. The 2D model therefore has a quantised transverse conductance σ trans = I trans /E = q 2 2π analogous to the Hall conductance in the Quantum Hall Effect (QHE). The time-periodicity of the Hamiltonian in Eq. 1 witĥ H(τ ) =Ĥ(τ + T ) allows us to use Floquet's theorem. Solutions of the time-dependent Schrödinger equation i ∂ τ |Ψ(τ ) = H(τ ) |Ψ(τ ) (S1) can thus be written as |Ψ(τ ) = e −i τ / |u(τ )(S2) with |u(τ + T ) = |u(τ ) and ∈ R. Due to the timeperiodicity of u(τ ) we expand it as a Fourier series, |u(τ ) = n e −iωnτ |u n ,(S3) where ω = 2π/T is the pump frequency. The change from the time-domain into the Fourier-domain is the key ingredient to map the 1D Thouless pump to a 2D tightbinding model. The index n is also called the photon number of the mode |u n . Using a multi-index α = (j, σ) we write the T -periodic 1D Hamiltonian for U = 0 in the Fourier-basis: H(τ ) = α,β h αβ (τ ) |α β| (S4) = α,β,m e −imωτ h m αβ |α β| with h m αβ = 1 T T 0 e imωτ h αβ (τ )dτ and |α corresponding to an atom localised on site j with spin σ. Likewise, we use Fourier decomposition to express the solutions to Eq. S1 as |Ψ(τ ) = e −i τ / n,α e −inωτ u n,α |α .(S5) where u n,α = α|u n . As a result, we obtain an eigenvalue equation for u n,α : u n,α = −n ωu n,α + β,m h m αβ u n−m,β ∀n, α(S6) which can be understood as a time independent Schrödinger equation of a 2D tight-binding model with a tilted potential energy along one axis. By explicitly evaluating the h m αβ , we get H 2D = H real + H synth + H diag + H V + H tilt ,(S7) with H real = −t j,n,σ (ĉ † j,n,σĉ j+1,n,σ + h.c.), H diag = − δ 0 2 j,n,σ e −iπj (ĉ †(S8) j,n,σĉ j+1,n+1,σ +ĉ † j,n,σĉ j+1,n−1,σ + h.c.), H real and H synth describe tunneling along the real (x) and synthetic (n) dimension, respectively. The diagonal tunnelling terms in H diag are crucial because they open a bandgap between the ground band and the first excited band, characterised by the topological Chern number C which is further related to the quantised Hall conductance via σ trans = q 2 2π C. The terms in H V describe the external potential along the real-space direction. H tilt corresponds to a linear tilt in potential energy along the synthetic dimension which can be thought of as originating from an electric field E = 2π qT pointing along n. H synth = − ∆ 0 2 j,n,σ e −iπj (iĉ † j,n,σĉ j,n+1,σ + h.c.), H V = j,n,σ V (j)ĉ † j, Edge modes and their reflection properties To illustrate the topological edge modes in the presence of an external potential, we evaluate the spectrum of H 2D in the adiabatic limit (ω → 0) for different potentials V (j) = 1 2 m(2πνa) 2 j κ (S9) with m being the mass of 40 K, trap frequency ν = 134 Hz, lattice spacing a = 532 nm, and lattice site j. The parameter κ, an even integer, characterises steepness of the trap; the limit κ → ∞ corresponds to the textbook case of infinitely sharp walls [28]. Fig. S3A-C shows the numerically calculated energy spectra, omitting states on localised to the left edge for clarity. Fig. S3A shows the spectrum for a box-like potential with κ = 24. In this case there is a family of topological edge states, marked in red, which connect the lower and the upper band (black), separated from topologically trivial states above 5 kHz (also in black). All red states are localised along the right edge in x-direction. The lower and upper band have Chern number 1 and −1, respectively. Considering the dynamics in this model, an applied electric field along n as defined in H tilt leads to Bloch oscillations with a period T along k n . At the same time, the center-of-mass of the atoms moves by one unit cell per Bloch oscillation period along x, which corresponds to the quantised bulk Hall drift [14,15]. This drift can be evaluated in the numerics by following the eigenstates in Fig. S3A in real space. Since the red edge states are gapped from the next higher-lying trivial (black) states, the atoms 'Bloch-oscillate' from the ground to the excited band via the red-marked edge modes over several periods. Once they are in the excited band they are transported backwards along x. Fig. S3B shows the situation for κ = 10. It behaves similarly to Fig. S3A, except that there are more localised states marked in red, compared to κ = 2. Likewise, these states are transported along x as they undergo Bloch oscillations. As before, this family of edge states is gapped from trivial states and connects right-moving to left-moving states, which leads to the reflection phenomenon. The experimentally relevant case is a harmonic trap with κ = 2 (see also refs. [23,[29][30][31]). Fig. S3C shows that for κ = 2 the number of localised sates outside of the bands is even larger than for κ = 10. As before, we adiabatically follow these localised states along k n shows the tiny avoided crossings which can lead to a slight period dependence of the observed reflection point (Fig. S1). (D) Energy spectrum for a linearly increasing staggered potential. The gapless, topological edge mode is marked in red. and evaluate their centre-of-mass along x. By numerically observing these drifts we confirm that the states describe quantised drifting in a large region, which manifests their nontrivial topological nature. Thus, the κ = 2 case is ideal to observe the reflection after long-distance quantised Hall drifts. However, the gaps between topological (right-moving and left-moving) and trivial (stationary) states become smaller, compared to the κ = 24 and κ = 10 cases, as shown in the insets of Fig. S3C (κ = 2). As a result, the reflection point for κ = 2 is spread out over several unit cells but the reflection itself remains intact. Faster pumping leads to non-adiabatic crossings of the energy gaps between right-moving and left-moving states, causing the reflection to happen later in time and further up in energy. We confirm this dependence experimentally in Fig. S1B, which shows a later reversal for smaller pump periods. Fig. S3D shows the spectrum for the linearly increasing staggered potential, described in the following sections. This potential allows a straightforward identification of the gapless edge mode (red line). The states corresponding to this gapless edge mode are localised around the topological boundary. x E FIG. S4. Linearly increasing staggered potential. To elucidate the topology in our system, a linearly increasing staggered potential is considered (blue): V (j) = jV0(−1) j with V0 = 1/2 × m(2πνa) 2 , as before. It is chosen such that the local tilt always equals the tilt from the harmonic potential (orange), but alternates in sign. The staggered potential allows a simple pictorial representation of the emergence of the topological boundary. In a local density approximation the trap linearly shifts the pump trajectory upwards in the ∆-δ plane, as depicted in the upper part of the figure. As soon as the trajectory ceases to enclose the critical point, a topological-trivial boundary develops. Staggered potential Another possibility to identify the topological boundary in our system makes use of a staggered potential. First, we consider a potential with uniform staggering, given by V c (j) = V (−1) j , where j indexes the lattice-site and 2V corresponds to the energy difference between adjacent sites. Adding such a potential to the Rice-Mele Hamiltonian (Eq. 1) changes its trajectory in the ∆-δ plane. The onsite energy in such a system is given by (∆(τ ) + V )(−1) j , which ranges from −∆ + V to ∆ + V . The tunnellings are unchanged. Therefore, the trajectory remains circular and it is simply shifted upwards by an amount V . A topological boundary emerges for a linearly increasing staggered potential, given by V (j) = jV 0 (−1) j , with V 0 = 1 2 m(2πνa) 2 as before. V (j) is chosen such that it has the same local tilt as the harmonic trap in the experiment. Within the local density approximation we assign a ∆-δ trajectory locally to each unit cell. The trajectories are thus linearly shifted upwards as function of j (Fig. S4), describing a change of topology in real space. We expect the local density approximation to be valid since the atomic eigenstates in the experiment are strongly localised. Similar models with linearly increasing staggered potential have been studied in refs. [29,32,38]. Local Chern marker The mathematical formulation of the Chern number as a topological invariant requires translational invariance, which does not apply to realistic experiments. Instead, we use a local quantity, known as Chern marker [8,33]. The local Chern marker depends on the real-space position and it is defined by: c(r γ ) = − 4π A c Im s=A,B r γs |PxQŷP |r γs , where r γ is the position of the unit cell γ with sub-latticesites at positions r γ A and r γ B , |r γs = c † γs |0 is the state localised on the corresponding lattice site , A c is the area of a real-space unit cell,Q = 1−P andP is the projector onto the ground band. DefiningP is not unambiguously possible in our system (Eq. S7) because of the energy shift from the harmonic confinement. Instead, we use a linearly increasing staggered potential, as described in the previous paragraph. This model leaves the bands intact and a ground band can be unambiguously defined. Experimentally, a local probe of the band topology is the velocity of the Hall drift, plotted in Fig. 2A. Theory and experiment agree approximately with one another. The local velocity is extracted from the atomic positions by fitting linear functions to groups of three adjacent datapoints in ten pump cycles. The resulting velocities are plotted against position and smoothed through a running average of width three (ten cycles). FIG. 1 . 1Reflection of quantised Hall drifts off a topological interface. (A) Topological Thouless pumping in the presence of confining potentials. In the non-interacting case (top), a harmonic trap gives rise to topological trivial (C = 0) and non-trivial (C = 0) regions, separated by a topological interface. The atoms exhibit a quantised drift until they are reflected at the interface. With repulsive on-site interactions (bottom), the reflection happens closer to the centre, accompanied by atoms still drifting in the original direction. (B) FIG. 2 . 2Measuring the reversal of a quantised Hall drift. (A) The time-trace of atomic in-situ images shows a quantised drift along x before the atoms are reflected at the topological boundary. Each density image is averaged over three individual measurements with the parameters V0 = 0.0148(9)t, ∆0 = 2.7(1)t, and T = 3 ms = 12.8(2) /t. The red dashed line indicates the topological boundary x edge / (2a) ≈ 1 2 ∆0/V0. The white dashed lines are linear fits to the atom drift, yielding slopes of 1.00(1) × 2a/T before, and −0.99(3) × 2a/T after the reflection. Cloud positions, averaged over the transverse direction, are fitted using Gaussians. The experimental Chern marker (lower panel, points) is determined by the velocity of the right-moving cloud at different positions. The theoretical Chern marker (line) is calculated in a staggered potential Vj = V0 (−1) j j which has the same local tilt ∆ext(j) = V0 j − 1 2 as the harmonic trap FIG. 3 . 3Reflection of quantised Hall drifts from an interacting topological edge. (A) An attractive Hubbard interaction of U = −3.48 . Experimental dependence of the reflection position. The reflection point is expected to depend linearly on the maximal site-offset per pump cycle ∆0 which is experimentally verified in (A). Deviations for large values of ∆0 can be explained by the finite waist of the laser beams forming the lattice. (B) shows the period dependence of the reflection position. Changing the pump period T over an order of magnitude only changes the reflection point by about 10 unit cells, which is a result of the smooth boundary of a harmonic potential. FIG. S3 . S3Energy spectra for different confining potentials. States localised to the left edge are omitted for clarity. Energy spectra of H2D in the limit ω → 0 for κ = 24 (A), κ = 10 (B), and κ = 2 (C). Topological edge modes which connect the two bands with Chern number 1 and −1 in (A) and (B) are marked in red. The upper inset in (C) marks the topological boundary where the reflection is observed as described in the main text. The inset in the center of (C) ACKNOWLEDGMENTSWe would like to thank Jason Ho, Gian-Michele Graf, Thomas Ihn, Fabian Grusdt, Fabian Heidrich-Meisner, Armando Aligia, and Eric Bertok for valuable discussions. We also thank Julian Léonard and NurÜnal for comments on an earlier version of the manuscript. We would like to thank Alexander Frank for his contributions to the electronic part of the experimental setup. 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[]
[ "Estimating the Jet Power of Mrk 231 During the 2017-2018 Flare", "Estimating the Jet Power of Mrk 231 During the 2017-2018 Flare" ]
[ "Cormac Reynolds \nCSIRO Astronomy and Space Science\nPO Box 11306102BentleyWAAustralia\n", "Brian Punsly [email protected] \nPhysics Department\nICRA\nPalos Verdes Estates CA, 90274: ICRANet, Piazza della Repub-blica 10, University La Sapienza1415, 65100Granvia Altamira, Pescara, RomaUSA, Italy, Italy\n", "Giovanni Miniutti \nCentro de Astrobiologia (CAB) ESA -European Space Astronomy Center (ESAC)\n\n", "Christopher P O&apos;dea \nDepartment of Physics and Astronomy\nUniversity of Manitoba\nR3T 2N2WinnipegMBCanada\n\nSchool of Physics & Astronomy\nRochester Institute of Technology\n14623RochesterNYUSA\n", "Natasha Hurley-Walker \nICRAR-Curtin University\nGPO Box U19876102PerthWestern AustraliaAustralia\n" ]
[ "CSIRO Astronomy and Space Science\nPO Box 11306102BentleyWAAustralia", "Physics Department\nICRA\nPalos Verdes Estates CA, 90274: ICRANet, Piazza della Repub-blica 10, University La Sapienza1415, 65100Granvia Altamira, Pescara, RomaUSA, Italy, Italy", "Centro de Astrobiologia (CAB) ESA -European Space Astronomy Center (ESAC)\n", "Department of Physics and Astronomy\nUniversity of Manitoba\nR3T 2N2WinnipegMBCanada", "School of Physics & Astronomy\nRochester Institute of Technology\n14623RochesterNYUSA", "ICRAR-Curtin University\nGPO Box U19876102PerthWestern AustraliaAustralia" ]
[]
Long-term 17.6 GHz radio monitoring of the broad absorption line quasar, Mrk 231, detected a strong flare in late 2017. This triggered four epochs of Very Long Baseline Array (VLBA) observations from 8.4 GHz to 43 GHz over a 10week period as well as an X-ray observation with NuSTAR. This was the third campaign of VLBA monitoring that we have obtained. The 43 GHz VLBA was degraded in all epochs with only 7 of 10 antennas available in three epochs and 8 in the first epoch. However, useful results were obtained due to a fortuitous capturing of a complete short 100 mJy flare at 17.6 GHz: growth and decay. This provided useful constraints on the physical model of the ejected plasma that were not available in previous campaigns. We consider four classes of models, discrete ejections (both protonic and positronic) and jetted (protonic and positronic). The most viable model is a "dissipative bright knot" in a faint background leptonic jet with an energy flux ∼ 10 43 ergs/sec. Inverse Compton scattering calculations (based on these models) in the ambient quasar photon field explains the lack of a detectable increase in X-ray luminosity measured by NuSTAR. We show that the core (the bright knot) moves towards a nearby secondary at ≈ 0.97c. The background jet is much fainter. Evidently, the high frequency VLBA core does not represent the point of origin of blazar jets, in general, and optical depth "core shift" estimates of jet points of origin can be misleading.1 The Arcminute Microkelvin Imager consists of two radio interferometric arrays located in the Mullard Radio Astronomical Observatory, Cambridge, UK(Zwart et al. 2008). Observations occur between 13.9 and 18.2 GHz in six frequency channels. The Small Array (AMI-SA) consists of ten 3.6 m diameter dishes with a maximum baseline of 20 m, with an angular resolution of 3 ′ , while the Large Array (AMI-LA) comprises eight 12.6 m diameter dishes with a maximum baseline of 110 m, giving an angular resolution of 0. ′ 5.
10.3847/1538-4357/ab72f0
[ "https://arxiv.org/pdf/2001.10697v2.pdf" ]
210,942,733
2001.10697
172188d9ea4ea8832d4061b3661f4d3b0198d002
Estimating the Jet Power of Mrk 231 During the 2017-2018 Flare 31 Jan 2020 Cormac Reynolds CSIRO Astronomy and Space Science PO Box 11306102BentleyWAAustralia Brian Punsly [email protected] Physics Department ICRA Palos Verdes Estates CA, 90274: ICRANet, Piazza della Repub-blica 10, University La Sapienza1415, 65100Granvia Altamira, Pescara, RomaUSA, Italy, Italy Giovanni Miniutti Centro de Astrobiologia (CAB) ESA -European Space Astronomy Center (ESAC) Christopher P O&apos;dea Department of Physics and Astronomy University of Manitoba R3T 2N2WinnipegMBCanada School of Physics & Astronomy Rochester Institute of Technology 14623RochesterNYUSA Natasha Hurley-Walker ICRAR-Curtin University GPO Box U19876102PerthWestern AustraliaAustralia Estimating the Jet Power of Mrk 231 During the 2017-2018 Flare 31 Jan 2020Subject headings: quasars: absorption lines -galaxies: jets -quasars: general -accretion, accretion disks -black hole physics Long-term 17.6 GHz radio monitoring of the broad absorption line quasar, Mrk 231, detected a strong flare in late 2017. This triggered four epochs of Very Long Baseline Array (VLBA) observations from 8.4 GHz to 43 GHz over a 10week period as well as an X-ray observation with NuSTAR. This was the third campaign of VLBA monitoring that we have obtained. The 43 GHz VLBA was degraded in all epochs with only 7 of 10 antennas available in three epochs and 8 in the first epoch. However, useful results were obtained due to a fortuitous capturing of a complete short 100 mJy flare at 17.6 GHz: growth and decay. This provided useful constraints on the physical model of the ejected plasma that were not available in previous campaigns. We consider four classes of models, discrete ejections (both protonic and positronic) and jetted (protonic and positronic). The most viable model is a "dissipative bright knot" in a faint background leptonic jet with an energy flux ∼ 10 43 ergs/sec. Inverse Compton scattering calculations (based on these models) in the ambient quasar photon field explains the lack of a detectable increase in X-ray luminosity measured by NuSTAR. We show that the core (the bright knot) moves towards a nearby secondary at ≈ 0.97c. The background jet is much fainter. Evidently, the high frequency VLBA core does not represent the point of origin of blazar jets, in general, and optical depth "core shift" estimates of jet points of origin can be misleading.1 The Arcminute Microkelvin Imager consists of two radio interferometric arrays located in the Mullard Radio Astronomical Observatory, Cambridge, UK(Zwart et al. 2008). Observations occur between 13.9 and 18.2 GHz in six frequency channels. The Small Array (AMI-SA) consists of ten 3.6 m diameter dishes with a maximum baseline of 20 m, with an angular resolution of 3 ′ , while the Large Array (AMI-LA) comprises eight 12.6 m diameter dishes with a maximum baseline of 110 m, giving an angular resolution of 0. ′ 5. Introduction Mrk 231 is a nearby quasar (redshift of z = 0.042) that has all of the extreme properties of the quasar population in one object. It has a high-luminosity accretion flow, a radio jet that can rival the power of other well-known nearby extragalactic radio sources and a broad absorption line (BAL) high velocity outflow. All of this combined in a nearby object provides an excellent laboratory to study the interplay between these phenomena. The radio jet makes this the brightest radio quiet quasar (RQQ) at high frequency. This means that we can capitalize on the ∼ 0.2 pc resolution of the Very Large Baseline Array (VLBA) to explore the interior of this RQQ, a circumstance unique to Mrk 231. This has motivated us to pursue a series 43 GHz observing campaigns with VLBA from 2006-2019 and almost continuous daily to weekly 17.6 GHz monitoring from 2013-2019 with AMI 1 . The third VLBA campaign is reported in this paper. This study concentrates on estimating the jet power during what was the strongest 15-20 GHz flare to date. The jet is extremely powerful for a RQQ during flare states with a kinetic luminosity crudely estimated to be Q ∼ 3 × 10 43 ergs s −1 for previous flares (Reynolds et al. 2009). A small number of RQQs that have also exhibited episodes of relativistic jet formation (Brunthaler et al. 2000;Blundell et al. 2003). However, Mrk 231 is more luminous at high frequency, more often, than other RQQs and can provide unique clues as to why some quasars are radio loud and some are radio quiet. Consequently, we have been monitoring the radio behavior at ∼ 20 GHz since 2009. Before 2013, the monitoring was very sporadic with the Very Large Array (VLA). Since 2011, we have detected 5 large blazar-like flares (see Reynolds et al. (2013) for a discussion) with flux densities ≥ 200 mJy at ∼ 20 GHz. The fourth of these major flares reached 350 mJy at the end of 2017 (see Figure 1). This initiated a target of opportunity four-epoch VLBA observation and one epoch observation with the NuSTAR X-ray telescope. Our primary aim is to improve on the crude two-epochbased estimate of Q quoted above. In Section 2, we will review our previous results of VLBA monitoring. In Section 3, we will discuss the radio light curve leading to the flare and during the flare. Section 4 collects the data obtained from our VLBA observation. We also describe the core spectra in terms of a synchrotron self-absorbed (SSA) power law. The simple spectral shape lends itself to a simple physical model of a uniform sphere that has been utilized previously for energy estimates of ejections. The details are reviewed in Section 5. The next two sections describe the solution space. In Section 8, we consider inverse Compton scattering in the ejection and compare this with our NuSTAR observation. Throughout this paper, we adopt the following cosmological parameters: H 0 =71 km s −1 Mpc −1 , Ω Λ = 0.73 and Ω m = 0.27. Past VLBA 43 GHz Observations The various VLBA epochs show two resolved components. There is a nuclear radio core with a highly variable spectrum. There is also a very steep spectrum, slowly variable, almost stationary secondary, K1. At 43 GHz, the core is more luminous during a flare. The 2006 Campaign The first 43 GHz observations were not triggered, but were obtained to see the benefit of higher resolution. They occurred in two epochs in 2006 separated by 3 months. In hind sight the observations were in a relatively low state for the core, ∼ 20 mJy at 43 GHz (compare to Table 2). The results were analyzed in Reynolds et al. (2009). Three observation frequencies were chosen: 15.3 GHz, 22.2 and 43.1 GHz. Using archival data, it was determined that time variability indicated that the line of sight, LOS, was restricted in order to avoid the inverse Compton catastrophe, < 25 • .6 +3 • .2 −2 • .6 . The core spectrum changed dramatically between the two epochs, with the 22 GHz flux density more than doubling. We were able to make a crude estimate of the jet power based on this spectral change: Q ≈ 3 × 10 43 erg/sec. It was also argued based on lower frequency VLBA in Ulvestad et al. (1999a) that the spectrum of K1 is free-free absorbed near 5.4 GHz and 1.6 GHz. Based on the emission measure, K1 was interpreted as a radio lobe that results from the jet emitted from the core being stopped by entrainment of the BAL wind. The 2015 Campaign It was determined that the 2006 campaign lacked the frequency coverage to properly describe the core spectrum, i.e. the SSA turnover. Thus, we proposed 8.4 GHz observations as well as 15.3 GHz, 22.2 GHz and 43.1 GHz observations. Furthermore, the observations were triggered by a strong flare at 17.6 GHz. Our primary goal was to detect a discrete ejection. We did find 1 clear and 1 marginal superluminal ejections at 43 GHz, but they were very faint, ∼ 1 − 2 mJy and faded very quickly (Reynolds et al. 2017). Thus, they were not Reynolds et al. (2013) Flare Studied in Reynolds et al. (2017) with VLBA and NuSTAR detectable in subsequent epochs 2-4 weeks later. Our three epochs of NuSTAR observations did not detect any X-ray evolution during the flare. The VLBA coverage stopped during the flare peak. There was not enough information to constrain the jet power. The spectra appeared to be SSA steep spectrum power laws (see Figure 4 for examples of SSA power laws in the current campaign), except in the last epoch. Curiously, in the last epoch the core spectrum changed to a flat spectrum power law with no evidence of SSA at frequencies lower than 8.4 GHz. The clear detection of the ejection of a weak superluminal component was used in conjunction with Doppler aberration arguments to restrict the LOS to < 23.5 • . Similar to the upper bound found in the 2006 campaign from the time variability Doppler argument (Reynolds et al. 2009). Reynolds et al. (2013) reported the methods of our radio monitoring with AMI at 13.5-18 GHz. Historical data before AMI can be found in Reynolds et al. (2013Reynolds et al. ( , 2017. Figure 1 shows the 17.6 GHz (the highest useable frequency) light curve from our more recent AMI monitoring. In November 2017, Mrk231 reached a historic high flux density in terms of our 17.6 GHz to 22 GHz monitoring. This triggered our 4 epoch 8.4 GHz,15.3 GHz,22.2 GHz and 43.1 GHz VLBA observations. The bottom frame of Figure 1 shows the 4 epochs, labeled A-D, superimposed on the flare profile. The flare amplitude is ≈ 100 mJy above the background level, reaching 350 mJy total flux density. Epoch A is during the rise of the flare. Epochs B and C, straddle the peak of the flare and Epoch D is in the tail of the flare decay. Thus, the four epochs captured almost the entire time evolution of the flare. This is exploited in Sections 4-7. Before the flare could completely decay, another more powerful flare emerged, reaching a peak of 400 mJy. The AMI Monitoring 2017/18 VLBA Observations The VLBA project, BR214, observed 4 epochs between 2017 November 6 and 2018 January 14. The data were correlated on the VLBA DiFX correlator (Deller et al. 2011) and calibrated with NRAO's Astronomical Imaging Processing System following the standard procedures described in the AIPS Cookbook and using the updated amplitude calibration strategy described in Walker (2014). All analysis was scripted in the ParselTongue interface (Kettenis et al. 2006). Model fitting of the data was done in the DIFMAP package (Shepherd et al. 1994(Shepherd et al. , 1995 Due to inclement weather and some equipment failures, we were unable to acquire data from all ten stations in any of the four epochs of observation. The St. Croix station was absent in all epochs due to hurricane damage and the Mauna Kea station was lost at 43 GHz in epochs B to D due to a problem with the Focus Rotation Module (FRM). These antennas provide the longest VLBA baselines and their loss significantly impacted the resolution of our observations. In addition freezing weather affected the performance of the FRM at a number of sites during all four epochs resulting in further data losses, disproportionately large at 43 GHz. The lost stations significantly reduced the resolution and sensitivity. The resulting u, v coverage was inadequate for resolving structures along the jet. We analyze the resulting data even though this circumstance made it impossible to directly monitor motion along the jet (a primary objective of the program) instead focusing on the spectral evolution of the flare. The observations were phase-referenced to J1302+5748 (1.3 • from Mrk 231) at 8.4, 15, 22, and 43 GHz, following the strategies described in Reynolds et al. (2009) for projects BR295 and BP124. J1311+5513 (∼ 80 mJy of unresolved flux density at 43 GHz) was used as a secondary calibrator to check the quality of the phase referencing, which appeared to work well at all frequencies and epochs. For each epoch of our VLBA monitoring, an 8 hour observation with almost continuous recording at 2 Gbps (256 MHz bandwidth, dual polarizations) was scheduled. Final on-source times and resultant image sensitivities (naturally weighted) for Mrk 231 are summarised in Table 1. Table 2 shows the results of our data reduction. It lists the fitted flux density of the two components in column (3) with the uncertainty in column (4). The absolute flux density uncertainties at 8.4, 15 and 22 GHz are 5%, 5% and 7%, respectively (Homan et al. 2002). At 43 GHz, we use the amplitude stability of the phase calibrator J1302+5748 over the campaigns in 2015 and 2017 (epochs A, B and D) as the measure of the flux density uncertainty. We estimate a flux density uncertainty of 8% at 43 GHz in this campaign, consistent with values used in our previous campaigns. The next four columns are the positional coordinates (relative to the assumed stationary component K1) and uncertainty. Columns (8)-(10) describe the Gaussian fit. The last column is the frequency. Our best radio images are from epoch A because we had 8 stations and the long baselines associated with Mauna Kea at 43 GHz. The other epochs B to D were missing Mauna Kea and had varying degrees of missed observing time at other stations due to freezing weather conditions (see Table 1). The 8.4 GHz to 22.2 GHz images from epoch A are shown in Figure 2. They are very similar to images from previous campaigns in Reynolds et al. (2009, 2017), but with lower resolution. The 43.1 GHz images in Figure 3 have the highest resolution and the most potential for elucidating the compact nuclear structure. Due to the missing stations described in Table 1, the 43.1 GHz observations during epochs B-D were significantly compromised. Epoch C had the most issues. Even though we were able to perform a self-calibration in epoch C, we measured "out of family" low flux densities in both the core and K1 in the following sense. K1 varies slowly and should be relatively constant from epochs B-D (Reynolds et al. 2009(Reynolds et al. , 2017. Furthermore, both of our calibrator sources had reduced amplitude as well. The main suspect is an error in the opacity correction which is large at 43.1 GHz but the confounding issue has not been identified. However, we can use the quasi-stationary calibrators as amplitude calibrators. The phase calibrator, J1302+5748, has significant flux at 43.1 GHz and is our most reliable absolute flux density calibrator. In order to re-scale the flux densities we averaged the flux density at 43 GHz for J1302+5748 in epochs B and D. This was 1.37 times larger than the flux density of J1302+5748 in epoch C. Thus, we use this factor to re-scale all the epoch C 43.1 GHz flux densities for Mrk231 in Table 2, as described in the table note. Core Spectra Even though a large amount of information was lost due to poorer than expected resolution, we did anticipate the additional information that can be found in the core spectrum with a multi-frequency observation plan. The spectra of the unresolved core in each epoch are shown in Figure 4. In Reynolds et al. (2009), we used the detailed nature of the spectral shape to argue that the spectrum of the radio core was most likely a power-law synchrotron spectrum that was seen through a SSA screen as opposed to a synchrotron power-law spectrum that was seen through an opaque screen caused by free-free absorption. The plots in Figure 4 show synchrotron self-absorbed (SSA) power law fits to the data. This is a simple spectral model that has been used to describe previous observations (Reynolds et al. 2009(Reynolds et al. , 2017. In Section 6, we will interpret these fits in terms of the physical models that emit these spectra. Figure 5 shows the other important piece of information that can be retrieved from (with respect to the phase reference calibrator) within the uncertainties of the VLBA over two decades (Ulvestad et al. 1999a,b;Reynolds et al. 2009Reynolds et al. , 2017. Thus, as in Reynolds et al. (2017), we use K1 as the origin for an astrometric description of core motion during the flare. There is more scatter in the 2017 data then we see in the 2015 data due to the loss of observing stations in the current analysis. We do not consider the poor epoch C observation in this analysis, due to the problems discussed earlier in this section. The apparent velocity is v app ≈ 0.97c based on the best fit line, directed towards K1. What we identify as the core appears to be a discrete ejection that dominates the total nuclear flux density at 43 GHz. The nuclear region and the ejected components are too small to be resolved based on the fits in Table 2. The displacements are about 1/3 of the beam width ( Figure 2) in our two best observations epochs A and B, and less in later epochs. The next section will present models of the physical nature of the discrete high brightness feature moving towards K1. Core Motion Synchrotron Self-Absorbed Homogeneous Plasmoids The core emission is unresolved at 43 GHz in all our campaigns including 2017/2018 (see Figure 3). Thus, there is no evidence of a preferred geometry. Consequently, there is no observational motivation to choose a model more complicated than a uniform spherical plasmoid. The simple homogeneous spherical volume model was shown by van der Laan (1966) to provide a very useful means of understanding the spectra and time evolution of astrophysical radio sources. Ostensibly, the invocation of a simple plasmoid seems less relevant than the notion of a continuous jet. A continuous jet has been inferred in extragalactic radio sources based on the need to replenish the distant radio lobes with energetic plasma in order to counteract synchrotron cooling (Kaiser & Alexander 1997). In a practical context blazar calculations typically reduce to plasmoid models. They are referred to as single zone homogeneous, spherical models (Ghisellini et al. 2010). But this is merely semantics. In order to appreciate the utility of these simple models, consider the following relevant circumstance. If there is a strong flare produced by a large explosive event in the jet (i.e., increase power injected at the base, a Parker instability at the point of origin or other instability) then the details of a much weaker background jet add no practical insight into the physical parameters responsible for the emission from an over-luminous finite region. This appears to be the case for Mrk 231 in which a weak background jet feeds K1 as evidenced by the large effects of synchrotron cooling that results in a very steep spectrum. The plasmoid is more versatile than a continuous jet model for our purposes. Because the explosive event responsible for the flare is so intense, we cannot assume its origin or nature based on the weak, invisible background jet. The plasmoid need not be similar to the rest of the jet. We need not assume whether it is protonic or leptonic, nor do we need to assume a bulk velocity or whether it is magnetically dominated or dominated by bulk kinetic energy. Instead, we will determine this from the data. The specific application of this model that is implemented in the following has been successful in describing a wide range of astrophysical phenomena. This spherical homogeneous discrete ejection model has been used to study the major flares in the Galactic black hole accretion system of GRS 1915+105 (Punsly 2012). It was applied to the neutron star binary merger that was the gravity wave source, GW170817, and associated gamma ray burst (GRB), GRB 170817A (Punsly 2019). It has been previously used in the study of quasar radio flares in Mrk 231 (Reynolds et al. 2009). The primary advantage of the method is that the SSA turnover provides information on the size that cannot be obtained from unresolved VLBA images. For example, our best radio image, epoch A, can only resolve a region of 2.5 × 10 17 cm or larger. Assuming a size equal to the resolution limit of the telescope generally results in plasmoid energy estimates that are off by one or more orders of magnitude due to the exaggerated volume of plasma (Fender et al. 1999;Punsly 2012). The first subsection will describe the underlying physics and the next subsection describes physical quantities of interest in the spherical plasmoids. The Underlying Physical Equations It is important to distinguish between quantities measured in the plasmoid frame of reference and those measured in the Earth observer frame of reference. The strategy will be to evaluate the physics in the plasma rest frame then transform the results to the observer's frame for comparison with observation. The underlying power law for the flux density is defined as S ν (ν = ν o ) = Sν −α o , where S is a constant. Observed quantities will be designated with a subscript, "o", in the following expressions. The observed frequency is related to the emitted frequency, ν, by ν o = δν. The bulk flow Doppler factor of the plasmoid, δ, δ = γ −1 1 − β cos θ , γ −1 = 1 − β 2 ,(1) where β is the normalized three-velocity of bulk motion and θ is the angle of the motion to the line of sight (LOS) to the observer. The SSA attenuation coefficient in the plasma rest frame is given by (Ginzburg & Syrovatskii 1969), µ(ν) = g(n) e 3 2πm e N Γ (m e c 2 ) 2α 3e 2πm 3 e c 5 1+2α 2 (B) (1.5+α) (ν) −(2.5+α) ,(2)g(n) = √3πN = Γmax Γ min N Γ Γ −n dΓ , n = 2α + 1 ,(3) where Γ is the ratio of lepton energy to rest mass energy, m e c 2 and Γ is the gamma function. B is the magnitude of the total magnetic field. The power-law spectral index for the flux density is α = (n − 1)/2. The low energy cutoff, E min = Γ min m e c 2 , is constrained loosely by the data in Table 3 and Figure 4. The fact that the core is very luminous at ν o = 8.4 GHz, means that the lepton energy distribution is not cutoff near this frequency. This is not a very stringent bound since it is just a consequence of the fact that 8.4 GHz is our lowest observing frequency. It would be a major coincidence if this were not to provide a loose upper bound on Γ min . Note that the SSA opacity in the observer's frame, µ(ν o ), is obtained by direct substitution of ν = ν o /δ into Equation (2). The uniform, homogeneous approximation yields a simplified solution to the radiative transfer equation (Ginzburg & Syrovatskii 1965;van der Laan 1966) S νø = S o ν −α o τ (ν o ) × 1 − e −τ (νo) , τ (ν o ) ≡ µ(ν o )L , τ (ν o ) = τ ν (−2.5+α) o ,(5) where τ (ν) is the SSA opacity, L is the path length in the rest frame of the plasma, S o is a normalization factor and τ is a constant. There are three unknowns in Equation (5), τ , α and S o . These are three constraints on the following theoretical model that are estimated from the observational data. These three constraints are used to establish the uniqueness and the existence of solutions in the following. The theoretical spectrum is parameterized by Equations (2)-(5) and the synchrotron emissivity that is given in Tucker (1975) as j ν = 1.7 × 10 −21 [4πN Γ ]a(n)B (1+α) 4 × 10 6 ν α ,(6)a(n) = 2 n−1 2 √ 3 Γ 3n−1 12 Γ 3n+19 12 Γ n+5 4 8 √ π(n + 1)Γ n+7 4 .(7) One can transform this to the observed flux density, S(ν o ), in the optically thin region of the spectrum using the relativistic transformation relations from Lind and Blandford (1985), S(ν o ) = δ (3+α) 4πD 2 L j ′ ν dV ′ ,(8) where D L is the luminosity distance and in this expression, the primed frame is the rest frame of the plasma. These are the basic equations needed to fit the data in Figure 4. Physical Quantities that Characterize Spherical Models The apparent velocity constrains the kinematics of the plasmoid (Rees 1966): v app /c = β app = β sin θ/(1 − β cos θ) ≈ 0.97 ,(9) where we used the result from Figure 5. The apparent motion and the LOS upper limit 25 • .6 +3 • .2 −2 • .6 noted in Section 2.1 constrain the dynamics. The validity of these numbers is a major assumption of this analysis. The models that produce the fits in Figure 4 are of interest if we can deduce the physical parameters in Equations (1)-(9) that are responsible for the spectra. Furthermore, we want to know the kinematic consequences of these parameters within a specific model. There are two basic classes of models. First are discrete ejections of plasma, a ballistic ejection of a plasmoid. This is essentially a blob of magnetized gas. The gas is turbulent and the magnetic field is randomized and tangled (Moffet 1975). In this model, the magnetic field strength behaves as an extra component of the internal gas pressure and acts as an expansive force on the plasmoid (Punsly 2008). Secondly, the discrete unresolved emission might be "knots" or regions of enhanced dissipation within a continuous jet. In the jet scenario, the magnetic field is predominantly an organized toroidal magnetic field, B ′ φ , in the rest frame of the plasma in contrast to the turbulent magnetic field of the discrete ejections (Blandford and Köingl 1979). The behavior of this magnetic field is much different than the turbulent magnetic field. It acts as a hoop stress that provides a confining force on the outflow (Punsly 2008). The Poynting Flux in a Strong Knot in the Jet We review the derivation of the dominance of the toroidal component of the magnetic field in order to derive an estimate of the poloidal Poynting flux. The dominance of the toroidal component of the magnetic field is a consequence of the perfect magnetohydrodynamic (MHD) assumption and approximate angular momentum conservation in the jet (Blandford and Köingl 1979). The total angular momentum flux per unit magnetic flux in the observer's coordinate system is (Punsly 2008), L ≈ kµr ⊥ γv φ − c 4π B φ r ⊥ , k ≡ Nγv P B P ; ,(10) where k is the perfect MHD conserved mass flux per unit poloidal magnetic flux, B P and v P are the poloidal magnetic field and the poloidal velocity, respectively. The quantity N is the number density in the plasma rest frame, µ is the specific enthalpy and r ⊥ is the cylindrical radius. The first term is the mechanical angular momentum flux per unit magnetic flux and the second term is the electromagnetic angular momentum flux per unit magnetic flux. The notion of describing the angular momentum per unit poloidal magnetic flux is physically motivated by angular momentum conservation and perfect MHD. In perfect MHD, angular momentum is transported conserved along each poloidal magnetic field line in a wind or jet (Punsly 2008). Jets expand as they propagate (Blandford and Köingl 1979). Thus, r ⊥ is much larger in the knot than at the jet base. Consider a jet that is highly magnetic, either Poynting flux dominated or at a minimum the mechanical angular momentum flux per unit magnetic flux and the electromagnetic angular momentum flux per unit magnetic flux are comparable. Then, by Equation (10), if L is approximately constant, in the limit of large r ⊥ , the following must be true for highly magnetic jets (assuming angular momentum outflow, i.e. L > 0, B P > 0, B φ < 0) v φ < L kµr ⊥ γ , 0.5 4π cr ⊥ L <| B φ |< 4π cr ⊥ L; .(11) The lower bound in the second relationship merely expresses the fact that by assumption (magnetic jet), the electromagnetic angular momentum flux is at least equal to the mechanical angular momentum flux. By poloidal magnetic flux conservation B P ∼ r −2 ⊥ , so by Equation (11), | B φ |≫ B P at large r ⊥ . To estimate the poloidal Poynting flux, S P , in the plasmoid, first transform fields to the observer's frame B φ = γB ′ φ E ⊥ = v P c γB ′ φ − v φ c γB P ≈ v P c γB ′ φ ,(12) where E ⊥ is the poloidal electric field orthogonal to the magnetic field direction. The poloidal Poynting flux in the observer's frame, S P , along the jet direction is (Punsly 2008): S P = c 4π E ⊥ B φ ≈ c 4π γ 2 β[B ′ φ ] 2 , B ′ φ ≈ B .(13) Mechanical Contributions to the Energy Flux The energy content is separated into two pieces. The first is the kinetic energy of the protons, K(protonic), K(protonic) = (γ − 1)Mc 2 ,(14) here M is the mass of the plasmoid. The other piece is named the lepto-magnetic energy, E(lm), and is composed of the volume integral of the leptonic internal energy density, U e , and the magnetic field energy density, U B . It is straightforward to compute the lepto-magnetic energy in a spherical volume, E(lm) = (U B + U e ) dV = 4 3 πR 3 B 2 8π + Γmax Γ min (m e c 2 )(N Γ E −n+1 ) d E . (15) The lepto-magnetic energy is often argued to be minimized in astrophysical sources (Hardcastle et al. 2004;Croston et al. 2005;Kataoka & Stawarz 2005). The relevance of this will be discussed in terms of the time evolution of various models of the flare in the next section. The leptons also have a kinetic energy analogous to Equation (14), K(leptonic) = (γ − 1)N e m e c 2 ,(16) where N e is the total number of leptons in the plasmoid. The other quantities of interest are the protonic and leptonic field aligned (jet axis aligned) poloidal energy fluxes in the frame of the observer. The protonic energy flux in the frame of the observer is approximately the kinetic energy flux density given E(proton) = N(γ − 1)γv P m p c 2 ,(17) where m p is the mass of the proton. The leptonic jet aligned poloidal energy flux density in the frame of the observer is ke(leptonic) = Nγv P γµc 2 ,(18) where k is the mass flux defined in Equation (10) and e is the specific energy of a lepton (Punsly 2008). From an energetics standpoint it is useful to subtract off the rest mass term which merely reflects the conservation of particle number from the source to the jet and cannot be converted into different forms of energy flux. This has been called the free thermo-kinetic energy flux and was defined in McKinney et al. (2012) as E(leptonic) = Nγv P γµc 2 − m e c 2 .(19) This would be appropriate for protonic outflows. However, for an electron-positron outflow, the mass is not conserved due to pair creation and in fact in the models considered the number of leptons in the plasmoid increases substantially over time. Thus, ke(leptonic) is the more appropriate energy flux in this study. The specific enthalpy decomposes as Nµ = U e + P ,(20) where the relativistic pressure, P ≈ (1/3)(U e − Nm e c 2 ) (Willott et al. 1999 Above the line, E(lm) is dominated by internal plasma energy. Below the line, E(lm) is magnetic energy dominated. Thus, the solution converts mechanical energy to magnetic energy. This seems inconsistent with magnetic acceleration of the plasmoid to β app = 0.97. Thus, the solution has no explanation of its existence. Fitting the Data with a Specific Spherical Model Mathematically, the theoretical determination of S ν depends on 7 parameters in Equations (2)-(8), N Γ , B, R (the radius of the sphere), α, δ, E min and E max , yet as there are only 3 constraints from the observation τ , α and S o , it is an under determined system of equations. Most of the particles are at low energy, so the solutions are insensitive to E max . In order to study the solution space, δ and E min are pre-set to a 2-D array of trial values. The reason for separating these two variables out in this fashion is that there is information that constrains these variables, apriori. First of all, Equations (1) and (9) combined with the upper bound on the LOS (from time variability considerations in Reynolds et al. (2009) and the superluminal ejection in Reynolds et al. (2017)) and the β app = 0.97 from Figure 5 greatly restricts the allowed values of δ. The results are shown in Figure 6. There has been no evidence of extreme blazar properties so we do not pursue extreme blazar-like LOS values (i.e < 10 • ). Thus, we expect δ to be in a fairly narrow range of ∼ 1.8 − 3. Using the formula for the observed frequency at which the peak of the synchrotron emission occurs (Tucker 1975) ν peak ≈ 3 × 10 6 E min m e c 2 2 δB(21) and noting from Table 3 there is strong core radiation at 8.4 GHz, the models quickly converge on an upper bound for Γ min of ≈ 25 − 30. For each trial pair of values, δ and E min , one has 4 unknowns, N Γ , B, R and α, but recall that there are only 3 constraints from the spectrum for each model. Thus, there is an infinite 1 dimensional set of solutions for each pre-assigned δ and E min that results in the same spectral output. First, a power-law fit to the high frequency optically thin synchrotron tail fixes S o and α in Equation (5). An arbitrary B is chosen in the spheroid. Then N Γ and the spheroid radius, R, are iteratively varied to produce this fitted S o and a value of τ that minimizes the least squares residuals of the SSA region at 15.2 GHz and 8.4 GHz. Another value of B is chosen and the process repeated in order to generate two new values of N Γ and R that reproduce the spectral fit. The process is repeated until the solution space of B, N Γ and R is spanned for the pre-assigned δ and E min . To this point, we have described the mathematics of how to fit the spectra and how to convert this to the physical parameters of a spherical volume. It is not known ahead of time what region of solution space is relevant to a realistic physical solution. In order to determine this, we consider various circumstances related to the jet of Mrk 231: 1. There needs to be a source for the long term (20 years) radio luminosity (∼ 10 41 ergs/sec) of the secondary source, K1, 2.5 lt-yrs away. (Reynolds et al. 2017). In order to describe the details of our current campaign, the highest contour levels are removed from the core. The size of the plasmoids is approximated by the jet solution in the lower left hand frame of Figure 11 with Γ min = 1, LOS = 14.5 • . The separation from the "nucleus" is somewhat arbitrary and is based on β app = 0.97 and the time lapse to the beginning of the flare rise in Figure 1. The separation from epochs A to D is based on β app = 0.97. The ellipse in K1 is the FWHM of the elliptical Gaussian fit to K1 (Reynolds et al. 2017). The overall geometry is very suggestive of a simple uniformly expanding conical jet with a half angle of ∼ 3 • . 2. There needs to be a mechanism that accelerates the jet away from the gravitational attraction of the central black hole. The only known mechanisms for launching jets to relativistic speed involve strong magnetic acceleration and the conversion of magnetic energy to bulk plasma energy (Parker 1958;Lovelace 1976;Blandford 1976;Blandford and Payne 1982). 3. At late times (i.e., epoch D), when the flare fades, the plasmoid likely tends toward the minimum energy configuration. This will be motivated more in the detailed discussions of the next section. These three constraints are essential for producing physically reasonable models in the following. Physical Models There are four types of emission regions that are possible. • A discrete ballistic ejection of a turbulent magnetized plasmoid made of electrons and positrons. This was the preferred solution for the major flares in GRS 1915+105 (Fender et al. 1999;Punsly 2012). This will be referred to as a leptonic plasmoid in the following. • A discrete ballistic ejection of a turbulent magnetized plasmoid made of protons and electrons. This was a possible solution for the ejection from the neutron star merger and gravity wave source GW170817 (Punsly 2019). This will be referred to as a protonic plasmoid in the following. • A high dissipation region or knot in an organized magnetic jet of leptonic plasma. This was the most likely source of the radio emission in GW170817 (Punsly 2019). This will be referred to as a leptonic jet in the following. • Another possibility is a high dissipation region or knot in an organized magnetic jet of protonic plasma. These types of physical models, all of which produce the fits in Figure 4, are described in this section. Figure 7 shows the dependence of E(lm) in Equation (15) on R for a wide span of pre-assigned values of δ and E min . The figure allows one to see how E(lm)(R) varies as one adjusts the pre-assigned δ and E min . More importantly, it shows how the plasmoid changes, independent of the pre-assigned δ and E min form epoch A to epoch D. The dashed line and the red dots in each panel show the time evolution of a possible physical solution. It is the solution found for the major flares in GRS 1915+105 (Punsly 2012). A blob of turbulent electron-positron plasma is ejected from the black hole accretion system. It obeys energy conservation and the radiation losses are negligible in the 69 days of monitoring (∼ 10 47 ergs). To the right of the minimum of each E(lm) curve, the stored plasmoid energy is more magnetic than mechanical and to the left of the minimum of E(lm) of each curve in Figure 7, the energy is more mechanical than magnetic. The dashed curve indicates a scenario in which the flare evolves from magnetically dominated to near equipartition at late times. This satisfies physical requirement (3) from Section 6. Magnetic energy is converted to mechanical form. This is consistent with physical requirement (2) of Section 6, magnetic forces can launch the jet and this energy is converted to bulk motion and internal leptonic energy. Leptonic Discrete Ejections The main concern is physical requirement 1 from Section 6. We can estimate the radio luminosity of K1 from the VLBA observations. For example, in our best observation, epoch A, we note that the K1 luminosity is down about 10% from its historic high in 2015. The data in Table 2 is well fit by a power law with α = 1.61. Most of the radio luminosity is at low frequency, but our data only goes down to 8.4 GHz. A previous, VLBA campaign in 1996 covered frequencies of 1.5 GHz, 2.3 GHz, 4.9 GHz, 8.4 GHz and 15.4 GHz (Ulvestad et al. 1999a). Even though the double is not resolved (even partially) below 8.4 GHz, the core was very weak at 8.4 and 17.6 GHz (∼ 15% of that in 2017). Thus, the low frequency emission can all be considered to be from K1 as a good approximation. It was argued that the powerlaw behavior began to be attenuated by free-free absorption below 8.4 GHz (Ulvestad et al. 1999a). The flux density peaked at 5 GHz. Thus, we estimate the radio luminosity from K1 in epoch A by integrating the power-law fit from 5 GHz to 50 GHz , ≈ 8.4 × 10 40 ergs/sec. Thus, from 2015 to 2018 this equates to ∼ 2.5 × 10 48 ergs/yr. The total energy supplied every year must be 2.5 × 10 48 /ǫ R ergs, where ǫ R is the radiative efficiency of the plasma from 5 GHz to 50 GHz in K1. For the case, LOS = 14.5 • and Γ min = 1, in Figure 7, there needs to be ≈ [2.5 × 10 48 ergs/yr/1.25 × 10 49 ergs]/ǫ R ≈ 0.2/ǫ R ejections per year. Since the light curve in Figure 1 shows at most 2 ejections per year. This implies a radiative efficiency of larger than 10% in K1 which would be a very extreme requirement for the radio lobe, but such a high efficiency might be consistent with a compact steep spectrum object (O'Dea 1998). Protonic Discrete Ejections Another possibility is a ballistic ejection of a blob of protonic plasma. Figure 8 plots K(protonic) as a function of E(lm) for the case LOS = 14.5 • and Γ min = 1. This is useful for understanding the time evolution of the system. For a flare that decays, we posit that the final state is near minimum energy, minimum E(lm), as in the discrete ejections in GRS 1915+105. If this is true then a solution associated with the dashed black line is indicated. The plasmoid is ejected with K(protonic) = 2.7 × 10 51 ergs, a constant total energy from epoch A to epoch D. Epoch D is relaxed to the minimum energy configuration. Figure 8 shows that the time evolution is one in which plasma internal energy is converted to magnetic turbulent energy. Based on Figure 1, the longest time frame in which energy is pumped into the plasmoid is the time lapse, T , from epoch A to the start of the flare (∼ 120 days, see Section 8 for more details). The instantaneous jet power required to energize and eject the plasmoid is therefore Q o ≈ K(protonic)/T > 2.7 × 10 44 ergs/sec. This is a large jet power by astrophysical standards. Theoretical efforts to explain the initiation of such powerful jets rely on some form of magnetic jet launching that requires the plasma dynamics to be dominated by magnetic forces (Lovelace 1976;Blandford 1976;Blandford and Payne 1982). Thus, it is hard to understand a mechanism in which inertial energy is converted to magnetic energy during the course of the plasmoid evolution. One would expect there to be an excess of magnetic energy in the early stages as required for relativistic jet initiation. The indicated time evolution does not satisfy physical requirement (2) of Section 6. Leptonic Jet Another possibility that is very common in extragalactic jet models is a magnetic jet filled with an electron-positron plasma (Blandford and Köingl 1979;Willott et al. 1999). In this scenario, the spherical plasmoid is a knot in the jet as indicated in Figure 9. The knot represents a region of enhanced dissipation. It might be related to a wavefront or shock signaling an increase in jet power from the source that is propagated down the conduit of the jet. In this circumstance, the following estimates of jet power represent a transient state of enhanced jet power. Alternatively, the enhanced dissipation might have arisen from interactions of the jet with the ambient medium. For example, it is likely that the jet is interacting with the very dense BAL wind (Reynolds et al. 2009). This provides a piston for launching strong dissipative waves such as shocks as well as a source of instabilities created from the jet/wind interface (Bicknell et al. 1990). In this instance, the jet power estimate is likely more indicative of the jet over a long time frame, but the interaction has allowed us to sample the quasi-steady jet (by increasing the dissipation) for a short period of time. The jet power in this model, Q, using Equations (13) and (18) is Q = [S P + ke(leptonic)]dA ⊥ + L r ,(22) where dA ⊥ is the cross sectional area element normal to the jet axis and L r is the energy flux lost to radiation. We constrain the solution space by assuming that the system approaches minimum E(lm) at late times (epoch D). This is motivated by the discrete ejections in GRS 1915+105 that approach minimum energy at late times and it has been argued that the hot spots (the terminal jet knot) and the surrounding radio lobes are near minimum energy in extragalactic radio sources (Punsly 2012;Hardcastle et al. 2004;Croston et al. 2005;Kataoka & Stawarz 2005). A common spheroid shape for the emission region is chosen to allow direct comparison between the four classes of models. However, it is not ideal for cross-sectional integration as in Equation (22) due to its nonuniform cross-sectional area. Thus, the evaluation of Q in Equation (22) is the average over all the cross-sections (i.e., πR 2 → (2/3)πR 2 ). Within the crude approximation of these simplified models this is not a huge source of uncertainty. The basic physics of the knot evolution is shown in Figure 10. The top frame plots Q versus E(lm). Epoch D is in the minimum energy configuration. The time evolution between epochs is indicated by the dashed line which is simply conservation of Q (radiation losses are negligible). The evolution makes sense from jet/wind launching theory, requirement (2) of Section 6. Magnetic energy flux is converted to inertial energy flux. The bottom frame of Figure 10 is a plot of the fraction of the jet power that is inertial energy flux: Inertial Fraction = ke(leptonic))dA ⊥ /Q . The dashed line is the same solution as in the top frame. In epoch A, the inertial component of Q is ∼ 5% of Q and increases steadily to epoch D where it is ∼ 40%. The dependence of jet power on the LOS is illustrated in the top left frame of Figure 11. A larger LOS equates to a more powerful jet as well as a stronger discrete ejection in the leptonic model of Section 7.1. The top right frame of Figure 11, shows that Q decreases as E min increases. It also shows that a protonic jet (see next section) is much more powerful. The lower left hand frame shows the time evolution of the spheroid radius. The solutions with Γ min = 1 were used to estimate the jet opening angle at K1 in Figure 9. The solution with Γ min = 1 and an LOS = 14.5 • has an expansion rate, 2(dR/dt) = 0.2c. The bottom right hand corner of Figure 11 shows that the magnetic field decreases as the knot propagates. This is expected from the conversion of magnetic to inertial energy indicated in Figure 10. Combining these results with Figure 6 and Equation (21) motivated our choice of Γ min = 30 as an upper limit for a viable lower energy cutoff. The solution has a smooth monotonic behavior after the peak of the flare at epoch B as might be expected for a plasmoid that relaxes to the minimum energy state. Evolution of Knot Radius in Leptonic Jet Models Protonic Jet It is possible that the jet is made of primarily protonic not positronic plasma. Then there are two inertial contributions to Q, one from the energized leptons and a dominant component from the bulk motion of protons. Equation (22) needs to be modified accordingly with Equation (17), Q protonic = [S P + ke(leptonic) + E(proton)]dA ⊥ + L r ≈ [S P + E(proton)]dA ⊥ . (24) The solutions that are plotted in the upper right hand frame of Figure 11 are computed assuming a minimum E(lm) state in epoch D, the same state as in all the other models. There is no concern with these solutions having enough power to energize K1 in accord with requirement (1) of Section 6. However, we are unable to find any solutions that satisfy physical requirements (2) and (3) that change monotonically and smoothly as the leptonic jet solutions of the previous subsection. Thus, our methods are not likely valid quantitatively, but might provide some qualitative insight into these solutions and will be discussed below. The biggest issue is the assumption of a constant velocity is likely grossly inaccurate. The top frame of Figure 12, shows the solution for Γ min = 10 and a LOS = 14.5 • that proceeds by converting magnetic energy to inertial energy. Based on the upper right hand frame in Figure 11, any value of Γ min < 10 is likely too powerful (A Fanaroff-Riley II, FR II, extragalactic radio source level luminosity). In order to explore protonic jets, it is useful to introduce the protonic energy density, U P ≈ Nm p c 2 .(25) The horizontal axis in the top frame of Figure 12 plots U P /U B , the ratio of inertial energy density to magnetic energy density. Even though there are numerous changes from our original leptonic jet models inserted by the introduction of protons, the leptons in the plasma still radiate the spectra in Figure 4 for all the solutions within Figures 11 and 12. There are solutions that proceed towards more magnetic energy over time, but by physical requirement (2) of Section 6 this is untenable. The dashed arrow shows the direction of time evolution, assuming a constant jet power. The end of the arrowhead is the minimum energy solution in epoch D. This fixes the solution. Without this assumption, the solution is unconstrained and our analysis is not useful. The only solutions that satisfy physical requirements (1)-(3) of Section 6 require a dramatic change between epochs C and D. This is The solution is smooth from epochs A to C and is consistent with our simple formalism. However, the knot needs a catastrophic change from epochs C to D in order to achieve conservation of energy and a relaxed state with E(lm) near a minimum at the end of the flare. There is a two orders of magnitude change in U B /U P . The bottom frame highlights the implications of the large change. There is massive entrainment of plasma and an abrupt halt to the steady expansion of the knot from epochs A to C. Such dynamics goes far beyond our simple model, but the implication is clear. Strong dissipative physical process are required to execute this plasma state transition. However, there is no evidence of this in the spectra, see Table 3 and Figure 4. apparent from the large gap in U P /U B from epoch C to epoch D in the top frame of Figure 12, two orders of magnitude. The bottom frame of Figure 12 indicates the required dynamics. From epochs A-C there is a smooth gradual expansion with modest entrainment of plasma. The jet power is so dominated by Poynting flux (97%) that the jet can be either positronic or protonic in the early stages without affecting the solution. However, the bottom frame of Figure 12, indicates intense protonic entrainment between epochs C and D. This is not unreasonable as deduced in Reynolds et al. (2009). At K1, there is evidence of overwhelming entrainment on parsec scales from the dense enveloping BAL wind. The solution described here is one of a powerful Poynting flux dominated jet in which all the power is converted into proton kinetic energy on the order of 1-2 light months from the source. There are a couple of issues with this solution that violate our basic assumptions. 1. Clearly, large entrainment would slow down the jet. Our models are not sophisticated enough to capture this, so the models are only qualitative. 2. In order to keep Q reasonable we had to choose an adhoc Γ min ≥ 10 based on the upper right hand frame of Figure 11. The bottom frame of Figure 12 shows an entrainment solution in which over 90% of the plasma enters the knot between epochs C and D. The plasma that surrounds the jet in the BAL wind is cold relative to the jet. This process should drastically lower Γ min . This complexity is far beyond our simple model and involves much uncertain physics and geometry. These shortcomings raise the question if our model is of any value in the analysis of this scenario. In order to explore this, we note that the model of a Poynting flux dominated jet with a steadily increasing radius and a slow and steady conversion of magnetic energy to mechanical energy seems qualitatively reasonable from a wind theory point of view (Blandford and Payne 1982;Punsly 2008). But, can we say anything about the notion of intense entrainment between epochs C and D that is implied by the prediction of the model? The model predicts its own failure in this time frame. First, we note that the knot in the Poynting jet is a very powerful dynamic object from epochs A-C. It propagates with β app = 0.97c, Q > 4 × 10 43 ergs/sec of pure Poynting flux and it is expanding (based on the top frame of Figure 12) at 2(dR/dt) = 0.16c. In a one month period, > 95% of the Poynting jet power is converted to protonic kinetic energy flux. This process also introduces 90% of the jet plasma. The Poynting flux fraction of Q changes from 97% to 3%. The radius stops expanding at 0.08c and contracts rapidly. This seems to be a very violent process involving dissipative instabilities and fast shocks (Kennel & Coroniti 1984). The introduction of new plasma and strong dissipative process should change the energy spectrum of the electrons. This should be independent of the entrainment details. Now, we look outside of the simple, spherical (likely invalid due to the strong entrainment induced changes) model for empirical data, in particular Table 3 and Figure 4. Note that α and n in the last two columns are very similar in epochs C and D, almost the same considering the crudeness of the model. The entrainment scenario should have produced a significant change unless there is a major coincidence that the process recreates the previous particle spectrum. By contrast, note the shift of the spectral peak and the decrease of peak flux intensity implied by Table 3 and Figure 4. This is the defining characteristic of cooling by adiabatic expansion: the same spectral shape with a shift of the spectral peak to lower frequency and lower intensity (Moffet 1975). This empirical (model independent) spectral change is consistent with the steady expansion of the leptonic jet model of the last subsection and not the violent contraction indicated for an the intense entrainment scenario of the protonic jet. For this reason, we consider a knot in a leptonic jet to be more likely to represent the physical solution than a knot in a protonic jet. We also note that the leptonic jet favors Γ min ≈ 1 and does not require the adhoc condition Γ min ≥ 10 needed for the protonic jet models. High Energy Observations The energetic flare at 7mm can manifest itself as enhanced X-ray emission. If so, this can provide a valuable tool for analyzing the possible physical nature of the flare. We performed a deep 80 ks NuSTAR observation during the flare rise on October 19, 2017 for this purpose. We were searching for evidence of a pronounced change from the NuSTAR observation on August 27, 2012, the only NuSTAR observation that occurred during a state of low radio jet activity. The older observation was for 41 ks and showed evidence of X-ray absorption indicative of an out-flowing wind with a velocity 17000 ± 4500 km/s (Feruglio et al. 2015;Reynolds et al. 2017). Thus, we proposed a deeper 80ks observation to clearly distinguish the high radio state X-ray spectrum from the low radio sate X-ray spectrum. This was demonstrated only marginally with the four ∼ 30ks NuSTAR observations during high radio states that were analyzed previously (Reynolds et al. 2017). Figure 13 clearly shows that the X-ray absorbing wind is suppressed during the high jet state. This corroborates the claim that the X-ray absorbing wind was suppressed during previous states of high jet activity based on shorter exposures (Reynolds et al. 2017). We note the related phenomenon, the disappearance of the high ionization ultraviolet broad absorption line wind during high radio states that was discussed in that study as well. As with the other high radio states, there is no pronounced increase in the X-ray luminosity. Similar to the other 5 NuSTAR epochs in Reynolds et al. (2017), the spectrum is fit with a photon number spectral index 1.50 ± 0.1 and a neutral hydrogen column density of is compared to the low radio state spectrum from August 2012 (black filled squares), with solid lines representing the respective best-fitting models. The X-ray continuum level is unchanged, but the X-ray ultra-fast outflow is suppressed in the high radio state. Although, in our analysis, we fit the NuSTAR data from the FPMA and FPMB detectors jointly, we show here the co-added spectra at each epoch for visual clarity; each spectrum is divided by the respective effective area to allow direct comparison. Confidence intervals of X-ray fluxes obtained by fitting jointly the NuSTAR FPMA and FPMB spectra at each epoch with a cross-normalization constant accounting for the different detector responses. 10 ± 5 × 10 22 cm −2 . Table 4 shows the observed flux for the 6 epochs. The flux is constant to within 20% over 5 years. The 10-30 keV column is the most robust for tracking flux variation, since it is not subject to changes in the intrinsic observing column. The observed 3-30 keV X-ray luminosity in the high radio state (2017) is ∼ 1.1 ×10 43 ergs/sec to 1.2 ×10 43 ergs/sec. In the remainder of this section, we estimate the expected X-ray flux in our plasmoid model and compare it to the continuum flux level. The epoch A observation/plasmoid model is the most suitable in time, being just 17 days after the NuSTAR observation. The energetic plasmoid is not only a source of synchrotron emission, but the energetic leptons can also upscatter soft background photons to the X-ray energies detectable with NuSTAR. There are a few possibilities relevant to blazars. There is synchrotron self-Compton scattering (SSC) of the synchrotron photons in the plasmoid as well as the external inverse Compton (EIC) scattering of the accretion disk photons, broad emission line photons and infrared photons from the dusty torus. In quasars, the EIC process dominates the SSC mechanism because the quasar is a very luminous source of soft photons. For highly relativistic jets, EIC from the photon fields of the broad emission lines or the torus tend to be dominant since they are extremely enhanced in the plasma frame of reference due to Doppler blue-shifting and conversely, the more luminous accretion disk photon field is significantly Doppler redshifted in the frame of reference of the plasma (Dermer and Schlickeiser 1993). However, in the plasmoid models of epoch A, the bulk Lorentz factor ∼ 1.6, so the dynamics are only transrelativistic and the Doppler effects are modest. Thus motivated, we cannot neglect the EIC luminosity from the copious accretion disk photons apriori in our analysis as shown below. Inverse Compton Scattering of Accretion Disk Photons It is not trivial to estimate the accretion disk luminosity. Mrk 231 has significant intrinsic extinction in the nuclear region (Smith et al. 1995;Lipari et al. 1994;Veilleux et al. 2013Veilleux et al. , 2016. This not only makes the direct observation of the accretion disk spectrum impossible, but it also makes the origin of the weak X-ray emission from October 2012 in Figure 13 very uncertain (Reynolds et al. 2017;Veilleux et al. 2013). An estimator of the accretion disk luminosity is the Hα broad emission line and this is very prominent in Mrk 231. The intrinsic extinction has been corrected with a LMC extinction law with E(B-V)=0.63, yielding a plausible intrinsic quasar spectrum in the optical band (Lipari et al. 1994;Bokensberg et al. 1977). This same extinction law does not work in the ultraviolet (Smith et al. 1995;Veilleux et al. 2016). This extinction correction for Hα is a factor of ∼ 4.5. There is so much intrinsic absorption (including that of Fe II) that Hβ is very weak and is difficult to use for estimating the bolometric luminosity of the accretion flow, L bol . The situation is even worse for the ultraviolet broad lines (Smith et al. 1995). The Hα line is a bonafide quasar broad line with an extinction corrected luminosity of L Hα = 4.7 × 10 43 ergs/sec and a full width half maximum (FWHM) of 2800 km/sec (Smith et al. 1995). L bol can be estimated with the formula (Greene & Ho 2007), L bol ≈ 2.34 × 10 44 L Hα 10 42 erg/sec 0.86 ergs/sec ≈ 6.45 × 10 45 erg/sec . Secondly, we can also estimate L bol from the reprocessed photons in the infrared from the dusty torus, but this is not simple in Mrk 231 either. The infrared band is very luminous and is actually dominated by starburst emission. The breakdown of the different components with detailed modeling indicates that the AGN luminosity at wavelengths longer than 3µm is L IR (AGN) ≈ 4 × 10 45 ergs/sec and the star burst luminosity is L IR (SB) ≈ 10 46 ergs/sec (Farrah et al. 2003). The AGN component is complicated by the existence of two strong components, a hotter component with the SED peaked at ≈ 4µm and a stronger cooler component peaked at ≈ 60µm. The second component is likely associated with the molecular disk imaged in the hydroxyl maser line with VLBI with an inner radius of 30 pc and an outer radius of 100 pc (Klöckner et al. 2003). Regardless of this complex nature we use the IR based bolometric estimators (Spinoglio et al. 1995 where λL λ (λ = 12µm) and λL λ (λ = 25µm) are from the AGN component of the fit to the IR SED (Farrah et al. 2003). The three estimation methods in Equations (26)-(28) closely agree with each giving us confidence in our estimate of L bol . The EIC luminosity, L EIC , and frequency, ν EIC , are related to the synchrotron luminosity, L synch and frequency, ν synch by (Tucker 1975) L EIC L synch ≈ U ph U B , ν EIC ν synch ∼ Γ 2 ,(29) where U ph is the energy density of the accretion disk photon field in the frame of reference of the plasmoid. The flux of the photon field of the accretion disk is diluted in the plasmoid in epoch A, by geometric dilution and also experiences Doppler redshifting. For the dissipative knot in a leptonic jet model (our preferred model in Section 7.3 and Figure 10), assuming a LOS = 14.5 • and Γ min = 1 for illustrative purposes, the relevant parameters are B = 0.41G U B = 6.69 × 10 −3 ergs/sec β = 0.816 . In order to compute U ph , taking into account geometric dilation, we need to know the distance the plasmoid was from the accretion disk during the NuSTAR observation. Based on the light curve in Figure 1, the plasmoid seems to have been ejected ∼ 100 days earlier. We illustrate this in Figure 14 which plots both the 17.6 GHz flux density and the spectral index from 14.6 GHz to 17.6 GHz at the beginning of the flare. The spectral index is important since as a plasmoid is ejected at the beginning of the flare it begins very compact and has large SSA at 17.6 GHz. Therefore, the flare begins at high frequency first and evolves to lower frequency (van der Laan 1966;Moffet 1975). For strong flares, even in the presence of the ∼ 120 mJy steep background flux density, the spectrum flattens as the new ejection emerges and can even invert (as in Figure 14) between 14.6 GHz and 17.6 GHz (Reynolds et al. 2017). These conditions occur near MJD 57940, thus this is a more likely flare ejection time than the other relative minimum near MJD 57974. We estimate ∼ 100 days interval between the flare ejection time and the NuSTAR observation. Assuming the velocity of Equation (30) is constant, it would have propagated D d−p ∼ 2.12 × 10 17 cm from the disk. In order to compute the relevant Doppler redshift, note that as the plasmoid moves away from the accretion disk, the distance from the disk become much larger than the radius of the bulk of most of the disk luminosity and the photons approach from behind to first approximation (Dermer and Schlickeiser 1993). Thus, the Doppler factor of the plasmoid relative to the disk would be computed from Equation (1) with θ = 180 • and β from Equation (30), δ d−p = 0.318. We can then compute the photon energy density in the plasmoid from Equation (8) and (Lightman et al. 1975;Tucker 1975) U ph ≈ δ 4 d−p L bol 4πcD 2 d−p = 3.93 × 10 −3 .(31) From Equation (21), (29) and (30), the range of Γ that results in EIC photons in the NuSTAR 2-30 keV band is 594 < Γ < 1170. These same leptons radiate synchrotron photons in the observer's frame in the far infrared, 1.36 × 10 12 Hz < ν o < 5.31 × 10 12 Hz. Based on the Epoch A fit in Table 3, these leptons radiate a synchrotron luminosity of L synch = 3.34×10 41 ergs/sec as observed at earth. From Equations (29) -(31), the expected EIC luminosity observed at earth in the NuSTAR band from these leptons is L EIC = 1.96 × 10 41 ergs/sec or a flux of F EIC = 4.48 × 10 −14 ergs/sec − cm 2 . This value is negligible compared to the fluxes in Table 4. Note that given the large distance from the accretion disk to the plasmoid, D d−p ∼ 2.12 × 10 17 cm, the broad line region is likely an order of magnitude closer to the black hole, and also illuminates the plasmoid primarily from behind (Greene & Ho 2007). Thus, the EIC flux is also weak, perhaps an order of magnitude less than that induced by the accretion disk photon field. Inverse Compton Emission of Infrared Photons The primary controlling factor in the determination of the EIC from the dusty torus is the geometric dilution. Thus, the closest significant component in the complex IR spectrum is the most relevant. This is the hot dust on the inner face of the torus. The hot dust is typically considered to be region in which graphite and silicate grains are sublimated with a temperature between ∼ 1500 K and ∼ 1800 K (Mor and Netzer 2012). It creates significant flux at wavelengths 1.6µm -2.2µm. There is also "hot dust" that radiates a similar luminosity at 3µm-4µm in Mrk 231 with a temperature ∼ 750 K (Lopez-Rodriguez et al. 2017). These components have the least geometric dilution of any of the IR components and contribute about > 1/3 of the total AGN IR luminosity (Farrah et al. 2003). Beside geometric dilution, one needs to know the relevant Doppler factors and that is based on the distribution of the hot dust. The primary piece of evidence is from the 3CRR catalog in which radio galaxy and radio loud quasar IR spectra are compared. The radio galaxy IR spectra are heavily attenuated at wavelengths short of 5µm. This implies that the hot dust is viewed through the optically thick torus (Hönig et al. 2011). The hot dust lies on the inner face of the torus and is restricted to a solid angle less than that subtended by the main body of the torus, less than ≈ 45 • (Barthel 1989). The covering factor, CF, of the hot dust has been estimated at 0.15-0.35 (Mor and Netzer 2012). We assume 0.15 < CF < 0.35 and use this to compute an upper limit on the EIC flux in the calculation below. We note that the distribution of hot dust at the inner edge of the torus is restricted to within 45 • of the equatorial plane. The IR spectrum from data in Lopez-Rodriguez et al. (2017) at wavelengths shorter than 4µm was fit with two blackbody components, T = 750 K as in Lopez-Rodriguez et al. (2017) and a hotter component T = 1650 K, Mor and Netzer (2012), with a luminosity of 1.06 × 10 45 ergs/sec and 1.65 × 10 45 ergs/sec, respectively. The radius at which these hot dust distributions reside are ≈ 0.35/CF 6.5 × 10 18 cm and ≈ 0.35/CF 1.7 × 10 18 cm, respectively. We assume that the dust is isotropically distributed relative to the equator in the range −45 • < θ < +45 • . If the hot dust is concentrated closer to the equatorial plane than 45 • then the Doppler factors are smaller and this calculation is an upper limit. For the hottest component, U ph ≈ 1 c 2π 0 θo −θo δ(β = 0.816, θ) 4 I BB (T = 1650K)dφ dθ = CF 0.35 (1.16 × 10 −2 ) ,(32) where θ o = 45 • . From Equations (29), (30) and (32), the EIC flux in the NuSTAR band from the 1650 K dust is (for 0.15 < CF < 0.35) 5.68 × 10 −14 < F EIC (T = 1650K) < 1.33 × 10 −13 . Similarly, for the other hot dust component 2.42 × 10 −15 < F EIC (T = 750K) < 5.55 × 10 −15 . Note how much more inefficient the T = 750K component is at inducing EIC emission in the plasmoid, since it is farther out. For completeness, we consider the remaining ∼ 3 × 10 45 ergs/sec of AGN IR emission from the SED that is peaked at 60µm (Farrah et al. 2003). Assuming a blackbody this would require a radius of 60 pc at T 40K (Farrah et al. 2003). The rotating molecular disk resolved with VLBI is roughly this size, 30-100 pc (Klöckner et al. 2003). It seems reasonable to assume that these are the same regions. The VLBI image reveals gas that is distributed in a region that is approximately orthogonal to the plasmoid velocity in Figure 5 as was the case for the hot dust (Lonsdale et al. 2003;Klöckner et al. 2003). There would be no strong Doppler enhancement for such a distribution. The geometric dilution is 10 4 times that of T = 1650K dust with only twice the luminosity. Thus, there is no evidence of a pole on distribution of luminous dust associated with the AGN. VLBI observations revealed that the AGN dominates the starburst emission within ∼ 100 pc of the quasar (Lonsdale et al. 2003). If we assume the most extreme upper bound of placing all the starburst emission along the jet axis 100 pc away and recompute Equation (31) with θ = 0, we get U ph < 3.93 × 10 −3 (10 4 ) 2×10 17 cm 3×10 20 cm 2 < 3 × 10 −5 . This is ∼ 10 −3 of the hot dust value in Equation (32). Of course, the starburst regions are probably more isotropically distributed and the contribution would be significantly less than this upper limit. Therefore, we do not consider starburst emission as a viable seed field for X-ray emission. In summation, the combined EIC flux from Sections 8.1 and 8.2 is < 8% of the observed continuum flux in Table 4. Summary and Concluding Remarks In this article we used a multi-frequency, multi-epoch VLBA campaign that suffered from significant degradation in resolution and sensitivity at 43 GHz in order to explore a luminous 17.6 GHz flare detected with AMI. The bad luck with lost stations was offset by a fortuitous time sampling that caught the rise, the peak and the decay of the flare. Our primary conclusion is that the central engine of Mrk 231, at the end of 2017, produced a strong moving knot in a Poynting flux dominated region of the jet. We determined that the knot was filled primarily with an electron-positron plasma that transports a power of Q ∼ 10 43 ergs/sec along the conduit of the jet. The existence of the simple SSA power-law spectrum of the unresolved radio core from 8.4 GHz to 43.1 GHz motivated a simple model, an optically thick uniform spherical volume for the emission region. There were four categories of models. If the spheroid is a discrete ejection, it can be positronic or protonic plasma. If the spheriod approximates a knot in a continuous jet it can be positronic or protonic plasma as well. This paper considered physical constraints that could narrow down the possible models. First, there were some hard constraints from observation. The LOS to the velocity of the ejected plasmoid was bounded to < 25 − 30 • in Reynolds et al. (2009Reynolds et al. ( , 2017) based on light curve variability and a superluminal ejection, respectively. In this paper, we found that the apparent core (the spheroid) was moving towards K1 in Figure 5 at β app ≈ 0.97. This constrained the kinematics, but did not favor one of the four categories over another. In order to differentiate amongst these 4 possibilities we established three more constraints in Section 6. The implications of the constraints introduced in Section 6 for the four classes of models were demonstrated in Section 7 and are summarized in Table 5. Clearly, the most robust model is a knot in a magnetic positron-electron jet. Figure 11 shows a gradual temporal evolution approaching adiabatic expansion at late times. The jet power based on Figure 11 is Q ∼ 10 43 ergs/sec. We also note that this estimate is more reliable (based on more data) than the Reynolds et al. (2009) estimate of a flare Q ∼ 3 × 10 43 ergs/sec. The discrete leptonic plasmoid ejections model is not ruled out. It would imply a long term time averaged power output, an order of magnitude less, with the impulsive power or launch power of Q impulsive ≈ E(lm)/T, with T loosely constrained by the data as less than the flare rise time of ∼ 120 days, Q impulsive > 1.25 × 10 49 ergs/1 × 10 7 sec > 1.25 × 10 42 erg/sec. In this estimate, it is assumed that Γ min = 1 and we used the top left hand frame of Figure 7 and Figure 1. The advantage of this model is that it is the same class of solution as was found for the major flares in GRS 1915+105 (Fender et al. 1999;Punsly 2012). Secondly, the source of the positronic plasma might explain the weak X-ray luminosity, an order of magnitude less than expected from a quasar of similar bolometric luminosity (Laor et al. 1997;Reynolds et al. 2017). The corona above the accretion disk that normally produces the X-ray luminosity is filled with pair plasma, but for some reason it is being episodically ejected as these radio plasmoids. This phenomenon has theoretical origins as a Parker instability or a twisted magnetic tower that acts as a piston or propellor, ejecting the plasmoid as it expands vertically relieving the magnetic stress (Lynden-Bell 2003). The continuous jet model, on the other hand, needs to be understood in terms of other extragalactic jets. First, we consider the "radio loudness" of Mrk 231. This determination is not a trivial or unambiguous calculation because of intrinsic extinction of the optical flux and the significant SSA of the radio emission. Radio loudness is typically defined by the ratio of the 5 GHz flux density, f(5), to the blue band flux density, f(B), R ≡ f (5)/f (B) (Kellermann et al. 1989;Stocke et al. 1992). A value of R > 10 is considered radio loud. Taking the values from the NASA Extragalactic Database, R = 280 mJy/13 mJy ≈ 22 which ostensibly seems to be radio loud. However, recall from Section 8 that there is strong attenuation of the continuum in this source. Based on the extinction law discussed in Section 8, we can compute an extinction corrected R, R * . The intrinsic extinction in B-band is a factor of ≈ 10. Thus, R * ≈ 280 mJy/[(10)(13 mJy)] ≈ 2 which is formally radio quiet. The jet in Mrk 231 feeds plasma to the luminous secondary, K1. In multiple campaigns, we have looked at wide field images at 8.4 GHz to see if there is any sign of emission emerging from the approximately stationary, K1. We have never detected any evidence of such a flow. The question is whether the jet is short because it is young or because its propagation is thwarted. The time it takes the jet to propagate to K1, 0.8 pc on the sky plane from the point of origin, is approximately 2.5 years based on our estimated v app = 0.97c, The core has been active for much longer than this, so this is not the origin of the short jet length. We conclude that the jet terminates at K1. As such, it should be considered a failed jet. We note that there is low frequency (1.4 GHz -5 GHz) emission on scales of 10-100 pc in Mrk 231. But, it is not aligned with the jet, nor does it connect with the jet termination point at K1, as noted above. This has been discussed in detail elsewhere (Ulvestad et al. 1999a). In spite of its radio quiet classification, Mrk 231 has an intrinsically powerful jet, ∼ 10 43 ergs/sec. It is comparable to some of the most powerful nearby, jets in active galactic nuclei such as M87 and 3C120 that propagate a few hundred kpc from the source (see below for a quantitative discussion). There are even radio quiet quasars with large scale radio structures hundreds of kpc in extent (Blundell et al. 2003). Yet, paradoxically, there is a failed jet in Mrk 231. To explore this, we quantify the magnitude of the jet truncation compared to extragalactic jets with similar power. In the extragalactic radio source lexicon, the Fanaroff-Riley classification of radio source morphology separates large radio sources into two classes FRI and FRII (Fanaroff & Riley 1974). The FRII sources are edge brightened with hot spots in their radio lobes and FRI are edge darkened and the lobes resembles plumes. Empirically, The top left hand panel shows the peak surface brightness as a compact core at the east end of a jet > 2mas. The top right panel shows a very bright knot ejected from the west end of the jet in December 2014, 0.25 mas from the feature that was previously defined as the core. The bottom left hand frame shows that in April 2015 the feature is 0.5 mas from the feature previously identified as the core, but now ∼ twice the brightness of the core. In September 2015, the ejected knot is > 0.7 mas from the core and more than 4 times as bright. This is a similar dynamic to what we see in Mrk 231 Figure 5, but on a scale two orders of magnitude larger. It would be very difficult or impossible estimate the point of origin with lower resolution. 2 1 0 −1 −2 −2 − there is an FRI/FRII luminosity divide as well, the FRIIs more luminous. In terms of long term time averaged jet power this divide occurs at Q ∼ 5 × 10 43 ergs/sec (Willott et al. 1999). Thus, we have estimated that Mrk 231 has the jet power of a strong FRI radio source. At redshifts similar to Mrk 231, FRI radio sources are usually narrow line radio galaxies (NLRGs) with a radio structure that extends well beyond galactic dimensions, with a linear size of ∼ 100 kpc or more (Willott et al. 1999). The jet in Mrk 231 ∼ 10 −5 the size of the FRI radio sources at similar redshift with similar jet power. In order to understand the origin of the termination of the powerful jet in Mrk 231, consider the very low accretion rates in these FRI NLRGs, 3-4 orders of magnitude less than a quasar (Chiaberge et al. 1999(Chiaberge et al. , 2002Hardcastle et al. 2009). However, there are the occasional FRI broad line galaxies such as the Seyfert 1 galaxy, 3C 120 that has a very prominent FRI morphology on a 400 kpc scale (Walker et al. 1987). Thus, the low accretion rate of FRI NLRGs is not the full explanation of the discrepancy with Mrk 231, but is likely related. One difference between 3C 120 and Mrk 231 is that Mrk 231 has a low ionization BAL wind (Lipari et al. 1994;Smith et al. 1995). In low radio states, it has also displayed evidence of a high ionization X-ray absorbing wind (Feruglio et al. 2015;Reynolds et al. 2017). There is also extreme amounts of intrinsic optical absorption in the galaxy itself from dusty gas (Lipari et al. 1994;Smith et al. 1995). All three circumstances point to a very dense nuclear environment through which the jet must propagate. Furthermore, in Reynolds et al. (2009), it was argued that the density of the free-free absorbing screen at K1 is consistent with the jet being stopped by the BAL wind at K1. By contrast, the extremely low, undetectable accretion of gas in FRI NLRGs is consistent with a very low density nuclear environment. The launching of the jet in Mrk 231 does not seem to be stopped by the dense nuclear environment, but its propagation does seem to be thwarted. The case of 3C 120 does not seem to be accommodated by this discussion. It is noted that there is no evidence of either a BAL wind or a high ionization X-ray absorbing wind in 3C 120 (Oke & Zimmerman 1979;Ballentyne et al. 2004). Thus, it might be the case that there is not an extremely dense nuclear environment near the source of the jet in 3C 120, hence jet propagation is not hindered. We cannot rule out the possibility that the BAL wind in Mrk 231 also diminishes the power of the jet launching mechanism as well as a providing a drag on its propagation. This conclusion does not exist in isolation. We apply the original definition of BAL quasars (BALQSOs) as quasars with UV absorbing gas that is blue shifted at least 5,000 km/s relative to the QSO rest frame and displaying a spread in velocity of at least 2,000 km s −1 , (Weymann et al. 1991). This definition was designed to exclude the "mini-BALQSOs," with the BALNicity index = 0 (Weymann 1997). This definition is preferred here since mini-BALQSOs tend to resemble non-BALQSOs more than BALQSOs in many spectral and broadband properties (Punsly 2006;Zhang et al. 2010;Bruni et al. 2013;Hayashi et al. 2013). It was found that the large scale radio emission (> 20 kpc in linear extent) in BALQ-SOs is strongly anti-correlated with the BALnicity index (Becker et al. 2000(Becker et al. , 2001. Dense outflows seem to prevent large scale jets from forming in BALQSOs. We note that Section 8 provided further support of the details of our model of a dissipative moving knot in a leptonic jet. We found that the plasmoid model predicts an X-ray flux < 8% of that of the historical levels detected by NuSTAR. Thus, no elevated NuSTAR flux levels are expected during the flare and none were observed. This shows consistency between the models and the high energy observations. The astrometry of the SSA core in Figure 5 provides a valuable laboratory for exploring the notion of a "core" in blazars. Mrk 231 has the advantage of the nearby stationary secondary for astrometric measurements, unparalleled in other blazars. In Mrk 231, the "apparent core" moves outward then eventually fades to the background surface brightness of the faint jet. This discovery might provide insight into more luminous blazar jets. Figure 15 shows an extreme example of "apparent core" motion in the blazar, 3C454.3, (z=0.859) observed with 43 GHz VLBA 2 . The ejected knot becomes much brighter than the feature previously identified as the core in earlier epochs. It appears to move ∼ 10.5 light years in 9 months, an apparent velocity of ∼ 13c. By contrast, the core in Mrk 231 moves only 0.2 light years in our observing campaign (see Figure 5) and even less in 2015. Such "small" displacements would be very difficult to detect without the low redshift of Mrk 231 and the precision astrometry made possible by the nearby stationary secondary, K1. Evidence of this phenomenon appears in simulations of blazar jets (Gomez et al. 1997). More importantly, there is evidence of this in the component motion of blazar jets. Some highly relativistic components appear to suddenly decelerate (Jorstad et al. 2017). This can be explained by a "core" that is actually a strong knot that moves downstream slightly (below the resolution limits of VLBA), before fading (A. Marscher private communication 2019). The lesson of Mrk 231 and 3C 454.3 in Figure 15 is that the ability of continuous jet models to predict the point of jet origin and the width of the jet at the point of origin is limited. They are too simplistic. The notion that the shift in the core position with frequency as a consequence of the change in the SSA opacity, Equation (2), can be extrapolated to infinite frequency to find the point of jet origin depends on the assumption that the low resolution observations are actually observing the core region (Blandford and Köingl 1979). Namely, the unresolved peak surface brightness is assumed to be the continuation of the jet to the frequency dependent photosphere of the launch region. However, the procedure can very likely be estimating the SSA shift of a bright (possibly moving) knot "near" the launch point and therefore estimates the location of an "arbitrary" point upstream of the launch point. In Mrk 231, it is 0.2 light years away from the point of origin, which is very large if this is the observational data that is input into a model of jet launching. The situation can be much more severe in a bright blazar like 3C 454.3 where the error in this method of estimating the point of jet origin can be 10 light years. In highly relativistic blazars, Doppler aberration can cause large swings in the apparent jet direction (Lind and Blandford 1985). Evidence of this wild jet bending near the base of the jet is the Event Horizon telescope image of 3C 279 (Event Horizon Telescope Collaboration 2019). In 3C 279, the brightest features are not likely to be the core, but knots in an apparently bent jet. In such circumstances, the accuracy of a "core shift" extrapolation to the point of jet origin is even more likely to be misleading. Fig. 1 . 1-The top frame is the historic AMI 17.6 GHz light curve. The bottom frame is the flare observed in this campaign with the four VLBA observation epochs A-D designated. Fig. 5 . 5-The position of the "core" relative to the astrometric origin, K1, as a function of time. There was too much degradation to the epoch C observation for it to be useful in this sensitive measurement. The similar plot fromReynolds et al. (2017) was added for historical reference. The better fit to a linear trajectory might have been the consequence of having all 10 VLBA stations in the 2015 campaign. Fig. 6 . 6-The Doppler and Lorentz factors as a function of the line of sight to the direction of plasmoid motion. A value of β app = 0.97 derived from Figure 5 is assumed in the calculations. Fig. 7 . 7-The time evolution of constant E(lm) discrete electron-positron ejection solutions described in Section 7.1. Note that there could be some protons, but their effect is negligible in the energetics by assumption. All of these solutions generate the fitted spectra inFigure 4. The red dots show the location of the constant E(lm) solution in each epoch on the infinite 1-D solution space plotted in the R-E(lm) plane. Magnetic energy is converted to plasma internal energy throughout the time evolution. The top two frames show the effect of varying the LOS and the bottom two frames show the effect of varying E min . Fig. 8 . 8-The discrete protonic plasmoid ejection model discussed in Section 7.2. It is a plot of the plasmoid kinetic energy as a function of E(lm). The solution has a constant kinetic energy as indicated by the dashed black line. Note the dashed red line called "equipartition". Fig. 9 . 9-This figure shows the geometry of the jet overlayed on our best (highest resolution and highest signal to noise) radio image from epoch 1 in 2015. The full image was published previously Fig. 10 . 10-The top frame shows the time evolution of a constant leptonic jet power, Q, solution for the flare from epoch A to epoch D in the E(lm) -Q plane. The conserved jet power solution is indicated by the dashed line. The bottom frame shows the rate at which Poynting flux is converted to mechanical energy flux as the propagating knot evolves from epoch A to epoch D. Fig. 11 . 11-The top left frame shows the dependence of Q on the assumed LOS for the leptonic jet model of Section 7.3. It also shows the dependence of E(lm) on the LOS for the end state (epoch D) which is also the conserved energy in the discrete leptonic ejection model of Section 7.1. The top right hand frame shows the dependence of Q on Γ min for both the leptonic jet model of Section 7.3 and the protonic jet model of Section 7.4. The bottom left hand frame shows the time evolution of the spheroid radius in the leptonic jet model of Section 7.3 for various LOS and Γ min choices. Notice that it is fairly steady. The bottom right hand frame shows the gradual decay of B in these models. Fig. 12 . 12-The dashed line in the top frame indicates a protonic jetted knot solution with the conversion of magnetic energy flux to protonic energy flux. Fig. 13 . 13-The high radio state NuSTAR spectrum from October 2017 (red empty circles) Fig. 14 . 14-The plasmoid ejection time is estimated from this plot of the spectral index and the 17.6 GHz flux density near the start of the flare. Since the spectrum flattens rapidly and eventually inverts near MJD 57940 this is a more viable starting point for the flare than the other relative minimum near MJD 57974. ), log[L bol ] = 0.942 log[λL λ (λ = 12µm)] + 3.642 ≈ 45.82 ,(27)log[L bol ] = 0.837 log[λL λ (λ = 25µm)] + 8.263 ≈ 45.75 , Fig. 15 . 15-An ejected peak surface brightness feature in the blazar 3C454.3 as seen with 43 GHz VLBA. Table 1 : 1Summary of BR214 VLBA Observations of Mrk 231 Note: BR = Brewster, MK = Mauna Kea, PT = Pie Town, SC = Saint CroixFrequency (GHz) Sensitivity (mJy/Beam) Missing Antennas Epoch A -2017 Nov 6 -MJD 58063 8.4 0.35 SC 15.4 0.33 SC 22.2 0.13 SC 43.3 0.19 SC,PT Epoch B -2017 Nov 20 -MJD 58077 8.4 0.36 SC 15.4 0.24 SC 22.2 0.10 SC 43.3 0.28 BR,MK,SC Epoch C -2017 Dec 15 -MJD 58102 8.4 0.26 SC 15.4 0.32 SC 22.2 0.14 SC 43.3 0.28 BR,MK,SC Epoch D -2018 Jan 14 -MJD 58132 8.4 0.27 SC 15.4 0.21 SC 22.2 0.24 SC 43.3 0.18 BR,MK,SC Table 2 : 2Summary of Model Fits to VLBA Observations a Due to an absolute flux calibration issue at 43 GHz in epoch C, the flux density at 43 GHz was scaled by the phase calibrator, J1302+5748. The flux density of J1302+5748 was scaled to its average value from epochs B and D. This produced a factor 1.37 re-scaling of the flux densities. After re-scaling, the flux density of K1 is constant at 43 GHz during epochs A-D within uncertainties as expected.1 2 3 4 5 6 7 8 9 10 11 12 Component MJD Flux Flux X X Y Y Major Axial PA Frequency Density Density σ σ σ Axis Ratio (Jy) (Jy) (mas) (mas) (mas) (mas) (mas) (deg) (GHz) Epoch A K1 58063 0.201 0.010 0.216 0.001 -0.081 0.002 0.54 0.48 63.5 8.4 Core 58063 0.100 0.005 1.189 0.002 0.383 0.003 0 - - 8.4 K1 58063 0.081 0.004 0.154 0.002 -0.145 0.002 0.32 0.63 72.0 15.3 Core 58063 0.156 0.008 1.185 0.001 0.342 0.001 0 - - 15.3 K1 58063 0.044 0.003 0.156 0.001 -0.097 0.001 0.30 0.86 66.3 22.2 Core 58063 0.131 0.009 1.236 0.001 0.389 0.001 0.10 0.45 61.7 22.2 K1 58063 0.014 0.001 0.151 0.003 0.383 0.003 0.28 0.67 83.3 43.1 Core 58063 0.082 0.007 1.22 0.001 -0.137 0.001 0.14 0.30 69.1 43.1 Epoch B K1 58077 0.203 0.010 0.120 0.002 -0.062 0.002 0.47 0.80 21.9 8.4 Core 58077 0.117 0.006 1.078 0.003 0.407 0.003 0 - - 8.4 K1 58077 0.088 0.004 0.183 0.001 -0.049 0.002 0.34 0.77 41.6 15.3 Core 58077 0.178 0.009 1.192 0.001 0.432 0.001 0 - - 15.3 K1 58077 0.050 0.004 0.192 0.001 -0.049 0.001 0.37 0.77 48.1 22.2 Core 58077 0.158 0.011 1.210 0.001 0.434 0.001 0.16 0.65 28.3 22.2 K1 58077 0.015 0.001 0.145 0.004 -0.003 0.004 0.28 0.84 23.5 43.1 Core 58077 0.100 0.008 1.199 0.001 0.496 0.001 0.18 0.42 25.9 43.1 Epoch C K1 58102 0.209 0.010 0.253 0.002 0.143 0.002 0.54 0.75 87.3 8.4 Core 58102 0.120 0.006 1.212 0.03 0.619 0.003 0 - - 8.4 K1 58102 0.086 0.004 0.264 0.002 0.081 0.002 0.38 0.88 59.9 15.3 Core 58102 0.179 0.009 1.264 0.001 0.578 0.001 0.40 0.40 55.4 15.3 K1 58102 0.049 0.003 0.217 0.002 -0.040 0.002 0.41 1 63.2 22.2 Core 58102 0.137 0.010 1.209 0.001 0.453 0.001 0.28 0.67 -33.8 22.2 K1 58102 0.014 a 0.002 0.219 0.007 -0.063 0.008 0.32 0.27 -74.3 43.1 Core 58102 0.077 a 0.006 1.227 0.001 0.457 0.002 0.14 0.75 -63.2 43.1 Epoch D K1 58132 0.234 0.012 0.235 0.002 -0.099 0.001 0.70 1 63.2 8.4 Core 58132 0.115 0.006 1.195 0.004 0.395 0.003 0 - - 8.4 K1 58132 0.088 0.004 0.096 0.002 -0.062 0.001 0.37 0.35 -18.6 15.3 Core 58132 0.148 0.008 1.084 0.001 0.416 0.001 0.366 0.55 54.0 15.3 K1 58132 0.057 0.004 0.157 0.002 -0.003 0.002 0.37 0.84 22.4 22.2 Core 58132 0.108 0.008 1.139 0.001 0.465 0.001 0.24 0.69 34.3 22.2 K1 58132 0.016 0.001 0.154 0.003 0.001 0.003 0.27 0.75 65.5 43.1 Core 58132 0.056 0.005 1.171 0.001 0.482 0.001 0.24 0.39 54.2 43.1 Table 2 2about the core dynamics during the flare. It has been shown that K1 is stationaryEpoch A, 8 GHz Epoch A, 15 GHz Lepto-Magnetic Energy vs. Spheroid Radius:LOS =14.5˚, Γ min = 5Lepto-Magnetic Energy vs. Spheroid Radius: LOS =14.5˚, Γ min = 302.0E+48 4.0E+48 6.0E+48 8.0E+48 1.0E+49 1.2E+49 1.4E+49 1.6E+49 1.8E+49 2.0E+49 3.5E+16 3.7E+16 3.9E+16 4.1E+16 4.3E+16 4.5E+16 4.7E+16 4.9E+16 5.1E+16 5.3E+16 5.5E+16 5.7E+16 5.9E+16 6.1E+16 6.3E+16 6.5E+16 6.7E+16 6.9E+16 7.1E+16 Magneto-Leptonic Energy (ergs) Spheroid Radius (cm) Lepto-Magnetic Energy vs. Spheroid Radius: LOS =14.5˚, Γ min = 1 Epoch A Epoch B Epoch C Epoch D Leptonic Plasmoid Solution 8.0E+48 1.3E+49 1.8E+49 2.3E+49 2.8E+49 3.3E+49 3.8E+49 4.3E+49 4.8E+49 4.5E+16 5.0E+16 5.5E+16 6.0E+16 6.5E+16 7.0E+16 7.5E+16 8.0E+16 8.5E+16 Magneto-Leptonic Energy (ergs) Spheroid Radius (cm) Lepto-Magnetic Energy vs. Spheroid Radius: LOS =32.8˚, Γ min = 1 Epoch A Epoch B Epoch C Epoch D Leptonic Plasmoid Solution 2.0E+48 4.0E+48 6.0E+48 8.0E+48 1.0E+49 1.2E+49 3.0E+16 3.6E+16 4.2E+16 4.8E+16 5.4E+16 6.0E+16 6.6E+16 Magneto-Leptonic Energy (ergs) Spheroid Radius (cm) Epoch A Epoch B Epoch C Epoch D Leptonic Plasmoid Solution 0.0E+00 2.0E+48 4.0E+48 6.0E+48 8.0E+48 3.0E+16 3.5E+16 4.0E+16 4.5E+16 5.0E+16 5.5E+16 6.0E+16 6.5E+16 Magneto-Leptonic Energy (ergs) Spheroid Radius (cm) Epoch A Epoch B Epoch C Epoch D Leptonic Plasmoid Solution ).Protonic Kinetic Energy vs. Lepto-Magnetic Energy: LOS =14.5˚, Γ min = 10.0E+00 5.0E+50 1.0E+51 1.5E+51 2.0E+51 2.5E+51 3.0E+51 3.5E+51 3.0E+48 5.0E+48 7.0E+48 9.0E+48 1.1E+49 1.3E+49 1.5E+49 1.7E+49 Kinetic Energy of Protonic Plasmoid (ergs) Lepto-Mangetic energy (ergs) Epoch A Epoch B Epoch C Epoch D Inertially Dominated Solutions Magnetically Dominated Solutions Table 3 : 3Fits to the Core SpectraEpoch Date Peak Frequency Peak Luminosity Spectral Index Number Index (MJD) (GHz) ergs/sec α n ν peak νL ν (ν = ν peak ) A 58063 13.9 8.08 × 10 40 0.83 2.66 B 58077 14.1 1.12 × 10 41 0.75 2.50 C 58102 13.2 1.05 × 10 41 0.97 2.94 D 58132 11.0 8.15 × 10 40 1.03 3.06 Ratio of Mechanical Energy Flux to Total JetPower: LOS =14.5˚, Γ min = 13.0E+42 4.0E+42 5.0E+42 6.0E+42 7.0E+42 8.0E+42 9.0E+42 1.0E+43 1.1E+43 1.2E+43 1.3E+43 3.0E+48 5.0E+48 7.0E+48 9.0E+48 1.1E+49 1.3E+49 1.5E+49 1.7E+49 Jet Power (ergs/sec) Lepto-Magnetic Energy (ergs) Jet Power vs. Lepto-Magnetic Energy : LOS =14.5˚, Γmin = 1 Epoch A Epoch B Epoch C Epoch D 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 8.0E+42 9.0E+42 1.0E+43 1.1E+43 1.2E+43 1.3E+43 Ratio of Mechanical Energy Flux to Jet Power Jet Power (erg/sec) Epoch A Epoch B Epoch C Epoch D Jet Power vs. Line of Sight : Γmin = 11.0E+49 1.5E+49 2.0E+49 2.5E+49 3.0E+49 3.5E+49 1.0E+43 1.1E+43 1.2E+43 1.3E+43 1.4E+43 1.5E+43 1.6E+43 1.7E+43 14 19 24 29 Lepto-Magnetic Energy (ergs) Jet Power (ergs/sec) Line of Sight to Jet Axis (Degrees) Jet Power Lepto-Magnetic Energy of the Discrete Ejection Model 42 42.5 43 43.5 44 44.5 45 1 6 11 16 21 26 31 Log [Jet Power (ergs/sec)] Γ min Jet Power as a Function of Γ min for LOS = 20L og of the Leptonic Jet Power vs Γ min Log of the Protonic Jet Power vs Γ min 3E+16 4E+16 5E+16 6E+16 7E+16 8E+16 0 10 20 30 40 50 60 70 Knot Radius (cm) Time Since Epoch A (Days) Table 4 : 4Long Term Evolution of NuSTAR Flux Date 3-10 keV Flux (10 −12 erg/sec-cm 2 ) 10-30 keV Flux (10 −12 erg/sec-cm 2 ) 90% Confidence Interval 90% Confidence Interval 2012 August 26 0.65-0.77 1.68-2.09 2013 May 09 0.76-0.89 1.36-1.75 2015 April 02 0.74-0.86 1.34-1.67 2015 April 19 0.61-0.74 1.25-1.66 2015 May 28 0.67-0.79 1.23-1.58 2017 October 19 0.90-0.98 1.77-2.00 Table 5 : 5Conformance of Models to ObservationEjection Requirement 1 Requirement 2 Requirement 3 Comments Assessment Type Q Sufficient Magnetic Evolves to to Power K1 Acceleration Minimum E(lm) Discrete Leptonic Marginal C C Requires > 10% Radiative Efficiency at K1 Marginal Discrete Protonic C NC C No Driving Mechanism Excluded Leptonic Knot in Jet C C C Conforms to Observation Protonic Knot in Jet C C C Requires Dissipative Entrainment Between Epochs C and D Conflicts with Constant α Excluded C means conformant and NC means nonconformant Clean I map. Array: BFHKLMNOS Clean I map. Array: BFHKLMNOPS 3C454.3 at 43.115 GHz 2015 Apr 12 Clean I map. Array: BFHKLMNOPS1 0 1 2 3C454.3 at 43.218 GHz 2009 Jul 27 Relative Declination (mas) Right Ascension (mas) Map center: RA: 22 53 57.748, Dec: +16 08 53.561 (2000.0) Map peak: 3.32 Jy/beam Contours %: −0.125 0.125 0.25 0.5 1 2 4 8 16 32 64 Beam FWHM: 0.33 x 0.14 (mas) at −10 o 0 0.5 1 1.5 2 2.5 3 Jy/beam 2 1 0 −1 −2 −2 −1 0 1 2 3C454.3 at 43.115 GHz 2014 Dec 29 Relative Declination (mas) Right Ascension (mas) Map center: RA: 22 53 57.748, Dec: +16 08 53.561 (2000.0) Map peak: 4.94 Jy/beam Contours %: −0.0625 0.0625 0.125 0.25 0.5 1 2 4 8 Contours %: 16 32 64 Beam FWHM: 0.33 x 0.14 (mas) at −10 o 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Jy/beam 2 1 0 −1 −2 −2 −1 0 1 2 Relative Declination (mas) Right Ascension (mas) Map center: RA: 22 53 57.748, Dec: +16 08 53.561 (2000.0) Map peak: 6.98 Jy/beam Contours %: 0.5 1 2 4 8 16 32 64 Beam FWHM: 0.33 x 0.14 (mas) at −10 o 0 1 2 3 4 5 6 Jy/beam The data were obtained with permission of the Boston University Blazar Monitoring program and was downloaded from http://www.bu.edu/blazars/VLBAproject.html This research has made use of data obtained with NuSTAR, a project led by Caltech, funded by NASA and managed by NASA/JPL, and has utilized the NUSTARDAS software package, jointly developed by the ASDC (Italy) and Caltech (USA). 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[ "RESTRICTED MEAN SURVIVAL TIME ESTIMATION USING BAYESIAN NONPARAMETRIC DEPENDENT MIXTURE MODELS A PREPRINT", "RESTRICTED MEAN SURVIVAL TIME ESTIMATION USING BAYESIAN NONPARAMETRIC DEPENDENT MIXTURE MODELS A PREPRINT" ]
[ "Parmar ; Royston ", "Uno " ]
[]
[]
Restricted mean survival time (RMST) is an intuitive summary statistic for time-to-event random variables, and can be used for measuring treatment effects. Compared to hazard ratio, its estimation procedure is robust against the non-proportional hazards assumption. We propose nonparametric Bayeisan (BNP) estimators for RMST using a dependent stick-breaking process prior mixture model that adjusts for mixed-type covariates. The proposed Bayesian estimators can yield both group-level causal estimate and subject-level predictions. Besides, we propose a novel dependent stick-breaking process prior that on average results in narrower credible intervals while maintaining similar coverage probability compared to a dependent probit stick-breaking process prior. We conduct simulation studies to investigate the performance of the proposed BNP RMST estimators compared to existing frequentist approaches and under different Bayesian modeling choices. The proposed framework is applied to estimate the treatment effect of an immuno therapy among KRAS wild-type colorectal cancer patients.
null
[ "https://export.arxiv.org/pdf/2305.14639v1.pdf" ]
258,865,556
2305.14639
4e836bf807996c36924a21c28bcb3e22bad59864
RESTRICTED MEAN SURVIVAL TIME ESTIMATION USING BAYESIAN NONPARAMETRIC DEPENDENT MIXTURE MODELS A PREPRINT Royston and ParmarCopyright Royston and Parmar2021. 2011. 2012. 2014. 2015. 2021. 2013. 2015 Parmar ; Royston Uno RESTRICTED MEAN SURVIVAL TIME ESTIMATION USING BAYESIAN NONPARAMETRIC DEPENDENT MIXTURE MODELS A PREPRINT Royston and Parmar2021. 2011. 2012. 2014. 2015. 2021. 2013. 2015Restricted Mean Survival Time · Bayesian Non-parametric Inference · Causal Inference Section 1 Introduction Restricted mean survival time (RMST) is an intuitive summary statistic for time-to-event random variables, and can be used for measuring treatment effects. Compared to hazard ratio, its estimation procedure is robust against the non-proportional hazards assumption. We propose nonparametric Bayeisan (BNP) estimators for RMST using a dependent stick-breaking process prior mixture model that adjusts for mixed-type covariates. The proposed Bayesian estimators can yield both group-level causal estimate and subject-level predictions. Besides, we propose a novel dependent stick-breaking process prior that on average results in narrower credible intervals while maintaining similar coverage probability compared to a dependent probit stick-breaking process prior. We conduct simulation studies to investigate the performance of the proposed BNP RMST estimators compared to existing frequentist approaches and under different Bayesian modeling choices. The proposed framework is applied to estimate the treatment effect of an immuno therapy among KRAS wild-type colorectal cancer patients. Section 1. Introduction In clinical research, one is often interested in quantifying covariate effect on a time-to-event, often observed with potential censoring. The Cox proportional-hazards model and the resulting hazard ratios (HRs) are the "go-to" approach for such analysis. However, interpreting an HR becomes difficult in presence of non-proportional hazards, which can occur when, for example, time-varying covariate effects exist. In recent years, the application of restricted mean survival time (RMST) in planning and analyzing randomized clinical trials (RCTs) with time-to-event endpoints have drawn the attention of many in the field of medical/clinical statistics (Tian et al., 2018;Freidlin et al., 2021;Royston and Parmar, 2011;Zhang and Schaubel, 2012;Uno et al., 2014Uno et al., , 2015Weir et al., 2021;Royston and Parmar, 2013;Wei et al., 2015;Tian et al., 2020). The popularity of RMST can be attributed to the potential benefits from using it as a measure of treatment effect in survival analysis over other conventional measures such as the HR. The interpretation of RMST is intuitive, clinically relevant, and is model-free in the sense that may not rely on assumptions such proportional hazards. Besides, RMST summarizes survival over a fixed follow-up time period and is of inherent interest in settings where cumulative covariate effects are appealing (Wang and Schaubel, 2018). There is an abundance of frequentist methods for estimating RMST and its associated variances in the literature. In an ideal RCT setting, RMST can be estimated consistently by the area under the Kaplan-Meier (KM) curve up to a specific time τ given that τ is less than or equal to the maximum observed event time and under non-informative censoring (Klein and Moeschberger, 2003;Tian et al., 2014). In addition to the nonparametric RMST estimators introduced by Irwin (1949) and Meier (1975), researchers have proposed various RMST regression methods. As summarized by Wang and Schaubel (2018), these approaches in general, estimate the regression parameters and baseline hazard from a Cox model, calculate the cumulative baseline hazard, which are transformed to obtain the survival function and, and integrate the survival function to obtain the RMST. Another category of RMST modeling approaches resembles the accelerated failure time model by assuming a linear relationship between covariates and E[log(T )] as the response variable Ambrogi et al., 2022). There are only a few works on Bayesian inference in RMST. Poynor and Kottas (2019) studied a related yet different problem of Bayesian inference for the mean residual life (MRL) function, defined as the expected remaining survival time given survival up to time τ . They developed a nonparametric Bayesian (BNP) inference approach for MRL functions by constructing a Dirichlet process (DP) mixture model for the underlying survival distribution. Zhang and Yin (2022) proposed BNP estimators for RMST, for both right and interval censored data, assuming mixture of Dirichlet process priors. In this article, we develop a Bayesian nonparametric dependent mixture (BNPDM) approach for regression modeling in RMST. We utilize these models to make inference about individual-level RMST difference (RMSTD), as well as population-level causal average treatment effect (ATE). We explore different prior choices for a dependent stick-breaking process (DSBP) mixture model, which includes: (i) A finite-dimensional predictor-dependent stick-breaking prior via sequential logistic regressions (Rigon and Durante, 2021;Ishwaran and James, 2001); (ii) the dependent probit-stick breaking process prior (Rodriguez and Dunson, 2011); (iii) our proposed novel shrinkage probit-stick breaking process prior, which is a data-adaptive stick-breaking prior based on a probit regression model. Research interest in RMST inference is often centered around comparing group differential RMST at one or multiple τ 's as a fixed-time analysis. However, only looking at RMST values at a single time point may not accurately reflect the totality of clinical effect and may be misleading about clinical significance of the experimental treatment (Freidlin and Korn, 2019). We provide point-wise Bayesian estimate and inference for the entire RMST curve. We evaluate and compare performance of the proposed BNPDM models with two existing non-Bayesian RMST approaches Ambrogi et al., 2022) through extensive simulation studies. The rest of this paper is organized as follows. In Section 2, we introduce our proposed BNPDM models for drawing RMST inference. In Section 3, we conduct simulation studies to examine the performance of the proposed BNPDM models both under different prior choices and compared to existing frequentist methods Ambrogi et al., 2022). In Section 4, we present an application of our proposed BNPDM models to analyze real data from a phase III colorectal cancer trial. In Section 5, we summarize our findings and give our thoughts on the characteristics of the proposed estimators. Section 2. Methodology Section 2.1 Nonparametric Bayesian Inference of Restricted Mean Survival Time Let T denote a random variable with non-negative support representing time from an appropriate time origin to a clinical event of interest, and assume that T is subject to non-informative right censorship due to either a random drop-out or reaching a maximum follow-up time. RMST is defined as µ(τ ) = E[min(T, τ )] = τ 0 S(t)dt where S(·) denotes the survival function of T . Thus, RMST can be interpreted as the average of all potential event times measured (from time 0) up to τ and mathematically measured as the area under the survival curve up to τ . For predictor w, we model the density of T as predictor-dependent mixtures of a predictor-dependent general kernel density as f (t | G w ) = K w (t | θ) dG w (θ) = L h=1 π h (w)K w (t | θ h ); t ∈ R + (1) where G w (·) = L h=1 v h (w) l<h 1 − v l (w) δ θ h (w) (·); 1 ≤ L ≤ ∞.(2) Here δ θ h (w) (·) is the Dirac measure at θ h (w), π h (w) = v h (w) l<h 1 − v l (w) , and π h (w) ≥ 0 are random functions of w such that L h=1 π h (w) = 1 a.s. for each fixed w ∈ W . v h (w), h ∈ N, are [0, 1]-valued (predictor- dependent) stochastic processes, independent from θ h (w), with index set W . v h (w) can be viewed as a transition kernel such that for all w ∈ W , v h (w, A ) is a probability measure, and for all A ∈ B(W ) (a Borel σ-field on W ), v h (w, A ) is measurable. There is some flexibility for choosing an appropriate transition kernel. For example, Dunson and Park (2008) chose v h (w) = v h K(w, Γ h ), v h ∼ Beta(1, λ), and Γ ∼ H is a location where K(·) : R × R → [0, 1] is a bounded kernel function. We propose the following formulation for the h th stick-breaking probability: v h (w) = g(ψ(w) ′ α h ), α h ∼ Q (3) for a link function g(·) : R → [0, 1], ψ(w) = (ψ 1 (w 1 ), . . . , ψ R (w R )) denoting R functions of covariate w = {w 1 , . . . , w R } and random measure Q is defined on R R . Subsequently, a predictor-dependent stick-breaking process can be defined as π 1 (w) = v 1 (w) π h (w) = 1 − v 1 (w) 1 − v 2 (w) . . . 1 − v h−1 (w) v h (w), h = 2, . . . , L − 1. π L (w) = 1 − L−1 h=1 π h (w) = 1 − v 1 (w) . . . 1 − v L−1 (w) .(4) For a finite L, the construction of the weights in (4) ensures that L h=1 v h = 1. By applying the linear form ψ(w) ′ α h , for certain function ψ(·), one can include mixed-type (both continuous and discrete) predictors {w r } such that v h (w) : R×, · · · , ×{0, 1, . . . , K r } × · · · → [0, 1] where, for example, a discrete values w r may have support on {0, 1, . . . , K r }. The link function g(·) can be chosen as an inverse logit link as for the case of logistic stick-breaking process priors Ren et al. (2011) and logit stick-breaking process priors (Rigon and Durante, 2021), or a probit link as for the case of probit stick-breaking process priors (Chung and Dunson, 2009;Rodriguez and Dunson, 2011;Pati and Dunson, 2014). Similarly, we model the kernel density function with predictor-dependent parameterizations. Suppose that K · | θ) with θ = (η, ω) is a two-parameter kernel density, for example, a lognormal density with parameters η and ω, or a Weibull density with scale η and shape ω, or a (two-parameter) Gamma density with rate η and shape ω. We model the h th cluster kernel density by incorporating predictor dependence on η such that K w (t | θ h ) = K t | η h (w) = ψ(w) T β h , ω h(5) where η h (w) is defined as a linear combination of ψ(w) T and atoms β h . Therefore, the atom sampling process in (2) is given by θ h (w) = ψ(w) T β h , ω h , θ h = (β h , ω h ), θ h ∼ P(6) for some random measure P . Under the above formulations (1)-(3), the survival function can also be represented in a constructive form as: S(t | G w ) = ∞ t f (t | G w )dt = ∞ t L h=1 π h (w)K w t | θ h dt = L h=1 π h (w) ∞ t K w t | θ h dt = L h=1 π h (w)S w t | θ h Similarly for the RMST function, RM ST t | G w = ∞ h=1 π h (w)RM ST w t | θ h(7) We show, in the supplemental materials, that the kernel RMST function is analytically tractable if the kernel density K(t | ·) assumes a Weibull or Gamma form. These convenient structures allow us to formulate individual and grouplevel BNP estimators in closed form expressions and result in improved computational efficiency. Alternatively, one can assume predictor dependence imposes only on either the mixing probabilities or the kernel density which defines a single-atoms predictor-dependent stick-breaking process mixture model pr a single-π linear dependent Dirichlet process (LDDP) mixture model, respectively. Section 2.2 Shrinkage Probit Stick-Breaking Process Prior We propose a novel DSBP prior that is inspired by (Rodriguez and Dunson, 2011;Ren et al., 2011;Rigon and Durante, 2021). Given a covariate matrix of N observations from R covariates W = {w 1,1 , . . . , w R,N } of both continuous and discrete type, suppose we consider a stick-breaking probability assignment mechanism defined by (3) where a multivariate normal prior is assumed for Q, namely, α h = (α h,1 , . . . , α h,R ) ∼ N R µ α , σ 2 α I R where (µ α , σ α ) is specified a priori. For this choice of Q, we have ψ(w i ) ′ α h ∼ N 1 R r=1 ψ r (w ir )µ αr . R r=1 ψ r (w ir ) 2 σ 2 αr , h = 1, . . . , L − 1.(8) We propose a DSBP prior based on the above form in which the stick-breaking probability of the i th observation for the h th cluster is modeled by a probit link as v h (w i ) = g ψ(w i ) ′ α h = Φ ψ(w i ) ′ α h − µ(W , α h ) σ(W , α h )(9) where Φ(·) is the CDF of a standard normal distribution, and the location and scale functions are specified as µ(W , α h ) = 1 N N i=1 ψ(w i ) ′ α h , and σ 2 (W , α h ) = N i=1 ψ(w) ′ i α h − µ(W , α h ) 2 /(N − 1)(10) We refer to the model in (9) based on the link g(s) = Φ (s − µ(W , α h ))/σ(W , α h ) as shrinkage probit model. The linear transformation function ψ(w i ) ′ α h ) is flexible and can accommodate mixed-type predictors. The SPSBP prior is distinct from (dependent) PSBP prior as the latter would assign stick breaking probabilities as v h (w i ) = Φ ψ(w i ) ′ α h The SPSBP prior is built on the basic structure of the (dependent) PSBP prior, yet it adds a feature of borrowing information from clusterings of predictors when assigning stick-breaking probabilities, an idea that was implemented using kernel functions in (Dunson et al., 2007;Dunson and Park, 2008;Ren et al., 2011). Consider a sample predictor matrix W N ×R and its linearly transformed components at the h th cluster: ψ(W ) ′ α h = {ψ(w i ) ′ α h , i = 1, . . . , N } where α h ∼ N R (µ, Σ), h = {1, . . . , L − 1}. Instead of assigning stick-breaking probabilities (π ih ) for ψ(w i ) ′ α h based on its location in the empirical cumulative distribution function of ψ(W ′ )α h with a standard normal probit model, the SPSBP analogously assigns π ih according to a probit model with mean and variance set equal to the sample first and second central moments of ψ(W ) ′ α h , i.e., the shrinkage probit model. The SPSBP prior utilizes the clustering information of ψ(W ) ′ α h around ψ(w i ) ′ α h , thus shrinking the difference between the empirical distribution of ψ(W ) ′ α h and the probit model that assigns π ih (for ψ(w i ) ′ α h ). On the other hand, no such shrinkage effect exists for the (dependent) PSBP prior since π ih are assigned uniformly based on Φ(·). Figure 1 shows density plots for sample observations from ψ(W ) ′ α h = {ψ(w i ) ′ α h , i = 1, . . . , N } (black solid lines), a standard normal (PSBP prior), and a normal distribution with mean and variance set according to (10) (SPSBP prior). Linearly transformed component ψ(w i ) ′ α h is defined in (10). Four covariates matrix (W ) generation scenarios (sample size N = 10, 000) are considered: (a) W ∼ N 4 (0, 0.1 · I 4 ); (b) W ∼ N 4 (0, 10 · I 4 ); (c) W ∼ N 4 (0, I 4 ); (d) same predictors generation scheme (RCT setting) as specified in the simulation studies section: W = {A, X 1 , X 2 , X 3 }, A ∼ Binomial(10, 000, 0.5), X 1 ∼ N (0, 1), X 2 ∼ Binomial(10, 000, 0.7), X 3 ∼ t(d.f. = 5). As shown by Figure 1, this shrinkage effect persists under various scenarios, e.g, when the scale of ψ(W ) ′ α h is either smaller, equal to, or larger than that of a standard normal, or in presence of a shifted location. Measuring the distance between ψ( w i ) ′ α h and ψ(w j ) ′ α h (where i ̸ = j) is realized through linear combinations and the multivariate Gaussian prior assumption on α h , which is motivated to accommodate mixed-type (both discrete and continuous) predictors. In replicated simulation studies, we observe that this shrinkage effect brings the benefit of obtaining 'shrunk credible intervals and smaller RMSEs, when estimating group-level ATE measured by RMST difference (RMSTD) compared to the PSBP prior. Therefore, the SPSBP prior (9) approximates a DP with precision parameter 1 marginally for each fixed w ∈ W given a sufficiently large N . In our application of the SPSBP prior for RMST inference, we assume that α h ∼ N R 0, σ 2 α I R , which leads to µ α1 + . . . + µ α R = 0. In this setting we can also conveniently apply Papageorgiou et al. (2015) for modeling spatially indexed data of mixed type followed a similar structure. Their model allows for observations that correspond to nearby areas to be more likely to have similar values for the component weights than observations from areas that are far apart. In their formulation, α h (w i ) are realizations of marginal Gaussian Markov random fields and the level of borrowing on clustering of covariates is controlled by pre-specified parameters of the random field. In comparison, the level of shrinkage in our proposed method in (10) is mostly data-dependent. v h (w i ) = Φ w ′ i α h σv h (W ,α h ) . The PSBP prior inψ(W ) ′ α h = {ψ(w i ) ′ α h , i = 1, . . . , N } (black solid lines), a standard normal (PSBP prior), and a normal distribution with µ and σ defined by 10 (SPSBP prior); ψ(w i ) ′ α h is defined in 10; four W generation scenarios (N = 10, 000) are considered: (a) W ∼ N 4 (0, 0.1 · I 4 ), (b) W ∼ N 4 (0, 10 · I 4 ), (c) W ∼ N 4 (0, I 4 ), (d) same predictors generation scheme (RCT setting) as specified in the simulation studies section: W = {A, X 1 , X 2 , X 3 }, A ∼ Binomial(10, 000, 0.5), X 1 ∼ N (0, 1), X 2 ∼ Binomial(10, 000, 0.7), X 3 ∼ t(d.f. = 5, ncp = 0). Another DSBP prior that can adjust for mixed-type predictors is the logit stick-breaking process (LSBP) prior introduced by Rigon and Durante (2021). Assuming a LSBP prior, the h th stick-breaking probability is given by v h (w i ) = 1/{1 − exp(ψ(w i ) ′ α h )}, α h ∼ N R (µ, Σ), h ∈ {1, . . . , L − 1} Section 2.3 A Bayesian Non-parametric Inference Framework Section 2.3.1 Subject-Level and Group-Level Estimands and Estimators In Section 2.1, we provided a general definition of RMST as the expected survival time of time-to-event T restricted to certain time point [0, τ ]. Conditional on the predictor matrix W and a fixed time point τ , a conditional RMST function can be defined as RM ST θ (t = τ | W ) = τ 0 S θ (t | W ) dt,(11) which relates to the (marginal) RMST function by RM ST (t = τ ) = E W RM ST (t = τ | W ) = ∞ −∞ τ 0 S θ (t | W )f W (w) dtdW(12) Tian et al. (2014) studied a class of frequentist regression models for estimating the conditional RMST given "baseline" covariate. In this section, we develop nonparametric Bayes conditional RMST inference, Let the observed i th time- to-event be (Y i , δ i ) where Y i = min(T i , C i ), δ i = I(T i ≤ C i ) and C i is the censoring variable including a maximum follow-up time of τ c . Also, W i = (A i , X i ) where X i is a vector of time-fixed covariate and A i is binary treatment group indicator. With a slight abuse of notation, let θ denote the combined parameter vector of both the stick-breaking process prior and the kernel density. The posterior mean conditional RMST function for the i th subject is RM ST (t = τ | w i ) = E θ τ 0 S θ (s | w i ) ds |Y , δ(13) which can be estimated based on Markov chain samples {θ 1 , . . . , θ L } and based on our model constructions in (1-7) as RM ST (t = τ | w i ) = E θ L h=1 π h ψ 1 (w i ) ′ α h τ 0 ∞ t K s | ψ 2 (w i ) ′ β h , ω h dsdt(14) where 1 ≤ L ≤ ∞, θ = {α, β, ω}. Here we distinguish the linear transformation function (ψ 1 (·)) applied in the stick-breaking probabilities from the one (ψ 2 (·)) applied in the kernel density with different subscripts. Conse- quently, a 100(1 − α)% credible interval (CI) for RM ST is [ RM ST (α/2) , RM ST (1−α/2) ] where RM ST (α/2) and RM ST (1−α/2) are calculated as the (α/2)th and (1 − α/2)th quantiles of the posterior distribution of RM ST . Chen and Tsiatis (2001) defined an average (marginal causal) treatment effect (ATE) in terms of the average group differential RMSTD under the counter-factual framework (Morgan and Winship, 2015) by, that is, ∆ = RM ST A1 (t = τ ) − RM ST A0 (t = τ ) = τ 0 E X S θ (t | A = 1, X) − E X S θ (t | A = 0, X) dt(15) We define marginal Bayesian estimators of the causal estimand ∆ defined in (15) using the empirical distribution of X as a nonparametric estimator of f X (x) ∆ = 1 N N i=1 E θ τ 0 S θ (t | A = 1, X i = x i ) − S θ (t | A = 0, X i = x i ) dt(16) Note that (27) does not adjust for potential confounding by censoring given our assumption that censoring is noninformative conditional on the covariate and treatment assignment, and that probability of censoring is positive. Given the constructions described in (1-7), we obtain ∆ = 1 N N i=1 E θ τ 0 S θ (t | A = 1, X i = x i ) − S θ (t | A = 0, X i = x i ) dt = 1 N N i=1 E θ L h=1 π h ψ A1,1 (w i ) ′ α h τ 0 ∞ t K s | ψ A1,2 (w i ) ′ β h , ω h dsdt − π h ψ A0,1 (w i ) ′ α h τ 0 ∞ t K s | ψ A0,2 (w i ) ′ β h , ω h dsdt(17) where 1 ≤ L ≤ ∞, θ = {α, β, ω}, and ψ ·,· (·) ′ is a linear transformation function. Consequently, a 100(1 − α)%-level credible interval (CI) for ∆ is given by [ ∆ (α/2) , ∆ (1−α/2) ] where ∆ (α/2) and ∆ (1−α/2) are calculated as the (α/2)th and (1 − α/2)th quantiles of the posterior distribution of ∆. The closed form equations of (14) assuming a Weibull and gamma kernel density, respectively, are shown in the appendix. Section 2.4 Prior Specifications for Posterior Sampling of Proposed Approaches Given sample observations o i = y i , w i = (a i , x i ), δ i : i = 1, . . . , N , we specify the likelihood and priors of the DSBP prior mixture models with covariates dependence on both the mixing probabilities and kernel densities in a hierarchical representation. For the h th cluster, y i | β i,h , ω h i.i.d. ∼ K y i | exp ψ(w i ) ′ β i,h , ω h , i = 1, . . . , N v h (w i ) = g(ψ(w i ) ′ α h ), π 1 (w i ) = v 1 (w i ) π h (w i ) = 1 − v 1 (w i ) 1 − v 2 (w i ) . . . 1 − v h−1 (w i ) v h (w i ), h ∈ {2, . . . , L − 1} α h = {α h,A , α h,1 , . . . , α h,R } ∼ N R+1 (µ α , Σ α ), h ∈ {1, . . . , L − 1} β h = {β h,A , β h,1 , . . . , β h,R } ∼ N R+1 (µ β , Σ β ), h ∈ {1, . . . , L} ω h ∼ unif (c low , c up ), h ∈ {1, . . . , L}(18) where (µ α , Σ α , µ β , Σ β , c low , c up ) are constants, and g(·) can be a standard probit (regression) model, a shrinkage probit model (9), or an inverse logit (regression) model. We can choose K(y i | ·) to be either a Weibull kernel density (scale= exp ψ(w i ) ′ β ih , shape= ω h ) or a gamma kernel density (rate= exp ψ(w i ) ′ β ih , shape= ω h ) whose support is on R + . We reparameterize ψ(w i ) ′ β ih on an exponential scale in order to select priors with support on R + . The α matrix is of dimension N × (L − 1) since π(w) has L − 1 degrees of freedom (π L (w) = 1 − L−1 h=1 π h (w)) . Let Q(·) denote the log joint density function, and let h(·) and H(·) denote the density and cumulative distribution function of the censoring r.v., respectively. Assuming a non-informative right censoring mechanism, Q θ = (α, β, ω), o = N i=1 f θ (y i ) 1 − H(y i ) δi S θ (y i )h(y i ) (1−δi) ∝ N i=1 L j=1 v j (w i ) l<j 1 − v l (w i ) · K y i | exp ψ(w i ) ′ β i,j , ω j δi · ∞ t K y i | exp ψ(w i ) ′ β i,j , ω j (1−δi) ∝ N i=1 L j=1 g(ψ(w i ) ′ α i,j ) l<j 1 − g(ψ(w i ) ′ α i,j ) · K y i | exp ψ(w i ) ′ β i,j , ω j δi · ∞ t K y i | exp ψ(w i ) ′ β i,j , ω j (1−δi) where α = {α i,j | i = 1, . . . , N, j = 1, . . . , L − 1}; α i,j = {α i,j,1 , . . . , α i,j,R }, β = {β i,j | i = 1, . . . , N, j = 1, . . . , L}; β i,j = {β i,j,1 , . . . , β i,j,R }, ω = {ω 1 , . . . , ω L }, o = {o 1 , . . . , o N }; o i = {y i , δ i , w i }. Section 3. Simulation Studies Section 3.1 Survival and Censoring Time Data Generation Models We consider two data generation settings: (i) randomized controlled trial (RCT) with a balanced design; (ii) observational study where treatment assignments are confounded by observed covariates. We focus on simulating time-to-event data using various data generative models at different sample size levels, e.g., N = {200, 500, 1, 000}. We consider a non-informative right-censoring mechanism with two components: (i) all patients are subject to random censoring/dropout after enrollment; (ii) all patients who don't experience an event are censored after a maximum follow-up time. Hence, we assign a time-to-censoring random variable with an exponential distribution C ∼ exp(rate = λ C = 0.05) and censor all observations beyond a maximum follow-up time of τ c = 10.5 years. We also incorporate a random recruitment mechanism (2-year period) following a uniform distribution of (0, 2). We independently generate three covariates of mixed-type (2 continuous and 1 discrete): X 1 ∼ N (0, 1), X 2 ∼ Binomial(N, p), and X 3 ∼ t(d.f. = 5, ncp = 0). Therefore, the observed data is of structure: o i = (a i , x i,1 , x i,2 , x i,3 , y i , δ i ), for i = 1, . . . , N ; o = {o 1 , . . . , o N } where y i = min(c i , t i , τ c ) and δ i = 1{t i ≤ c i }. For notational convenience, we denote the tuple of covariates and treatment assignment for the i th patient by w i = (a i , x i ). For the RCT setting, we randomly assign patients, with equal probability of 50% (A ∼ Binomial(N, p = 0.5)), to one of the two treatment groups: a test and a control group denoted by A 1 and A 0 , respectively. For the observational study setting, we specify treatment assignment probabilities as a linear combination of covariates values under a logit transformation: logit(p i ) = x T i β i where x i = {x i,1 , x i,2 , x i,3 } and β = {β 1 = 1, β 2 = 1, β 3 = 1} for i = 1, . . . , N such that A i ∼ Binom(N, p = x ′ i β). Under the Weibull survival time generation model, the data generating model for survival time T is a Weibull distribution with baseline shape parameter γ 0 , baseline scale parameter λ 0 and multiplicative covariates effects. For the i th patient, its survival function is specified as: S i (t) = exp{−λ 0 (t γ0 )exp(w ′ i β W eibull )}(19) where w i = {a i , x i,1 , x i,2 , x i,3 }, β W eibull = {β a = −2.5, β x1 = 1.5, β x2 = 2.5, β x3 = 1.5}, λ 0 = 1.5, and γ 0 = 1.5. The true RMST value for the i th patient is evaluated by integrating (19) from t = 0 to τ given the patient's covariates values and coefficients of covariates effect (β W eibull ). Under the lognormal survival time generation model, the survival time T is assigned a lognormal distribution with a fixed standard deviation (σ = 1) and mean parameter that is a linear combination of covariates (µ = w ′ β). For the ith subject, t i i.i.d. ∼ Lognormal log(µ i ) = w ′ i β lognormal , log(σ i ) = 1(20) where w i = {a i , x i,1 , x i,2 , x i,3 }, and β lognormal controls covariates' effect on survival. We assign β 1 lognormal = {β a = 2.5, β x1 = 1.5, β x2 = 2.5, β x3 = 1.5} for a positive treatment effect (default setting), β x1 = −1.5, and β x1 = 0 for a negative and null treatment effect, respectively. The true RMST value for the i th patient is evaluated by τ 0 1 − Φ (t i − µ i ) dt i given the patient's covariates values and coefficients of covariates effect (β lognormal ). Additionally, we consider a two-component Weibull mixture survival time generation model, which allows for more flexible baseline hazard functions. The two-component mixture Weibull distributions are additive on the survival scale, with a mixing proportion parameter ρ, i.e. S(t) = ρS 1 (t) + (1 − ρ)S 2 (t). The survival function for the i th patient is defined as S i (t) = ρ · e −λ 1 0 (t γ 1 0 ) + (1 − ρ) · e −λ 2 0 (t γ 2 0 ) exp{w ′ i β 2W eibull }(21) where ρ denotes the mixture proportion; (λ 1 0 , γ 1 0 ), (λ 2 0 , γ 2 0 ) denote the baseline scale and shape parameters for the two mixture distributions; w i = {a i , x i,1 , . . . , x i,R } denotes observed treatment assignment and covariates for the i th subject; β 2W eibull denotes the coefficients of covariates effect. The true RMST value for the i th patient is evaluated by integrating 21 from t = 0 to τ given (w i , β 2W eibull ) where the coefficients β 2W eibull are set at the same numerical values as β W eibull and β 2 lognormal . Section 3.2 Simulation Scenarios For a comprehensive evaluation of our inferential tools, we consider a total of 6 data generation scenarios: {3 data generation models: Weibull, lognormal, and two-components Weibull mixture}×{2 study settings: randomized controlled trial and observational study}. The lognormal model has two extra covariates settings: negative and null treatment effects in addition to the default positive treatment effect setting shared by the Weibull and two-component Weibull mixture models. We evaluate performance of the proposed BNP estimators of RMST, and compare with two frequentist methods: For each simulation scenario, we randomly generate and fix a sample of covariates (W ) at a given sample size. Then for each simulation replication, we randomly generate a sample of outcomes (Y , δ) given W and make inferences given observed data O = (W , Y , δ). In an oncology study setting, 5 year is usually considered a mile-stone for treatment evaluation. Hence for fixed-time analysis, our simulation studies evaluate RMST estimations at τ = 5 years. Besides, we conduct RMST curve estimations on a grid of time points. For prior specifications, we set µ α = 0, Σ α = 400 · I 4 , µ β = 0, Σ β = 400 · I 4 , c low = {0.01, 0.1}, c up = {10, 15} (18) such that α and β each follows a four-dimensional Gaussian distribution with an independent covariance structure and equal standard deviation of 20 for each predictor. We use a minimum burn-in iteration size of 2, 000 and a minimum posterior sampling iteration size of 1, 000 for all NUTS runs. Initial values are provided by Stan as a default setting. Figure 2 shows estimation results on ATE by BNPDM models assuming a SPSBP prior and a PSBP prior, both with a Weibull kernel density, where survival data are generated by a lognormal model under a RCT setting. Dots and bars denote point estimates and credible intervals, respectively. A bar and a dot are colored red together if the credible interval covers the true/population ATE. Given a moderate sample size (500), biases (averaged over 100 replications) of both models are near zero. However, credible intervals and RMSE estimated under the SPSBP prior are much tighter and smaller compared to those of the PSBP prior. Specifically, the average credible interval (CI) length of the SPSBP prior is less than half of that of the PSBP prior (0.33/0.74 ≈ 0.45). The (default) PSBP prior model give volatile estimates while the modified PSBP prior (SPSBP prior) model stabilize estimates and results in "shrinked" credible intervals in comparison. Figure 3 shows results estimated by the LSBP prior and the LDDP prior models, both with a Weibull kernel density, under the same lognormal-RCT data generation setting. The LSBP and LDDP prior models result in higher biases and RMSEs, yet slightly smaller average CI length compared to the SPSBP prior (model). However, their CPs are very low (52% and 21%) compared to SPSBP prior and PSBP prior (91% and 90%) though the PSBP prior's CP may be inflated due to its extra wide credible intervals. Regarding subject-level RMST inference, all DSBP prior (SPSBP, PSBP, and LSBP) models have superior performance compared to their frequentist counterparts Ambrogi et al., 2022) under the two-components Weibull mixture data generation model as shown in Figure F1-F5 (supplemental materials). Numerical results (for τ = 5; Figure F1-F2) in Table 1 show that, with a sample size of 500, RMSE given by the SPSBP prior is less than half of that by Tian et al. (2014)'s method (0.25 compared to 0.57). For subject-level CP, which is defined by the proportion of time the credible or confidence interval covers the true individual RMSTD value, the LSBP prior yields 0.88. In comparison, CP by Tian et al. (2014) and Ambrogi et al. (2022) are both 0.48. Furthermore, this higher CP is not achieved with increased interval width. On the contrary, the average credible interval length of the LSBP prior is 0.42 compared to 0.75 (average confidence interval length) given by the frequentists' methods. Restricted mean survival time is a function of restricted time, and estimating RMST at only a single time point τ does not tell the whole story of temporal survival relationship. Therefore, we expand the time horizon to evaluate the performance of BNPDM models for RMST inference on a grid of time points (τ s). We calculate point estimates and point-wise credible intervals. On the other hand, given τ and posterior sample θ drawn from P r(θ | Y , δ, W ), the RMST function is just a deterministic function of these quantities. Therefore, we can attain an entire RMST curve estimate with a single NUTS run. We estimated RMST curves under 100 data replications and show their results in Figures F11-F20 (supplemental materials). The corresponding numerical results are summarized in Table 2. Section 3.3 Simulation Results Section 4. Real Data Applications The epidermal growth factor receptor (EGFR) has been proven to be a clinically meaningful target for monoclonal antibodies (mAbs) with efficacy established in treatment of metastatic colorectal cancer (mCRC) (Cunningham et al., 2004;Bokemeyer et al., 2009). Panitumumab (Pmab) is a (fully) human anti-EGFR that was approved as monotherapy for patients with chemotherapy-refractory mCRC (Giusti et al., 2007). A randomized phase III study was designed and conducted to evaluate the efficacy and safety of Pmab plus infusional fluorouracil, leucovorin, and oxaliplatin (FOLFOX4) versus FOLFOX4 alone as an initial treatment for mCRC in patients with previously untreated mCRC according to tumor KRAS status (Douillard et al., 2010). The presence of activating KRAS mutations was identified as a potent predictor of resistance to EGFR-directed antibodies (e.g., cetuximab and panitumumab) (Heinemann et al., 2009). In this real data application example we only focus on RMSTD inference among KRAS wild type (WT) patients for its clinical relevance. This phase III study was designed as an open-label, randomized, phase III trial to compare the treatment effect of adding Pamb to FOLFOX4 in patients with WT KRAS tumors and also in patients with mutant (MT) KRAS tumors (Douillard et al., 2010). A primary analysis of log-rank tests, stratified by random assignment factors, were conducted on the progression-free survival (PFS) and overall survival (OS) endpoints among the WT KRAS and MT KRAS patient stratums (Douillard et al., 2010). A prespecified final analysis, which included OS, was later reported in (Douillard et al., 2014). We conduct a reanalysis of the selected study to estimate group differential treatment effect on the OS endpoint. The maximum observed event time is approximately 3.8 years, and we evaluate RMSTD up to 3.5 years. We include BMI and age as predictive covariates, which are both prognostic in mCRC (Lieu et al., 2014). With all incomplete records removed, the study population has a total number of 652 observations. We apply BNPDM models with SPSBP and LSBP priors both assuming a Weibull and Gamma kernel densities. We obtained both point estimates and credible intervals from month 1 to month 45 with an increasing step size of 1 month. We compare results under both BNPDM models with those given by Tian et al. (2014)'s method in Figure 4 and show a numerical summary in Table T1 (supplemental materials). Section 5. Discussions In this article, we constructed a BNP estimation framework for estimating treatment effect measured by both average group differential RMST and subject-level RMST. Zhang and Yin (2022) proposed a BNP RMST estimator by putting mixture of Dirichlet process priors on the cumulative distribution function of the survival time random variable. Taking a different route, we treat the density of survival function hierarchically as a mixture of kernel densities where the mixtures have a (dependent) stick-breaking process prior. Our modeling approach is analogous to that of a dependent DP mixture (DDPM) model but with a more flexible stick-breaking probability assignment mechanism. A major advantage of our approach is to enable adjustments of mixed-type covariates/predictors. While many works on modeling spatial data focus on dependent structures that adjust for continuous covariates (Reich et al., 2007;Ren et al., 2011;Diana et al., 2020), mixed-type covariates are more often seen in clinical settings. When the data generation model is a function of covariates, both group-level and subject-level RMST inference could be less efficient or inconsistent, without properly adjusting for the observed covariates. We proposed a novel dependent stick-breaking process prior: the SPSBP (shrinkage probit stick-breaking process) prior, which is inspired by Rodriguez and Dunson (2011)'s (dependent) PSBP (probit stick-breaking process) prior. The SPSBP prior results in less variable estimates (e.g., narrower credible intervals) given a small or moderate sample size compared to the PSBP prior. This shrinkage effect is achieved through a more efficient stick-breaking probability assignment process that utilizes the sample first and second (central) moments of the empirical density function of the linearly transformed covariates. However, since the level of shrinking is controlled by sample variance and inversely proportional to the observed sample size, having a huge sample size could results in assigning the entire unit length towards the first few sticks, which results in a less discrete realization of the stick-breaking process. Fortunately, we did not experience such issue when modeling data up to 2, 000 observations and 10 clusters. In fact, we found out through simulation studies that modeling with a smaller cluster size (3 or 5) often result in better performance, compared to using say 10 clusters, under a sample size between 200 and 2, 000. Our simulation studies show decent performance by BNPDM models on group-level RMSTD inference giving ignorable biases and CPs up to 90% for a 95% nominal under Weibull and lognormal data generation models (RCT setting). In comparison, the two frequentist methods Ambrogi et al., 2022) give consistent point estimates and CPs that attain the nominal level of 95% under the same data generation settings (Weibull-RCT or lognormal-RCT). Credible sets of infinite-dimensional Bayesian models are not automatically frequentist confidence sets, and it is not automatically true that they contain the truth with the probability at least the credible level (Szabó et al., 2015). The less efficiency of certain BNP models is well studied in the literature. For example, see (Cox, 1993;Diaconis and Freedman, 1997). However, we found that BNPDM models' subject-level RMST prediction results are better than those of frequentist methods Ambrogi et al., 2022) when the underlying data generation model is a two-component mixture of Weibulls. Besides, we found our BNPDM models have more robust performances against cases where treatment assignment is confounded by observed covariates compared to Tian et al. (2014); Ambrogi et al. (2022)'s methods. Appendix Assuming a Weibull kernel density (scale=ψ(W ) ′ β, shape=ω), (14) has a closed form RM ST (t = τ | w i ) = E θ L h=1 π h ψ(w i ) ′ α h · τ · S i,h (τ ) + ψ(w i ) ′ β h · γ 1 ω h + 1, ψ(w i ) ′ β (−ω h ) h τ ω h where γ(s, x) = x 0 t s−1 e −t dt is the lower incomplete gamma function and S i,h (τ ) = exp − (τ /ψ(w i ) ′ β h ) ω h . Assuming a Gamma kernel density (rate=ψ(W ) ′ β, shape=ω), (14) has a closed form RM ST (t = τ | w i ) = E θ L h=1 π h ψ(w i ) ′ α h · τ + Γ(ω h ) −1 ω h · γ ω h , ψ(w i ) ′ β h τ − ψ(w i ) ′ β h τ ω h e − ψ(wi) ′ β h τ ψ(w i ) ′ β h − τ · γ ω h , ψ(w i ) ′ β h · τ given the property that γ(s + 1, x) = sγ(s, x) − x s e −x where γ(s, x) = x 0 t s−1 e −t dt is the lower incomplete gamma function and Γ(·) denotes the gamma function. Supplemental Materials Performance Evaluations Single-τ Group-Level Results and Subject-Level Results Figure 5 shows results for group-level ATE inference under the Weibull data generation model (RCT setting), all four models (with Weibull kernel density) give unbiased estimates and satisfying CPs (from 88% to 94%). Models with SPSBP prior, LSBP prior, and LDDP prior show similar properties and have narrower credible intervals compared to that of PSBP prior. The LDDP prior (model) shows the best results probably due to the fact that the analysis model (kernel density part) matches the data generation model. Figure 6 shows results under the two-components Weibull mixture data generation model. The LSBP prior shows unstable results with several credible interval lower bounds down to zero. The PSBP prior still has volatile performance with possible signs of divergence. All priors (except for the PSBP prior) show small to moderate biases (from −0.08 to −0.15). Figure 7 is a (points) scatter plot showing individual-level RMST prediction bias for each subject, under the twocomponents Weibull data generation model with various approaches. The two frequentist methods Ambrogi et al., 2022) result in large negatively biased estimates and dense small positively biased estimates on average. The LDDP prior's results are partially positively biased but mostly condensed on the zero bias line. In contrast, the results given by three DSBP prior models have the least biases and RMSEs (See Table 1 in the manuscript). For all three models (SPSBP, PSBP, and LSBP), the bias scatters are narrowly and evenly distributed around the zero bias line. Real Data Analysis Results Figure 9 shows survival probabilities estimated for the two treatment groups of PRIME trial data on the OS endpoint among KRAS WT patients. A Cox proportional hazards model (Therneau and Grambsch, 2000) that adjusts for body mass index (BMI) and age was fitted. Both point estimates (colored solid lines) and 95% confidence intervals (colored shades) are presented, and median survival time is marked by black dotted lines. Figure 9: Estimated survival probabilities A Cox proportional hazards model (Therneau and Grambsch, 2000) that adjusts for bmi and kras is fitted to estimate survival probabilities; both point estimates (colored solid lines) and 95% confidence intervals (colored shades) are presented; median survival time is marked by black dotted lines. Additional Results Linear Dependent Dirichlet Process Mixture Models We can model a density function through a Dirichlet process mixture (DPM) model: f (y) = K θ (y) dG(θ) where G ∼ DP (α, G 0 ). Alternatively, we can incorporate covariates dependence on (certain) parameter(s) of the kernel density, which results in a LDDP mixture model: f (y) = K y | θ(w) dG θ(w) ; G ∼ DP (α, G 0 )(22) where θ(w) = ψ(w) T β, ω . The DP prior is assigned on β and ω: (β, ω) ∼ G where β = {β 1 , . . . , β R } for adjusting R predictors. In the above formulation, covariates dependence are only introduced on the point masses (atoms), which categorize it as a single-π DDP model Quintana et al. (2022). For the h th cluster, y i | β i,h , ω h i.i.d. ∼ K y i | exp ψ(w i ) tr β i,h , ω h , i = 1, . . . , N v h ∼ Beta(1, M ), π 1 = v 1 π h = 1 − v 1 1 − v 2 . . . 1 − v h−1 v h , h ∈ {2, . . . , L − 1} β h = {β h,A , β h,1 , . . . , β h,R } ∼ N R+1 (µ β , Σ β ), h ∈ {1, . . . , L} ω h ∼ unif (c low , c up ), h ∈ {1, . . . , L}(23) where (M, µ β , Σ β , c low , c up ) are constants. For more flexibility, one could model M ∼ Gamma(a α , b α ) given constants (a α , b α ). As previously stated, we can choose K(y i | ·) to be either a Weibull kernel density (scale= exp ψ(w i ) tr β ih , shape= ω h ) or a gamma kernel density (rate= exp ψ(w i ) tr β ih , shape= ω h ) whose support is on R + . Subsequently, its log joint density is Q θ = (β, ω, M ), o = N i=1 f θ (y i ) 1 − H(y i ) δi S θ (y i )h(y i ) (1−δi) ∝ N i=1 f θ (y i ) δi S θ (y i ) (1−δi) ∝ N i=1 L j=1 π j · K y i | exp ψ(w i ) tr β i,j , ω j δi · ∞ t K y i | exp ψ(w i ) tr β i,j , ω j (1−δi) ∝ N i=1 L j=1 v j l<j 1 − v l · K y i | exp ψ(w i ) tr β i,j , ω j δi · ∞ t K y i | exp ψ(w i ) tr β i,j , ω j (1−δi)(24) where v j ∼ Beta(1, M ), β = {β i,j | i = 1, . . . , N, j = 1, . . . , L}; β i,j = {β i,j,1 , . . . , β i,j,R }, ω = {ω 1 , . . . , ω L }, o = {o 1 , . . . , o N }; o i = {y i , δ i , w i }. Single-Atoms Dependent Stick-Breaking Process Prior Mixture Models We can model the RMST function by assigning a single-atoms DSBP prior . This modeling approach assumes predictor dependence only on the mixing probabilities, which resembles a single-atoms dependent Dirichlet process prior mixture model. With a slight abuse of notations, let ω = {ω 1 , ω 2 } denote the parameters of a two-parameter kernel density. For this approach, we model both parameters {ω 1 , ω 2 } assuming a (generalized) stick-breaking process prior, without incorporating covariates-dependence on the kernel densities. For the h th cluster, y i | ω 1,h , ω 2,h i.i.d. ∼ K y i | ω 1,h , ω 2,h , i = 1, . . . , N v h (w i ) = g(ψ(w i ) tr α h ), π 1 (w i ) = v 1 (w i ) π h (w i ) = 1 − v 1 (w i ) 1 − v 2 (w i ) . . . 1 − v h−1 (w i ) v h (w i ), where (µ α , Σ α , c low , c up , d low , d up ) are constants. For more flexibility, one could model ω 1 ∼ Gamma(a 1 , b 1 ); ω 2 ∼ Gamma(a 2 , b 2 ) given constants (a 1 , b 1 , a 2 , b 2 ). The log joint density is Assuming a Weibull kernel density (scale=ψ(W ) ′ β, shape=ω), (14) has a closed form Q θ = (α, ω 1 , ω 2 ), o = N i=1 f θ (y i ) 1 − H(y i ) δi S θ (y i )h(y i ) (1−δi) ∝ N i=1 f θ (y i ) δi S θ (y i ) (1−δi) ∝ N i=1 L j=1 π j (w i ) · K y i | ω 1,j , ω 2,j δi · ∞ t K y i | ω 1,j , ω 2,j (1−δi) ∝ N i=1 L j=1 v j (w i ) l<j 1 − v l (w i ) · K y i | ω 1,j , ω 2,j δi · ∞ t K y i | ω 1,j , ω 2,j (1−δi) ∝ N i=1 L j=1 g(ψ(w i ) tr α i,j ) l<j 1 − g(ψ(w i ) tr α i,j ) · K y i | ω 1,j , ω 2,j δi · ∞ t K y i | ω 1,j , ω 2,j (1−δi)(26)RM ST (t = τ | w i ) = E θ L h=1 π h ψ(w i ) ′ α h · τ · S i,h (τ ) + ψ(w i ) ′ β h · γ 1 ω h + 1, ψ(w i ) ′ β (−ω h ) h τ ω h where γ(s, x) = x 0 t s−1 e −t dt is the lower incomplete gamma function and S i,h (τ ) = exp − (τ /ψ(w i ) ′ β h ) ω h . Assuming a Gamma kernel density (rate=ψ(W ) ′ β, shape=ω), (14) has a closed form RM ST (t = τ | w i ) = E θ L h=1 π h ψ(w i ) ′ α h · τ + Γ(ω h ) −1 ω h · γ ω h , ψ(w i ) ′ β h τ − ψ(w i ) ′ β h τ ω h e − ψ(wi) ′ β h τ ψ(w i ) ′ β h − τ · γ ω h , ψ(w i ) ′ β h · τ given the property that γ(s + 1, x) = sγ(s, x) − x s e −x where γ(s, x) = x 0 t s−1 e −t dt is the lower incomplete gamma function and Γ(·) denotes the gamma function. Define the causal RMSTD estimand and a BNP estimator as follows: ∆(τ ) = RMST A1 (t = τ ) − RMST A0 (t = τ ) = τ 0 E X S θ (t | A = 1, X) − E X S θ (t | A = 0, X) dt ∆ = E X E θ τ 0 S θ (t | A = 1, X i = x i ) − S θ (t | A = 0, X i = x i ) dt = 1 N N i=1 E θ τ 0 S θ (t | A = 1, X i = x i ) − S θ (t | A = 0, X i = x i ) dt = 1 N N i=1 E θ L h=1 π h ψ A1,1 (w i ) ′ α h τ 0 ∞ t K s | ψ A1,2 (w i ) ′ β h , ω h dsdt − π h ψ A0,1 (w i ) ′ α h τ 0 ∞ t K s | ψ A0,2 (w i ) ′ β h , ω h dsdt(27) Figure 1 : 1SPSBP versus PSPB prior: differences between with shrinkage effect and without shrinkage effect in assigning stick-breaking probabilities Density plots for sample observations from (i) Tian et al. (2014)'s direct (RMST) regression method; (ii) Ambrogi et al. (2022)'s pseudo-values method. We make evaluations in terms of bias, root mean square error (RMSE), and coverage probability (CP). We also evaluate Bayesian models' performance under different choices of DSBP priors and kernel densities. Figure 2 : 2Single time point analysis scenario 1: ATE point estimates with 95% credible intervals comparing SPSBP prior with PSBP prior (lognormal data generation model under a RCT setting; τ = 5) (a) SPSBP prior-Weibull kernel: CP = 0.91, Average CI Length=0.33, bias = 0.05, RMSE = 0.1 (b) PSBP prior-Weibull kernel: CP = 0.9, Average CI Length=0.74, bias = 0.01, RMSE = 0.2 CP denotes coverage probability; ACredIntL denotes average credible intervals' length; sample size N = 500; RMSTD evaluated at τ = 5 years; coefficients of predictor effects: (βA = 2.5, βW 1 = 1.5, βW 2 = 2.5, βW 3 = 1.3); black solid lines and blue dashed lines mark the true RMSTD value and average RMSTD point estimates, respectively. Figure 3 : 3Single time point analysis scenario 2: ATE point estimates with 95% credible intervals comparing LSBP prior with LDDP prior (lognormal data generation model under a RCT setting; τ = 5) (a) LSBP prior-Weibull kernel: CP = 0.52, Average CI Length=0.26, bias=0.1, RMSE=0.18 (b) LDDP prior-Weibull kernel: CP = 0.21, Average CI Length=0.24, bias=0.18, RMSE=0.2CP denotes coverage probability; ACredIntL denotes average credible intervals' length; sample size N = 500; RMSTD evaluated at τ = 5 years; coefficients of predictor effects: (β A = 2.5, β W1 = 1.5, β W2 = 2.5, β W3 = 1.3); black solid lines and blue dashed lines mark the true RMSTD value and average RMSTD point estimates, respectively. Figure 4 : 4RMSTD curve estimation: real data analysis-comparing average treatment effect difference between FOLFOX4+Panitumumab versus FOLFOX 4 using PRIME(Douillard et al., 2010) data (unit: months, sample size: Figure 5 : 5Single time point analysis scenario S1: ATE point estimates with 95% credible intervals comparing 4 prior models (Weibull data generation model under an RCT setting; τ = 5) (a) SPSBP prior-Weibull kernel (b) PSBP prior-Weibull kernel (c) LSBP prior-Weibull kernel (d) LDDP prior-Weibull kernel RMSTD evaluated at τ = 5 years; coefficients of predictor effects: (β A = 2.5, β W1 = 1.5, β W2 = 2.5, β W3 = 1.3); black solid lines and blue dashed lines mark the true RMSTD value and average RMSTD point estimates, respectively. Figure 6 : 6Single time point analysis scenario S1: ATE point estimates with 95% credible intervals comparing 4 prior models (2-components Weibull data generation model under an RCT setting; τ = 5)(a) SPSBP prior-Weibull kernel (b) PSBP prior-Weibull kernel (c) LSBP prior-Weibull kernel (d) LDDP prior-Weibull kernel RMSTD evaluated at τ = 5 years; coefficients of predictor effects: (β A = 2.5, β W1 = 1.5, β W2 = 2.5, β W3 = 1.3); black solid lines and blue dashed lines mark the true RMSTD value and average RMSTD point estimates, respectively. Figure 7 :Figure 8 : 78Subject Subject level RMST predictions scenario 1-C: w tr i β 2W eibull (i = {1, . . . , N }) versus bias; data are generated under the two-components Weibull mixture model; τ = 5 h ∈ {2, . . . , L − 1} α h = {α h,A , α h,1 , . . . , α h,R } ∼ N R+1 (µ α , Σ α ), h ∈ {1, . . . , L − 1} ω 1,h ∼ unif (c low , c up ), ω 2,h ∼ unif (d low , d up ), h ∈ {1, . . . , L} where α = {α i,j | i = 1, . . . , N, j = 1, . . . , L − 1}; α i,j = {α i,j,1 , . . . , α i,j,R }, ω = {ω 1 , ω 2 }; ω 1 = {ω 1,1 , . . . , ω 1,L }; ω 2 = {ω 2,1 , . . . , ω 2,L }, o = {o 1 , . . . , o N }; o i = {y i , δ i , w i }. Table 1 : 1SUBJECT-LEVEL RMST PREDICTIONS SCENARIO 1-AMethod Bias RMSE Coverage Probability Average Credible/ Confidence Interval Length SPSBP-Weibull 0.08 0.25 0.556 0.3121 PSBP-Weibull 0.01 0.19 0.886 0.3830 LSBP-Weibull 0.04 0.21 0.880 0.4164 LDDP-Weibull 0.18 0.37 0.402 0.2239 Tian et al. (2014) 0.00 0.57 0.484 0.7477 Ambrogi et al. (2022) 0.00 0.57 0.484 0.7448 Table 2 : 2Subject-level RMSTD Curve Inference Results; Data generation model is a 2-components Weibull mixture modelτ Average Absolute Bias RMSE SPSBP Weibull LSBP Weibull LSBP Gamma Tian et al. (2014) Ambrogi et al. (2022) SPSBP Weibull LSBP Weibull LSBP Gamma Tian et al. (2014) Ambrogi et al. (2022) 1 0.67 0.04 0.10 0.08 0.08 1.2 0.06 0.13 0.10 0.10 2 0.53 0.06 0.11 0.17 0.17 1.04 0.10 0.17 0.21 0.21 3 0.42 0.08 0.14 0.28 0.28 0.86 0.14 0.24 0.34 0.34 4 0.34 0.10 0.16 0.37 0.37 0.70 0.18 0.30 0.47 0.47 5 0.28 0.12 0.18 0.44 0.44 0.55 0.23 0.36 0.57 0.57 6 0.24 0.14 0.20 0.50 0.50 0.42 0.27 0.41 0.67 0.67 7 0.21 0.15 0.21 0.55 0.55 0.33 0.31 0.46 0.75 0.75 8 0.19 0.16 0.23 0.58 0.59 0.29 0.36 0.51 0.83 0.83 9 0.21 0.18 0.24 0.61 0.62 0.32 0.40 0.55 0.90 0.90 10 0.23 0.19 0.25 0.64 0.65 0.39 0.44 0.59 0.96 0.96 Table 3 : 3REAL DATA RMSTD CURVE ANALYSISMethod τ (years) 0.5 1 1.5 2 2.5 3 3.5 LSBP Weibull 95% CI Lower 0.00 0.00 0.00 0.00 0.00 0.00 -0.01 Point Estimate 0.01 0.02 0.04 0.08 0.11 0.14 0.15 95% CI Upper 0.01 0.04 0.09 0.16 0.22 0.27 0.31 SPSBP Weibull 95% CI Lower 0.00 -0.01 -0.02 -0.03 -0.04 -0.06 -0.07 Point Estimate 0.00 0.02 0.04 0.06 0.09 0.11 0.12 95% CI Upper 0.01 0.04 0.10 0.16 0.22 0.26 0.30 PSBP Weibull 95% CI Lower 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Point Estimate 0.01 0.03 0.07 0.13 0.19 0.24 0.29 95% CI Upper 0.02 0.07 0.15 0.26 0.38 0.49 0.59 LSBP Gamma 95% CI Lower 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Point Estimate 0.00 0.02 0.04 0.07 0.11 0.14 0.17 95% CI Upper 0.01 0.04 0.09 0.15 0.22 0.28 0.34 SPSBP Gamma 95% CI Lower 0.00 -0.01 -0.01 -0.02 -0.02 -0.03 -0.03 Point Estimate 0.00 0.02 0.04 0.07 0.10 0.13 0.15 95% CI Upper 0.01 0.05 0.10 0.16 0.23 0.30 0.37 PSBP Gamma 95% CI Lower -0.01 -0.03 -0.06 -0.10 -0.15 -0.20 -0.25 Point Estimate 0.00 0.01 0.03 0.05 0.07 0.09 0.11 95% CI Upper 0.01 0.05 0.10 0.17 0.24 0.31 0.38 Tian et al. 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[ "Generalized Planning as Heuristic Search: A new planning search-space that leverages pointers over objects", "Generalized Planning as Heuristic Search: A new planning search-space that leverages pointers over objects" ]
[ "Javier Segovia-Aguas \nArtificial Intelligence\nUniversitat Pompeu Fabra\nUniversitat Politècnica de València\nUniversitat Pompeu Fabra\n\n", "Sergio Jiménez \nArtificial Intelligence\nUniversitat Pompeu Fabra\nUniversitat Politècnica de València\nUniversitat Pompeu Fabra\n\n", "Anders Jonsson \nArtificial Intelligence\nUniversitat Pompeu Fabra\nUniversitat Politècnica de València\nUniversitat Pompeu Fabra\n\n" ]
[ "Artificial Intelligence\nUniversitat Pompeu Fabra\nUniversitat Politècnica de València\nUniversitat Pompeu Fabra\n", "Artificial Intelligence\nUniversitat Pompeu Fabra\nUniversitat Politècnica de València\nUniversitat Pompeu Fabra\n", "Artificial Intelligence\nUniversitat Pompeu Fabra\nUniversitat Politècnica de València\nUniversitat Pompeu Fabra\n" ]
[]
Planning as heuristic search is one of the most successful approaches to classical planning but unfortunately, it does not extend trivially to Generalized Planning (GP). GP aims to compute algorithmic solutions that are valid for a set of classical planning instances from a given domain, even if these instances differ in the number of objects, the number of state variables, their domain size, or their initial and goal configuration. The generalization requirements of GP make it impractical to perform the state-space search that is usually implemented by heuristic planners. This paper adapts the planning as heuristic search paradigm to the generalization requirements of GP, and presents the first native heuristic search approach to GP. First, the paper introduces a new pointer-based solution space for GP that is independent of the number of classical planning instances in a GP problem and the size of those instances (i.e. the number of objects, state variables and their domain sizes). Second, the paper defines a set of evaluation and heuristic functions for guiding a combinatorial search in our new GP solution space. The computation of these evaluation and heuristic functions does not require grounding states or actions in advance. Therefore our GP as heuristic search approach can handle large sets of state variables with large numerical domains, e.g. integers. Lastly, the paper defines an upgraded version of our novel algorithm for GP called Best-First Generalized Planning (BFGP), that implements a best-first search in our pointer-based solution space, and that is guided by our evaluation/heuristic functions for GP.Author / 00 (2023) 1-33 2 to have polynomial algorithmic solutions [9]. In other words, one can compute a single compact general solution that exploits some common structure of the different classical planning instances in a given domain. Generalized plans are then not sequences of actions, but algorithmic solutions that supplement planning actions with control-flow constructs. For example, a generalized plan that solves any classical planning instance from the blocksworld domain [10] can be compactly specified as follows: put all the blocks on the table and then, in a proper order, move each block to its goal placement. This generalized plan is able to solve any blocksworld instance, no matter the actual number, or identity of the blocks, and no matter the initial and goal configuration of the blocks. Note however that the knowledge represented in a given input set of classical planning instances may not be enough to specify an algorithmic solution that solves them all. For example, instances of the classical planning blocksworld domain do not include representation features for specifying whether all blocks are on the table, or for specifying the proper order for moving the blocks to their goal placements. A big challenge in GP is then to automatically discover the representation features that are key for computing a compact and general solution for a given set of planning instances. With this regard, researchers have proposed different languages for compactly represent GP solutions, and associated algorithms for computing a GP solution in a given language.Automated planning has not achieved the level of integration with common programming languages, like C, Java, or Python, that is achieved by other forms of problem solving, such as constraint satisfaction or operational research[11,12,13]. An important reason is the low-level representations traditionally used in planning for specifying problems and solutions[14,15,16,17,18]. Given the algorithmic kind of GP solutions, GP is a promising research direction to bridge the current gap between automated planning and programming. However, most of the work on GP still inherits the Strips representation, in which states are represented specifying the properties and relations of a set of objects, and where actions represent object manipulations. In this work we provide a pointer-based representation for GP problems and solutions, that is closer to common programming languages, and that applies also to the object-centered problems that traditionally are addressed in the automated planning community. In addition we show that our pointer-based representation allows us to adapt the planning as heuristic search paradigm to GP: given a GP problem that comprises a finite set of classical planning instances from a given domain, our GP as heuristic search approach implements a combinatorial search to find a program that solves the full set of input instances. With our new pointer-based representation we are able to solve challenging programming tasks that were out of reach for previous top-down GP solvers.Heuristic search is one of the most successful approaches to classical planning[19,20]. The winners of the International Planning Competition (IPC) are often heuristic planners[21], and the workshop on Heuristics and Search for Domain-Independent Planning (HSDIP) is one of the discussion forums with the longest tradition at the International Conference on Automated Planning and Scheduling (ICAPS), the major international conference for research on automated planning. Briefly, the planning as heuristic search approach addresses the computation of sequential plans as a combinatorial search in the space of the states reachable from a single given initial state. This combinatorial search is usually implemented as a forward search, guided by heuristics that are automatically extracted from the declarative representation of the planning problem. There is a wide range of different heuristics for classical planning, but most of them are based on the notion of relaxed plan[22]. The relaxed plan is a solution to a relaxation of the classical planning problem, which is computed assuming that goals (and action preconditions) are independent. The cost of the relaxed plan is an informative estimate of the actual cost-to-go for many classical planning problems, and its computation is much cheaper than the computation of the actual solution to the planning problem.In the last two decades a wide landscape of effective search algorithms, and heuristic functions, have been developed for classical planning[23,24,25,26,27,28]. Unfortunately, search algorithms and heuristic functions from classical planning cannot be directly extended to GP. The computation of relaxed plans, as it is implemented by offthe-shelf heuristic planners, requires a pre-processing step for grounding states and actions[29]. On the other hand, GP solutions must be able to generalize to (possibly infinite) sets of classical planning instances, with different sets of state variables (i.e. state variables with different domain sizes and/or different number of state variables) as well as with different initial states and goals. These particular generalization requirements of GP make it impractical to ground states and actions and hence, to apply the state-space search and the cost-to-go estimates of heuristic planners.With respect to previous work on GP, our heuristic search approach to GP introduces the following contributions:1. A pointer-based representation for GP problems and solutions. Our representation formalism is closer to common programming languages, and it also applies to object-centered representations (like Strips) that are tradi-Related WorkHere we first review previous work on GP according to the following three dimensions: problem representation, solution representation, and computational approach. Then, we connect the research work on GP with other relevant areas in AI, such as program synthesis, deep learning, and (deep) reinforcement learning. Last, we list the features that distinguish our GP as heuristic search approach from the reviewed related work.Regarding problem representation, there are two different approaches for the specification of the set of classical planning instances that are comprised in a GP problem. The explicit approach, that enumerates every classical planning instance in a GP problem[34], and the implicit approach, that defines the constraints that hold for the set of classical planning instances of a GP problem. The implicit approach is of interest because it allows to compactly specify infinite sets of classical planning instances (e.g. the infinite set of the classical planning instances that belong to the 3Author / 00 (2023) 1-33 4 blocksworld domain) [35,36,6]. In addition to the set of classical planning instances, extra background knowledge can also be specified in a GP problem with the aim of reducing the space of hypothetical solutions. For instance, plan traces/demonstrations on how to solve some of the input instances[37,38,39], the full state space [8], the particular subset of state features that can be used for computing a generalized plan[40,41], negative examples that specify undesired behavior for the targeted GP solutions[42,8], or state invariants that any state in a given domain must satisfy[43].With respect to solution representation, different formalisms appeared in the planning literature to represent solutions that are valid for a set of classical planning instances; sequential plans are used in conformant planning [44], conditional tree-like plans are used in contingent planning[20], or policies are used in FOND planning, as well as in MDP/POMDP planning[45]. In all these planning settings, a set of different classical planning instances, with different initial states, can be implicitly represented as a disjunctive formulae over the state variables. Different goals can also be considered by coding them as part of the state representation, e.g. using static state variables. Since the early days of AI planning, hierarchies, LTL formulae, and policies, are also used to specify sketches of general solutions[46]. In the planning literature these solution sketches are often called domain-specific control knowledge, since they are traditionally used to control the planning process, and they apply to the entire set of classical planning instances that belong to a given domain[47,48,37,38]. Last but not least, algorithmic solutions, represented either as lifted policies, finite automata, or as programs with control-flow constructs for branching and looping, are used to represent GP solutions [35,49,40,50,51,52,1,34,7].Regarding to the computation of generalized plans, there are two main approaches for addressing GP problems. The top-down/offline approach considers the entire set of classical planning instances in a given GP problem as a single batch, and computes a solution plan that is valid for the full batch at once. A common approach for the offline computation of generalized plans is compiling the GP problem into another form of problem solving, and using an off-the-shelf solver to work out the compiled problem. With this regard, GP problems have been compiled into classical planning problems[52,34], conformant planning problems[40], LTL synthesis problems[53], FOND planning problems[54,6] or MAXSAT problems [8]. The compilation approach is appealing because it allows to leverage the latest advances of other well-founded scientific communities, with robust and scalable solvers. In addition, the computational complexity of some of these tasks is theoretically characterized with respect to structural features of the input problems, which may provide insights on the difficulty of the addressed GP problem. A weak point of the compilation approach is however the size of the compiled problems to be solved; solvers are usually sensitive to the size of the input problems. On the other hand, the bottom-up/online approach incrementally processes the set of classical planning instances in a GP problem [1, 3]. Given a classical planning instance, a solution to that instance is computed and then, the solution is merged with solutions computed for the previous instances. The online approach is then appealing for handling GP problems that comprise large sets of classical planning instances. The main drawback of online approaches is dealing with the overfitting produced by the individual processing of the different classical planning instances in a GP problem.As noted by previous work on GP, the aims of GP are connected to program synthesis[34,6,53,33]. Program synthesis is a task traditionally studied by the computer-aided verification community[55], and that aims the computation of programs such that they satisfy a given correctness specification[56,57,58]. Program synthesis follows the functional programming paradigm. This means that a program is a function composition, where each function in the composition is a mapping of its input parameters into a single output, and where loops are implemented using recursion. Work on program synthesis is classified according to how the correctness specification of a program is formulated. The programming by example (PbE) paradigm specifies the desired program behaviour with a finite and non-empty set of ground input/output examples. This approach is related to the explicit representation of GP problems; a ground input/output example can be understood as the initial/goal state pair that represents a classical planning instance, and the instruction set of the functional programming language can be understood as the available actions for transforming an initial state into a goal state. Program synthesis also allows the implicit representation of the input correctness specifications, e.g. using fist-order formulae specified in SMTLIB, the formal language for SAT-Modulo Theories (SMT)[59]. The mainstream approach for program synthesis is to specify a formal grammar that allows to incrementally enumerate the space of possible programs, and to leverage the satisfiability machinery of SMT solvers to validate whether a candidate program is actually a solution. With this regard, work on theorem proving is also related to program synthesis, specially since SMT solvers allow the representation and satisfaction of first-order logic formulae[60]. Lastly, another popular trend in program synthesis is Programming by sketches that addresses program 4
10.48550/arxiv.2301.11087
[ "https://export.arxiv.org/pdf/2301.11087v1.pdf" ]
256,274,690
2301.11087
f4722e9dabc8904dcac876001f7feb9df2b7f0e2
Generalized Planning as Heuristic Search: A new planning search-space that leverages pointers over objects Javier Segovia-Aguas Artificial Intelligence Universitat Pompeu Fabra Universitat Politècnica de València Universitat Pompeu Fabra Sergio Jiménez Artificial Intelligence Universitat Pompeu Fabra Universitat Politècnica de València Universitat Pompeu Fabra Anders Jonsson Artificial Intelligence Universitat Pompeu Fabra Universitat Politècnica de València Universitat Pompeu Fabra Generalized Planning as Heuristic Search: A new planning search-space that leverages pointers over objects 00 (2023) 1-33Generalized planningclassical planningheuristic searchplanning and learningdomain-specific control knowledgeprogram synthesisprogramming by example Planning as heuristic search is one of the most successful approaches to classical planning but unfortunately, it does not extend trivially to Generalized Planning (GP). GP aims to compute algorithmic solutions that are valid for a set of classical planning instances from a given domain, even if these instances differ in the number of objects, the number of state variables, their domain size, or their initial and goal configuration. The generalization requirements of GP make it impractical to perform the state-space search that is usually implemented by heuristic planners. This paper adapts the planning as heuristic search paradigm to the generalization requirements of GP, and presents the first native heuristic search approach to GP. First, the paper introduces a new pointer-based solution space for GP that is independent of the number of classical planning instances in a GP problem and the size of those instances (i.e. the number of objects, state variables and their domain sizes). Second, the paper defines a set of evaluation and heuristic functions for guiding a combinatorial search in our new GP solution space. The computation of these evaluation and heuristic functions does not require grounding states or actions in advance. Therefore our GP as heuristic search approach can handle large sets of state variables with large numerical domains, e.g. integers. Lastly, the paper defines an upgraded version of our novel algorithm for GP called Best-First Generalized Planning (BFGP), that implements a best-first search in our pointer-based solution space, and that is guided by our evaluation/heuristic functions for GP.Author / 00 (2023) 1-33 2 to have polynomial algorithmic solutions [9]. In other words, one can compute a single compact general solution that exploits some common structure of the different classical planning instances in a given domain. Generalized plans are then not sequences of actions, but algorithmic solutions that supplement planning actions with control-flow constructs. For example, a generalized plan that solves any classical planning instance from the blocksworld domain [10] can be compactly specified as follows: put all the blocks on the table and then, in a proper order, move each block to its goal placement. This generalized plan is able to solve any blocksworld instance, no matter the actual number, or identity of the blocks, and no matter the initial and goal configuration of the blocks. Note however that the knowledge represented in a given input set of classical planning instances may not be enough to specify an algorithmic solution that solves them all. For example, instances of the classical planning blocksworld domain do not include representation features for specifying whether all blocks are on the table, or for specifying the proper order for moving the blocks to their goal placements. A big challenge in GP is then to automatically discover the representation features that are key for computing a compact and general solution for a given set of planning instances. With this regard, researchers have proposed different languages for compactly represent GP solutions, and associated algorithms for computing a GP solution in a given language.Automated planning has not achieved the level of integration with common programming languages, like C, Java, or Python, that is achieved by other forms of problem solving, such as constraint satisfaction or operational research[11,12,13]. An important reason is the low-level representations traditionally used in planning for specifying problems and solutions[14,15,16,17,18]. Given the algorithmic kind of GP solutions, GP is a promising research direction to bridge the current gap between automated planning and programming. However, most of the work on GP still inherits the Strips representation, in which states are represented specifying the properties and relations of a set of objects, and where actions represent object manipulations. In this work we provide a pointer-based representation for GP problems and solutions, that is closer to common programming languages, and that applies also to the object-centered problems that traditionally are addressed in the automated planning community. In addition we show that our pointer-based representation allows us to adapt the planning as heuristic search paradigm to GP: given a GP problem that comprises a finite set of classical planning instances from a given domain, our GP as heuristic search approach implements a combinatorial search to find a program that solves the full set of input instances. With our new pointer-based representation we are able to solve challenging programming tasks that were out of reach for previous top-down GP solvers.Heuristic search is one of the most successful approaches to classical planning[19,20]. The winners of the International Planning Competition (IPC) are often heuristic planners[21], and the workshop on Heuristics and Search for Domain-Independent Planning (HSDIP) is one of the discussion forums with the longest tradition at the International Conference on Automated Planning and Scheduling (ICAPS), the major international conference for research on automated planning. Briefly, the planning as heuristic search approach addresses the computation of sequential plans as a combinatorial search in the space of the states reachable from a single given initial state. This combinatorial search is usually implemented as a forward search, guided by heuristics that are automatically extracted from the declarative representation of the planning problem. There is a wide range of different heuristics for classical planning, but most of them are based on the notion of relaxed plan[22]. The relaxed plan is a solution to a relaxation of the classical planning problem, which is computed assuming that goals (and action preconditions) are independent. The cost of the relaxed plan is an informative estimate of the actual cost-to-go for many classical planning problems, and its computation is much cheaper than the computation of the actual solution to the planning problem.In the last two decades a wide landscape of effective search algorithms, and heuristic functions, have been developed for classical planning[23,24,25,26,27,28]. Unfortunately, search algorithms and heuristic functions from classical planning cannot be directly extended to GP. The computation of relaxed plans, as it is implemented by offthe-shelf heuristic planners, requires a pre-processing step for grounding states and actions[29]. On the other hand, GP solutions must be able to generalize to (possibly infinite) sets of classical planning instances, with different sets of state variables (i.e. state variables with different domain sizes and/or different number of state variables) as well as with different initial states and goals. These particular generalization requirements of GP make it impractical to ground states and actions and hence, to apply the state-space search and the cost-to-go estimates of heuristic planners.With respect to previous work on GP, our heuristic search approach to GP introduces the following contributions:1. A pointer-based representation for GP problems and solutions. Our representation formalism is closer to common programming languages, and it also applies to object-centered representations (like Strips) that are tradi-Related WorkHere we first review previous work on GP according to the following three dimensions: problem representation, solution representation, and computational approach. Then, we connect the research work on GP with other relevant areas in AI, such as program synthesis, deep learning, and (deep) reinforcement learning. Last, we list the features that distinguish our GP as heuristic search approach from the reviewed related work.Regarding problem representation, there are two different approaches for the specification of the set of classical planning instances that are comprised in a GP problem. The explicit approach, that enumerates every classical planning instance in a GP problem[34], and the implicit approach, that defines the constraints that hold for the set of classical planning instances of a GP problem. The implicit approach is of interest because it allows to compactly specify infinite sets of classical planning instances (e.g. the infinite set of the classical planning instances that belong to the 3Author / 00 (2023) 1-33 4 blocksworld domain) [35,36,6]. In addition to the set of classical planning instances, extra background knowledge can also be specified in a GP problem with the aim of reducing the space of hypothetical solutions. For instance, plan traces/demonstrations on how to solve some of the input instances[37,38,39], the full state space [8], the particular subset of state features that can be used for computing a generalized plan[40,41], negative examples that specify undesired behavior for the targeted GP solutions[42,8], or state invariants that any state in a given domain must satisfy[43].With respect to solution representation, different formalisms appeared in the planning literature to represent solutions that are valid for a set of classical planning instances; sequential plans are used in conformant planning [44], conditional tree-like plans are used in contingent planning[20], or policies are used in FOND planning, as well as in MDP/POMDP planning[45]. In all these planning settings, a set of different classical planning instances, with different initial states, can be implicitly represented as a disjunctive formulae over the state variables. Different goals can also be considered by coding them as part of the state representation, e.g. using static state variables. Since the early days of AI planning, hierarchies, LTL formulae, and policies, are also used to specify sketches of general solutions[46]. In the planning literature these solution sketches are often called domain-specific control knowledge, since they are traditionally used to control the planning process, and they apply to the entire set of classical planning instances that belong to a given domain[47,48,37,38]. Last but not least, algorithmic solutions, represented either as lifted policies, finite automata, or as programs with control-flow constructs for branching and looping, are used to represent GP solutions [35,49,40,50,51,52,1,34,7].Regarding to the computation of generalized plans, there are two main approaches for addressing GP problems. The top-down/offline approach considers the entire set of classical planning instances in a given GP problem as a single batch, and computes a solution plan that is valid for the full batch at once. A common approach for the offline computation of generalized plans is compiling the GP problem into another form of problem solving, and using an off-the-shelf solver to work out the compiled problem. With this regard, GP problems have been compiled into classical planning problems[52,34], conformant planning problems[40], LTL synthesis problems[53], FOND planning problems[54,6] or MAXSAT problems [8]. The compilation approach is appealing because it allows to leverage the latest advances of other well-founded scientific communities, with robust and scalable solvers. In addition, the computational complexity of some of these tasks is theoretically characterized with respect to structural features of the input problems, which may provide insights on the difficulty of the addressed GP problem. A weak point of the compilation approach is however the size of the compiled problems to be solved; solvers are usually sensitive to the size of the input problems. On the other hand, the bottom-up/online approach incrementally processes the set of classical planning instances in a GP problem [1, 3]. Given a classical planning instance, a solution to that instance is computed and then, the solution is merged with solutions computed for the previous instances. The online approach is then appealing for handling GP problems that comprise large sets of classical planning instances. The main drawback of online approaches is dealing with the overfitting produced by the individual processing of the different classical planning instances in a GP problem.As noted by previous work on GP, the aims of GP are connected to program synthesis[34,6,53,33]. Program synthesis is a task traditionally studied by the computer-aided verification community[55], and that aims the computation of programs such that they satisfy a given correctness specification[56,57,58]. Program synthesis follows the functional programming paradigm. This means that a program is a function composition, where each function in the composition is a mapping of its input parameters into a single output, and where loops are implemented using recursion. Work on program synthesis is classified according to how the correctness specification of a program is formulated. The programming by example (PbE) paradigm specifies the desired program behaviour with a finite and non-empty set of ground input/output examples. This approach is related to the explicit representation of GP problems; a ground input/output example can be understood as the initial/goal state pair that represents a classical planning instance, and the instruction set of the functional programming language can be understood as the available actions for transforming an initial state into a goal state. Program synthesis also allows the implicit representation of the input correctness specifications, e.g. using fist-order formulae specified in SMTLIB, the formal language for SAT-Modulo Theories (SMT)[59]. The mainstream approach for program synthesis is to specify a formal grammar that allows to incrementally enumerate the space of possible programs, and to leverage the satisfiability machinery of SMT solvers to validate whether a candidate program is actually a solution. With this regard, work on theorem proving is also related to program synthesis, specially since SMT solvers allow the representation and satisfaction of first-order logic formulae[60]. Lastly, another popular trend in program synthesis is Programming by sketches that addresses program 4 Introduction Generalized planning (GP) addresses the representation and computation of solutions that are valid for a set of classical planning instances from a given domain [1,2,3,4,5,6,7,8]. In the worst case, each classical planning instance may require a completely different solution. In practice, however, many classical planning domains are known tionally used in automated planning. 2. A tractable solution-space for GP. We leverage the computational models of the Random-Access Machine [30] and the Intel x86 FLAGS register [31] to define an innovative pointer-based search space for GP. Interestingly our new search space for GP is independent of the number of input planning instances in a GP problem, and the size of these instances (i.e. the number of objects, state variables, and their domain sizes). 3. Grounding-free evaluation/heuristic functions for GP. We define several evaluation and heuristic functions to guide a combinatorial search in our solution space for GP. Evaluating these functions does not require to ground states/actions in advance, so they apply to GP problems where state variables have large domains (e.g. integers). 4. A heuristic search algorithm for GP. We present the BFGP algorithm for GP that implements a best-first search in our GP solution-space, and that is guided by our evaluation and heuristic functions. 5. A translator from the Strips fragment of PDDL to our pointer-based representation for GP. We automate the representation change from PDDL to pointer-based, and show several solutions to planning domains from the International Planning Competition (IPC) [32] which are validated on large random instances. A preliminary description of our GP as heuristic search approach previously appeared at an ICAPS conference paper [33]. In this work we review and extend the seminal ideas presented in the conference paper, and provide a more exhaustive evaluation of our GP as heuristic search approach. Compared to the conference paper, the present paper includes the following novel material: • We formalize the notion of pointer over the objects of a planning problem, and introduce a pointer-based formalization for planning action schemes, classical planning problems and solutions. We show that our pointerbased formalization directly applies to object-centered planning problems that are traditionally addressed in automated planning. • We introduce the notion of partially specified planning program, that refers to the sketch of an algorithmic planning solution, and that allows to explain better our search algorithm and heuristics functions for GP. We also implemented new evaluation functions for guiding our GP as heuristic search approach. • We provide theoretical results of our heuristic search algorithm for GP, that include termination, soundness, completeness, and complexity proofs. We also extend the empirical evaluation, including more results at a wider landscape of planning domains, to characterize better the performance of our GP as heuristic search approach. The paper is structured as follows: Section 2 presents the planning models we rely on in this work (namely the classical planning model and the GP planning model) and also presents planning programs and the Random Access Machine, the formalisms we leverage for the representation of our algorithmic planning solutions. Section 3 shows how to extend the classical planning model with a set of pointers over objects, and the corresponding primitive operations for manipulating these pointers. This extension allows us to define, in an agnostic manner, a set of features and a set of actions for computing planning programs that can solve any instance from a given planning domain. Section 4 describes our GP as heuristic search approach; the section provides details on our solution space, evaluation functions, and heuristic search algorithm for GP. Section 5 presents the empirical evaluation of our approach and its comparison with the classical planning compilation for GP, that serves as a baseline. Finally, Section 6 wraps-up our work and discusses open issues and future work. synthesis in the particular setting where a partially specified solution is provided as input [61]. Besides computational methods for formal verification and logic satisfaction, optimization methods (that are predominant in Machine Learning [62]) have also been applied to the computation of planning solutions that generalize. For instance, off-the-shelf Deep Learning (DL) tools, have been successfully applied to the computation of generalized policies for classical and probabilistic planning domains [63,64,65]. Generalized policies are a powerful solution representation formalism whose applicability goes beyond classical planning. Generalized policies can represent planning solutions that can deal with non-deterministic actions [66], and whose aim is not to satisfy a given goal condition but to optimize a given utility function [67]. The aims of GP are also related to Reinforcement Learning (RL) [68]; while the cited DL approaches can be viewed as off-line optimization approaches to GP, the RL paradigm can be viewed as an online optimization approach to GP. RL methods incrementally compute policies, by iteratively addressing a set of sequential decision-making episodes. In RL learning experience is however not given beforehand (learning experience is collected by the autonomous exploration of the state space), and RL assumes that there is an explicit notion of reward function (which helps to guide exploration towards the most promising portions of the state-space). Note that DL and DRL approaches learn policies, without requiring a symbolic representation of the state and the action space. This means that it is possible to compute policies (deep policies) that generalize from raw sensor data (e.g. sequences of images) [69,70]. The main disadvantage of computing solutions represented as deep policies is that they are black-box models that lack transparency and explanation capacity, which makes it difficult to interpret the produced solutions. This is a strong requirement in application areas that require humans in the loop, such as health, law, or defense [71]. With regard to the reviewed related work, our GP as heuristic planning approach is framed as follows: • Numeric state variables. Previous work on generalized planning mainly followed the object-centered Strips representation. Addressing programming tasks with such representation is unpractical since it requires to encode all values in the domain of a state variable as objects. Other approaches, such as Qualitative Numeric Planning (QNP) [72,73], handle large numeric state variables qualitatively with propositions to denote whether a variable is equal to zero. In this work we handle GP problems with integer state variables, which allow to naturally address diverse programming tasks as if they were GP problems. • Explicit problem representation. In this work, a GP problem comprises the explicit enumeration of a finite set of classical planning instances to be solved. Interestingly our experimental results show that, in several domains, solving a small set of a few randomly generated classical planning instances, is enough to obtain a solution that generalizes to the infinite set of problems that belong to a given domain. • No background knowledge. Our approach does not require any additional help such as state invariants, plans/traces/demonstrations, negative examples, or the specification of the subset of features to appear in the generalized plans. • Generalized plans represented as structured programs. Structured programming provides a white-box modeling paradigm that is widely popular. In this work we focus on generalized plans represented as structured programs, with control flow constructs for branching and looping the program execution flow. The application of a generalized plan on a particular instance is then a deterministic matching-free process, which makes it easier to define effective evaluation and heuristic functions. Further, the asymptotic complexity of structured programs can be assessed from their structure, which is also helpful to establish preferences on different possible generalized plans. • Off-line satisfiability approach. This work follows an off-line approach to GP that aims to compute, at once, a generalized plan that exactly solves all the classical planning instances that are given as input. Because many heuristic search algorithms are easily extended to online versions, we believe that our GP as heuristic search approach is a stepping stone towards online approaches that can deal with larger sets of classical planning instances. • Native heuristic search for GP. By native heuristic search, we mean that we defined a search space, evaluation/heuristic functions, and a search algorithm, that are specially targeted to GP. Our GP as heuristic search approach is related to an existing classical planning compilation for GP [34]. Our approach overcomes however 5 the main drawback of the compilation whose search space grows exponentially with the number and domain size of the state variables; in practice, this limits the applicability of the compilation to planning instances of small size since the performance of off-the-shelf classical planners is sensitive to the size of the input instances. Our experiments support this claim, and show that our BFGP algorithm significantly reduces the CPU-time required to compute and validate generalized plans, compared to the classical planning compilation approach to GP [34]. Background This section introduces the necessary notation to formalize our GP as heuristic search approach. First, the section formalizes the classical planning model and the generalized planning model. Then the section formalizes planning programs, our formalism for the compact representation of planning solutions, and that applies to both classical planning and generalized planning. Lastly the section formalizes the Random Access Machine given that, to define a tractable solution space for GP, our GP as heuristic planning approach borrows several mechanisms from this abstract computation machine. Classical Planning Our formalization of the classical planning model is similar to the abstract planning framework called Finite Functional Planning, that was introduced for the theoretical analysis of different ground languages for classical planning [74]. Let X be a set of state variables, where each variable x ∈ X has a domain D x . A state s is a total assignment of values to the set of state variables, i.e. s = x 0 = v 0 , . . . , x N = v N , such that ∀ 0≤i≤N v i ∈ D x i . For a subset of the state variables X ⊆ X, let D[X ] = × x∈X D x denote its joint domain. The state space is denoted as S = D[X] . Given a state s ∈ S , and a subset of variables X ⊆ X, let s |X = x i = v i x i ∈X be the projection of s onto X i.e. the partial state that is defined by the values assigned by s to the subset of state variables in X . The projection of s onto X defines the subset {s | s ∈ S , s |X ⊆ s} of the states that are consistent with the corresponding partial state. Last, let us define a state-constraint C as a Boolean function C : S → {0, 1} over the state variables, that implicitly defines the subset of states S C ⊆ S that are consistent with that constraint. Let A be a set of deterministic actions such that each action a ∈ A is characterized by two functions; an applicability function ρ a : S → {0, 1} and a successor function θ a : S → S . An action a ∈ A is applicable in a given state s ∈ S iff ρ a (s) equals 1. The execution of an applicable action a ∈ A, in a state s ∈ S results in the successor state s = s ⊕ a. Note that our definition of deterministic actions generalizes actions with conditional effects [75], common in GP since their state-dependent outcomes allows the adaptation of generalized plans to different classical planning instances. A classical planning instance is a tuple P = X, A, I, G , where X is a set of state variables, A is a set of actions, I ∈ S is an initial state, and G is a constrain on the value of the state variables that induces the subset of goal states S G = {s | s G, s ∈ S }. Given P, a plan is an action sequence π = a 1 , . . . , a m whose execution induces a trajectory τ = s 0 , a 1 , s 1 , . . . , a m , s m such that, for each 1 ≤ i ≤ m, a i is applicable in s i−1 and results in the successor s i = s i−1 ⊕ a i . A plan π solves P if and only if the execution of π in s 0 = I finishes in a goal state, i.e. s m ∈ S G . We say π is optimal if |π| = m is minimal among the set of all the plans that solve P. Planning languages, such as PDDL [76], can compactly represent the infinite set of classical planning instances of a given domain using a finite set of functions and a finite set of action schemes. Given a finite set of objects Ω, and a finite set of functions Φ defined over that set of objects, we assume that each state variable x ∈ X stands for a function interpretation x ≡ φ( − → o ), where φ ∈ Φ is a function with arity ar(φ), and − → o ∈ Ω ar(φ) is a vector of objects comprised in the Cartesian product space of Ω ar(φ) . For keeping compact the number of state variables, objects and function signatures can by typed so the number of possible function interpretations is constrained. Functions in Φ can be Boolean e.g. to represent PDDL predicates, or numeric e.g. to represent PDDL numeric fluents. Likewise, given a set of action schemes Ξ, we assume that each action a ∈ A is built from an action schema ξ ∈ Ξ by substituting each variable in the action scheme with an object from Ω. An action scheme ξ ∈ Ξ is a tuple ξ = name(ξ), par(ξ), pre(ξ), e f f (ξ) ; where name(ξ) is the identifier of the action schema, par(ξ) is its list of variables (again these variables can be typed so they can only be substituted by objects of the same type), pre(ξ) is a FOL Boolean formula defined over par(ξ) that compactly represents the subset of states where the corresponding ground actions are applicable, and eff(ξ) is list of FOL constraints that compactly represents the updates of the state variables caused by the application of the corresponding ground actions. Generalized Planning Generalized planning is an umbrella term that refers to more general notions of planning [7]. This work builds on top of the inductive formalism for GP, where a GP problem is defined as a finite set of classical planning instances that share a common structure [5,53]. In this work we assume that the finite set of classical planning instances in a GP problem belong to the same domain. These instances share then a common structure since they are built from the same sets of functions Φ and action schemes Ξ. Definition 1 (GP problem). A GP problem P = {P 1 , . . . , P T } is a finite and non-empty set of T classical planning instances P 1 = X 1 , A 1 , I 1 , G 1 , . . . , P T = X T , A T , I T , G T such that at each instance P t ∈ P, 1 ≤ t ≤ T , may differ in the set of state variables, actions, initial state, and goals, but the corresponding set of state variables X t is induced from the common set of functions Φ, and the set of actions A t from the common set of action schemes Ξ. There are diverse representations for GP solutions that range from generalized polices [35,49], to finite state controllers [40,50], formal grammars [51], hierarchies [77,52], or programs [1,34]. Each representation has its own expressiveness capacity, as well as its own validation complexity and computation complexity. In spite of this representation diversity, we can define a common condition under which a generalized plan is considered a solution to a GP problem. First, let us define exec(Π, P) = a 1 , . . . , a m as the sequential plan that is produced by the execution of a generalized plan Π on a classical planning instance P. Definition 2 (GP solution). A generalized plan Π solves a GP problem P = {P 1 , . . . , P T } iff, for every classical planning instance P t ∈ P, 1 ≤ t ≤ T , it holds that the sequential plan exec(Π, P t ) solves P t . A GP solution Π for a given GP problem P is optimal iff, for every P t ∈ P, the sequential plan exec(Π, P t ) induced by Π for solving the classical planing instance P t is an optimal plan for that instance. Example. Figure 1 shows the initial state and goal of two classical planning instances, P 1 = X, A, I 1 , G 1 and P 2 = X, A, I 2 , G 2 , for sorting two six-element lists. In this example the two instances share the same set of state variables X = {x i ≡ vector(o i )|0 ≤ i ≤ 5} that is built with the one-arity function Φ = {vector} and the set of objects Ω 1 = Ω 2 = {o 0 , . . . , o 5 }, and where ∀ x∈X D x = N 0 . The two classical planning instances also share the set of deterministic actions A, with 6×5 2 actions swap(o i , o j ), that swap the content of two list positions i < j, and that are induced from the single action scheme Ξ = {swap(x, y)}. An example solution plan for P 1 is π 1 = swap(o 0 , o 5 ), swap(o 1 , o 2 ), swap(o 1 , o 3 ) while π 2 = swap(o 0 , o 2 ), swap(o 3 , o 5 ) is an example of a sequential plan that solves P 2 . Note that the set P = {P 1 , P 2 } is a GP problem since they are two classical planning instances that are built using the same set of functions Φ and action schemes Ξ. Figure 2 shows an example of a generalized plan that solves the GP problem P = {P 1 , P 2 }, and that is represented as a sorting network. The sorting network is illustrated using two different types of items (namely the wires and the comparators). For each state variable, there is a wire that carries the value of that variable from left to right in the network. On the other hand, comparators connect two different wires, corresponding to a pair of variables (x i , x j ), such that i < j. When a pair of values traveling through a pair of wires (i, j), encounters a comparator, then the comparator applies the action swap (o i , o j ) iff vector(o i ) ≥ vector(o j ), which in turn is x i ≥ x j . The sorting network of Figure 2 can actually solve any instance for sorting the content of any six-element list, no matter its initial content. This solution is however not valid for sorting lists with different lengths. In this paper we will show how to represent and compute planning solutions that leverage indirect memory addressing to generalize no matter the number of objects, and corresponding state variables. Initial State Goal State P 2 3 2 1 6 5 4 1 2 3 4 5 6 P 1 6 3 4 2 5 1 1 2 3 4 5 6 Figure 1: Example of two classical planning instances for sorting the content of two six-element lists by swapping the list elements. x 5 x 4 x 3 x 2 x 1 x 0 Figure 2: Example of a generalized plan, represented as a sorting network that solves any classical planning instances for sorting the content of a six-element list, no matter its initial content. Planning programs In this work we represent planning solutions as planning programs [34]. Unlike sequential plans, planning programs include a control flow construct which allows the compact representation of solutions to classical planning problems and to GP problems. Formally a planning program is a sequence of n instructions Π = w 0 , . . . , w n−1 , where each instruction w i ∈ Π is associated with a program line 0 ≤ i < n, and it is either: • A planning action w i ∈ A. • A goto instruction w i = go(i , !y), where i is a program line 0 ≤ i < i or i + 1 < i < n, and y is a proposition. • A termination instruction w i = end. The last instruction of a planning program is always a termination instruction, i.e. w n−1 = end. The execution model for a planning program is a program state (s, i), i.e. a pair of a planning state s ∈ S and program counter 0 ≤ i < n. Given a program state (s, i), the execution of a programmed instruction w i is defined as: • If w i ∈ A, the new program state is (s , i + 1), where s = s ⊕ w i is the successor when applying w i in s. • If w i = go(i , !y), the new program state is (s, i + 1) if y holds in s, and (s, i ) otherwise 1 . Proposition y can be the result of an arbitrary expression on state variables, e.g. a state feature [78]. • If w i = end, program execution terminates. To execute a planning program Π on a classical planning instance P = X, A, I, G , the initial program state is set to (I, 0), i.e. the initial state of P and the first program line of Π. A program Π solves P iff the execution terminates in a program state (s, i) that satisfies the goal condition, i.e. w i = end and s ∈ S G . Otherwise the execution of the program fails. If a planning program fails to solve the planning instance, the only possible sources of failure are: 1. Inapplicable program, i.e. executing action w i ∈ A fails in program state (s, i) since w i is not applicable in s. 2. Incorrect program, i.e. execution terminates in a program state (s, i) that does not satisfy the goal condition, i.e. (w i = end) ∧ (s S G ). 3. Infinite program, i.e. execution enters into an infinite loop that never reaches an end instruction. In this work we model instructions w i ∈ A as if they were always applicable but that their effects only update the current state iff the preconditions of the action hold in the current planning state. Formally, when executing w i in (s, i), the new program state is (s , i + 1) iff w i is applicable, otherwise it is (s, i + 1). Therefore, in this work the execution of a program on a classical planning instance will never return an inapplicable program, and only incorrect or infinite program are possible sources of failure. This particular action modeling is common in Reinforcement Learning [68], and in conformant planning [44], because it allows to deliver compact solutions that apply to sets of different sequential decision-making problems (typically with different initial states). The Random-Access Machine The Random-Access Machine (RAM) is an abstract computation machine, in the class of register machines, that is polynomially equivalent to a Turing machine [79]. The RAM machine enhances a multiple-register counter machine [80] with indirect memory addressing. The indirect memory addressing of RAM machines is useful for defining RAM programs that access an unbounded number of registers, no matter how many there are. With this regard, a register in a RAM machine is then a memory location with both a content (a single natural number), and an address (a unique identifier that works as a natural number). Let be r the address of a RAM register, and [r] the content of that register. Diverse base instructions sets, that are Turing complete, can be defined on the RAM registers. Our GP as heuristic search approach builds on the instructions of the Base set 1 and the Base set 3, that are briefly defined as follows: • copy(r 1 , r 2 ) copies the content of r 1 in r 2 , i.e. [r 1 ] → [r 2 ]. • jmp(z, r) jumps to instruction z if register r is zero, else the RAM execution continues to the next instruction. • halt terminates the RAM execution. Auxiliary dedicated registers allow to reduce the size of the RAM instruction set. For instance, the number of registers called out by the RAM instructions can be reduced by using an accumulator dedicated register. WLOG in this work we extend Base sets 1 and 3 with two FLAGS registers, the zero and the carry registers, that are dedicated to store the outcome of three-way comparisons [81] between two registers, as well as to condition jump instructions. On the other hand, the RAM instruction set can be extended with extra instructions, that can actually be built as blocks of the base RAM instructions, but that allow the definition of more compact RAM programs. For instance, an extra instruction for clearing (setting to zero) the value of a register. Planning with a RAM Inspired by the computational model of the RAM machine, this section extends the classical planning model with a set of pointers, defined over the set of objects used to build the state variables and actions of a classical planning instance, and with the set of primitive operations for manipulating these pointers. Extending the classical planning model with a set of pointers, and their primitives, allows the agnostic definition of a set of state features, and a set of actions, that are shared for the different classical planning instances in a given domain and that can be leveraged for the computation of generalized plans. For instance, the selection sort algorithm is able to solve any sorting instance, no matter the content or the length of the input list, because it is provided with mechanisms for indexing the list positions and for operating over those indexes. First, the section shows how to compactly represent a transition system using pointers. Then the section formalizes our extension of the classical planning model with a set of pointers, and their corresponding primitive operations. The last part of the section shows that our pointer-based formalism applies also to the object-centered representations traditionally used in classical planning. Representing transition systems with pointers Formally, a transition system is a pair (S , →), where S is a set of states, and → denotes a relation of state transitions S × S . Unlike finite-state automata, the set of states and the set of state transitions of a transition system are not necessarily finite, and no initial/goal states are specified. State constraints allow the compact representation of (possibly infinite) sets of states. For instance given the set of state variables X = {x 0 , x 1 , x 2 , x 3 , x 4 , x 5 } of the example in Figures 1 and 2, the state constraint x 0 ≤ x 1 ≤ x 2 ≤ x 3 ≤ x 4 ≤ x 5 defines the subset of states where the content of these variables is sorted in increasing order and it applies no matter the domain of the state variables (that can actually be infinite). In object-centered transition systems, states are factored and each state variable is addressed by a function φ ∈ Φ fed with a list of objects − → o , i.e. φ( − → o ) s.t. − → o ∈ Ω ar(φ) ; where Ω is the set of objects and ar(φ) denotes the arity of φ. For instance in the previous example, given the one-arity function vector and the six-objects set Ω = {o 0 , o 1 , o 2 , o 3 , o 4 , o 5 }, each state variable x i ∈ X may be defined as x i ≡ vector(o i ). Definition 3 (Pointer). Given a set of objects Ω, we define a pointer as a finite-domain variable z whose domain is D z = [0, . . . , |Ω| − 1], where |Ω| denotes the number of objects. Pointers are then variables for indexing the objects of a transition system that, in combination with function symbols, are useful to define state constraints that produce not only compact, but general representations of a possibly infinite set of states. By general we mean that a constraint represents a set of states that share some common structure, no matter the actual number of objects, and corresponding state variables. For instance, Figure 3 shows the Boolean function constraint sorted, that implements a global constraint for checking whether the content of the vector of state variables is sorted in increasing order. The constraint sorted function is procedurally defined, leveraging a single pointer i, and it applies to any number of objects, and to any domain size of the corresponding state variables. Besides the compact and general definition of (possibly infinite) sets of states, pointers over objects also enable the compact and general definition of (possibly infinite) sets of state transitions via action schemes. Definition 4 (Action schema with pointers). Given a set of X state variables, an action schema with pointers is a tuple name, params, pre, eff where: • name is the symbol that uniquely identifies the action schema. • params is a finite set of pointers Z defined over the set Ω of objects. • pre is a state constraint where state variables are indirectly addressed via the function symbols and the pointers in params, i.e. x ≡ φ( − → z ) such that φ ∈ Φ and − → z ∈ Z ar(φ) . The pre state constraint implicitly represents the subset of states where the action schema is applicable. • eff is a partial assignment of the state variables where state variables are indirectly addressed via the function symbols and the pointers in params. The eff partial assignment implicitly represents the successor state that results from the execution of the action schema at a given state. To illustrate our pointer-based definition of an action schema, Figure 4 shows a procedural representation of the preconditions (pre) and the effects (eff) of the swap action schema. When applicable, the swap action schema exchanges the value of the state variables induced by its two parameters (the pointers i and j). The swap action schema is succinct, because it compactly defines an infinite set of different state transitions that share a common structure. The swap action schema is also general, because it applies to any sorting instance, no matter the number of state variables (i.e. the length of the vector of state variables) or the domain size of these variables. What is more, the execution of the schema swap pre and schema swap eff procedures is a deterministic matching-free process since the input pointers do always index an object in Ω. Classical planning with pointers Here we extend the classical planning model (introduced in Section 2) with a set of pointers and their corresponding primitive operations. We formalize the set of primitive operations over pointers leveraging the notion of Random-Access Machine. In more detail, we extend the classical planning model with a RAM machine of |Z| + 2 registers; |Z| pointers that reference the planning objects, plus two FLAGS registers (the zero and the carry). The two FLAGS registers store the outcome of three-way comparisons, between two different pointers or between two state variables that are indirectly addressed via function symbols and pointers. Given a classical planning instance P = X, A, I, G , such that the state variables and actions are generated with the set of functions Φ and action schemes Ξ of a given domain, and the set of objects Ω, then the extended classical planning instance with a RAM machine of |Z| + 2 registers is defined as P Z = X Z , A Z , I Z , G , where: • The new set of state variables X Z comprises: -The state variables X of the original planning instance, such that each state variable x i ∈ X is x i ≡ φ( − → o ) with φ ∈ Φ and − → o ∈ Ω ar(φ) , as defined above. -Two Boolean variables Y = {y z , y c }, that play the role of the zero and carry FLAGS registers, respectively. -The pointers Z, a set of extra state variables s.t. each z ∈ Z has finite domain D z = [0, . . . , |Ω| − 1]. -A set of derived state variables X Z = { φ( − → z ) | φ ∈ Φ, − → z ∈ Z ar(φ) } whose value is given by the interpretations of the functions of the domain with the corresponding pointers. Given a fixed number of pointers, Y, Z, and X Z , are a subset of state variables that is shared by all the instances that belong to the same domain, no matter the number of objects of each instance. • The new set of actions A Z will represent the set of actions that is shared for the different classical planning instances in a given domain, and it includes: -The planning actions A that result from reformulating each action scheme ξ ∈ Ξ into its corresponding pointer-based version. The reformulation is a two-step procedure that requires that Z contains, at least, as many pointers as the largest arity of an scheme in Ξ: (i), each parameter in par(ξ) is replaced with a pointer in Z and (ii), preconditions and effects are rewritten to refer to these pointers. -The RAM actions that implement the following sets of RAM instructions {inc(z 1 ), dec(z 1 ), cmp(z 1 , z 2 ), set(z 1 , z 2 ) | z 1 , z 2 ∈ Z} over the pointers in Z, and {test(φ( − → z 1 )), cmp(φ( − → z 1 ), φ( − → z 2 )) | − → z 1 , − → z 2 ∈ Z ar(φ) } over the lists of pointers in Z ar(φ) for each function symbol φ ∈ Φ. Respectively, these RAM instructions increment/decrement a pointer by one, compare two pointers, set the value of a pointer z 2 to another pointer z 1 , test whether φ( − → z 1 ) is greater than zero, and compare 2 the value of φ( − → z 1 ) and φ( − → z 2 ). Each RAM action also updates the Y = {y z , y c } flags, according to the result of the corresponding RAM instruction (which is denoted here by res): inc(z 1 ) =⇒ res := z 1 + 1, dec(z 1 ) =⇒ res := z 1 − 1, cmp(z 1 , z 2 ) =⇒ res := z 1 − z 2 , set(z 1 , z 2 ) =⇒ res := z 2 , test(φ( − → z 1 )) =⇒ res := φ( − → z 1 ), cmp(φ( − → z 1 ), φ( − → z 2 )) =⇒ res := φ( − → z 1 ) − φ( − → z 2 ),y z := (res == 0), y c := (res > 0). • The new initial state I Z is the initial state of the original planning instance, but extended with all pointers set to zero and the two FLAGS set to False. The goals are the same as those of the original planning instance. Example. Here we extend the classical planning instance P 1 = X, A, I 1 , G 1 (illustrated in Figure 1) with a RAM machine of |Z| + 2 registers, where Z = {i, j} is a set of two pointers. According to this extension, our pointer-based representation of the sequential plan π 1 = swap(o 0 , o 5 ), swap(o 1 , o 2 ), swap(o 1 , o 3 ) is the follow- ing sequence of thirteen actions π 1 = inc( j) 5 , swap(i, j), inc(i), dec( j) 3 , swap(i, j), inc( j), swap(i, j) ; where superscripts refer to the number of times that an instruction is sequentially repeated, and where swap(i, j) refers to the pointer-based action schema defined in Figure 4. Likewise, our pointer-based version of the sequential plan π 2 = swap(o 0 , o 2 ), swap(o 3 , o 5 ) , that solves the classical planning problem P 2 illustrated in Figure 1, is the ten-action sequence π 2 = inc( j) 2 , swap(i, j), inc(i) 3 , inc( j) 3 , swap(i, j) . Note that any sequential plan for solving a classical planning instance from the vector sorting domain, no matter the number of state variables and no matter the domain size of these state variables, can be built using exclusively actions from the following set: {inc(i), inc( j), dec(i), dec( j), swap(i, j)}. 3.2.1. Theoretical properties Theorem 1. Given a classical planning instance P, its extension P Z , with a RAM machine of |Z| pointers, preserves the solution space of P. Proof. ⇒: Let π = a 1 , . . . , a m be a plan that solves P, an equivalent plan π that solves P Z is built as follows; for each action a i ∈ π, A contains a pointer-based action schema a i that replaces each parameter in par(a i ) with a pointer z ∈ Z. For each such pointer z, the plan repeatedly applies RAM actions inc(z) or dec(z) until they reference the associated vector of objects − → o , and then it applies a i . The resulting plan π has exactly the same effect as π on the original planning state variables in X, and since the goal condition of P Z is the same as that of P, it follows that π solves P Z . ⇐: Let π = a 1 , . . . , a m be a plan that solves P Z . Identify each action in A among those of π , and execute π to identify the assignment of objects to pointers when applying each action in A . Construct a plan π corresponding to the subsequence of actions in A from π , replacing each action schema a i ∈ A by an original action a i ∈ A and choosing as parameters of a i the objects referenced by the pointers of a i at the moment of execution. Hence a i has the same effect as a i on the state variables in X, implying that π has the same effect as π on X. Since the goal condition of P is the same as that of P Z , it follows that π solves P. Our extension of a classical planning problem with a RAM machine of |Z| + 2 registers preserves the solution space of the original problem. Sequential plans in the extended planning model are however longer when pointers must be incremented (or decremented) multiple times to access the corresponding objects, before the corresponding action schema is executed. For instance, our pointer-based version of plan π 1 required thirteen steps while the original sequential plan only had three steps. Likewise our pointer-based version of plan π 2 required ten steps while the original sequential plan only had two steps. On the other hand, our extension does not require explicit action grounding; it is the planner itself who determines the values of the pointers that now feed the action schemas, so in a sense the planner is in charge of doing a partial instantiation. Further, our extension separates the instantiation of each parameter of an action schema. The computation of sequential plans with our pointer-based formalism is however out of the scope of this paper. We will exclusively address the computation of planning solutions represented as planning programs. Theorem 2. The new set of actions A Z is independent of the number of objects Ω, state variables X, and their domain size. Proof. The number of actions of a classical planning instance, extended with a RAM machine of |Z| pointers, is |A Z | = 2|Z| 2 + φ∈Φ |Z| 2ar(φ) + |A |.(1) This number exclusively depends on the number of pointers in Z and on the arity of the functions in Φ and the action schemes in Ξ. First, the increment/decrement instructions induce 2|Z| actions, the set instructions over pointers induce |Z| 2 − |Z| actions, and comparison instructions of pointers induce |Z| 2 − |Z| actions. The comparison instructions can compare two pointers but for symmetry breaking, we only consider the single parameter ordering (z i , z j ) where i < j, i.e. we consider cmp(z 1 ,z 2 ) but not cmp(z 2 ,z 1 ). Second, test instructions are defined over each function symbol and list of pointers with the same size as its arity, inducing φ |Z| ar(φ) actions, and comparison of predicates with pointers induce φ (|Z| 2ar(φ) − |Z| ar(φ) ) actions. Therefore, the total number of RAM instructions are 2|Z| + 2(|Z| 2 − |Z|) + φ (|Z| ar(φ) + |Z| 2ar(φ) − |Z| ar(φ) ) = 2|Z| 2 + φ |Z| 2ar(φ) which only depends on the number of pointers in Z and the arity of each function symbol φ. Last, as defined by our abstraction procedure, the number of actions in A is given by the number of parameters of the actions schemes Ξ and the number of pointers in Z to replace these parameters. This means that the size of A is upper bounded by |A | ≤ ξ∈Ξ |Z| |par(ξ)| . As a consequence it follows that A Z , whose size is given by |A Z | = 2|Z| 2 + φ |Z| 2ar(φ) + |A |, it is also independent of the number of objects Ω, state variables in X and their domain size. Strips with pointers Since the early 70's, the Strips representation formalism is widely used for research in automated planning [82]. Even today, Strips is an essential fragment of PDDL [76], the input language of the International Planning Competition, and most planners support the Strips representation features. Here we show that our pointer-based view of planning problems and solutions applies also to object-centered planning formalisms, such as Strips planning. In fact, our pointer-based formalism can be understood as an instantiation of F-Strips [83], where the single level of indirection of pointers over objects is enough to represent Strips problems with constant memory access. Strips is an object-centered planning formalism that compactly represents the set of states of a transition system using a finite set of objects, and a finite set of FOL predicates, that indicate the properties of the objects and their relations. Likewise, Strips compactly represents the space of the possible state transitions using FOL operators, which are defined as a tuple op = name(op), args(op), pre(op), eff − (op), eff + (op) and where, name(op) is a unique identifier of the operator, args(op) is a set of variable symbols specifying the arguments of the operator, and pre(op), eff − (op), eff + (op) are sets of FOL predicates, with variables exclusively taken from args(op), and that respectively specify the preconditions, negative effects and positive effects. The representation of a Strips problem is completed specifying an initial state, that defines the initial situation for all the objects, and the aimed set of goal states, which is typically specified as a partial state. State representation. In our pointer-based formalism for Strips problems, each state variable x ∈ X has domain D x = {0, 1}, and it is built as a FOL Strips predicate φ ∈ Φ grounded by a vector of objects − → o ∈ Ω ar(φ) . Figure 5 shows the representation of a blocksworld state using the Strips formalism as well as using our formalism. (handempty)) (not (on ?x ?y))))) unifying all the previous predicate and object tuple valuations into a vector. The length of the vector of state variables is then upper bounded by |X| ≤ k≥0 n k |Ω| k , where n k is the number of first-order predicates with arity k. For instance, the X vector contains at most |Ω| 2 + 3|Ω| + 1 state variables for the blocksworld domain. State-invariants (e.g. in the blocksworld a block cannot be on top of two different blocks simultaneously) can also be leveraged to save space for the memory allocation of the state variables. Action representation. Given a FOL Strips operator op = name(op), args(op), pre(op), eff − (op), eff + (op) , our pointer-based formalism produces its corresponding pointer-based action schema name, params, pre, eff : • The name of the action schema is name(op), the name of the given FOL Strips operator. • For each argument in args(op), the action schema has a pointer that indexes an object o ∈ Ω. • The set pre(op) is transformed into a conjunctive arithmetic-logic expression with conditions of two kinds: (i) for each pointer in the parameters of the action schema, the conditions asserting that the pointer is within its domain and (ii), for each precondition in pre(op) a condition asserting that the state variable addressed by the pointers content equals to some specific value of its domain. • Each negative effect in eff − (op) is transformed into an indirect variable assignment that sets the corresponding state variable to 0. Likewise, each positive effect in eff + (op) is transformed into an indirect variable assignment that sets the corresponding state variable to 1. Figure 7 shows our pointer-based definition for the unstack action schema from the blocksworld that implements the corresponding operator represented in the Strips fragment of PDDL of Figure 6. The action schema of Figure 7 is implemented using two pointers (i and j), and it applies to any blocksword instance, no matter the number of blocks or their identity. In this implementation the state variables are global, meaning that they can be accessed from any of the action schemas. Again the execution of these procedures is a deterministic matching-free process since the input pointers do always have a block assigned. Problem representation. We complete our pointer-based representation of a Strips problem with the init and goal procedures. Figure 8 shows the init and goal procedures for the planning problem of unstacking the 3-block tower of Figure 5. They have the same formal structure as the procedures we use for our pointer-based representation of the preconditions and effects of an action schema, but with no arguments. In more detail: • The init procedure is a write-only procedure, that implements a total variable assignment of the state variables for specifying the initial state of the Strips problem. • The goal procedure is a read-only Boolean procedure, that encodes the state-constraint that specifies the subset of goal states. 14 Plan representation. Following our pointer-based representation, a sequential plan π = a 1 , . . . , a n is a sequence of transformations of the state variables using: (i) our pointer-based action schemas that encode the FOL Strips operators and (ii), the subset of RAM actions {inc(z), dec(z)| z ∈ Z} for achieving the aimed binding for each parameter of an action schema. Our pointer-based representation of the four-action plan π = unstack(b1,b2), putdown(b1), unstack(b2,b3), putdown(b2) for unstacking the three-block tower of Figure 5, is the following sequence of actions π = inc( j), unstack(i, j), putdown(i), inc(i), inc( j), unstack(i, j), putdown(i) that leverages two pointers Z = {i, j}, that are defined over the set of blocks Ω. In spite of its popularity, the Strips representation is too low-level for many interesting applications [14,15,17]. Our pointer-based representation naturally extends beyond Strips to more expressive object-oriented representations. For instance, state variables can also comprise numeric variables (e.g. integers or reals) to implement numeric fluents as in PDDL2.1 [84] or in RDDL [16]. Object typing can also be implemented in a straightforward way (e.g. specializing pointers to the number of objects of a particular type) to compact the size of the vector of state-variables and to optimize the implementation of quantified preconditions/effects/goals [85]. Last, our pointer-based representation also supports conditional effects [86], e.g. an action schema can specify multiple variable assignments conditioned by the different values of the state variables. Generalized planning as heuristic search First the section shows how we build, in an agnostic manner, a GP problem from a set of classical planning instances of a given domain. Then, the section describes in detail our GP as heuristic search approach: our search space for GP, the evaluation/heuristic functions that we use for guiding the search, and the particular details of our search algorithm. From a set of classical planning instances to a GP problem Generalized plans leverage relevant subsets of shared state variables (features) and actions whose execution is well-defined for any possible value of the state variables of the classical planning instances to be solved. Building a GP problem from a set of classical planning instances of a given domain is not trivial because these two ingredients may not be given in the representation of the classical planning instances. On the one hand given a classical planning domain, the specification of a set of features that is (i), expressive enough to represent a polynomial solution valid for any instance in the domain and (ii), compact enough for the effective computation of that solution, is a complex task that requires expert knowledge on both the domain and the aimed solution. In fact, the automatic specification of expressive and compact features for a planning domain is a challenging research question that is investigated since the early days of automated planning [87]. On the other hand the set of ground actions for the different instances of a given domain, is usually different since it depends on the number of objects. Back to the sorting example illustrated in Figures 1 and 2, the classical planning instances for sorting a vector of length six induced 6×5 2 swap(o i , o j ), i < j actions, while instances for sorting a vector of length seven would induce a set of 7×6 2 swap(o i , o j ) actions. Given a finite and non-empty set of T classical planning instances from a given domain, our approach for automatically building a GP problem is to extend the instances with a RAM machine of |Z| pointers. The result is a GP problem P = {P 1 , . . . , P T }, where each instance P t ∈ P, 1 ≤ t ≤ T may differ in the actual set of objects, initial state, and goals, but all instances necessarily share the subset of state variables X Z ∩ Y ∩ Z and the same set of actions A Z . Formally, P 1 = X Z 1 , A Z , I Z 1 , G 1 , . . . , P T = X Z T , A Z , I Z T , G T where ∀ 1≤t≤T X Z t ⊂ X Z t , ∀ 1≤t≤T Y ⊂ X Z t , ∀ 1≤t≤T Z ⊂ X Z t , and ∀ a ∈A Z par(a ) ∈ Z ar(a ) . The number of pointers |Z| is a parameter that indicates how many pointers are used in the extension of the classical planning instances 3 . Our extension with a RAM machine of |Z| pointers automatically defines a minimalist but general set of features for the set of a classical planning instances from a given domain. Definition 5 (The feature language). We define the feature language as the four possible joint values of the two Boolean variables Y = {y z , y c }, and we denote this language as L = {(¬y z ∧ ¬y c ), (y z ∧ ¬y c ), (¬y z ∧ y c ), (y z ∧ y c )}. We say L is minimalist because it only contains four elements, and we say L is general because it is independent of the number of objects and hence, of the domain of the state variables and the number of state variables. Note that our features are a function of (i) the state variables and (ii) the last executed action, since they all may affect the value of Y = {y z , y c }. Such notion of feature is related to the notion of state observation in the POMDP formalism, where observations depend on the current state and the action just taken [88]. With this regard it can be understood that our GP approach computes, at the same time, a generalized plan and an observation function useful for that generalized plan. Our feature language is also related to Qualitative numeric planning [72,73,6] which leverages propositions to abstract the value of numeric state variables. Given that our FLAGS Y = {y z , y c } depend on the last executed action, and considering that only RAM instructions update the variables in Y, we have an observation space of 2 |Y| × (2|Z| 2 + φ |Z| 2ar(φ) ) state observations implemented with only |Y| Boolean variables. The four joint values of {y z , y c } model then a large space of observations, e.g. = 0, 0, < 0, > 0, ≤ 0, ≥ 0 as well as relations =, , <, >, ≤, ≥ on pairs of state variables. Likewise, our extension with a RAM machine of |Z| pointers automatically defines the shared set of actions A Z , that is well-defined for the set of a classical planning instances from a given domain. Because the set of pointers Z is fixed for the T input classical planning instances we have that, after our extension, all the instances share the same set of actions A Z . The execution of the actions in A Z is well-defined over the subset of state variables Z, no matter the actual number of objects, or the corresponding number and domain size of the state variables; we recall the reader that the set of actions A Z exclusively depends on the number of pointers |Z| and the arity of actions and functions (Theorem 2). The search space Briefly, our GP as heuristic search approach implements a combinatorial search in the solution space of the possible planning programs. Next we provide more details on how we implemented a tractable search space for GP. Definition 6 (Partially specified planning program). A partially specified planning program is a planning program such that the content of some of its program lines may be undefined. Each node of our search space is a partially specified planning program which is binary encoded as follows. Given a set of state variables X, a set of actions A, a maximum number of program lines n such that the last instruction is w n−1 = end, and defining the propositions of goto instructions as (x = v) atoms where x ∈ X and v ∈ D x , we have that the space of possible planning programs is represented by the following bit-vectors: 1. The action vector of length (n − 1) × |A|, indicating whether an action a ∈ A is programmed on line 0 ≤ i < n − 1. 2. The transition vector of length (n − 1) × (n − 2), indicating whether a go(i , * ) instruction is programmed on line 0 ≤ i < n − 1. 3. The proposition vector of length (n − 1) × x∈X |D x |, indicating whether a go( * , ! x = v ) instruction is programmed on line 0 ≤ i < n − 1. A partially specified planning program is then encoded as the concatenation of these three bit-vectors and the length of the resulting bit-vector is: (n − 1)        |A| + (n − 2) + x∈X |D x |        .(2) The binary encoding allows us to quantify the similarity of two partially specified planning programs (e.g. the Hamming distance of their corresponding bit-vector representation) and more importantly, to systematically enumerate the space of all the possible planning programs with a maximum of n lines. Let us define the empty program as the particular partially specified planning program whose instructions are all undefined (i.e. all bits of its bit-vector representation are set to False). Starting from the empty program, we can enumerate the entire set of possible planning programs with two search operators: • program(i,a), that programs an action a ∈ A at line i of a program • program(i,i',x,v), that programs a goto(i , ! x = v ) instructions at line i of a program. These two search operators are only applicable when i is an undefined program line (i.e. in the bit-vector representation the bits corresponding to the encoding of the program line i are set to False). Given the bit-vector representation of a partially specified planning program, the application of the program(i,a) or program(i,i',x,v) search operators set to True the corresponding bits. With this regard, the partially specified planning program of a given search node is at Hamming distance 1 from its parent, when programming a planning action with program(i,a), or at Hamming distance 2, when programming a goto instruction with program(i,i',x,v). In fact, this is the search space leveraged by the classical planning compilation approach for computing planning programs with an off-theshelf classical planner [34]. Equation 2 reveals that the number of planning programs with n lines depends on the number of grounded actions |A|, the number of state variables x ∈ X, and their domain D x . This dependence causes an important scalability issue, limiting the applicability of the cited compilation to planning instances of contained size. Definition 7 (The GP search space). Given a GP problem P, that is built extending a set {P 1 , . . . , P T } of classical planning instances from a given domain with a RAM machine of |Z| pointers. Our GP search space is the set of partially specified planning programs that can be built with n program lines, the set of planning actions A Z , and the set of goto instructions that are exclusively conditioned on a feature in L. Definition 7 leverages our minimalist feature language L to build a tractable solution space for GP. We represent GP solutions as planning programs where goto instructions can exclusively be conditioned on a feature in L. Limiting the conditions of goto instructions to any of the four features in L greatly reduces the number of planning programs with n lines, specially when state variables have large domains (i.e. integers); the proposition vector required to encode a planning program becomes now a vector of only (n−1)×4 bits (one bit for each of the four features in L). Equation 2 simplifies then to: (n − 1) |A Z | + (n − 2) + 4 .(3) Equation 3 shows that the size of our new solution space for GP is independent of the number of objects and hence the number of original state variables and their domain size; Theorem 2 already showed that A Z no longer grows with the number of objects. This novel GP solution space can now scale to planning problems where state variables have large domains (e.g. integers) and that have a large number of state variables. 17 Last but not least, since the Y = {y z , y c } FLAGS store the outcome of three-way comparisons, the fourth case (y z ∧ y c ) ∈ L can actually never happen as a result of a comparison. This fourth case is however useful for representing unconditional gotos. Example. Figure 9 shows two examples of planning programs that can be found by our BFGP algorithm, that searches in our solution search-space for GP: (left) a generalized plan for reversing a list, and (right) a generalized plan for sorting a list. Note that in both planning programs goto instructions are exclusively conditioned on a feature in L, and that both planning programs are actually solutions for an infinite set of classical planning problems; they generalize with a swap action schema of arity 2 and a vector predicate symbol of arity 1, no matter the number of objects Ω and no matter the state variables content, i.e. x i ≡ vector(o i ) such that o i ∈ Ω, x i ∈ X and D x i ⊆ N 0 . In the planning program for reversing a list (left), line 0 sets the pointer j to the last element of the list. Then, line 1 swaps in the vector the element pointed by i (initially set to zero) and the element pointed by j, pointer j is decremented, pointer i is incremented, and this sequence of instructions is repeated until the condition on line 5 becomes false, i.e when j > i, which means that reversing the list is finished. The planning program for sorting a list (right) is actually an implementation of the selection-sort algorithm. In this program, pointers j and i are used for inner (lines 5-7) and outer (lines 8-11) loops respectively, and min to point to the minimum value in the inner loop (lines 3-4); ¬y z ∧ ¬y c on line 2 represents whether the content of vector( j) is less than the content of vector(min), while y z ∧ ¬y c on line 7 represents whether j == length (resp. i == length on line 11). Theoretical properties Theorem 3. The space of planning programs that exclusively branch over the features in L preserves the solution space of planning programs. Proof. ⇒: Given a GP problem P and a planning program Π, that solves P and whose goto instructions are exclusively conditioned on the features in L. An equivalent planning program whose execution flow branches with the original goto(i, !(x = v)) instructions is built by defining a set of four Boolean state variables; one Boolean variable for each feature in L, that are mutually exclusive, and whose value is actually given by the joint value of the Y = {y z , y c } variables. With these new Boolean state variables defined, we can replace each feature in the condition of a goto instruction by a !(x = 1) condition checking whether its corresponding Boolean variable equals 1. ⇐: Given a GP problem P and a planning program Π that solves P. An equivalent planning program, that exclusively branches over any of the features in L, is built replacing each goto(i, !(x = v)) instruction in Π, where x ≡ φ( − → o ) s.t. φ ∈ Φ and − → o ∈ Ω ar(φ) , by a finite block of instructions that: (i) increments/decrements a vector of auxiliary pointers − − → z aux , with size ar(φ), until they indirectly address objects − → o , (ii) given auxiliary static state variables for each possible value, i.e. ∀ v∈D x x v , and a dedicated object for each new state variable o v such that x v ≡ φ(o v ), increments/decrements another auxiliary pointer z static in a function φstatic(z static ) until it reaches object o v such that x v ≡ φstatic(o v ) which equals v, (iii) compares the content of these two state variables with a cmp(φ( − − → z aux ), φstatic(z static )) instruction and (iv), jumps to the i target line when the state variables differ in their content with a goto(i, !(y z ∧ ¬y c )) instruction. 18 Note however that our GP solution space may still be incomplete in the sense that either the bound n on the maximum number of program lines, or the maximum number of pointers available |Z|, may be too small to accommodate a solution to a given GP problem. In that case, a higher-level combinatorial search can be implemented to incrementally find a suitable number of required program lines and pointers. For instance, like is done in SAT-planning where the planning horizon is iterative incremented until it is large enough to accommodate a solution plan [89]. The evaluation functions Here we define two different families of evaluation functions, that exploit two different sources of information, to guide a combinatorial search in our GP solution space of partially specified planning programs: • The program structure. Given a partially specified planning program Π, we define a set of evaluation functions f (Π), that establish different kinds of preferences/priors on the structure of the aimed generalized plans. For instance, following the Occam's razor principle a structural function can prefer generalized plans of simpler complexity or it can prefer generalized plans with more programmed lines so program execution failures can be detected earlier in the search. f 1 (Π), the number of goto instructions in Π. f 2 (Π), the number of undefined program lines in Π. f 3 (Π), the number of repeated actions in Π, f 7 (Π), the max number of nested goto instructions in Π. A goto instruction jumps from an origin program line to a destination program line. We say that a goto instructions is nested when it appears within the origin and destination lines of another goto instruction. • The empirical performance of the program. Given a partially specified planning program Π and a GP problem P = {P 1 , . . . , P T }, we define a set of evaluation functions f (Π, P) that assess the performance of Π on P, executing Π on each of the classical planning instances P t ∈ P, 1 ≤ t ≤ T . Section 2 defined the execution of a planning program on a classical planning instance as a deterministic procedure that terminates either succeeding to solve that instance or failing it. Likewise the execution of a partially specified planning program is a deterministic procedure that introduces a new termination case, reaching an unspecified program line. When the program execution terminates because an unspecified program line is reached, f (Π, P) functions can be used to assess the cost of that program execution, as well as to estimate how far is the program from solving the given GP problem, which define cost and heuristic functions for GP respectively. f 4 (Π, P) = n − max P t ∈P f 4 (Π, P t ), where f 4 (Π, P t ) returns the number of the undefined program line eventually reached after executing Π on the classical planning instance P t ∈ P. f 5 (Π, P) = P t ∈P f 5 (Π, P t ), where f 5 (Π, P t ) = x∈X t (v x − G t (x)) 2 . Here, v x ∈ D x is the value eventually reached, for the state variable x ∈ X t , after executing Π on the classical planning instance P t ∈ P, and G t (x) is the value for this same variable as specified in the goals of P t . Note that for Boolean variables the squared difference becomes a simple difference. This means that for Strips planning problems, where all the state variables are Boolean, f 5 (Π, P t ) is actually a counter of how many atomic goals in G t are still not true. f 6 (Π, P) = P t ∈P |exec(Π, P t )|, where exec(Π, P t ) is the sequence of actions induced from executing the planning program Π on the planning instance P t . f 8 (Π, P) = f 5 (Π, P) + f 6 (Π, P) is the sum of an estimation to the goal and the total accumulated cost, akin to an evaluation function for A * searching algorithm. f 9 (Π, P) = W · f 5 (Π, P) + f 6 (Π, P) is similar to f 8 but the estimation to the goal is multiplied by a factor W, which is set to 5 by default, akin to an evaluation function for WA * searching algorithm. All these functions are evaluation functions (i.e. smaller values are preferred). The structural functions f 1 (Π), f 2 (Π), f 3 (Π) and f 7 (Π), are all computed in linear time by traversing the bit-vector representation of Π. On the other hand, the computation complexity of the three empirical functions f 4 (Π, P), f 5 (Π, P), f 6 (Π, P), f 8 (Π, P) and f 9 (Π, P) is given by the complexity of the partially specified program Π. Performance-based functions accumulate a set of T costs (one for each classical planning instance in the GP problem) that could actually be expressed as a combination of different aggregation functions, e.g. sum, max, average, weighted average, etc. Functions f 4 (Π, P) and f 5 (Π, P) are the only cost-to-go heuristic functions; they provide an estimate on how far is a partially specified planning program from solving a GP problem. With this regard, f 5 (Π, P) requires that the goal condition of the classical planning instances in a GP problem is specified as a partial state. On the other hand f 4 (Π, P) does not post any particular requirement on the structure the goal condition, so they can even be a black-box Boolean procedure over the state variables. Example. We illustrate how our evaluation functions work on the following partially specified program Π = 0.swap(i,j) 1.inc(i) 2.dec(j) 3.goto(2,!(y z ∧ ¬y c )) 4... 5.end, where only line 4 is not programmed yet. The value of the evaluation functions for this partially specified program is f 1 (Π) = 1, f 2 (Π) = 5 − 4 = 1, f 3 (Π) = 0, f 7 (Π) = 1. Given the GP problem P = {P 1 , P 2 } that comprises the two classical planning instances illustrated in Figure 1, and pointers i and j starting at the first and last object indexes, respectively, we can compute f 4 and f 5 to evaluate how far Π is from solving the GP problem of sorting lists, the accumulated cost f 6 , and evaluation functions f 8 and f 9 that combine heuristic-like functions with accumulated cost. In this case f 4 (Π, P) = 5−4 = 1, f 5 (Π, P) = 32, f 6 (Π, P) = 14 + 14 = 28, f 8 (Π, P) = 32 + 28 = 60 and f 9 (Π, P) = 32 + 5 · 28 = 172. The search algorithm Here we describe our heuristic search algorithm for generalized planning. This algorithm implements a Best-First Search (BFS) in our GP solution space of the possible partially specified planning programs with n program lines, and a RAM machine with |Z| pointers. Algorithm 1 shows the pseudo-code of our Best-First Search for Generalized Planning (BFGP). Algorithm 1: Best-First Generalized Planning (BFGP) Data: A generalized planning problem P, a number of pointers |Z|, a number of program lines n Result: A generalized plan Π that solves P Open ← {Π empty } ; while Open ∅ do Π ← extractBestProgram(Open) ; ChildrenPrograms ← expandProgram(Π, |Z|, n) ; for Π ∈ ChildrenProgams do evaluateProgram(Π , P); if isGoal(Π , P) then return(Π ); if not isDeadEnd(Π , P) then Open ← insertProgram(Open,Π ); end end BFGP is a frontier search algorithm meaning that, to reduce memory requirements, BFGP stores only the open list of generated nodes but not the closed list of expanded nodes [90]. Initially the open list of the BFGP algorithm only contains the search node that corresponds to the empty program of n lines, which is denoted as Π empty in Algorithm 1. The empty program is then the root node of the search-tree developed by BFGP. The node extraction and node insertion procedures of the BFGP algorithm are implemented as in a regular BFS search. Next we provide more details on the particular expansion and evaluation mechanisms that are implemented by our BFGP algorithm. The BFGP algorithm sequentially expands the best node in the open list, that is implemented as a priority queue, and that is sorted according to the value of our evaluation/heuristic functions explained above. Let Π be the partially specified program that corresponds to the best node extracted by BFGP from the open list. BFGP expands that node generating one successor node for each partially specified program that result from 20 programming the maximum undefined program line that is reached after executing Π on all the instances in P. In other words given a partially specified program Π, only its max P t ∈(P) f 4 (Π, P t ) line is programmable. BFGP implements this particular node expansion procedure because it guarantees that duplicate successors are not generated in the BFGP search-tree. In addition, this node expansion procedure induces a tractable branching factor of (|A Z | + (n − 2) × 4); at a given program line BFGP can only program a planning action in A Z or a goto instruction that can jump to n − 2 different destination program lines, and that is conditioned by any of the four different features in L. The depth of the search tree developed by the BFGP algorithm is the number of program lines n, since only an undefined line can be programmed. Before a new search node is inserted into the open list, the corresponding partially specified program Π is executed on all the classical planning instances in P. This execution is implemented by the node evaluation procedure of the BFGP algorithm, and it can result in the three following different outcomes: 1. Π is a solution for P. If the execution of Π solves all the instances P t ∈ P, then search ends, and Π will be returned as a valid solution for the GP problem P. 2. Π fails to solve P. If the execution of Π on a given instance P t ∈ P fails, this means that the search node corresponding to the partially planning program Π is a dead-end. The search node will be discarded, so Π is not inserted into the open list. 3. Π may still be a solution for P. This means that the execution of Π on some of the classical planning instances in P reached an undefined program line (Π might solve some of the instances in P). As a consequence Π is inserted into the open list, at its corresponding position according to the value of our evaluation/heuristic functions explained above. Example. Let us recover from the previous example the GP problem P = {P 1 , P 2 }, and the partially specified program Π = 0.swap(i,j) 1.inc(i) 2.dec(j) 3.goto(2,!(y z ∧ ¬y c )) 4... 5.end, where lines [0, 3] are programmed and only line 4 is unspecified. Imagine now that BFGP extracts this program from the open list because it has the best evaluation value. In this case, the previous execution of Π on the classical planning instances P 1 and P 2 , implemented by the node evaluation procedure, ended in both instances at the undefined program line 4. This means that the only programmable line is 4. Assuming that two pointers are available (i.e. Z = {i, j}) we can program any of following twelve actions in line 4. {inc(i), inc( j), dec(i), dec( j), cmp(i, j), set(i, j), set( j, i), test(vector(i)), test(vector( j)), cmp(vector(i), vector( j)), swap(i, j), swap( j, i) }. A goto can only be programmed after a RAM action, which is not the case of line 4, since line 3 has another goto instruction. In other words the search node corresponding to the partially specified program from the previous example would have a total of twelve children nodes that could be added to the open list. In the hypothetical case that previous line 3. would contain a RAM action, a goto instruction for jumping to lines [0, 3] conditioned by the corresponding four features in L could also be programmed at line 4. Theoretical properties Theorem 4 (Termination). Given a generalized planning problem P, a finite set of pointers Z, and a finite number of program lines n, the execution of the BFGP algorithm always terminates. Proof. By definition of the expansion procedure of the BFGP algorithm (i), only unspecified lines can be programmed and (ii), any children program always has one more line programmed than its parent. This means that BFGP increases the number of programmed lines, until all lines are programmed. When all lines are programmed BFGP necessarily terminates, either by succeeding to solve P, or by failing to solve some of the classical planning instances in P. The only possible cause for the non-termination of the BFGP algorithm would be that BFGP could generate duplicate search nodes, allowing the infinite re-opening of an already discarded node. Again by definition of the expansion procedure of the BFGP algorithm, the re-opening of an already discarded node is impossible; BFGP only allows programming the maximum undefined program line that is reached after the execution of that program on all the instances in P. Theorem 5 (Completeness). Given a GP problem P, a maximum number of pointers |Z|, and maximum number of program lines n, if there is a planning program Π within these bounds that solves P, then the BFGP algorithm can compute it. Proof. The BFGP algorithm implements a complete enumeration of the entire space of planning programs with a maximum number of pointers |Z| and maximum number of program lines n except: (i), a search node was identified as a dead-end or (ii), the ancestor of a search node was identified as a dead-end. BFGP is safe because it only discards a search node when its corresponding partially specified planning program failed to solve an input planning instance (which is actually the definition for not being a GP solution). Furthermore, if a partially specified planning program failed to solve an input planning instance, any planning program that can be built programming the remaining undefined program lines will also fail to solve that problem. Theorem 6 (Soundness). If the execution of the BFGP algorithm on a GP problem P outputs a generalized plan Π, this means that Π is a solution for P. Proof. The BFGP algorithm runs until: (i) the open list is empty, which means that search space is exhausted without finding a solution and no generalized plan is output or (ii), BFGP extracted from the open list a planning program whose execution, in all the classical planning instances P t ∈ P, resulted successful. This is actually the definition of a solution for a GP problem. Proof. The BFGP algorithm is an implementation of a BFS, whose memory and time complexity are characterized as O(b d ), where b denotes the branching factor and d denotes the depth of the corresponding search tree. The branching factor of the search tree induced by the BFGP algorithm is the number of different instructions that can be programmed at an undefined program line, which is b ≤ |A Z | + (n − 2) × 4; gotos can only be programmed after RAM operations. The depth of the search tree induced by the BFGP is given by the maximum number of program lines n. With respect to solution quality BFGP does not guarantee that the computed planning programs are optimal. BFGP can however compute optimal planning programs when run in anytime mode. In this case we can use f 6 (Π, P) to rank GP solutions according to their execution cost in the classical planning instances that are comprised in the given GP problem (e.g. to prefer a sorting program with smaller computational complexity). Evaluation This section evaluates the empirical performance of our GP as heuristic search approach. All experiments are performed in an Ubuntu 20.04 LTS, with AMD® Ryzen 7 3700x 8-core processor × 16 and 32GB of RAM 4 . Benchmarks We report results in nine different domains; two propositional domains and seven numerical domains. In the propositional domains the functions Φ that induce the state variables are Boolean. In the numerical domains these functions are positive numeric functions. Next we provide more details on the nine domains: • Gripper, a robot must pick all balls from room A and drop them in room B. • Visitall, starting from the bottom-left corner of a squared grid, an agent must visit all cells. • Corridor, an agent moves from an arbitrary initial location to a destination location in a corridor. • Fibonacci, compute the n th term of the Fibonacci sequence. • Find, counts the number of occurrences of a specific value in a list. • Reverse, for reversing the content of a list. • Select, find the minimum value of a list. • Sorting, for sorting the values of a vector. • Triangular Sum, compute the n th triangular number. Gripper and Visitall are propositional, the remaining seven domains are numeric. For each domain, we build a GP problem that comprises ten randomly generated classical planning instances 5 . In the case of the gripper domain, instances go from 2 to 11 balls in room A to be dropped in room B; in visitall instances are squared grids ranging from 2 × 2 to 11 × 11 cells; corridor have instances from length 3 to 12; fibo and triangular sum instances range from the 2 nd to the 11 th number in the sequence; last find, reverse, select and sorting have instances with vectors from size 2 to 11 that are initialized with random content. The result of arithmetical operations in these domains is bounded to 10 2 in the synthesis of GP solutions, and to 10 9 in the validation of GP solutions. All domains include the base RAM instructions {inc(z 1 ), dec(z 1 ), cmp(z 1 , z 2 ), set(z 1 , z 2 ) | z 1 , z 2 ∈ Z}, such that z 1 and z 2 are pointers of the same sort, and the RAM instructions {test(φ( − → z 1 )), cmp(φ( − → z 1 ), φ( − → z 2 )) | − → z 1 , − → z 2 ∈ Z ar(φ) }, for each function φ ∈ Φ and where function parameters and pointers must also be of the same sort. We recall that compare instructions are only defined for numeric functions. In addition, each domain contains the regular planning action schemes that do not affect the FLAGS. • Propositional domains. The gripper domain includes the following three action schemes; move(z 1 , z 2 ) to denote the robot is moving from the room pointed by z 2 to the one pointed by z 1 , pick(z 1 , z 2 , z 3 ) to pick the ball pointed by z 1 , at room pointed by z 2 , and with the gripper pointed by z 3 , and drop(z 1 , z 2 , z 3 ), to drop ball z 1 at room z 2 with gripper z 3 . Visitall only needs one action schema to visit a cell addressed by two pointers, of type row and column respectively, e.g. visit(z 1 , z 2 ). • Numerical domains. The triangular sum and Fibonacci domains include the action schemes vector-inc(z 1 ) and vector-dec(z 1 ), to increase and decrease by one the content of the vector at z 1 , and the action scheme vector-add(z 1 , z 2 ) for adding the value at z 2 to the content of the vector at z 1 . Similarly, corridor needs two action schemes, vector-left(z 1 ) and vector-right(z 1 ), to increase or decrease by one the location of the agent. Select only requires the base RAM instructions. Find includes the accumulate(z 1 ) action schema for counting the number of occurrences of the target element. Reverse and Sorting include the swap(z 1 , z 2 ) action scheme to swap the values addressed by z 1 and z 2 . We model the regular planning actions as they are always executable but that their effects only update the current state iff their preconditions hold in the current state. Otherwise the execution of an action has no effect. Synthesis of GP Solutions Here we present two different experiments to evaluate the performance of the BFGP algorithm in the given benchmark. First, we asses every evaluation/heuristic function f i by running BFGP( f i ). Second we search for the best combination of two evaluation/heuristic functions, where one is structured-based and the other performance-based. Table 1 details the results of the first synthesis experiment where the BFGP algorithm uses each of our nine different evaluation/heuristic functions (the computation bounds are 3, 600 seconds of CPU-time and 32GB of memory and best results are shown in bold). Regarding structure-based functions f 2 dominates in all domains and metrics (except in the find domain where f 3 is faster) and it also has the highest coverage solving 8 out of 9 domains ( f 1 , f 3 and f 7 have lower coverage failing in the same three problems, namely corridor, gripper and sorting). Regarding performance-based functions, there is not a strictly dominant one since the best scores are shared among f 4 , f 5 and f 9 . Function f 4 has the lowest memory consumption but could not solve gripper and sorting; f 9 is the best for solving corridor, but it is unable to solve gripper within the time bound; and f 5 is the function with the least number of expanded nodes in more than half of domains, in addition to the best coverage solving all problems. functions. In the rest of domains, there are at least 4 or more functions that do not solve them, such as gripper which is the least solved domain (only f 2 and f 5 solve it), and sorting which is solved by f 5 , f 8 and f 9 but it is the hardest in terms of time average. Figure 10 shows the programs computed by BFGP( f 5 ). In Corridor there are two pointers, i pointing to the agent location and j pointing to the target location; the solution moves the agent right until it surpasses the target location, then it moves the agent left until it reaches the target location. In Fibonacci, pointers a and b are used to compute the n-th Fibonacci number, where a addresses the F a number to which F a−1 and F a−2 are added using b as an auxiliary pointer; and finishes when a reaches the n-th element (the last one). In Find, there is a pointer i to iterate over a vector, a pointer t which targets a value in the vector, and a counter pointer a whose address content is only increased by one every time an occurrence of the target t is found in the vector (Lines 1-2). The process repeats until i reaches the end of the vector. Performance of BFGP( f i ) The synthesized solutions The Gripper solution uses one pointer for balls (b 1 ), two for rooms (r 1 and r 2 ) and one for grippers g 1 ; for each ball b 1 , the agent will pick it up from room r 1 (always room A) using gripper g 1 (always left gripper), sets r 2 to room B, moves from A to B, drop ball b 1 at room B, goes back to room A, and continues with the next ball. The Reverse domain uses two pointers i and j and finds a solution with O(n 2 ) complexity of a vector of size n; it moves all values from j to n − 1 indexes one location to right and places the last element in the j-th location, using i as an auxiliary pointer; then increases j by one until it reaches the end of the vector. The Select domain has two pointers a and b; it iterates over the vector with pointer a, and assigns a to b every time the value pointed by a is smaller than the one pointed by b. The Sorting solution is succinct but complex to interpret; j always points to the first location, so all swaps are done with this location; then, two cases may happen when reaching Line 3: either the first element was wrongly sorted in the previous swap and detected because the i-th value is larger than the first, so all values from the i-th to the n-th location are shifted one place to their right, the first element is placed in the i-th location (now is correctly order with respect to i + 1 value) and the last is placed first (defined in Lines 0-6), and continues again in Line 3; or the largest value of the vector is in the first location and the rest are sorted in increasing order, so the problem can be solved by shifting all values once to their left applying instructions at Lines 3, 4, 7, 8, and 9 in sequence. In Triangular Sum, b points to the i th number in the sequence and a to value n, then a is added to b, a is decreased by one, and the whole process repeats until the value a is pointing is 0. The last domain, Visitall, has two pointers i and j that are used for rows and columns, respectively; since the agent always starts in the bottom-left corner, it visits all i-th cells for a given j; then it moves back to the first row, increases the columns ( j) by one, and repeats the procedure until all columns have been explored. Validation of the synthesized solutions Here we validate the BFGP( f 5 ) solutions of Figure 10 with a larger and harder set of instances. Table 3 reports the CPU time, and peak memory, yield when running the solutions synthesized by BFGP( f 5 ) on a validation set. All the solutions synthesized by BFGP( f 5 ) were successfully validated, besides Reverse with infinite detection mode that ran out of memory. The largest CPU-times and memory peaks correspond to the configuration that implements the detection of infinite programs, which requires saving states to detect whether they are revisited during execution. Skipping this mechanism allows to validate terminating programs much faster [42]. In the validation set, each state variable from the planning domain is bounded by 10 9 , instead of 10 2 which was the synthesis bound. Corridor and Gripper are validated over 1,000 instances, where for each n ∈ [12, 1,011], the first has random initial and goal locations below n, and the second n balls initially in room A. Fibonacci has a validation set of 33 instances, ranging from the 11 th Fibonacci term to the 43 rd , i.e. the integer 701,408,733 (the 44 th number would overflow the validation bound). The solutions for Reverse, Select, and Find domains, are validated on 102 instances each, with vector sizes ranging from 1,000 to 11,100, and random integer elements bounded by 10 9 . Similarly, Sorting has 100 validation instances with vectors of random integers, but their sizes range from 12 to 111. The solution for Triangular Sum is validated over 44,709 instances, the last one corresponding to the 44,720 th term in the sequence, i.e. the integer 999,961,560 (as in Fibonacci, the next number would overflow). In Visitall, there are 50 validation instances with squared grids range from 12 × 12 to 61 × 61. Performance of BFGP with function combinations Interestingly, the base performance of BFGP with a single evaluation/heuristic function is improved combining both structural and cost-to-go information; we can guide the search of BFGP with a cost-to-go heuristic function and break ties with a structural evaluation function, and vice versa. Thus, we run all configurations of BFGP( f i , f j ) and BFGP( f j , f i ) such that f i ∈ { f 1 , f 2 , f 3 , f 7 } and f j ∈ { f 4 , f 5 , f 6 , f 8 , f 9 }, and select the configuration that solves all domains and with the best average time. There are 40 BFGP( f i , f j )/BFGP( f j , f i ) configurations, but only BFGP( f 5 , f 3 ) and BFGP( f 5 , f 7 ) are able to solve all domains. The performance of these two configurations is then compared against BFGP( f 5 ), since it is the only single evaluation/heuristic function that solve all domains in the previous experiment. that combining goal-oriented functions with structural-based functions that measure the syntactic complexity of a program, in that specific order, is the best configuration. We also compared the performance of BFGP( f 5 , f 7 ), in terms of CPU-time, with the compilation-based approach for GP [91,34]. The compilation-based approach, that we named PP, computes planning programs, following a topdown strategy, with the planner LAMA-2011 (first solution setting) to solve the classical planning problems that result from the compilation. Table 5 summarizes the results of this comparison. There are 3 domains where PP is faster than BFGP( f 5 , f 7 ), but in these domains the GP problems addressed by PP are easier: i) Gripper in PP has the same action move for both directions and picks are only available for the next ball, while in BFGP actions are parameterized with pointers, so it first needs to find that pointers r 1 and r 2 point to rooms A and B respectively, pick balls only from room r 1 , move from r 1 to r 2 to drop the ball, and move backwards from r 2 to r 1 ; ii) Reverse in PP has one of the pointers in the last position of vector from the initial state, reproducing this setting in BFGP a program of 6 lines is found in less than 1 second after expanding 260 nodes and evaluating 2,560 nodes, however, it is more interesting to us a solution that synthesizes where to place and how to use each pointer, even though it is a harder problem; and iii) Triangular Sum in PP just accumulates one variable to another one, while in BFGP the pointers should point to the right variables, then use them. In the rest of domains, BFGP dominates PP, even though programs are larger, BFGP must reason on how to use the pointers, and BFGP uses more instances with larger values, being able to solve domains where PP dies because of the grounding among other reasons. Validation of GP solutions for more complex domains Here we present several GP benchmarks, with known polynomial time solutions, but that result too complex for our current BFGP algorithm (within the given time and memory bounds). Our aim is showing that our approach is expressive enough to represent solutions to GP problems coming from IPC planning domains, noise-free supervised classification tasks, and numerical domains. These solutions are succinctly represented as planning programs, instead of long sequences of grounded actions for large problems, and validated efficiently without being affected by the grounding methods of planners. • Blocks Ontable, towers of blocks where all blocks must be placed on the table. • Grid, an agent has to move from an arbitrary location to a destination one in a 2D grid. • Miconic, is an elevator problem where passengers at origin, wait for the elevator to enter, and then served at their destination floor. • Michalski Trains, is a classic of relational supervised machine learning. A binary noise-free classification task with 10 trains that either go east or west, and multiple features such as the number of wheels, wagons, or their shape for each train among others. The goal is to learn the features that classify all trains in the right direction. • Satellite, consists of taking images of different targets with instruments that are boarded in satellites. In addition, instruments need to be calibrated and in specific modes for taking each image; and each satellite has only power for one instrument at a time, so it needs to switch the current instrument off, switch on the next and calibrate it, before using a new instrument for taking images. • Sieve of Erathostenes, is a method to find prime numbers up to a certain bound using only additive and iterative mechanisms. • Spanner, consists of tighten all loose nuts at the end of a corridor, with the picked spanners along the corridor. Spanners can only be used once, and when the agent moves to the next room it can not go back, so if there are unpicked spanners in visited rooms the task could become unsolvable. Figure 11 shows the hand-coded solutions for these benchmarks. In Blocks Ontable, given n blocks the complexity of the solution is cubic, i.e. O(n 3 ), where it searches n times, every o1 block that is on top of an o2 block, then unstack and put o1 down on the table. In Grid, the agent moves to the bottom left corner, then each coordinate is increased by one while they are smaller than their goal, visiting the resulting coordinate. In Spanner, an agent picks up all available spanners in location l1, walks to the next l2 location and repeats the process until it reaches the last location (the gate), collecting all spanners on its path; once in the gate, it tightens each loose nut with a spanner. The solution to Michalski Trains is summarized as, each train t1 will go east if it has a car which is closed and short, otherwise it will go west. In Sieve of Eratosthenes all numbers are initially classified as primes, and it should decide whether they are not; so it iterates over i and uses j and k as auxiliary pointers, where the first acts as a counter that ranges from 0 to i, and second adds up to the next multiple of i, i.e. k % i = 0; then every k-th number will be set to no prime, i is increased by one and the process repeats until i reaches the last element. In Miconic, the elevator always starts in the first floor f 1, so for every floor it boards and departs passenger p1 whether possible; once it reaches the last floor, all passengers are either served or in the elevator, so it will serve all possible passenger in each floor while it goes down, until the first floor is reached again. The last domain, Satellite, is the most complex because it requires iteration over multiple variable types, i.e. satellites, instruments, modes and directions. The solution to this domain consists of switching off all instruments and turning all satellites to the first direction; then for each satellite, the i1 instrument is switched on, 28 calibrated with its calibration target direction d2, and used to take images of every direction d2 in every mode m1; once it finishes, the satellite turns to the first direction d1 again, switches off the current instrument, and continues with the next one, until all satellites have used all their instruments. We get one main take away lesson from the analysis of Figure 11 solutions; solutions have common high-level structures, that either iterate over all combinations of variable types (i.e. Blocks, Miconic, Satellite, . . . ) or build a complex logic query (i.e. Michalski Trains). This suggests that planning programs may be synthesized more efficiently using predefined structures (such as FOR or IF-THEN-ELSE constructs) although this is out of the scope of this paper. Table 6 shows the validation results in complex domains, where validation without infinite detection scales much better again, and all domains are successfully validated (besides Satellite with infinite detection mode that runs out of memory). Blocks Ontable can be solved with 13 lines and 3 pointers, and the validation set consists of 20 instances that range from 12 to 31 blocks. Grid requires 14 lines of code and 2 pointers, and it is validated with 248 instances with grids between 5 × 5 and 66 × 66 size. Miconic needs 20 lines and 3 pointers, and 20 instances that validates from 12 floors and 18 passengers to 31 floors and 46 passengers. Michalski Trains uses 15 lines and 6 pointers to classify all the trains in the unique classical task with 10 trains and their features. Satellite is by difference the most complex in terms of required lines and pointers, which are 43 and 5, respectively. Its validation set consist of 20 instances, starting with 12 satellites, 24 instruments and modes, and 48 directions, and finishing with 31 satellites, 62 instruments and modes and 124 directions. Sieve of Erathostenes requires 16 lines and 3 pointers to classify either as prime or non-prime, all the numbers comprised in the first 111 natural numbers. Spanner, uses 14 lines and 5 pointers to solve all 20 instances of the validation set, that range from 18 spanners and nuts and a corridor with 14 locations, to 46 spanners and nuts and a corridor with 33 locations. Table 6: Validation of complex domains, CPU-time (secs) and memory peak for program validation, with/out infinite program detection. ME stands for memory exceeded. Best results in bold. Conclusions We believe this work is a step-forward towards building a stronger connection between the areas of automated planning and programming. The work presented a formalization of classical planning as a vector transformation task, which is a common programming task. According to this formalism, computing a sequential plan for this tasks is computing a composition of vector transformation operations. Likewise computing a generalized plan is computing an algorithmic expression of the vector transformations. With the aim of building more bridges between automated planning and programming, we are exploring the extension of our approach to GP problems that include real state variables. We believe that we can address this kind of GP problems by introducing the notion of precision for the comparison of real variables, and redefining accordingly our mechanism for the update of the FLAGs registers. Another interesting research direction is the extension of our GP as heuristic search approach for computing generalized plans starting from different input settings. For instance, the computation of generalized plans from a set of plan traces that demonstrates how to solve several planning problems. We are also interested on exploring the application of our GP as heuristic search approach to planning problems that are not goal-oriented, where the objective is to maximize a given utility function [92]. In this particular setting, ideas from approximated policy iteration [93], and reinforcement learning [68], could be incorporated to our framework. On the other hand, the BFGP algorithm starts from the empty program, but nothing prevents us from starting search from a partially specified generalized plan [94] with the aim of developing online approaches to GP. In fact, local search approaches have already shown successful for planning [95] and program synthesis [96,57]. Our cost-to-go heuristics are still less informed than the current heuristics for classical planning, in the sense that our heuristics only consider the goals that are explicitly provided in the problem representation. A clear example is f 5 (Π, P t ), that builds on top of the Euclidean distance, and that for Strips planning problems is actually a goal counter. We believe that better estimates may be obtained by building on top of the powerful ideas of modern planning heuristics [97,26,98]. In more detail, a promising approach for the development of more informative heuristics for GP is to consider sub-goals, that are not explicit given in the problem representation [99]. For instance sets of sub-goals can be discovered as a pre-processing step, without grounding, regarding the set of relevant atoms that are traversed by the polynomial IW(1) algorithm, when achieving individual goals [100]. Since we are approaching GP as a classic tree search, a wide landscape of effective techniques, coming from heuristic search and classical planning, can actually improve the base performance of our approach. We mention some of the more promising ones. Large open lists can be handled more effectively splitting them in several smaller lists [26]. Delayed duplicate detection could be implemented to manage large closed lists with magnetic disk memory [101]. Further, more sophisticated mechanism can be implemented for handling closed nodes. For instance, once a search node is cancelled (e.g. because f i (Π, P) identified that the planning program fails on a given instance), any program equivalent to this node should also be cancelled, e.g. any program that can be built with transpositions of the causally-independent instructions. Given that the dept of the search-tree is bounded, techniques coming from SAT/CSP/SMTs, such a non-chronological backtracking, limited discrepancy search [102], or taboo search [103], might also result effective to improve our approach. Last, SATPLAN planners exploit multiple-thread computing to parallelize search in solution spaces with different bounds [89]. This same idea could be applied to multiple searches for GP solutions with different program sizes. • inc(r) increments the content of the register r by 1, i.e. ([r] + 1) → [r]. Likewise, dec(r) decrements the content of the register r, i.e. ([r] − 1) → [r]. Figure 3 : 3Boolean function constraint sorted that implements a constraint for validating whether the vector of state variables is sorted in increasing order. The constraint is implemented leveraging the single pointer i over the objects in Ω; vector(i) is interpreted as vector(o i ) ≡ x i ∈ X. Figure 4 : 4Pointer-based representation of the preconditions and the effects of the swap action schema. When applicable, the swap action schema exchanges the value of the state variables indexed by its two parameters, the pointers i and j. Figure 6 : 6The unstack Strips operator from the blocksworld domain represented in the PDDL language. Figure 7 : 7The unstack action schema from blocksworld defined with two pointers (i and j). Figure 8 : 8ontable(b1)=1 and ontable(b2)=1 and ontable(b3)=1); } The init and goal procedures for representing the Strips planning problem of unstacking the three-block tower ofFigure 5. Figure 9 : 9Two examples of generalized plans: (left) for reversing a list; (right) for sorting a list with the selection-sort algorithm. Theorem 7 ( 7Time and Memory). The time and memory consumption of the BFGP algorithm are characterized by the big-Oh expression O((|A Z | + (n − 2) × 4) n ). Figure 10 : 10Solutions computed by BFGP( f 5 ). In this state there are three blocks, Ω = {b1, b2, b3}, that are stacked in a single tower. Predicates clear(?x), holding(?x), and ontable(?x), are encoded as three different Boolean functions that map each vector of objects to either 0 or 1 in the current state. Omitted state variables are assumed to be zero valued. Our vector X of state variables is the result of 13 Figure 5: Example of a three-block state from the blocksworld (left), and its corresponding representation as a vector of bits (right).1 2 3 State representation Predicate Strips Boolean functions (clear ?x) (clear b1) clear(b1)=1 (handempty) (handempty) handempty()=1 (holding ?x) - - (on ?x ?y) (on b1 b2) (on b2 b3) on(b1,b2)=1, on(b2,b3)=1 (ontable ?x) (ontable b3) ontable(b3)=1 (:action unstack :parameters (?x ?y) :precondition (and (clear ?x) (handempty) (on ?x ?y)) :effect (and (holding ?x) (clear ?y) (not (clear ?x)) (not Table 2 2summarizes the results from Table 1, grouping results by domains and averaging the metrics by the total number of functions that solved each domain. There are 6 domains which are solved by all the nine evaluation/heuristicDomain n, |Z| f 1 f 2 f 3 Time Mem. Exp. Eval. Time Mem. Exp. Eval. Time Mem. Exp. Eval. Corridor 8, 2 TO - - - 1,367 4 4.2M 4.2M TO - - - Fibonacci 7, 2 779 715 2.9M 5.1M 115 4 0.5M 0.5M 1,960 1,118 8.0M 10.0M Find 6, 3 32 18 0.2M 0.2M 31 4 0.2M 0.2M 23 23 0.1M 0.1M Gripper 8, 4 TO - - - 2,968 4 13.1M 13.1M TO - - - Reverse 7, 2 317 192 1.4M 2.1M 10 4 47.9K 48.0K 224 235 0.3M 1.5M Select 7, 2 192 96 0.8M 1.1M 15 4 82.8K 82.8K 98 97 0.2M 0.6M Sorting 11, 2 TO - - - TO - - - TO - - - T. Sum 6, 2 40 58 0.2M 0.4M 2 4 14.7K 14.7K 38 75 61.2K 0.4M Visitall 8, 2 1,631 219 2.0M 2.8M 38 4 66.1K 66.2K 408 112 0.2M 0.7M Average 498.5 216.3 1.2M 1.9M 568.3 4.0 2.3M 2.3M 458.5 276.7 1.5M 2.2M f 4 f 5 f 6 Corridor 8, 2 1,521 5 4.2M 4.2M 970 80 1.9M 2.1M TO - - - Fibonacci 7, 2 147 4 0.5M 0.5M 206 203 0.2M 1.2M 2,798 1,779 11.0M 11.0M Find 6, 3 39 4 0.2M 0.2M 34 23 82.9K 0.2M 41 31 0.2M 0.2M Gripper 8, 4 TO - - - 10 18 9.9K 72.9K TO - - - Reverse 7, 2 13 4 48.3K 48.4K 690 356 2.5M 2.5M 651 380 2.5M 2.5M Select 7, 2 21 4 85.5K 86.5K 17 12 43.1K 68.3K 228 171 1.0M 1.1M Sorting 11, 2 TO - - - 2,693 110 1.5M 1.5M TO - - - T. Sum 6, 2 3 4 14.7K 14.7K 15 7 72.6K 78.1K 84 99 0.6M 0.6M Visitall 8, 2 53 5 67.5K 68.1K 3 5 683 2.0K 2,474 365 2.8M 2.8M Average 61.1 4.3 0.7M 0.7M 515.3 90.4 0.7M 0.9M 1,046.0 470.8 3.0M 3.0M f 7 f 8 f 9 Corridor 8, 2 TO - - - 1,317 368 2.1M 2.8M 857 94 1.8M 2.0M Fibonacci 7, 2 789 716 2.9M 5.1M 251 264 0.2M 1.5M 157 195 0.1M 1.1M Find 6, 3 32 18 0.2M 0.2M 39 22 0.2M 0.2M 34 21 0.2M 0.2M Gripper 8, 4 TO - - - TO - - - TO - - - Reverse 7, 2 336 197 1.5M 2.1M 662 339 2.5M 2.5M 677 339 2.5M 2.5M Select 7, 2 200 99 0.8M 1.1M 225 123 1.0M 1.1M 17 12 57.2K 68.4K Sorting 11, 2 TO - - - 2,711 95 1.5M 1.5M 2,820 95 1.5M 1.5M T. Sum 6, 2 39 58 0.2M 0.4M 15 8 72.6K 78.1K 15 8 72.6K 78.1K Visitall 8, 2 846 196 1.2M 1.7M 2,714 344 2.8M 2.8M 60 10 41.2K 67.3K Average 373.7 214.0 1.1M 1.8M 991.8 195.4 1.3M 1.6M 579.6 96.8 0.8M 0.9M Table 1: We report the number of program lines n, and pointers |Z| per domain, and for each evaluation/heuristic function, CPU (secs), memory peak (MBs), and the numbers of expanded and evaluated nodes. TO stands for Time-Out (>1h of CPU). Best results in bold. Table 2 : 2We report for each domain, the time (secs), memory peak (MBs), and expanded and evaluated nodes averaged by the number of functions that solved the domain in Table 1. Table 4 4summarizes that comparison, showing that BFGP( f 5 ) is improved in every domain either by BFGP( f 5 , f 3 ) or BFGP( f 5 , f 7 ). Furthermore, BFGP( f 5 , f 7 ) has the best average performance in all four metrics, empirically provingDomain Instances Time ∞ Mem ∞ Time Mem Corridor 1,000 0.54 6.5MB 0.43 6.3MB Fibonacci 33 0.01 4.8MB 0.01 4.7MB Find 102 1,542.85 2.2GB 1.77 0.3GB Gripper 1,000 93.5 0.5GB 4.77 0.5GB Reverse 102 ME ME 3,553.16 0.5GB Select 102 1,407.54 2.5GB 1.87 0.3GB Sorting 100 230.98 0.5GB 17.19 8.8MB Triangular Sum 44,709 3,244.84 0.1GB 2,357.56 0.1GB Visitall 50 42.21 0.2GB 0.33 48MB Table 3 : 3Validation set, CPU-time (secs) and memory peak for program validation, with/out infinite program detection. ME stands for memory exceeded. Best results in bold.Domain BFGP( f 5 , f 3 ) BFGP( f 5 , f 7 ) BFGP( f 5 ) Time Mem. Exp. Eval. Time Mem. Exp. Eval. Time Mem. Exp. Eval. Corridor 675 55 1.8M 2.0M 658 64 1.7M 1.9M 970 80 1.9M 2.1M Fibonacci 990 553 2.7M 4.3M 55 68 43.4K 0.4M 206 203 0.2M 1.2M Find 29 16 68.5K 0.1M 38 14 0.2M 0.2M 34 23 82.9K 0.2M Gripper 9 17 9.7K 67.4K 7 13 8.7K 50.1K 10 18 9.9K 72.9K Reverse 702 217 2.5M 2.5M 676 184 2.5M 2.5M 690 356 2.5M 2.5M Select 14 10 32.0K 54.0K 17 9 47.6K 68.3K 17 12 43.1K 68.3K Sorting 2,484 39 1.3M 1.4M 2,710 82 1.5M 1.5M 2,693 110 1.5M 1.5M T. Sum 15 7 72.6K 78.0K 15 6 72.6K 78.1K 15 7 72.6K 78.1K Visitall 2 5 275 605 3 5 582 1.7K 3 5 683 2.0K Average 546.7 101.9 0.9M 1.2M 464.3 49.4 0.7M 0.8M 579.6 96.8 0.8M 0.9M Table 4 : 4For each domain we report, CPU time (secs), memory peak (MBs), num. of expanded and evaluated nodes of BFGP( f 5 , f 3 ), BFGP( f 5 , f 7 ) and BFGP( f 5 ). TO means time-out (> 1h of CPU). Best results in bold. DomainPP in sec. BFGP( f 5 , f 7 ) in sec.Corridor - 658 Fibonacci 3,570 55 Find 274.86 38 Gripper 1 7 Reverse 87.86 676 Select 204.20 17 Sorting - 2,710 Triangular Sum 0.85 15 Visitall - 3 Table 5 : 5Computing CPU-time (secs) for solving domains in the GP compilation approach (PP) and BFGP( f 5 , f 7 ). We adopt the convention of jumping to line i whenever y is false, following the JMP instructions in the Random-Access Machine that jump when a register equals zero.8 cmp(φ( − → z 1 ), φ( − → z 2 )) instructions are only defined for numeric functions. At least Z must contain as many pointers as the largest arity of the functions Φ and action schemes Ξ of the given domain. The source code, evaluation scripts, and used benchmarks can be found at: https://github.com/aig-upf/best-first-generalized-planning. For reproducibility reasons we fix the random seed to generate the classical planning instances in the GP problems. min , i ) 1. cmp ( vector ( j ) , vector ( min )). ( min , i ) 1. cmp ( vector ( j ) , vector ( min )) ¬(¬y z ∧ ¬y c )) 3. set ( min , j ) 4. swap (i , min ) 5. inc ( j ) 6. cmp ( length , j ) 7. goto (1 , ¬(y z ∧ ¬y c )) 8. inc ( i ) 9. set (j , i ) 10. cmp ( length , i ) 11. goto (0 , ¬(y z ∧ ¬y cgoto (5 , ¬(¬y z ∧ ¬y c )) 3. set ( min , j ) 4. swap (i , min ) 5. inc ( j ) 6. cmp ( length , j ) 7. goto (1 , ¬(y z ∧ ¬y c )) 8. inc ( i ) 9. set (j , i ) 10. cmp ( length , i ) 11. goto (0 , ¬(y z ∧ ¬y c )) cmp ( vector ( i ) , vector ( j )). cmp ( vector ( i ) , vector ( j )) cmp ( vector ( i ) , vector ( j )). cmp ( vector ( i ) , vector ( j )) dec ( b ) 2. vector -add (a , b ) 3. set (b , a ) 4. inc ( a ) 5. goto (0 ,¬(y z ∧ ¬y c )) 6. end FIND 0. cmp ( vector ( i ) , vector ( t-add (a , b ) 1. dec ( b ) 2. vector -add (a , b ) 3. set (b , a ) 4. inc ( a ) 5. goto (0 ,¬(y z ∧ ¬y c )) 6. end FIND 0. cmp ( vector ( i ) , vector ( t )) accumulate ( a ) 3. inc ( i ) 4. goto (0 ,¬(y z ∧ ¬y c )) (i , j ) 1. swap (i , j ) 2. inc ( i ) 3. goto (1 ,¬(y z ∧ ¬y c )) 4. inc ( j ) 5. goto (0 ,¬(y z ∧ ¬y c )) 6. end SELECT 0. 2inc ( b ) 1. cmp ( vector ( a ) , vector ( b )goto (3 ,¬(y z ∧ ¬y c )) 2. accumulate ( a ) 3. inc ( i ) 4. goto (0 ,¬(y z ∧ ¬y c )) (i , j ) 1. swap (i , j ) 2. inc ( i ) 3. goto (1 ,¬(y z ∧ ¬y c )) 4. inc ( j ) 5. goto (0 ,¬(y z ∧ ¬y c )) 6. end SELECT 0. inc ( b ) 1. cmp ( vector ( a ) , vector ( b )) cmp ( vector ( i ) , vector ( j )). cmp ( vector ( i ) , vector ( j )) -add (b , a ) 2. vector -dec ( a ) 3. test ( vector ( a )). -add (b , a ) 2. vector -dec ( a ) 3. test ( vector ( a )) visit (i , j ) 1. inc ( i ) 2. goto (0 ,¬(y z ∧ ¬y c )) 3. dec ( i ) 4. goto (3 ,¬(y z ∧ ¬y c )) 5. inc ( j ) 6. goto (0 ,¬(y z ∧ ¬y c )) ( i ) 1. goto (0 ,¬(y z ∧ ¬y c )) 2. dec ( j ) 3. 5. end VISITALL 0goto (0 ,¬(y z ∧ ¬y c )) 4. test ( goal -xpos ( i )goto (0 ,¬(y z ∧ ¬y c )) 5. end VISITALL 0. visit (i , j ) 1. inc ( i ) 2. goto (0 ,¬(y z ∧ ¬y c )) 3. dec ( i ) 4. goto (3 ,¬(y z ∧ ¬y c )) 5. inc ( j ) 6. goto (0 ,¬(y z ∧ ¬y c )) ( i ) 1. goto (0 ,¬(y z ∧ ¬y c )) 2. dec ( j ) 3. goto (0 ,¬(y z ∧ ¬y c )) 4. test ( goal -xpos ( i )) ¬(y z ∧ ¬y c )) 6. inc ( i ) 7. goto (4 ,¬(y z ∧ ¬y c )) 8. test ( goal -ypos ( j )). goto (8 ,¬(y z ∧ ¬y c )) 6. inc ( i ) 7. goto (4 ,¬(y z ∧ ¬y c )) 8. test ( goal -ypos ( j )) ¬(y z ∧ ¬y c )) 12. visit (i , j ) 13. end SPANNER 0. walk ( l1 , l2 , m1 ) 1. set ( l1 , l2 ) 2. pi ckup_sp anner ( l1 , s1 , m1 ) 3. inc ( s1 ) 4. goto (2 ,¬(y z ∧ ¬y c )) 5. dec ( s1 ) 6. goto. 10. inc ( j ) 11. goto (812inc ( l2 ) 8. goto (0 ,¬(y z ∧ ¬y cgoto (12 ,¬(y z ∧ ¬y c )) 10. inc ( j ) 11. goto (8 ,¬(y z ∧ ¬y c )) 12. visit (i , j ) 13. end SPANNER 0. walk ( l1 , l2 , m1 ) 1. set ( l1 , l2 ) 2. pi ckup_sp anner ( l1 , s1 , m1 ) 3. inc ( s1 ) 4. goto (2 ,¬(y z ∧ ¬y c )) 5. dec ( s1 ) 6. goto (5 ,¬(y z ∧ ¬y c )) 7. inc ( l2 ) 8. goto (0 ,¬(y z ∧ ¬y c )) tighten_nut ( l1 , s1 , m1 , n1 ) 10. inc ( s1 ) 11. inc ( n1 ) 12. goto. tighten_nut ( l1 , s1 , m1 , n1 ) 10. inc ( s1 ) 11. inc ( n1 ) 12. goto (9 ,¬(y z ∧ ¬y c )) ¬(¬y z ∧ y c )) 7. inc ( j ) 8. cmp (i , j ) 9. goto (5 ,¬(y z ∧ ¬y c )) 10. set -no -prime ( k ) 11. cmp (i , j ) 12. goto (3 ,¬(¬y z ∧ y c )) 13. inc ( i ) 14. goto (2 ,¬(y z ∧ ¬y c )) 15. end MICONIC 0. board ( p1 , f1 ) 1. depart ( p1 , f1 ) 2. inc ( p1 ) 3. goto (0 ,¬(y z ∧ ¬y c )) 4. dec ( p1 ) 5. goto (4 ,¬(y z ∧ ¬y c )) 6. inc ( f2 ) 7. up ( f1 , f2 ) 8. inc ( f1 ) 9. goto (0 ,¬(y z ∧ ¬y c )) 10. dec ( f2 ) 11. down ( f1 , f2 ) 12. depart ( p1 , f2 ) 13. inc ( p1 ) 14. goto. 15. dec ( p1 ) 16. goto (15OF ERATHOSTENES 0. inc ( i ) 1. inc ( i ) 2. set (k , i ) 3. dec ( j ) 4. goto (3 ,¬(y z ∧ ¬y c )) 5. inc ( k ) 6. goto. 13dec ( f1 ) 18. goto (10 ,¬(y z ∧ ¬y c )) ( i1 , s1 ) 14. test ( cal_target ( i1 , d2 )OF ERATHOSTENES 0. inc ( i ) 1. inc ( i ) 2. set (k , i ) 3. dec ( j ) 4. goto (3 ,¬(y z ∧ ¬y c )) 5. inc ( k ) 6. goto (13 ,¬(¬y z ∧ y c )) 7. inc ( j ) 8. cmp (i , j ) 9. goto (5 ,¬(y z ∧ ¬y c )) 10. set -no -prime ( k ) 11. cmp (i , j ) 12. goto (3 ,¬(¬y z ∧ y c )) 13. inc ( i ) 14. goto (2 ,¬(y z ∧ ¬y c )) 15. end MICONIC 0. board ( p1 , f1 ) 1. depart ( p1 , f1 ) 2. inc ( p1 ) 3. goto (0 ,¬(y z ∧ ¬y c )) 4. dec ( p1 ) 5. goto (4 ,¬(y z ∧ ¬y c )) 6. inc ( f2 ) 7. up ( f1 , f2 ) 8. inc ( f1 ) 9. goto (0 ,¬(y z ∧ ¬y c )) 10. dec ( f2 ) 11. down ( f1 , f2 ) 12. depart ( p1 , f2 ) 13. inc ( p1 ) 14. goto (12 ,¬(y z ∧ ¬y c )) 15. dec ( p1 ) 16. goto (15 ,¬(y z ∧ ¬y c )) 17. dec ( f1 ) 18. goto (10 ,¬(y z ∧ ¬y c )) ( i1 , s1 ) 14. test ( cal_target ( i1 , d2 )) turn_to ( s1 , d2 , d1 ) 17. calibrate ( s1 , i1 , d2 ) 18. turn_to ( s1 , d1 , d2 ) 19. inc ( d2 ) 20. goto (14 ,¬(y z ∧ ¬y c )) 21. set ( d2 , d1 ) 22. take_image ( s1 , d2 , i1 , m1 ) 23. inc ( m1 ) 24. goto (22 ,¬(y z ∧ ¬y c )) 25. dec ( m1 ) 26. 16goto (25 ,¬(y z ∧ ¬y cgoto (19 ,¬(¬y z ∧ y c )) 16. turn_to ( s1 , d2 , d1 ) 17. calibrate ( s1 , i1 , d2 ) 18. turn_to ( s1 , d1 , d2 ) 19. inc ( d2 ) 20. goto (14 ,¬(y z ∧ ¬y c )) 21. set ( d2 , d1 ) 22. take_image ( s1 , d2 , i1 , m1 ) 23. inc ( m1 ) 24. goto (22 ,¬(y z ∧ ¬y c )) 25. dec ( m1 ) 26. goto (25 ,¬(y z ∧ ¬y c )) inc ( d1 ) 30. goto (22 ,¬(y z ∧ ¬y c )) 31. dec ( d1 ) 32. goto (31 ,¬(y z ∧ ¬y c )) 33. turn_to ( s1 , d1 , d2 ) 34. set ( d2 , d1 ) 35. switch_off ( i1 , s1 ) 36. inc ( i1 ) 37. goto (13 ,¬(y z ∧ ¬y c )) 38. dec ( i1 ) 39. goto. 27inc ( d2 ) 28. turn_to ( s1 , d2 , d1 ) 29.. inc ( s1 ) 41. goto (13 ,¬(y z ∧ ¬y c27. inc ( d2 ) 28. turn_to ( s1 , d2 , d1 ) 29. inc ( d1 ) 30. goto (22 ,¬(y z ∧ ¬y c )) 31. dec ( d1 ) 32. goto (31 ,¬(y z ∧ ¬y c )) 33. turn_to ( s1 , d1 , d2 ) 34. set ( d2 , d1 ) 35. switch_off ( i1 , s1 ) 36. inc ( i1 ) 37. goto (13 ,¬(y z ∧ ¬y c )) 38. dec ( i1 ) 39. goto (38 ,¬(y z ∧ ¬y c )) 40. inc ( s1 ) 41. goto (13 ,¬(y z ∧ ¬y c )) Learning domain-specific planners by example. 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[ "https://github.com/aig-upf/best-first-generalized-planning." ]
[ "Stopping time detection of wood panel compression: A functional time series approach", "Stopping time detection of wood panel compression: A functional time series approach" ]
[ "Han Lin Shang \nDepartment of Actuarial Studies and Business Analytics Macquarie University\nDepartment of Statistics and Actuarial Science\nDepartment of Statistics and Actuarial Science\nSimon Fraser University\nUniversity of Waterloo\n\n", "Jiguo Cao \nDepartment of Actuarial Studies and Business Analytics Macquarie University\nDepartment of Statistics and Actuarial Science\nDepartment of Statistics and Actuarial Science\nSimon Fraser University\nUniversity of Waterloo\n\n", "Peijun Sang \nDepartment of Actuarial Studies and Business Analytics Macquarie University\nDepartment of Statistics and Actuarial Science\nDepartment of Statistics and Actuarial Science\nSimon Fraser University\nUniversity of Waterloo\n\n" ]
[ "Department of Actuarial Studies and Business Analytics Macquarie University\nDepartment of Statistics and Actuarial Science\nDepartment of Statistics and Actuarial Science\nSimon Fraser University\nUniversity of Waterloo\n", "Department of Actuarial Studies and Business Analytics Macquarie University\nDepartment of Statistics and Actuarial Science\nDepartment of Statistics and Actuarial Science\nSimon Fraser University\nUniversity of Waterloo\n", "Department of Actuarial Studies and Business Analytics Macquarie University\nDepartment of Statistics and Actuarial Science\nDepartment of Statistics and Actuarial Science\nSimon Fraser University\nUniversity of Waterloo\n" ]
[]
We consider determining the optimal stopping time for the glue curing of wood panels in an automatic process environment. Using the near-infrared spectroscopy technology to monitor the manufacturing process ensures substantial savings in energy and time. We collect a time series of curves from a near-infrared spectrum probe consisting of 72 spectra and aim to detect an optimal stopping time. We propose an estimation procedure to determine the optimal stopping time of wood panel compression and the estimation uncertainty associated with the estimated stopping time. Our method first divides the entire data set into a training sample and a testing sample, then iteratively computes integrated squared forecast errors based on the testing sample. We then apply a structural break detection method with one breakpoint to determine an estimated optimal stopping time from a univariate time series of the integrated squared forecast errors. We also investigate the finite-sample performance of the proposed method via a series of simulation studies.
10.1111/rssc.12572
[ "https://arxiv.org/pdf/2204.13197v1.pdf" ]
248,426,950
2204.13197
4185d8a48ef24c2cd1bee82469f4708dc11aa0fa
Stopping time detection of wood panel compression: A functional time series approach Han Lin Shang Department of Actuarial Studies and Business Analytics Macquarie University Department of Statistics and Actuarial Science Department of Statistics and Actuarial Science Simon Fraser University University of Waterloo Jiguo Cao Department of Actuarial Studies and Business Analytics Macquarie University Department of Statistics and Actuarial Science Department of Statistics and Actuarial Science Simon Fraser University University of Waterloo Peijun Sang Department of Actuarial Studies and Business Analytics Macquarie University Department of Statistics and Actuarial Science Department of Statistics and Actuarial Science Simon Fraser University University of Waterloo Stopping time detection of wood panel compression: A functional time series approach 1 arXiv:2204.13197v1 [stat.ME] 27 Apr 2022functional principal component analysisintegrated squared forecast errorlong-run covariance functionstructural changewood panel NIR spectra We consider determining the optimal stopping time for the glue curing of wood panels in an automatic process environment. Using the near-infrared spectroscopy technology to monitor the manufacturing process ensures substantial savings in energy and time. We collect a time series of curves from a near-infrared spectrum probe consisting of 72 spectra and aim to detect an optimal stopping time. We propose an estimation procedure to determine the optimal stopping time of wood panel compression and the estimation uncertainty associated with the estimated stopping time. Our method first divides the entire data set into a training sample and a testing sample, then iteratively computes integrated squared forecast errors based on the testing sample. We then apply a structural break detection method with one breakpoint to determine an estimated optimal stopping time from a univariate time series of the integrated squared forecast errors. We also investigate the finite-sample performance of the proposed method via a series of simulation studies. Introduction Functional time series consist of random functions observed over time. Each function, denoted by {X t (u), t ∈ Z}, is a realization of a stochastic process X (u) where u ∈ I ⊂ R represents a continuum bounded within a finite interval I that is a subset of the real line R. There has been a surge of interest in studying functional time series that take values in an infinitedimensional space in recent years. Examples of functional time series include intraday stock price curves with each functional observation defined as a pricing function of time points within a day (e.g., Horváth et al. 2014), and age-specific fertility rate curves with each functional observation defined as a function of different ages for a particular calendar year (e.g., Li et al. 2020). In our wood panel compression data, the continuum ranges near-infrared (NIR) spectrum wavelengths. The analysis of NIR curves has been attracting extensive attention in chemometrics and less so in statistics. However, in statistics, Goutis (1998) considered a scalar-on-function regression with a roughness penalty imposed on the second-order derivative of a functional variable and applied the scalar-on-function regression to predict protein content from a set of wheat spectra. In a scalar-on-function regression, Brown et al. (2001) considered a wavelet transformation of the discretized curves and applied a Bayesian variable selection method to the multivariate regression of predictands on wavelet coefficients. Further, they applied the scalar-on-function regression to predict fat, sugar, and water content from the NIR spectrum of biscuit doughs. Ferraty & Vieu (2002) and Ferraty (2014) considered a nonparametric framework to capture the relationship between a scalar-valued response and a function-valued predictor. They applied the nonparametric regression to predict the fat content of a meat sample based on its NIR absorbance spectrum. Further, Ferraty & Vieu (2003) extended the nonparametric framework from regression to curve discrimination and classified those spectra that have the fat content larger than 20% from a set of NIR spectrum of meat samples. In a scalar-on-function regression, Hooker & Shang (2021) found that the second derivative of the NIR curves is the optimal covariate for analyzing the fat content. In the existing literature, the NIR spectra curves were independent and identically distributed (i.i.d.) and were often considered an explanatory variable in a regression that involves at least a functional variable. There is yet work on analyzing a functional time series of NIR curves to the best of our knowledge, with an application to wood panel compression in the lumber industry. We aim to fill the gap by contributing a novel methodology for determining the optimal stopping time of wood panel compression. Our developed method is of practical importance to the lumber industry worldwide, potentially saving manufacturing and labor costs. In particular, we address an important problem from FPInnovations (https://web.fpinnovations.ca), a not-for-profit R&D private organization that focuses on creating solutions to boost the growth of the Canadian forest section. As advocated by FPInnovations, using the NIR spectroscopy technology to monitor the manufacturing process ensures substantial savings in energy and time. It increases productivity, thus enhancing the overall competitiveness of the Canadian wood processing industry. Finally, the proposed methodology can be applied to other densely observed functional time series where spectral signals can be extracted to reflect the dynamic process of interest. Subtle changes in experimental conditions, such as temperature and pressure in a laboratory, and moisture content in wood, are expected to impact the spectroscopy curves. Ideally, we would like to observe multiple functional time series of NIR spectra and find an optimal stopping time for each of the functional time series. Due to the data availability, we focus on a single functional time series. With a time series of 72 spectra acquired from a near-infrared spectrum probe, we apply a functional time series forecasting method to estimate the optimal stopping time of wood panel compression, along with the estimation uncertainty of the stopping time. Our method first uses functional principal component analysis for the functional time series to reduce dimensionality. Secondly, we model each set of principal component scores using an autoregressive integrated moving average to obtain the forecast scores. By multiplying the forecast scores with functional principal components, we obtain forecast curves. We iteratively compute integrated squared forecast errors (ISFEs) of holdout functional time series. Based on a univariate time series of the forecasting errors, we then apply a simple structure break procedure to determine a breakpoint. Our work is aligned with many papers in functional change-point analysis literature. There have been many attempts to detect one or more change points and the corresponding different regimes. When there is an abrupt change, it is commonly referred to as the "change point" for the point in time where the change takes place. The change occurs when a stochastic process exhibits a shift in mean, variance, or distribution. Berkes et al. (2009) and Aue et al. (2009) developed statistical methods to test the null hypothesis of no structural break against the alternative of a single break in the mean function, assuming that the errors are i.i.d. curves. By reducing the infinite dimension of the functional time series to a finite dimension via the classic functional principal component analysis, change-point detection methods developed for multivariate time series can be applied to identify breaks (see, e.g., Aue et al. 2009, Berkes et al. 2009, Aston & Kirch 2012, Torgovitski 2015. Zhang et al. (2011) studied a structural break detection method for serially correlated functional time series, and the method is based on a self-normalization approach of Shao & Zhang (2010). In a time domain, Aue et al. (2018) considered a fully functional approach for detecting structural breaks in functional time series data without dimension reduction. The fully functional approach avoids possible information loss caused by the dimension reduction, thus may have more reliable numerical performance. In a frequency domain, Aue & van Delft (2020) considered smooth deviations from stationarity of functional time series. When there is a smoothly changing situation, it is commonly referred to as a "regime switch" for a different regime after the change point. The detection of regime switch can be thought of as change-point detection with one change point. This is the focus of our paper. Unlike those existing papers, our proposed method considers a functional changepoint problem from a forecasting perspective. A standard change point method can be applied to a univariate time series of forecasting errors. The remainder of this paper is structured as follows. In Section 2, we describe the motivating data set consisting of the NIR spectrum of wood panels. In Section 3, we introduce a functional principal component analysis for reducing the dimensionality of a functional time series. In Section 4, we introduce a functional time series forecasting method and implement a forecasting scheme to determine forecasting errors, such as ISFEs, for some holdout samples. In Section 5, we present a sieve bootstrap method to obtain bootstrapped functional time series forecasts, from which we obtain a distribution of optimal stopping time to quantify uncertainty. In Section 6, we present a simulation study. In Section 7, we present the results for the motivating data set. From the ISFEs, we apply a structural break detection method to identify the optimal stopping time. Conclusions are drawn in Section 8, along with some ideas on how the methodology can be further extended. Wood panel NIR spectra This data set consists of 72 sets of spectra acquired from a NIR probe inserted into the wood panel matrix before panel pressing. Spectra were automatically collected during the panel processing. All spectra were acquired using an Analytical Spectral Devices field portable spectrometer. The acquisition range was from 350 to 2300nm, with a 1nm spectral resolution (i.e., 1951 points per spectrum). The actual time of acquisition for each spectrum is embedded. Therefore, the time interval between spectra can be computed as the difference between consecutive spectra times. In Figure 1, we present a perspective plot of NIR spectra curves. NIR spectra can be viewed as a set of dense functional data. While the perspective plot does not reveal the optimal stopping time, it shows a temporal change in the NIR spectra curves at different wavelengths. The optimal stopping time is one among 72 time points where the wood panel compression completes. We aim to determine the earlier termination time. A functional time series forecasting method We do not restrict our considerations to a particular functional time series forecasting method. Many of the functional predictors applied in the statistical literature fit our stopping time detection algorithm in Section 4. We elaborate on some examples: 1) functional autoregressive of order one (Bosq 2000), where a functional predictor can be lagged observations of a functional response; 2) nonparametric functional regression (Antoniadis et al. 2006 Functional principal component analysis From the sample X (u) = {X 1 (u), . . . , X n (u)}, we compute the sample mean function and sample covariance functions, which are defined by X (u) = 1 n n ∑ t=1 X t (u), C(u, v) = 1 n n ∑ t=1 {[X t (u) − X (u)][X t (v) − X (v)]}. Via Mercer's lemma, the empirical covariance function can be decomposed as C(u, v) = ∞ ∑ k=1 λ k φ k (u) φ k (v), where λ 1 > λ 2 > · · · ≥ 0 are the sample eigenvalues, and [ φ 1 (t), φ 2 (t), . . . ] are the corresponding orthogonal sample eigenfunctions. The realizations of the stochastic process X can be written as X t (u) = X (u) + ∞ ∑ k=1 β t,k φ k (u), t = 1, 2, . . . , n,(1) where β t,k is the k th estimated principal component score for the t th time period. In practice, we truncate (1) into the first K < n terms and a model residual term ζ t (u). This leads to X t (u) = X (u) + K ∑ k=1 β t,k φ k (u) + ζ t (u),(2) where expansion (2) facilitates dimension reduction as the first K terms often provide a good approximation to the infinite sums, and thus the information contained in X (u) can be properly summarized by the long vector ( β 1 , . . . , β K ) , where β k = ( β 1,k , . . . , β n,k ) and denotes matrix transpose. When a functional data set is i.i.d., the functional principal component analysis is an adequate dimension-reduction technique. However, in the presence of temporal dependence, long-run covariance plays an essential role in modeling functional time series (Rice & Shang 2017). The long-run covariance can be defined as C(u, v) = ∞ ∑ =−∞ γ (u, v), where γ (u, v) denotes a symmetric and nonnegative definite autocovariance for any . In practice, the long-run covariance function can be estimated from a functional time series. Given its definition as a bi-infinite sum, a natural estimator of C is C ϕ,κ (u, v) = ∞ ∑ =−∞ W κ ϕ γ (u, v), where ϕ is known as the bandwidth parameter, and γ (u, v) =            1 n n− ∑ t=1 X t (u) − X (u) X t+ (v) − X (v) , ≥ 0 1 n n ∑ t=1− X t (u) − X (u) X t+ (v) − X (v) , < 0, is an estimator of γ l (x, u), and W κ is a symmetric weight function with bounded support of order κ. The estimation accuracy of this estimator crucially depends on the bandwidth parameter, which can be selected by a plug-in algorithm of Rice & Shang (2017). With the estimated long-run covariance, a set of dynamic functional principal components and their associated scores may be obtained. Selection of the number of components Although it can be a research topic on its own, there are several approaches for selecting the number of retained terms K: 1) scree plots or the fraction of variance explained by the first few functional principal components (Chiou 2012); 2) pseudo-versions of Akaike information criterion and Bayesian information criterion (Yao et al. 2005); 3) predictive cross-validation leaving out one or more curves (Yao et al. 2005); 4) bootstrap methods (Hall & Vial 2006); and 5) eigenvalue ratio test (Ahn & Horenstein 2013, Lam & Yao 2012). In this article, we determine K by a modified eigenvalue ratio criterion introduced in Li et al. (2020). The estimated value of K is determined as the integer minimizing ratios of two adjacent empirical eigenvalues given by K = argmin 1≤k≤k max λ k+1 λ k × 1 λ k λ 1 ≥ θ + 1 λ k λ 1 < θ , where k max is a pre-specified positive integer, θ is a pre-specified small positive number, and 1(·) is the binary indicator function. When without prior information about a possible maximum of K, it is unproblematic to choose a relatively large k max , e.g., k max = # k λ k ≥ ∑ n k=1 λ k /n, k ≥ 1 (Ahn & Horenstein 2013). Given that the small empirical eigenvalues λ k for some K < k < k max are close to zero, we adopt the threshold constant θ = 1/ ln[max( λ 1 , n)] to ensure consistency of K. Univariate time series forecasting methods The forecasts of these principal component scores can be obtained via either a multivariate or univariate time series forecasting method. Among the multivariate time series forecasting methods, the vector autoregressive moving average model is commonly used (see, e.g., Aue et al. 2015). However, these models often require stationarity and do not have an automatic order selection algorithm. In contrast, a univariate time series forecasting method, such as the autoregressive integrated moving average (ARIMA) model, can handle the non-stationarity of the principal component scores. Also, we use the automatic algorithm of Hyndman & Khandakar (2008) to choose the optimal orders of autoregressive p, moving average q, and difference order d. For each set of principal component scores, an ARIMA(p, d, q) model is built, which has autoregressive components of order p, moving average components of order q, and degree of a difference d needed to achieve stationarity. The model can be expressed as ∆ d β t = c + p ∑ l=1 η l ∆ d β t−l + π t + q ∑ ν=1 ψ ν π t−ν , t = max(p, q) + 1, . . . , n, where ∆ d β t represents the stationary time series after applying the difference operator of order d, c is the drift term; {η 1 , . . . , η p } represents the coefficients of the autoregressive components; {ψ 1 , . . . , ψ q } represents the coefficients of the moving average components; and π t is a sequence of i.i.d. random variables with mean zero and a finite variance. In the automatic ARIMA model, d is selected based on successive Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit root tests (Kwiatkowski et al. 1992). KPSS tests are used for testing the null hypothesis that an observable time series is stationary around a deterministic trend. We test the original data (i.e., the first set of principal component scores) for a unit root; if the test result is significant, we test the differenced data for a unit root. The procedure continues until we obtain our first insignificant result. After determining the value of d, the orders of p and q are selected based on the optimal Akaike information criterion with a correction for finite sample sizes (Akaike 1974). Having identified the optimal ARIMA(p, d, q) model, the maximum likelihood method can estimate the parameters. Forecasting functional time series From the estimated long-run covariance function, we can extract estimated dynamic functional principal components B = { φ 1 (u), . . . , φ K (u)}. Conditioning on the past functions X (u) and the estimated functional principal components B, the h-step-ahead point forecast of X n+h (u) can be expressed as X n+h|n (u) = E [X n+h (u)|X (u), B] = X (u) + K ∑ k=1 β n+h|n,k φ k (u), where β n+h|n,k denotes univariate time series forecasts of the k th principal component scores and h denotes a forecast horizon. Let us consider our motivating example, consisting of 72 NIR spectra observed over time. Suppose we observe the first 71 NIR spectra, and we aim to produce a one-step-ahead curve forecast. Through a dynamic functional principal component analysis, we obtain estimated functional principal components and their associated scores in Figure 2. In the forecasting scheme, we produce one-step-ahead point forecasts using the first three functional observations initially. We re-estimate the parameters in a functional time-series model using the second to fourth functional observations. Forecasts from the estimated models are then produced for one-step-ahead. We iterate this process by increasing the sample size by one observation until the end of the sample period. This process produces 69 one-step-ahead forecasts. We compare these forecasts with the holdout samples (i.e., 4 th to 72 th curves) to determine the point forecast accuracy. To evaluate the point forecast accuracy, we use the integrated squared forecast error (ISFE) (see also Hyndman & Ullah 2007), which measures how close the forecasts are compared to the actual values of the variable being forecast. It can be written as γ+1 = I X γ+1 (u) − X γ+1|γ (u) 2 du, γ = 3, . . . , (n − 1), where X γ+1 (u) represents the (γ + 1) th holdout sample in the forecasting scheme, while X γ+1|γ (u) represents the one-step-ahead point forecasts for the holdout sample. Regression-based approach Since our goal is to determine the optimal stopping time, the number of breakpoints is one. To estimate the breakpoint, we follow the methodology as set out in Bai & Perron (2003) (refer to Zeileis et al. 2003, Zeileis & Kleiber 2005, for a detailed description of an implementation). Suppose we observe a univariate time series of ISFEs, denoted by γ+1 for γ = 3, . . . , (n − 1). We estimate a random walk with a piecewise constant drift for the time-dependent variables: ∆ γ+1 =    ς 1 + ε γ+1 γ + 1 ≤ γ * ς 2 + ε γ+1 γ + 1 > γ * where ς 1 and ς 2 denote the respective mean terms before and after a breakpoint, and ε γ+1 is the error term. We estimate this model using ordinary least squares by minimizing the sum of squared residuals (SSR): SSR (γ * ) = γ * −1 ∑ γ=3 ∆ γ+1 − ς 1 2 + n−1 ∑ γ=γ * ∆ γ+1 − ς 2 2 . This model specification identifies one breakpoint, which divides the univariate time series of the forecast errors up into two regimes with different shifts. The optimal stopping time is determined as the one that minimizes the SSR. Bootstrapping functional time series forecasts Bootstrapping has been receiving increasing attention in the functional time series literature as a way of quantifying uncertainty. Politis & Romano (1994) Using a univariate time series forecasting method, we can obtain one-step-ahead forecasts for each set of the estimated principal component scores, { β 1,k , . . . , β n,k } for k = 1, . . . , K. Let the one-step-ahead forecast errors be given by ϑ t−1,k = β t,k − β t|t−1,k for t = 2, . . . , γ and γ = 3, . . . , (n − 1). These can then be sampled with replacement to give a bootstrap sample of β γ+1,k : β (b) γ+1|γ,k = β γ+1|γ,k + ϑ (b) * ,k , b = 1, . . . , B, where B = 1, 000 symbolizes the number of bootstrap replications and ϑ (b) * ,k are sampled with replacement from { ϑ 1,k , . . . , ϑ γ−1,k }. Assuming the first K principal components approximate the original functional time series relatively well, the model residual should contribute nothing but random noise. Consequently, we can bootstrap the model residuals in (2) by sampling with replacement from the model residual term { ζ 1 (u), . . . , ζ γ−1 (u)}. Adding two components of variability, we obtain B variants for X γ+1 (u), X (b) γ+1|γ (u) = K ∑ k=1 β (b) γ+1|γ,k φ k (u) + ζ (b) γ+1 (u). With the bootstrapped { X (1) γ+1|γ (u), . . . , X (B) γ+1|γ (u)}, we fit a functional time series model in Section 3, where the retained number of principal components is estimated from eigenvalue ratio criterion and is allowed to be different in bootstrap samples. We obtain a one-step-ahead forecast by conditioning the estimated mean function and functional principal components in each bootstrap sample. Then, we compute the one-step-ahead bootstrap forecast errors between the bootstrapped forecasts and actual holdout samples, (b) γ+1 = I X γ+1 (u) − X (b) γ+1|γ (u) 2 du. With each bootstrapped errors { (b) 4 , . . . , (b) n }, we apply the structure break detection algorithm in Section 4.2 to find the estimates of the optimal stopping time. Simulation studies We consider two data generating processes. In Section 6.1, we consider a stationary functional time series process, where there is a presence of abrupt change at a pre-fixed location. In Section 6.2 and 6.3, we consider a non-stationary functional time series process, where there is a presence of abrupt and gradual change at a randomly assigned location, respectively. An abrupt change in the mean of a stationary functional time series We utilize Monte Carlo simulation to evaluate the performance of our method. The data generating process is a pointwise FAR(1), given by X 1 (u) = 10 × u × (1 − u) + ω × B 1 (u) X t (u) = (ρ + c)X t−1 (u) + ω × B t (u), t = 2, . . . , n Y t (u) = |X t−1 (u) − X t (u)| |X t−1 (u) + 0.1| , where {B t (u), t = 1, . . . , n, u ∈ [0, 1]} denote i.i.d. standard Brownian motions. In practice, we discretize continuum u on 101 equally spaced grid points. We consider three values of ω = 0.1, 0.5, 0.9 to reflect three levels of noise to signal. The coefficients satisfy |ρ| < 1 and |ρ + c| < 1 in order to ensure the stationarity. Let ρ = 0.2 and c = 0 for curves from two to τ = n/2 ; while c = 0.7 for curves from n/2 + 1 to n. An example of simulated curves with a sample size n = 400 is presented in Figure 3, where the actual stopping time is 200. In each replication, we apply our regression-based approach to estimating the optimal stopping time. With the estimated stopping time points from 1,000 replications, we present cumulative distribution functions (CDFs) for the regression-based approach with three different noise to signal levels in Figure 4. In Table 1, we tabulate the number of times among 1,000 replications when the estimated stopping time is greater than or equal to the actual stopping time. As sample size n increases, the chance of covering the holdout stopping time generally decreases, but the median of the the ω value is, the larger the signal-to-noise ratio is. From Table 1, the mean and median of the estimated stopping time are not distant from the actual stopping time. Table 1 For three sample sizes, we determine the mean and median of the estimated change points and the number of times out of 1,000 replications when the estimated stopping time is greater than or equal to the actual stopping time. From a conservative aspect, it is essential not to miss the actual stopping time. Out of 1,000 replications, each number in the table reflects a probability where the actual stopping time is within the estimated stopping time. An abrupt change in the mean of a non-stationary functional time series We consider another data generating process for simulating functional time series. We begin with simulating a time series of error functions [ 1 (u), 2 (u), . . . , n (u)] given below: t (u) = K ∑ k=1 β t,k φ k (u) + ζ t (u),(3) where [φ 1 (u), φ 2 (u), . . . , φ K (u)] are randomly sampled with replacement from K = 21 Fourier basis functions, and ζ t (u) denotes innovation term that can be independent over t. We consider 101 equally-spaced grid between 0 and 1. Let β t = (β t,1 , β t,2 , . . . , β t,K ) be a K-dimensional vector. We generate β t from a vector autoregression of order 1 (VAR(1)) model, β t = Aβ t−1 + ξ t , t = 2, . . . , n, where A = (a ij ) K×K is the VAR(1) coefficient matrix, and ξ t denotes the error term of the VAR(1) model at time t. Following Li et al. (2020), we consider two possible structures for A: 1) a diagonal matrix with diagonal elements drawn from a U(−0.5, 0.5) and ξ t is generated by a K-dimensional normal distribution with mean zero and power-decay covariance structure cor(ξ i t , ξ j t ) = ρ |i−j| , where ρ denotes a correlation parameter, such as ρ = 0.5. 2) Alternatively, A is a banded autoregressive matrix with a i,j indpendently drawn from a U(−0.3, 0.3) when |i − j| ≤ 3 and a ij = 0 when |i − j| > 3, and ξ t is independently generated by a K-dimensional normal distribution with mean zero and identity covariance matrix. To specify a change-point location for the population, we draw a value from a U(0.25 × n, 0.75 × n). The lower and upper bounds of the uniform distribution are purposely chosen so that the location of a change point does not lie on the boundary of a sample. We could divide the change-point location τ by sample size n to compute a probability p ∈ (0.25, 0.75). Following Aue et al. (2018), a class of break functions was given by δ * k (u) = 1 √ k k ∑ w=1 φ w (u), k = 1, 2, . . . , K, δ k (u) = δ * k (u) × √ c, where the normalization is required to ensure δ * k (u) has unit norm. For a population, δ 1 (u) is the case of a break only in the leading eigendirection, while δ K (u) is the case of a break that affects all eigendirections (see, e.g., Aue et al. 2018). The value of c controls the magnitude of the break, and it links to the signal-to-noise ratio SNR = c × p(1 − p) tr( C ) ,(4) where tr( C ) denote the trace of the estimated long-run covariance of the error term in (3). We can compute the value of c for a given SNR value. With a chosen eigendirection k, such as k = 1, we simulate n samples of a non-stationary functional time series as follows: X t (u) = δ k (u) × 1{t > τ} + t (u), Y t (u) = X (u) + X t (u), where X (u) = 1 n ∑ n t=1 X t (u). In Table 2, we present some summary statistics of our estimated change points for three sample sizes and two SNRs. As SNR increases from 0.01 to 0.1, the mean and median of the estimated change points are generally closer to those of the actual change points for the same sample size. The probability that the actual stopping time is within the estimated stopping time also increases. Table 2 For three sample sizes, we determine the mean and median of the actual and estimated change points and the number of times out of 1,000 replications when the estimated stopping time is greater than or equal to the actual stopping time. True change point τ Estimated change point τ SNR = 0.1 SNR = 0.01 A n Mean Median Mean Median #( τ >= τ) Mean Median #( τ >= τ) A gradual change in the mean of a non-stationary functional time series While Sections 6.1 and 6.2 present stationary and non-stationary functional time series processes with an abrupt change in mean, in this section, we modify the data generating process of Section 6.2 from an abrupt change to a gradual change in mean. With a chosen eigendirection k, such as k = 1, we simulate n samples of a non-stationary functional time series as follows: X t (u) = √ t × n α √ n × δ k (u) × 1{t > τ} + t (u), Y t (u) = X (u) + X t (u), where t is a time index representing a gradual change. Also, α ∈ (0, 1 2 ) is a constant, and together with c control the magnitude of the change point. As α or c increases, the magnitude of the change point increases; thus, it is easier to be detected. In Table 3, we present some summary statistics of our estimated change points for three sample sizes and two SNRs. As α increases for the same SNR and sample size, the mean and median of the estimated change points are closer to those of the actual change points. As SNR increases from 0.01 to 0.1, the mean and median of the estimated change points are closer to those of the actual change points for the same sample size and same α value. As a result, this new process monitoring tool ensures substantial savings in energy and time. Using the eigenvalue ratio criterion, the number of retained components is determined to be one. With the functional time series forecasting method, we compute the ISFEs in Figure 5. From a univariate time series of these ISFEs, we then implement a structural change detection method described in Section 4.2 to identify one breakpoint. This breakpoint is also our estimated optimal stopping time. The breakpoint occurs at time point 27 using the rolling window scheme. As a result, time point 27 may be the optimal stopping time using the regression-based approach. The point estimate of the optimal stopping time is limited without an adequate assessment of probabilistic uncertainty associated with the point estimate. In computing the forecast errors, forecast uncertainty stems from systematic deviations (e.g., due to parameter and model uncertainties) and random fluctuations (e.g., due to the model error term). Thus, it is essential to provide an interval forecast as well as a point forecast of the optimal stopping time to (1) assess the future uncertainty level; (2) enable different strategies to be planned for a range of possible outcomes indicated by the interval forecast; (3) compare forecasts from different methods more thoroughly; and (4) explore different scenarios based on various assumptions. The uncertainty associated with our estimated stopping time may vary largely due to the forecast errors in the forecasting schemes. We apply a functional time series forecasting method that utilizes the principal components and their associated scores to compute these forecast errors. By bootstrapping the scores and model residuals, we obtain a set of bootstrap functional time series conditional on the estimated mean function, estimated functional principal components, and observed functional time series. We compute the ISFEs and determine the optimal stopping time for each bootstrap replication using the bootstrap series. In turn, bootstrapping provides us with a distribution of the estimated optimal stopping time, from which we can determine the mode (frequently occurring value). The mode is time point 29 using the regression-based approach. Out of 1,000 bootstrap replications, there are only seven times where the stopping time is time point 30. The detected time points 29 and 30 obtained from the 1,000 bootstrap samples differ from the time point 27 identified by the same approach applied to the original functional time series. An explanation is that the original functional time series are some realizations of an unknown stochastic process. The bootstrap samples generally mimic the temporal dependence exhibited in the original functional time series, but the bootstrap samples contain a random noise with a practically unknown signal-to-noise ratio. In turn, the detected time points may differ from the one from the original data. The economic benefit of this empirical application is substantial. Among 72 time series of NIR spectra, our estimated optimal time point indicates that we do not need to observe the entire sample to uncover when the glue curing process may be completed. In turn, this finding could save energy, time, and labor costs. For comparison, we also implemented a structural break method of Aue et al. (2018) and detected the change point at time point 22 using both fully-function and functional principal component analysis-based approaches. The optimal stopping time is recommended at time point 30 for this data set. It is better to overestimate than underestimate the stopping time from an applied perspective since the process may not be completed if stopped too early. Conclusion We provide a solution for determining the optimal stopping time for phenolic glue curing of wood panels in an automated process environment from a time series of NIR spectra. This solution, based on the NIR spectroscopy technology, provides a novel alternative to monitoring the process. With a more accurate evaluation of the curing process's stopping time, the proposed solution leads to substantial savings in energy, time, and labor costs, thus enhancing the overall competitiveness of the Canadian wood industry. Furthermore, the proposed methodology can be applied to other applications such as qualitative analysis and quality control, where spectral signals can be employed to analyze a dynamical process of interest. The essence of our methodology is to identify breakpoints by applying a structural change method to iterative one-step-ahead forecast errors. These forecast errors are obtained by computing the ISFEs between the holdout functional time series and their one-step-ahead forecasts. Our approach is general enough to apply other structural break methods and functional time series forecasting methods. There are at least three ways in which the present paper can be extended: 1) We implement a regression-based approach to detect a single change point due to the nature of our problem. However, one could apply other change-point methods, including those that can detect multiple change points (see, e.g., Qian et al. 2019, Wu & Zhou 2020, Xiong & Cribben 2021. Then, the stopping time is estimated by the point that has the longest homogenous segment to the end of the time points. 2) A challenging and important research direction is the change-point detection for high-dimensional functional time series. In this study, we observe only one time series of near-infrared spectroscopy curves. As the experimental conditions can change from one to another, the stopping time can vary among different wood products. If we observe multiple Figure 1 1A perspective plot of 72 NIR spectra curves with the wavelength from 350 to 2300nm. ); 3) functional principal component regression where scores are modeled and forecast via a multivariate time series forecasting method (Aue et al. 2015); and 4) functional principal component regression where scores are modeled and forecast via a univariate time series forecasting method (Hyndman & Shang 2009). We consider a functional principal component regression where scores are modeled and forecast via a univariate time series forecasting method. The univariate time series forecasting method can model a possible non-stationary series of scores. Figure 2 2By applying a dynamic functional principal component analysis to the observed NIR spectra from time points 1 to 71, we obtain the estimated mean function, the first estimated functional principal component, and its associated scores. Using the eigenvalue ratio criterion, the number of retained components is one. By forecasting the scores with the ARIMA(0, 2, 0) model, we obtain the one-step-ahead forecast curve conditional on the estimated functional principal component and mean function.4 Stopping time detection 4.1 Rolling window schemeWe consider a rolling window scheme of a functional time series forecasting method commonly used to assess model and parameter stabilities over time. It assesses the constancy of a model's parameter by computing parameter estimates and their corresponding forecasts over a rolling window of a fixed size through the entire sample. obtained weak convergence results for approximate sums of weakly dependent, Hilbert space-valued random variables.Dehling et al. (2015) also obtained weak convergence results for Hilbert space-valued random variables, which are assumed to be weakly dependent in the sense of near-epoch dependence and showed consistency of a non-overlapping block bootstrap procedure.Nyarige (2016) andFranke & Nyarige (2019) proposed a residual-based bootstrap for functional autoregressions. They showed that the empirical distribution of the centered sample innovations converges to the distribution of the innovations with respect to the Mallows metric. Pilavakis et al. (2019) established theoretical results for the moving block and the tapered block bootstrap. Shang (2019) applied a maximum entropy bootstrap procedure. Paparoditis (2018) considered the functional autoregressions and derived bootstrap consistency as the sample size and order of functional autoregression both tend to infinity. From a nonparametric viewpoint, Ferraty & Vieu (2011) applied a residual-based bootstrap procedure to construct confidence intervals for the regression function. Zhu & Politis (2017) proposed a kernel estimation of the first-order nonparametric functional autoregression model and its bootstrap approximation. Bootstrapping helps with statistical inference and forecast uncertainty quantification. Paparoditis & Shang (2021) proposed a sieve bootstrap for constructing prediction bands. Since the sieve bootstrap method can handle model misspecification, it improves the calibration of interval forecasts. However, the sieve bootstrap relies on stationarity in a functional time series, where our NIR data set does not hold. Instead, we consider a nonparametric bootstrap method of Hyndman & Shang (2009). In the principal component decomposition, at least three sources of uncertainty need to be considered. These are truncation errors in the principal component decomposition (concerning the estimated number of retained principal components), estimated mean function and functional principal components, and forecast errors in the forecast principal component scores. Since principal component scores are regarded as surrogates of the original functional time series, these principal component scores capture the temporal dependence structure inherited in the original functional time series (Paparoditis 2018, Shang 2018). By adequately bootstrapping the forecast principal component scores, we can generate a set of bootstrapped curve forecasts conditional on the estimated mean function and estimated functional principal components from the observed functional time series. Figure 3 3One replication of n = 400 simulated curves {Y 1 (u), . . . , Y 400 (u)}. Figure 4 4With 1,000 replications, we compute the empirical CDFs of the stopping time using the regression-based approach with a sample size of n = 400 for a noise-to-signal ratio ω = 0.1, 0.5, 0.9. estimated stopping time becomes more accurate. For the three ω values, the possibility of covering the holdout stopping time generally decreases from ω = 0.1 to ω = 0.9. The smaller Figure 5 5Based on the ISFEs, we determine the estimated stopping time using the regressionbased approach. ( high-dimensional) time series of such functional time series for various wood products, it would be helpful to propose a hypothesis test to check if there is a change point in any of the functional time series. If so, we may develop a change-point detection method to identify those functional time series that have a change point. 3) By bridging a gap between academia and industry partners, our change-point detection method can be applied by employees in the wood panel industry to verify the effectiveness of the stopping-time estimation for a range of wood products. We aim to enrich the collaboration with a developed computer code in . Table 3 3We determine the mean and median of the estimated and actual change points from 1,000 replications for three sample sizes and two SNRs. Mean Median Mean Median Mean Median Mean Median monitored, considerable energies must be consumed to guarantee the resin's curing completion.True change point τ Estimated change point τ α = 0.05 α = 0.25 α = 0.45 A SNR n Application to the wood panel data setAccording toBekhta et al. 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Engl i sh ti tl e: R ealel l i pti c surfaces and R agsdal e-V i ro i nequal i ty A bstract. O n a realregularellipticsurface w ithoutm ultiple ber,the B ettinum ber h 1 and the H odge num ber h 1;1 are related by h 1 h 1;1 . W e prove that it's alw ays possible to deform such algebraic surface to obtain h 1 = h 1;1 . Furtherm ore,w e can im pose that each hom ology class can be represented by a realalgebraic curve. W e use a realversion ofthe m odular construction ofelliptic surfaces.R esum e. P our une surface elliptique r eelle r eguli ere etsans bre m ultiple,le nom bre de B ettih 1 et le nom bre de H odge h 1;1 sont li es par l'in egalit e h 1 h 1;1 . O n m ontre qu'on peut toujours d eform er une telle surface alg ebrique pour obtenir h 1 = h 1;1 . D e plus, on peut im poser que chaque classe d'hom ologie soit repr esentable par une courbe alg ebrique r eelle. O n utilise une version adapt ee au cas r eelde la construction des surfaces elliptiques m odulaires. 1991 M athem atics Subject C l assi cation. 14J27 14C 25 14P 25. K ey w ords and phrases. A lgebraic cycles,Topology ofR ealalgebraic surfaces,M odularE lliptic
10.1007/s002090000132
[ "https://export.arxiv.org/pdf/math/9807020v1.pdf" ]
117,188,567
math/9807020
3b90c08fabacb814ecc1a89a693cb3a2fc0ab731
3 Jul 1998 3 Jul 1998arXiv:math/9807020v1 [math.AG] SU R FA C E S E LLIP T IQ U E S R E E LLE S E T IN E G A LIT E D E R A G SD A LE -V IR O Engl i sh ti tl e: R ealel l i pti c surfaces and R agsdal e-V i ro i nequal i ty A bstract. O n a realregularellipticsurface w ithoutm ultiple ber,the B ettinum ber h 1 and the H odge num ber h 1;1 are related by h 1 h 1;1 . W e prove that it's alw ays possible to deform such algebraic surface to obtain h 1 = h 1;1 . Furtherm ore,w e can im pose that each hom ology class can be represented by a realalgebraic curve. W e use a realversion ofthe m odular construction ofelliptic surfaces.R esum e. P our une surface elliptique r eelle r eguli ere etsans bre m ultiple,le nom bre de B ettih 1 et le nom bre de H odge h 1;1 sont li es par l'in egalit e h 1 h 1;1 . O n m ontre qu'on peut toujours d eform er une telle surface alg ebrique pour obtenir h 1 = h 1;1 . D e plus, on peut im poser que chaque classe d'hom ologie soit repr esentable par une courbe alg ebrique r eelle. O n utilise une version adapt ee au cas r eelde la construction des surfaces elliptiques m odulaires. 1991 M athem atics Subject C l assi cation. 14J27 14C 25 14P 25. K ey w ords and phrases. A lgebraic cycles,Topology ofR ealalgebraic surfaces,M odularE lliptic Introduction U ne vari et e projecti ve (ou quasi -projecti ve) X est di te r eel l e ou d e ni e sur R l orsque X est m uni e d' une i nvol uti on anti -hol om orphe X appel ee structure r eel l e. D ans ce cas,on appel l e parti e r eel l e de X et on note X (R ),l ' ensem bl e des poi nts xes de X . Par conventi on,une surface est projecti ve l i sse. U ne surface est di te el l i pti que s' i lexi ste une appl i cati on hol om orphe surjecti ve :X ! o u est une courbe l i sse com pacte etl a bre g en eri que de estune courbe de genre 1. U ne tel l e surface estdi te el l i pti que r eel l e l orsque X et sontd e ni es surR et X = o u est l a structure r eel l e de . D ' apr es un r esul tat de V . K harl am ov, [ K ha] , une surface el l i pti que r eel l e X r egul i ere (i . e. tel l e que H 1 (X ;O X )= f0g) et sans bre m ul ti pl e,v eri e l ' i n egal i t e de R agsdal e-V i ro (1. 1) h 1 (X (R )) h 1;1 (X ) o u h 1 (X (R )) est l e rang du prem i er groupe d' hom ol ogi e H 1 (X (R );Z=2) de X (R ) et h 1;1 (X ) = di m C H 1 (X ; 1 X ). M ai ntenant,on note h alg 1 (X (R )) l e rang du sousgroupe de H 1 (X (R );Z=2)engendr e parl escl asses fondam ental es de courbes al g ebri ques r eel l es (voi r [ B H ] ).O n a al ors h alg 1 (X (R )) h 1 (X (R )). Le nom bre h 1;1 esti nvari antpard eform ati on etl e r esul tatpri nci palde ce travai l est : (1.2) T h eor em e. Toute surface ell iptique r egul i ere sans bre m ul tipl e peutêtre d eform ee sur C en une surface ell iptique r eell e tell e que h alg 1 = h 1 = h 1;1 . Rem arques. U ne surface d' Enri quesX estel l i pti que r egul i ere avec deux bresm ulti pl es. Pour une tel l e surface,on a h 1;1 (X ) = 10,m ai s i ln' exi ste aucune surface d' Enri ques r eel l e tel l e que h alg 1 = h 1 = 10. En revanche, i l exi ste des surfaces d' Enri ques r eel l es tel l es que h alg 1 = 11 et h 1 = 12,cf. [ D K h]et [ M avH ] . O n ne peux pasesp ererune versi on du th eor em e 1. 2 avec d eform ati on surR . En e et,mêm e sih 1 (X (R ))= h 1;1 (X ),i lpeut n' exi ster aucune d eform ati on sur R de X tel l e que h alg 1 = h 1;1 ,voi r a ce sujet l e com portem ent des surfaces K 3,[ M a] . Le poi nt cl e dans l a preuve du th eor em e 1. 2 est l a constructi on d' une sui te fX k g k 1 de surfaces el l i pti ques r eel l es qui v eri ent h alg 1 (X k (R )) = h 1;1 (X k ), cf. th eor em e 5. 3. Les surfaces X k sont des surfaces el l i pti ques m odul ai res au sens de T .Shi oda [ Sho]dont on a adapt e l es constructi ons au cas r eel . Jusqu' a pr esent,l es surfaces el l i pti ques m odul ai res ne sem bl ent pas avoi r et e uti l i s ees en g eom etri e al g ebri que r eel l e. Le pl an de cet arti cl e est organi s e com m e sui t : l a secti on 2 est consacr ee aux pr el i m i nai res sur l es courbes de genre 1 r eel l es, l es r ef erences sont [ Si 3] et [ Si l ] . D ans l a secti on 3, on m ontre com m ent uti l i ser l es constructi ons anal yti ques de K .K odai ra [ K o]en r eel . G râce a l a cl assi cati on des bressi ngul i eresdespi nceaux r eel s de courbes el l i pti ques due a R . Si l hol [ Si 1] , on r edui t al ors l e probl em e a l a constructi on de surfaces el l i pti ques avec bres si ngul i eres donn ees a pri ori . La secti on 4 est consacr ee aux surfaces el l i pti ques m odul ai res r eel l es. O n adapte l es travaux de T .Shi oda [ Sho]et M .N ori[ N o]pour r edui re l e probl em e a l a recherche de groupes Fuchsi ens ari thm eti ques parti cul i ers. En n, en secti on 5, on donne l es dom ai nes fondam entaux d' une sui te de groupes k qui servent de base a l a constructi on des surfaces X k . Je ti ens a rem erci er V . K harl am ov pour m ' avoi r i ndi qu e l e r esul tat (1. 1) et P.Schm utz Schal l erpourson ai de dansl a constructi on desdom ai nesfondam entaux de l a secti on 5. 2;R ) ! SL(2;R ), a b c d 7 ! a b c d , cl ai rem ent S i ndui t une i nvol uti on sur PSL(2;R ) que l ' on notera encore S. Soi t z 2 H , Soi t un groupe Fuchsi en (i . e. un sous-groupe di scret de PSL(2;R )), H i ndui t une structure r eel l e sur l e quoti ent H = siet seul em ent si H = H i . e. siet seul em entsi eststabl eparS.D anstoutel a sui te,cesontl essous-groupesd' i ndi ce nidu groupe m odul ai re PSL(2;Z) PSL(2;R ) quivont nous i nt eresser. Soi t C une courbe projecti ve l i sse de genre 1, al ors i l exi ste 2 H tel que C = C =(Z + Z). R eci proquem ent, soi t 2 H ,on note C ( ) l a courbe quoti ent C =(Z + Z). La foncti on m odul ai re el l i pti que est not ee j:H ! C . Le si gne = si gni e i som orphe sur C . La proposi ti on sui vante est cl assi que. (2.1) P roposition. Soit( ; 0 )2 H H ,l es trois assertions suivantes sont equival entes. (1) C ( ) = C ( 0 ) ; (2) j( )= j( 0 ) ; (3) ilexiste A 2 PSL(2;Z),telque A : = 0 . O n note donc j(C )= j( ) pour quel conque telque C ( ) = C . La foncti on j:H ! C estd e ni esurR ,c' est-a-di reque8 2 H ,j( )= j( H ( )). (2.2) P roposition. SoitC une courbe projective l isse de genre 1,l es trois assertions suivantes sont equival entes. (1) C peutêtre d e nie sur R ; (2) j(C )2 R ; (3) 9 2 H = C ( ) = C et2< ( )2 Z. Soi t 2 H v eri ant2< ( )2 Z,l a structure r eel l e i ndui te surC ( )= C =(Z + Z) par l a conjugai son com pl exe de C est tel l e que C ( )(R )6 = ;. (2.3) P roposition. Lorsque C estm unie d'une structure r eell e,on a si # C (R )= 2,al ors j(C ) 1 et9 2 H = C ( ) = C et< ( )2 Z si # C (R ) = 1, al ors j(C ) 1 et 9 2 H = C ( ) = C et 2< ( ) est un entier im pair. (2.4)Rem arque. Le casj(C )= 1 correspond aux courbesy 2 = x 3 + x ety 2 = x 3 x quisont i som orphes sur C par (x;y) 7 ! (ix; ),o u 2 = i. La prem i ere courbe poss ede une parti e r eel l e connexe,l a parti e r eel l e de l a seconde courbe poss ede deux com posantes connexes. Surfaces elliptiques r eelles O n consi d ere une surface el l i pti que r eel l e :X ! . O n suppose que poss ede au m oi ns une bre si ngul i ere, est sans bre m ul ti pl e et qu' aucune courbe excepti onnel l e n' est contenue dans une bre. Par hypoth ese, est m uni e d' une i nvol uti on anti -hol om orphe . O n note C u = 1 fug l a bre au dessus de u 2 . O n consi d ere un ensem bl e ni stabl e par telque C u est l i sse pour tout u 2 0 = n . C om m e est stabl e par , 0 m uni e de l a restri cti on de est encore r eel l e. L' i nvari ant foncti onnelde est l a foncti on m erom orphe J : 0 ! C ,u 7 ! j(C u ) (cf. Secti on 2). Par constructi on J est d e ni e sur R , i . e. J = J. O n peut prol onger J en une foncti on hol om orphe J : ! P 1 (C )quiv eri e J = conj J M ai ntenant X j 0 = 1 ( 0 ) ! 0 est un br e di erenti el en tores, donc l es groupesd' hom ol ogi e des bresfH 1 (C u (C );Z)g u2 0 form entun fai sceau l ocal em ent constant au dessus de 0 . O n peut etendre ce fai sceau a , [ K o,x7] . C e fai sceau etendu G est l ' i nvari ant hom ol ogi que de . D e mêm e que J,G est d e nisur R . SiJ est non constante,on peut etendre l ' ensem bl e ni ,avec ( )= ,pour obteni r 8u 2 0 , J(u) 6 2 f0;1;1 g. O n note 0 : 1 ( 0 ) ! SL(2;Z) l ' hom om orphi sm e de m onodrom i e associ e. Le m orphi sm e 0 est une repr esentati on de 1 ( 0 ) quid eterm i ne et est d eterm i n e par l e fai sceau G . C om m e est r eel l e,pour tout 2 1 ( 0 ), on a 0 ( ( )) = S( 0 ( )) o u est l ' i nvol uti on i ndui te par sur 1 ( 0 ) et S est d e ni e au d ebut de l a secti on 2. Soi t u 2 et soi t 2 1 ( 0 ) l ' el em ent repr esent e par un l acet si m pl e tournant dans l e sens posi ti f autour de u. Le poi nt u est un pôl e de J siet seul em ent si l ' ordre de 0 ( ) est i n ni . D ans ce cas, 0 ( ) est conjugu ee dans SL(2;Z) a une m atri ce d' une des form es (3. 1) 1 m 0 1 ou 1 m 0 1 avec m > 0 cf. [ K o x9 ] . D ans l e prem i er cas,on di t que l a bre C u est du type I m et dans l e second cas,du type I m . Siu estun pôl e de J appartenant a (R ),C u estr eel l e etl estypesr eel spossi bl es de C u (R )sont cl assi es par l a tabl e (3. 2)ci -dessous extrai te de [ Si 2,T h. V II(1. 5)] . O n suppose que adm et une secti on r eel l e,soi t u 0 un poi nt de 0 (R ) voi si n de u. A l ors l a bre C u 0 est l i sse et C u 0( R ) poss ede une ou deux com posantes connexes. D ans l a deuxi em e col onne de l a tabl e, on a i ndi qu e l e nom bre de com posantes connexesde C u 0( R )s' i lreste constantl orsque u 0 vari eau voi si nage de u dans 0 (R ). O n a i ndi qu e " " si # C u 0( R )change au voi si nage de u. Lescol onnes3 et4 donnent l a charact eri sti que d' Eul ertopol ogi que de C u (R )etC u . La derni ere col onne donne l e nom bre (C u ) 1 de com posantes i rr educti bl es de C u quine rencontrent pas l a secti on. (4.2) D e nition. Si 1 6 2 0 ( 1 ( 0 )),une tel l e surface :X ! est appel ee surface el l i pti que m odul ai re r eel l e. Pour si m pl i er, et ceci correspond a l a restri cti on sur l es types de bres si ngul i eresd ej a fai te en secti on 3,on se restrei ntau caso u estsanspoi ntel l i pti que i . e. est sans torsi on,on a al ors (cf. [ Shm ]ou [ Sho (4. O n note fu l g = J 1 f0;1;1 g avec pour 1 l t,J(u l ) = 1 . A l ors l es bres si ngul i eresde :X ! sontl est bresC u l = 1 fu l g,l t(cf. [ Sho] )etl e type de l a bre C u l est determ i n e par 0 ( l ) o u l 2 1 ( 0 )est l ' el em ent repr esent e par un l acet si m pl e tournant dans l e sens posi ti fautour de u l . Soi t l t,choi si ssons un poi nt z 2 Q [ f1 g repr esentant u l . C om m e u l est un cusp,l e g en erateur du stabi l i sateur de z dans est conjugu e dans PSL(2;Z) a 1 m l 0 1 m od f 1g, m l > 0 etl a bre C u l estdu type I m l ou I m l . Les i nvari antsnum eri ques de X sont al ors donn es par (4.6) E xem ple. O n consi d ere l e groupe de congruence (2)= a b c d 2 SL(2;Z)= a d 1 m od 2;b c 0 m od 2g =f 1g F ig. 1 O n a repr esent e sur l a gure 1 un dom ai ne fondam entalpour (2). Ilest l i m i t e parl esg eod esi ques 1 ; 1 ;~ 1 ;~ 1 ,avec l esi denti cati onsl l pourune g eod esi que l. q(X )= 0 12 (O X )= 6 + 6 (I ) C e quii m pose que l e nom bre de bres de type I est i m pai r donc egal a 1 ou 3. D ans l e prem i er cas,l es troi s bres sont I 2 ;I 2 ;I 2 . A l ors (O X ) = 1 et X est une surface rati onnel l e r eel l e tel l e que h alg 1 (X (R )) = h 1 (X (R )) = h 1;1 (X ) = 10. D ans l e deuxi em e cas,l es bres si ngul i eres sonttroi sI 2 , (O X )= 2 etX estune surface K 3 r eel l e tel l e que h alg 1 (X (R ))= h 1 (X (R ))= h 1;1 (X )= 20. C onstruction de surfaces extr em ales Pour une surface el l i pti que X r egul i ere,on a h 1;1 (X )= di m C H 2 (X ;C ) 2di m C H 2 (X ;O X ): L' hypoth ese de r egul ari t e nous donne di m C H 2 (X ;C )= 12 (O X ) 2 (voi raussi (5.2) D e nition. U ne surface el l i pti que r egul i ere :X ! P 1 est di te norm al i s ee siet seul em ent si (1) aucune courbe excepti onnel l e n' est contenue dans une bre ; (2) est sans bre m ul ti pl e. (5.3) T h eor em e. Pour tout entier k > 0, ilexiste une surface ell iptique r eell e norm al is ee :X k ! P 1 quiv eri e (O X k )= k et h alg 1 (X k (R ))= 10k A vant de prouver l e th eor em e 5. 3,on m ontre com m ent on peut d edui re de ce r esul tat l e th eor em e 1. 2 de l ' i ntroducti on. (5.4) T h eor em e (K odaira). Toute surface ell iptique r egul i ere X provientd'une surface ell iptique norm al is ee Y par une succession de ecl atem ents de points etde transform ations l ogarithm iques d'ordres respectifs n 1 :::;n . D eux surfaces ell iptiques r egul i eres X et X 0 sont d eform ations l 'une de l 'autre sietseul em entsi (1) Y etY 0 sontd eform ations l 'une de l 'autre (2) = 0 (3) = 0 etn l = n 0 l ,8l2 f1;:::; g C et enonc e est ti r e de [ Pe,p. 306] . Le th eor em e sui vant se trouve en prem i ere page de [ K as] . K as (5. 5), X k est d eform ati on de Y . N otons X 0 l a surface obtenue a parti r de X k apr es ecl atem ent de poi nts de X k (R ). A l orsX 0 estune d eform ati on de X par(5. 4)eth alg 1 (X 0 (R ))= h 1;1 (X 0 ). D ' apr es (1. 1), com m e X 0 est une surface el l i pti que r eel l e r egul i ere et sans bre m ul ti pl e,on a h alg 1 (X 0 (R )) h 1 (X 0 (R )) h 1;1 (X 0 ): d' o u l e r esul tat. Le cas k = 1 est trai t e dans l ' exem pl e 4. 6. Soi t k 2, a parti r du dom ai ne fondam entalD [ e D de (2),cf. Fi g.1,on construi tun nouveau dom ai ne obtenu en adjoi gnantau tri angl e D [ e D k 2 transl at esde D a droi te etl eurssym etri quespar rapportau dem i -axe i m agi nai re.Le nouveau dom ai ne estbord e parl esg eod esi ques k 1 ; k 1 ; k 2 ;::: 1 ,~ 1 ;:::;~ k 1 ;~ k 1 i denti ees deux a deux par l l . Sik est pai r,6k 6 n' est pas di vi si bl e par 12 et n ecessai rem ent l ' une des bres si ngul i ere est de type I . D onc i lexi ste encore un rel evem ent de telque toutes l es bres si ngul i eres soi ent de type I . O n peut donc obteni r pour tout k 2 une surface not ee X k dont l a l i ste des bres si ngul i eres est I m 1 ;:::;I m k + 1 avec P l m l = 6k 6. O n a al ors 1;1 Par constructi on, l es k + 1 cusps apparti ennent a k (R ) donc l es k + 1 bres de type I m l sont r eel l es. C om m e k (2),l es cusps sont chacun conjugu e (dans PSL(2;Z)) a un cusp 1 m l 0 1 avec m l pai r. O n d edui tal orsdel a tabl e3. 2 quel enom bredecom posantesconnexedel a parti e r eel l e d' une bre r eel l e de est constant au voi si nage de chaque bre si ngul i ere. D onc toutesl es bresl i ssesde X k (R )ontl e mêm e nom bre de com posantesconnexe. Parai l l eurs,sion note l ' i m agedu dem i -axei m agi nai reparl ' appl i cati on canoni que H ! k = (H [ P 1 (Q ))= k ,on a k (R ). D ' apr es l es proposi ti ons 2. 2 et 2. 3, on a J(u) > 1 pour presque tout l es poi nts u de donc # C u (R ) = 2. D ' apr es ce quipr ec ede,on a pl us g en eral em ent # C u (R )= 2 pour toute l es bres r eel l es l i sses de . Par constructi on, X k (R ) est connexe et h 1 (X k (R )) = 2 P l (C u l (R )). D e nouveau a parti r de l a tabl e (3. 2), m ai ntenant qu' on connai t l e nom bre de composantes au voi si nage de chaque cusp u l ,on d edui t (C u l (R ))= m l 4. D e pl us, d' apr es [ Si 2,V II. 1] ,l es com posantes i rr educti bl es r eel l es d' une bre de type I qui ne rencontrent pas l a secti on sont i nd ependantes. D e l a (C u l (R ))= (C u ) 1 et h alg 1 (X k (R ))= 2 + X l m l + 4(k + 1)= 10k La proposi ti on sui vantesert a d eterm i nerl etypetopol ogi quedessurfacesX k (R ). donc X (R )6 = ; car n ecessai rem ent i lexi ste un poi nt de S \ K X dans X (R ). O n a al ors w 1 (X (R ))6 = 0 d' apr es (5. 7) et X (R ) est non ori entabl e. O n note S g l a surface (topol ogi que) ori entabl e de genre g et V q = # q P 2 (R ) l a 2. P r elim inaires : C ourbes de genre 1 r eelles O n note H = fz 2 C = = (z) > 0g l e dem i -pl an sup eri eur. L' acti on du groupe PSL(2;R ) = SL(2;R )=f 1g sur H est not ee z 7 ! A : z o u A : z = az+ b cz+ d si A est repr esent e par a b c d , ad bc = 1. L' i nvol uti on H :z 7 ! z de H est antihol om orphe. O n note S : SL( m pl i er,et com m e cel a su ra pour l a sui te,on a l ai ss e de côt e l es bres si ngul i eres quipourrai ent appara^ tre en dehors des pôl es de J. R em arquons que P t charact eri sti que d' Eul er topol ogi que vaut 12 foi s l a charact eri sti que d' Eul er hol om orphe (O X ). Sitoutesl es bres si ngul i eres de sont de type I ou I ,on ti re de l ' avant-derni ere col onne de l a tabl e deg(J) et (I ) est l e nom bre de bres du type I . A u vu de l a tabl e 3. 2,on n' a pas de form ul e aussig en eral e pour l es i nvari ants r eel s,voi r l ' exem pl e 4. 6 et l a preuve du th eor em e 5. 3.4. Surfaces elliptiques m odulaires r eellesSoi t PSL(2;Z) un sous-groupe d' i ndi ce ni ,en parti cul i er est un groupe Fuchsi en ari thm eti que. C om m e op ere sur P 1 (Q ),l e quoti ent = (H [ P 1 (Q ))= est bi en d e ni . Par hypoth ese,l e nom bre de cl asses de conjugai sons parabol i ques de est ni donc est une courbe projecti ve l i sse cf. [ Shm , x1. 3 et x1. 5] . Si 0 est un sous-groupe,al ors l ' appl i cati on canoni que de H = sur H = 0 s' etend en une appl i cati on hol om orphe de sur 0. En parti cul i er,on a une appl i cati on hol om orphe J : ! P 1 en prenant 0 = PSL(2;Z) et en i denti ant 0 avec P 1 vi a l a foncti on j, cf. secti on 2.Sion suppose de pl usque eststabl e parS, etJ sontnaturel l em ent d e ni s sur R . L' appl i cati on J est ram i ee seul em ent au dessus des troi s poi nts 0,1 et 1 ,cf. [ N o,prop. 2. 1] . D e pl us,l e degr e de J est egal a l ' i ndi ce de dans PSL(2;Z) degJ = [ PSL(2;Z): ] N otons 0 = n J 1 f0;1;1 g et p:H ! 0 l e revêtem ent uni versel . Par constructi on,i lexi ste un i som orphi sm e w :H ! H quifai tcom m uterl e di agram eri e w H = H w . D e l a,i lexi ste une uni que repr esentati on : 1 ( 0 )! PSL(2;Z) :tel l e que 8 2 1 ( 0 ),8ũ 2 H , ( ): w (ũ)= w ( :ũ) et ( ( ))= S( ( )). M ai ntenant chaque rel evem ent 0 : 1 ( 0 ) ! SL(2;Z) de v eri e encore 0 ( ( )) = S( ( )). Ilest faci l e d' adapter au cas r eell a constructi on de K odai ra[ K o,p. 578] , pour m ontrer qu' i lexi ste une surface el l i pti que :X ! g est l e genre de ,test l e nom bre de cusps et = [ PSL(2;Z): ] . 1 m l et (I ) est l e nom bre de bre de type I . (4. 5 ) 5Lem m e. Si est sans torsion et g( ) = 0, il existe exactem ent t 1 rel evem ents distincts 0 :1 ( 0 )! SL(2;Z) de : 1 ( 0 )! PSL(2;Z) quiv eri ent 0 ( l )= 1 si ( l )= 1o u t estl e nom bre de cusps de .C ' est un cas parti cul i er de[ N o,prop. 2. 3] . C e dom ai ne est stabl e par H . D e l a,l a courbe quoti ent = (H [ P 1 (Q ))= (2)est m uni e de l a structure r eel l e quoti ent et sa parti e r eel l e (R ) est repr esent ee dans H parl a r euni on desg eod esi ques 1 [ 1 etdu dem i -axe i m agi nai re.En parti cul i er, l es troi s cusps u 1 ;u 2 ;u 3 de ,repr esent es par 0;1 et 1 apparti ennent a (R ). R em arquons au vu des i denti cati ons que g( )= 0 i . e. = P 1 (C ). C onsi d eronsune surface m odul ai re :X ! associ ee a un rel evem ent,cf. (4. 1), 0 : 1 ( n fu l g)! SL(2;Z) La brati on poss ede exactem ent troi s bres si ngul i eres du type I m ou I m et d' apr es (4. 4), ( 5 . 51) h 1;1 (X )= 10 (O X ): En e et,d' apr es l e th eor em e de d ecom posi ti on de H odge, em e (K as). D eux surfacesell iptiquesnorm al is eesY etY 0 sontd eform ations l 'une de l 'autre sietseul em entsi (O Y )= (O Y 0) Preuve du th eor em e 1.2. Soi tV une surface r eel l e et W ! V un ecl atem entcentr e en un poi nt de V (R ),al ors W est une surface r eel l Soi t X une surface el l i pti que r egul i ere, d' apr es (5. 4), X provi ent d' une surface norm al i s ee Y par une sui te de d' ecl atem ents et transform ati ons l ogari thm i ques. Si de pl us X est sans bre m ul ti pl e, on a = 0. Posons k = (O Y ), et consi d erons une surface X k v eri ant l es hypoth eses du th eor em e 5. 3, al ors h alg 1 (X k (R )) = h 1;1 (Y ) et d' apr es l e th eor em e de F ig. 2 C 2' est l e dom ai ne fondam ental d' un sous-groupe k (2) d' i ndi ce ni dans PSL(2;Z),cf. e. g.[ D R ] .Parconstructi on,l a courbe quoti ent k = (H [ P 1 (Q ))= k est sans poi nt el l i pti que et poss ede k + 1 cusps.Le dom ai ne fondam ental est stabl e par H et l a courbe k est m uni e de l a structure r eel l e quoti ent. Sa parti e r eel l e k (R ) est repr esent ee dans H par l a r euni on des g eod esi ques k 1 [ k 1 [ k 2 [ ::: 1 et du dem i -axe i m agi nai re.A u vu des i denti cati ons,on a g( k )= 0 et d' apr es (4. 3) ( k )= 6(k 1):Sui vant l a secti on 4, on a une appl i cati on J k : k ! P 1 et on note 0 k = k n J 1 k f0;1;1 g.O n peut construi re pour chaque rel evem ent 0 : 1 ( 0 k )! SL(2;Z) de : 1 ( 0 k )! k PSL(2;Z)une surface el l i pti que m odul ai re r eel l e avec secti on r eel l e :X ! P 1 d' i nvari ants (J k ; 0 ). Par constructi on, X poss ede exactem ent k + 1 bres si ngul i eres et ces bres sont de type I m ou I m . D ' apr es l es form ul es (4. 4),l a surface X v eri e : I ) est l e nom bre de bres de type I m . En parti cul i er,quel que soi t l e choi x de 0 ,X est une surface el l i pti que r egul i ere et h 1;1 (X )= 10 (O X ).Sik est i m pai r,l e nom bre de bres si ngul i eres est pai r et d' apr es l e l em m e 4. 5, i lexi ste un rel evem ent 0 telque toutes l es bres si ngul i eres soi ent de type I . (5. 6 ) 6P roposition. SoitX ! P 1 une surface ell iptique r eell e norm al is ee, si (O X ) estpair etX (R )6 = ;,al ors X (R ) estorientabl e. si (O X ) estim pair etX adm etune section d e nie sur R ,al ors X (R )6 = ; etX (R ) estnon orientabl e. Preuve. Sousl eshypoth esesconsi d er ees,un di vi seurcanoni que de X estdonn e par (cf. e. g. [ B PV ,p. 162] ) : K X = ( (O X ) 2)F o u F est une bre quel conque. La cl asse w 1 (X (R )) 2 H 1 (X (R );Z=2), dual e de Poi ncar e de l a 1 ere cl asse de Sti efel -W hi tney,est repr esent ee par K X (R ) et pour tout di vi seur r eelD de X ,on a (5. 7) D : K X D (R )\ K X (R ) m od 2 D ans l e cas o u (O X ) est pai r, K X 0 m od 2 d' o u w 1 (X (R )) = 0 et X (R ) est ori entabl e. M ai ntenant,supposons que (O X ) esti m pai ret notons S l a courbe i m age dans X d' une secti on d e ni e sur R ,on a S: K X 1 m od 2 Soit X k une surface v eri ant l es hypoth eses du th eor em e 5.3, al ors sik estpair,X k (R )= S 5k etsik estim pair,X k (R )= V 10k . Parhypoth ese,on a h 1 (X k (R ))= 10k etl. C Orollaire, C orollaire. Soit X k une surface v eri ant l es hypoth eses du th eor em e 5.3, al ors sik estpair,X k (R )= S 5k etsik estim pair,X k (R )= V 10k . Parhypoth ese,on a h 1 (X k (R ))= 10k etl ' ori entabi l i t eestdonn ee parl a proposi - . # X (R ) h. 1X (R )) h. 1 (X ) 2(r(X ) 1# X (R ) h 1 (X (R )) h 1;1 (X ) 2(r(X ) 1) X )d esi gne l e nom bre de cl asses r eel l es i nd ependantes dans l e groupe de N eron-Severide X . C om m e X k est norm al i s ee,on a h alg 1 (X k (R )) r(X k ) h 1;1 (X k ). voi r par exem pl e [ Si 2,III(1. 10)val abl e pour une surface r eel l e X quel conque o u r(X )d esi gne l e nom bre de cl asses r eel l es i nd ependantes dans l e groupe de N eron-Severide X . C om m e X k est nor- m al i s ee,on a h alg 1 (X k (R )) r(X k ) h 1;1 (X k ),voi r par exem pl e [ Si 2,III(1. 10)] , C P V ] W .B Arth, A Eters, En, C om pact com pl ex surfaces,E rgebnisse der M athem atik. B erlin H eidelbergSpringerP V ] W .B arth,C .P eters,A .V an de V en,C om pact com pl ex surfaces,E rgebnisse der M athe- m atik,Springer,B erlin H eidelberg,1984. La cl asse d'hom ol ogie fondam ental e d'un espace anal ytique. E Orel, A , B ul. Soc.M ath.France. 83E .B orel,A .H ae iger,La cl asse d'hom ol ogie fondam ental e d'un espace anal ytique,B ul. Soc.M ath.France 83 (1961),461{513. ] A K H, V Egtyarev, Topol ogicalcl assi cation of realE nriques surfaces. 35K h] A .D egtyarev,V .K harlam ov,Topol ogicalcl assi cation of realE nriques surfaces,Topol- ogy 35 (1996),no.3,711{730. Sur l es pol ygones g en erateurs de groupes Fuchsiens. D R ] G .De R Ham, L'ens.M ath. 171D R ] G .de R ham ,Sur l es pol ygones g en erateurs de groupes Fuchsiens,L'ens.M ath.17 (1971), no.1,49{61. O n the deform ation types of regul ar ell iptic surfaces, C om plex analysis and algebraic geom etry. A , W .B aily,T .ShiodaC am bridge university pressC am bridge London N ew Y orkA . K as, O n the deform ation types of regul ar ell iptic surfaces, C om plex analysis and algebraic geom etry (W .B aily,T .Shioda, eds.),C am bridge university press,C am bridge London N ew Y ork,1977. S , Fuchsian groups,C hicago Lectures in M athem atics,T he U niversity ofC hicago P ress,C hicago. S.K atok,Fuchsian groups,C hicago Lectures in M athem atics,T he U niversity ofC hicago P ress,C hicago,1992. O n the structure of com pact com pl ex anal ytic surfaces II & III, A m er. K Odaira, J. M ath. 88K . K odaira, O n the structure of com pact com pl ex anal ytic surfaces II & III, A m er. J. M ath.88 & 90 (1966 & 1968),682{721 & 56{83. C ycl es al g ebriques sur l es surface K 3 r eell es, M ath. F , Z. 2254F.M angolte,C ycl es al g ebriques sur l es surface K 3 r eell es, M ath.Z.225 (1997), no.4, 559{576. F Angolte, J , A l gebraic cycl es and topol ogy of real E nriques surfaces, C om pos.M ath. 110F. M angolte, J. van H am el, A l gebraic cycl es and topol ogy of real E nriques surfaces, C om pos.M ath.110 (1998),no.2,215{237. M Ori, O n certain ell iptic surfaces w ith m axim alP icard num ber. 24M .N ori,O n certain ell iptic surfaces w ith m axim alP icard num ber,Topology 24 (1984), 175{186. U Ersson, H orikaw a surfaces w ith m axim al P icard num ber, M ath. A nn. 259U . P ersson, H orikaw a surfaces w ith m axim al P icard num ber, M ath. A nn. 259 (1982), 287{312. Introduction to the arithm etic theory of autom orphic functions, P rinceton U niv. G Shim, P ressG . Shim ura, Introduction to the arithm etic theory of autom orphic functions, P rinceton U niv.P ress,1971. O n ell iptic m odul ar surfaces. T Shioda, J.M ath.Soc.Japan. 24T .Shioda,O n ell iptic m odul ar surfaces,J.M ath.Soc.Japan 24 (1972),20{59. R Silhol, R ealal gebraic surfaces w ith rationalor ell iptic berings,M ath. 186R .Silhol,R ealal gebraic surfaces w ith rationalor ell iptic berings,M ath.Z.186 (1984), 465{499. R eal al gebraic surfaces. R Silhol, Lectures notes in M ath. 1392SpringerR . Silhol, R eal al gebraic surfaces, Lectures notes in M ath., vol. 1392, Springer, B erlin H eidelberg N ew Y ork,1989. C om pacti cations of m odul i spaces in real al gebraic geom etry. R Silhol, Invent.M ath. 107R . Silhol, C om pacti cations of m odul i spaces in real al gebraic geom etry, Invent.M ath. 107 (1992),151{202. Silverm an, T he arithm etic of ell iptic curves. J , G raduate Texts in M ath. -V erlag,N ew Y orkSpringer106J. H . Silverm an, T he arithm etic of ell iptic curves, G raduate Texts in M ath., vol. 106, Springer-V erlag,N ew Y ork,1986. . Niversit E De Savoie ; L E B Ourget, T Rance, El, 33) 4 79 75 87 42 E -m ailaddress: [email protected] aboratoire de M ath em atiques, U niversit e de Savoie,F -73 376 L e B ourget du L ac C edex, F rance, T el: (33) 4 79 75 86 60, Fax: (33) 4 79 75 87 42 E -m ailaddress: [email protected]
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[ "arXiv:hep-ph/0001022v2 21 Sep 2000 One-particle inclusive CP asymmetries", "arXiv:hep-ph/0001022v2 21 Sep 2000 One-particle inclusive CP asymmetries" ]
[ "Xavier Calmet \nInstitut für Theoretische Teilchenphysik\nLudwig-Maximilians-Universität, Sektion Physik\nTheresienstraße 37D-80333MünchenGermany\n", "Thomas Mannel \nUniversität Karlsruhe\nD-76128KarlsruheGermany\n", "Ingo Schwarze \nUniversität Karlsruhe\nD-76128KarlsruheGermany\n" ]
[ "Institut für Theoretische Teilchenphysik\nLudwig-Maximilians-Universität, Sektion Physik\nTheresienstraße 37D-80333MünchenGermany", "Universität Karlsruhe\nD-76128KarlsruheGermany", "Universität Karlsruhe\nD-76128KarlsruheGermany" ]
[]
One-particle inclusive CP asymmetries in the decays of the type B → (−) D ( * ) X are considered in the framework of a QCD based method to calculate the rates for one-particle inclusive decays.
10.1103/physrevd.62.096014
[ "https://export.arxiv.org/pdf/hep-ph/0001022v2.pdf" ]
2,281,131
hep-ph/0001022
2d78a069acf99ea84b5658253547a4b7a2b7f2fa
arXiv:hep-ph/0001022v2 21 Sep 2000 One-particle inclusive CP asymmetries 21 September 2000 Xavier Calmet Institut für Theoretische Teilchenphysik Ludwig-Maximilians-Universität, Sektion Physik Theresienstraße 37D-80333MünchenGermany Thomas Mannel Universität Karlsruhe D-76128KarlsruheGermany Ingo Schwarze Universität Karlsruhe D-76128KarlsruheGermany arXiv:hep-ph/0001022v2 21 Sep 2000 One-particle inclusive CP asymmetries 21 September 2000To be published in Physical Review D62, 0960xx (Received 5 January 2000; to be published 1 November 2000)PACS numbers: 1130Er, 1325Hw One-particle inclusive CP asymmetries in the decays of the type B → (−) D ( * ) X are considered in the framework of a QCD based method to calculate the rates for one-particle inclusive decays. Introduction One of the main goals in B physics is a detailed study of flavor mixing, which is encoded in the Cabibbo-Kobayashi-Maskawa (CKM) matrix of the standard model. In particular, the violation of the CP symmetry, which the standard model describes by a nontrivial phase in the CKM matrix or equivalently by the angles of the unitarity triangle, will be investigated. Typically CP asymmetries are expected to be large in some of the exclusive nonleptonic B decays which, however, have only small branching ratios. Examples are the determination of β from B → J/ψ K s and of α from B → ππ. In addition, in these exclusive nonleptonic decays it is very hard to obtain a good theoretical control over the hadronic uncertainties, in particular due to the presence of strong phases. On the other hand, inclusive decays have large branching fractions but typically smaller CP asymmetries than exclusive decays [1]. One may use parton hadron duality to obtain a good theoretical description. This has been studied by Beneke, Buchalla and Dunietz who set up a theoretically clean method to calculate the CP asymmetries in inclusive B decays [2]. They still find sizable CP asymmetries, but their measurement would require to identify charmless final states inclusively, which is not an easy task. One-particle inclusive decays lie somehow between these two cases. This class of decays still has large branching fractions and some of the expected CP asymmetries are sizable. Furthermore, a measurement of these decays is feasible. For one-particle inclusive decays of the type B → (−) D ( * ) X, a QCD based description has been developed recently, exploiting factorization and the heavy mass limit for both the b and the c quark [3]. Since the expansion parameters are Λ QCD /(m b − m c ), 1/N C and α s (m c ), corrections to the leading term could be fairly large, in the worst case of the order of 30%. Using this method, which unfortunately is not completely model independent, we compute mixing induced time-dependent and time-integrated CP asymmetries in the framework of the standard model. In view of the considerable uncertainties due to an unknown strong phase, our method cannot yet be used for a competitive determination of the CP violation parameters, in particular compared to a measurement of sin(2β) in the "gold-plated" channel B → J/ψ K s . However, it can be used as an estimate of the one-particle inclusive CP asymmetries, for which we shall use present central values of the CP angles β and γ [4]. Compared to fully inclusive methods, the advantage is that we can predict asymmetries for the various spins and charges of the ground-state charmed mesons separately. This is certainly a worthwhile task, in particular since we are not aware of any previous prediction for these asymmetries, not even in the context of quark models. After introducing our notations for B mixing in Sec. 2, we calculate the relevant matrix elements in Sec. 3 and model the form factors in Sec. 4. The numerical results are given in Sec. 5. 2 CP asymmetries in B → (−) D ( * ) X In Wigner Weisskopf approximation the time evolution of an initially pure B 0 or B 0 , B 0 phys (t) = g + (t) B 0 − q p g − (t) B 0 , B 0 phys (t) = g + (t) B 0 − p q g − (t) B 0 ,(1) is determined by the time-dependent functions g + (t) = e −iM t− 1 2 Γt cosh ∆Γt 4 cos ∆Mt 2 + i sinh ∆Γt 4 sin ∆Mt 2 g − (t) = e −iM t− 1 2 Γt sinh ∆Γt 4 cos ∆Mt 2 + i cosh ∆Γt 4 sin ∆Mt 2 ,(2)q p = ∆M − i 2 ∆Γ 2 M 12 − i 2 Γ 12 = M * 12 |M 12 | 1 − 1 2 a + O(a 2 ) , a = Im Γ 12 M 12 .(3) In fact, Γ 12 /M 12 = O(m 2 b /m 2 t ) is very small and hence q/p is to a good approximation a phase factor. The time-dependent rate for the decay of a B meson into a set of final states |f = i |f i can be written as Γ[B(t) → f ] = 1 2m B i dφ i (2π) 4 δ 4 (p B − p f i ) B(t) |H eff | f i f i |H eff | B(t) = 1 2m B d 4 x B(t) |H eff (x)Π f H eff (0)| B(t) ,(4) where dφ i is the phase space element of the state |f i and Π f = i dφ i |f i f i |(5) is the projector on the set of final states. Note that both an exclusive final state as well as inclusive states can be treated in this way. Even differential distributions can be considered if the phase spaces dφ i are not fully integrated. The CP asymmetries we are going to consider are of the type A CP (t) = Γ(B 0 (t) → f ) − Γ(B 0 (t) → f ) Γ(B 0 (t) → f ) + Γ(B 0 (t) → f )(6) which involves the CP conjugate set |f > of final states. Up to here the discussion is completely general. In the following we shall use the above formalism to compute the CP asymmetries for one-particle inclusive final states, for which the projector reads Π f = X |XY XY | ,(7) where Y can be a D or a D meson. Since the sum runs over all possible states X, the CP conjugate of the projector is Π f = X XY XY .(8) Inserting the time-dependent states (1) we obtain Γ[B(t)→Y X] = |g + (t)| 2 Γ BB Y + q p g − (t) 2 Γ B B Y − 2Re q p g * + g − (t)T BB Y , Γ[B(t)→Y X] = |g + (t)| 2 Γ B B Y + p q g − (t) 2 Γ BB Y − 2Re p q g * + g − (t)T BB Y ,(9) where the matrix elements are defined by Γ BB Y = 1 2m B d 4 x B |H eff (x)Π Y H eff (0)| B , T BB Y = 1 2m B d 4 x B |H eff (x)Π Y H eff (0)| B .(10) The ∆B = 2 transition matrix elements representing the interference between the mixed and the unmixed amplitudes are related by CP T symmetry, such that T Y := T BB Y = T BB Y * .(11) The direct CP asymmetries in these processes are expected to be tiny. In fact, using the method described in Ref. [3], they turn out to be of higher order in the 1/m expansion. Hence we have Γ Y := Γ BB Y = Γ B B Y = Γ(B → Y X), Γ Y := Γ BB Y ,(12)T Y = T Y .(13) Inserting the time-dependent decay rates in Eq. (9) and neglecting both the width difference and a, such that q/p becomes a phase factor, we obtain for the time-dependent CP asymmetries A CP (t) = sin (∆Mt) Im q p T Y cos 2 ∆M t 2 Γ Y + sin 2 ∆M t 2 Γ Y ,(14) from which we get the time-integrated asymmetry A CP = 2 x Im q p T Y (2 + x 2 ) Γ Y + x 2 Γ Y ,(15) where x = ∆M/Γ is measured to be x = 0.73 [5]. Transition matrix elements In order to compute the CP asymmetries, one has to evaluate the matrix elements in Eq. (10). The total rates Γ Y have already been discussed in Ref. [3], so we only need to calculate the interference term T Y . The relevant pieces of the effective Hamiltonian contributing to this inter- ference are (ub) V −A (dc) V −A and (cb) V −A (du) V −A interfering with each other and (cb) V −A (dc) V −A interfering with itself, so T Y is a sum of the two contri- butions T Y = T c + T u ,(16)T q = 1 2m B G 2 F 2 V cb V * qd V qb V * cd |C 1 | 2 X (2π) 4 δ 4 (p B − p D − p X ) (17) B 0 (qb) V −A (dc) V −A DX DX (dq) V −A (cb) V −A B 0 . Fierzing the operators into the form (db ) V −A (uc) V −A , (db) V −A (cu) V −A and (db) V −A (cc) V −A one can reproduce the inclusive results of Ref. [2]. In order to evaluate the interference term for the one-particle inclusive case, we use the method developed in Ref. [3]. It is based on factorization, which holds to leading order in the 1/N C expansion, where N C is the number of QCD colors. Thus we can write the interference terms as products of two tensors T q = 1 2m B G 2 F 2 V cb V * qd V qb V * cd |C 1 | 2 d 4 Q (2π) 4 K µν (p B , Q) dφ D P µν q (p D , Q) (18) with K µν (p B , Q) = X (2π) 4 δ 4 (p B − p X − Q) (19) B 0 (p B ) (dγ µ (1−γ 5 )b) X X (dγ ν (1−γ 5 )b) B 0 (p B ) , P µν q (p D , Q) = X ′ (2π) 4 δ 4 (Q − p D − p X ′ ) (20) 0 |(qγ µ (1−γ 5 )c)| D ( * ) (p D )X ′ D ( * ) (p D )X ′ |(cγ ν (1−γ 5 )q)| 0 . The tensor K µν (p B , Q) is fully inclusive and one can perform a standard short distance expansion. The resulting ∆B = 2 matrix element can be parameterized by the decay constant f B of the B meson and the bag factors B and B s for the axial vector and the scalar current, respectively. The other tensor P µν q (p D , Q) involves a projection on a one-particle inclusive charmed meson state and hence we cannot perform a short distance expansion. We proceed along the same lines as in Ref. [3], where the rates for wrong charm decays have been modeled. Heavy quark symmetry yields the Dirac matrix structure P µν q (p D , Q) ∝ H D ( * ) (p D )γ µ (1−γ 5 ) ⊗ γ ν (1−γ 5 )H D ( * ) (p D ),(21) where the representation matrices for the charmed mesons are H D = √ m D 1 + / v D 2 γ 5 , H D * = √ m D * 1 + / v D * 2 / ǫ.(22) In principle, all possible contractions of the light quark indices may contribute, giving rise to several form factors. For a first estimate, it is sufficient to use only the simplest one of these contractions, P µν q (p D ,Q) = 2π δ (Q−p D ) 2 −m 2 q Tr / p D γ µ (1−γ 5 ) ( / Q−/ p D ) γ ν (1−γ 5 ) f qY ,(23) corresponding to a replacement of the D ( * ) X final state by a pair of free quarks, rescaled by an operator-and decay-channel-specific form factorf qY , where Y is one of the ground state D mesons. In the following, we call this contraction "partonic." Using this ansatz and the heavy mass limit, the transition matrix elements read T c = − G 2 F m 3 B f 2 B 24π (V cb V * cd ) 2 |C 1 | 2 √ 1−4z[(1−4z)B +2(1+2z)B S ]f cY ,(24)T u = − G 2 F m 3 B f 2 B 24π V cb V * ud V ub V * cd |C 1 | 2 (1−z) 2 [(1−z)B +2(1+2z)B S ]f uY ,(25) where z = (m c /m b ) 2 and C 1 is the Wilson coefficient of the effective Hamiltonian in the notation of Ref. [3]. Equations (24) and (25) correspond to the expression for the width difference of neutral heavy meson systems [6]. In the standard CKM parametrization, the phases of the transition matrix elements are arg(T c ) = 0, (26) arg(T u ) = arg(−V ub ) = −γ,(27)arg(q/p) = arg(−V 2 td ) = −2β,(28) such that Im q p T Y = sin(2β) |T c | + sin(2β + γ) |T u | .(29) Modeling the form factors We assume that the form factorsf qY do not vary strongly over the accessible phase space and hence we approximate them by constants. For the case q = c, these constants have been fitted to the wrong charm yield in B decays [3]. Operators analogous to the case q = u are Cabibbo suppressed when calculating wrong charm rates, so they did not appear in Ref. [3]. Assuming that all charm quarks eventually hadronize to D mesons, we usẽ f uD 0 +f uD + = 1.(30) To resolve the spin and charge counting, we first discuss the heavy mass limit where the pseudoscalar and vector charmed mesons form a degenerate ground state doublet. The decay of vector to pseudoscalar mesons will be discussed below. In the following, D dir refers to those D mesons that do not result from D * decays, and D ( * ) can be either D dir or D * . As long as the light quark spin indices of the D ( * ) meson representation matrices are contracted with each other, Eq. (21) reproduces the naive spin countingf Different contractions yield results of comparable size. The experimental spin counting factor appears to be smaller by roughly a factor of two [3]. Since this effect is not yet understood, we treat it as an uncertainty. qD * 0 = 3f qD 0 dir ,f qD * + = 3f qD + dir .(31) Concerning charge counting, we argued by isospin symmetry [3] that in the case q = c we havef cD ( * )0 =f cD ( * )+ . In the case q = u, two topologies can contribute to the decay amplitude: the charm quark can either hadronize with the u quark from the weak effective current, in which case the isospin of the state |X is I X = 0, or with a u or d quark from vacuum, which contains both I X = 0 and I X = 1 contributions. In the case I X = 0, both amplitudes can interfere, so there are three contributions to the decay ratẽ f uD ( * )0 = |a 1 + a 2 | 2 = |a 1 | 2 + |a 2 | 2 + 2Re {a * 1 a 2 } f uD ( * )+ = |a 2 | 2 ,(33) see Figs. 1-3. One might doubt whether using the partonic contraction given in Eq. (23) is justified for all the topologies, as it appears to correspond to the topology in Fig. 2, while the topology in Fig. 1 should rather be described by the contraction P µν q (p D , Q) ∝ Tr H D ( * ) (p D )γ µ (1−γ 5 ) Tr γ ν (1−γ 5 )H D ( * ) (p D ) .(34) This is not a problem for three reasons. First, we do not claim to be able to accurately model the matrix element, but we only give the simplest possible ansatz by rescaling the partonic result. In particular, it is clearly not yet feasible to model particular contributions individually. We only use the three topologies to estimate the integrated relative magnitudes of the two main contributions and to bound the magnitude of their interference term. Secondly, neither the time-dependent nor the time-integrated asymmetries depend on the choice of the contraction unless studied differentially in the momentum of the charmed meson, which so far we do not attempt to do. Finally, as noted in Ref. [3], the choice of the wrong charm contraction appeared to have little influence even on differential observables. The topologies in Figs. 1 and 2 also occur in wrong charm production in B decays. Figure 1 corresponds to the process B → D ( * )+ s X, Fig. 2 to the process B → D ( * ) X, where D ( * ) can be either D ( * )0 or D ( * )+ . Both contributions are experimentally known to be of similar size, i.e., (10 ± 2.5)% [5] and (7.9 ± 2.2)% [7], respectively, such that |a 1 | 2 = 2 |a 2 | 2 .(35) The relative phase of the two contributions is unknown. Therefore, although it may be large, we have to treat the interference part as a theoretical uncertainty. This is acceptable since the q = u contribution is smaller than the q = c contribution according to T u T c = V ub V cb V ud V cd (1 − z) 2 (1 + z) √ 1 − 4zf ũ f c ∝ V ub V cb (1 + z) |V cd | ≈ 0.4.(36) Off the heavy mass limit, D * → D decay has to be taken into account. In the same way as in Ref. [3], we get f qD + =f qD + dir + Br D * + → D + X f qD * + f qD 0 =f qD 0 dir +f qD * 0 + Br D * + → D 0 X f qD * + .(37) The coefficients obtained from Eqs. (30)-(37) and Ref. [3] are summarized in Table 1. The ranges given result from varying the spin counting factor in Eq. (31) from 3 down to 3/2 and the interference in Eq. (33) from the central value of vanishing interference to full constructive and destructive interference. Table 1: Operator-and channel-specific form factors. Results We have computed the parameters for the time-dependent CP asymmetries as well as the time-integrated asymmetries. We have inserted recent values for sin 2β = 0.75 and γ = 68 • [4]. In addition, we use V cb = 0.04, V ub = 0.08 V cb , z = 0.09, x = 0.73, f B = 180 MeV, Br (D * + → D 0 Y ) = 1 − Br (D * + → D + Y ) = 0.683 and C 1 = B = B S = 1. The results of the calculations can be found in Fig. 4 and Table 2. To assess the uncertainties involved in Fig. 4, note that according to Eq. (14) the shapes of the time-dependent asymmetries are determined by the ratios of the wrong to right charm rates Γ Y /Γ Y . We checked numerically that the shapes would hardly change even if these ratios were off by 30%. The dominant contribution to the uncertainty of the amplitudes arises from the transition matrix elements T Y and is directly proportional to the uncertainties of the time-integrated asymmetries given in Table 2. Suppose N perfectly tagged B 0 decays are recorded in an experiment. In order to establish the asymmetry in a channel with a branching ratio b on the 3σ level, A 3 ≥ ∆A = 1 √ 2bN(38) has to be satisfied. The necessary numbers of tagged B 0 decays are given in the last column of Table 2. Since the asymmetry tends to be roughly inversely proportional to the branching ratio by Eq. (15), we obtain from Eq. (38) N ∝ 1 A 2 b ∝ b,(39) such that rare channels are advantageous for observing one-particle inclusive asymmetries. Table 2: Branching ratios, integrated CP asymmetries and numbers of necessary tagged B 0 decays for the one-particle inclusive B 0 → (−) D ( * ) X decay channels. Concerning D * 0 , see the text. The channel B 0 → D * 0 X deserves a further comment. Looking at Fig. 4, there is an obvious problem at small proper decay times. The reason for this problem is that we have discussed all the rates only to leading order in the combined 1/N C and 1/m Q expansions. However, this leading term vanishes for the channel B 0 → D * 0 X and thus subleading terms become relevant. On the other hand, the numerator T Y of the CP asymmetries is given by a matrix element of a dimension six operator and hence is suppressed compared to the leading terms of most of the rates. In other words, while in most of the rates the asymmetries are of subleading order f 2 B /m 2 B , this is not the case for the channel B 0 → D * 0 X. Unfortunately we cannot compute this possibly large asymmetry, since this would involve to compute subleading terms for the decay rate. Hence we try to estimate the asymmetry by varying Br B 0 → D * 0 X in Eq. (15) and show the reaction of the asymmetry in Fig. 5 and of the necessary number of tagged B 0 events in Fig. 6. The wrong charm asymmetry is practically unaffected by Br B 0 → D * 0 X since the pole occurs near four average lifetimes where most of the B mesons have already decayed, but the right charm asymmetry turns out to be extremely sensitive. Therefore we cannot predict the latter quantitatively, but it can be as large as several percent, and it will be measurable with a few 100 000 tagged B 0 events. Conclusion Motivated by the work on fully inclusive CP asymmetries and the question how to measure them, we studied one-particle inclusive CP asymmetries. In the final state only a (−) D ( * ) meson has to be identified and thus they are experimentally more easily accessible than the fully inclusive CP asymmetries. We have used a similar method as in in Ref. [3] to calculate the timedependent and time-integrated CP asymmetries for one-particle inclusive B → (−) D ( * ) X decays. It turns out that, as in Ref. [3], one cannot avoid to introduce some model dependence. Furthermore, there is also some dependence on an unknown relative phase, which we treat as an uncertainty. Due to these uncertainties we cannot expect our method to compete with proposed methods using "gold-plated" channels for determining CKM parameters, but we can still give estimates for the expected CP asymmetries of the different ground state (−) D mesons. For most of the asymmetries we find results of a few 10 −3 , but some are expected to be as large as several percent. These effects should be observable at the B factories. The channels involving right and wrong charm neutral vector mesons turn out to be most promising: they are expected to have the largest asymmetries, and the theoretical method yields the best results for the production rates and spectra of the vector mesons [3]. where ∆M = M H − M L > 0 and ∆Γ = Γ H − Γ L < 0 are the mass and width differences between the mass eigenstates |B H > = p|B 0 > + q|B 0 > and |B L > = p|B 0 > − q|B 0 >. The quantity q/p is given in terms of the off-diagonal elements of the Hamiltonian H = M − iΓ/2 of the neutral B meson system Figure 1 :Figure 2 : 12Topology yielding |a 1 | 2 in Eq. (33). Topology yielding |a 2 | 2 in Eq. (33). Figure 3 : 3Interference topology for Eq. (33). Figure 4 : 4Time-dependent CP asymmetries in B 0 → (−)DX for pseudoscalar (above), vector (below), charged (left), neutral (right), right charm (solid), and wrong charm (dashed) Figure 5 : 5Time Figure 6 : 6Necessary number of tagged B 0 events in B 0 → (−) D * 0 X as a function of Br(B 0 → D * 0 X). AcknowledgmentsThe authors thank Thomas Gehrmann for fruitful discussions. This work (X. C. during his time in Karlsruhe, T. M. and I. S.) was supported by the DFG Graduiertenkolleg "Elementarteilchenphysik an Beschleunigern" and by the DFG Forschergruppe "Quantenfeldtheorie, Computeralgebra und Monte-Carlo-Simulation." . I Dunietz, Eur. Phys. J. C. 7197I. Dunietz, Eur. Phys. J. C 7, 197 (1999). . M Beneke, G Buchalla, I Dunietz, Phys. Lett. B. 393132M. Beneke, G. Buchalla, and I. Dunietz, Phys. Lett. B 393, 132 (1997). . X Calmet, T Mannel, I Schwarze, Phys. Rev. D. 61114004X. Calmet, T. Mannel, and I. Schwarze, Phys. Rev. D 61, 114004 (2000). . M Ciuchini, E Franco, L Giusti, V Lubicz, G Martinelli, Nucl. Phys. 573201M. Ciuchini, E. Franco, L. Giusti, V. Lubicz, and G. Martinelli, Nucl. Phys. B573, 201 (2000). . C Caso, Eur. Phys. J. C. 3Particle Data GroupParticle Data Group, C. Caso et al., Eur. Phys. J. C 3 (1998). . J S Hagelin, Nucl. Phys. 193123J. S. Hagelin, Nucl. Phys. B193, 123 (1981). . T E Coan, CLEO CollaborationPhys. Rev. Lett. 801150CLEO Collaboration, T.E. Coan et al., Phys. Rev. Lett. 80, 1150 (1998).
[]
[ "Leveraging Unlabeled Data for Crowd Counting by Learning to Rank", "Leveraging Unlabeled Data for Crowd Counting by Learning to Rank" ]
[ "Xialei Liu [email protected] ", "Joost Van De Weijer ", "Andrew D Bagdanov [email protected] ", "\nComputer Vision Center\nComputer Vision Center\nMICC\nBarcelona, BarcelonaSpain, Spain\n", "\nUniversity of Florence Florence\nItaly\n" ]
[ "Computer Vision Center\nComputer Vision Center\nMICC\nBarcelona, BarcelonaSpain, Spain", "University of Florence Florence\nItaly" ]
[]
We propose a novel crowd counting approach that leverages abundantly available unlabeled crowd imagery in a learning-to-rank framework. To induce a ranking of cropped images , we use the observation that any sub-image of a crowded scene image is guaranteed to contain the same number or fewer persons than the super-image. This allows us to address the problem of limited size of existing datasets for crowd counting. We collect two crowd scene datasets from Google using keyword searches and queryby-example image retrieval, respectively. We demonstrate how to efficiently learn from these unlabeled datasets by incorporating learning-to-rank in a multi-task network which simultaneously ranks images and estimates crowd density maps. Experiments on two of the most challenging crowd counting datasets show that our approach obtains state-ofthe-art results.
10.1109/cvpr.2018.00799
[ "https://arxiv.org/pdf/1803.03095v1.pdf" ]
3,787,969
1803.03095
8ca1128ebd20eb5a346d9adb4cb427aef61f3552
Leveraging Unlabeled Data for Crowd Counting by Learning to Rank Xialei Liu [email protected] Joost Van De Weijer Andrew D Bagdanov [email protected] Computer Vision Center Computer Vision Center MICC Barcelona, BarcelonaSpain, Spain University of Florence Florence Italy Leveraging Unlabeled Data for Crowd Counting by Learning to Rank We propose a novel crowd counting approach that leverages abundantly available unlabeled crowd imagery in a learning-to-rank framework. To induce a ranking of cropped images , we use the observation that any sub-image of a crowded scene image is guaranteed to contain the same number or fewer persons than the super-image. This allows us to address the problem of limited size of existing datasets for crowd counting. We collect two crowd scene datasets from Google using keyword searches and queryby-example image retrieval, respectively. We demonstrate how to efficiently learn from these unlabeled datasets by incorporating learning-to-rank in a multi-task network which simultaneously ranks images and estimates crowd density maps. Experiments on two of the most challenging crowd counting datasets show that our approach obtains state-ofthe-art results. Introduction Crowd counting and crowd density estimation techniques aim to count the number of persons in crowded scenes. They are essential in video surveillance [3], safety monitoring, and behavior analysis [29]. Person counting and density estimation are instances of a broader class of classical counting problems in computer vision. Counting semantic image features is important in medical and biological image processing [18], vehicle counting [25], and numerous other application contexts. Despite the attention the crowd counting problem has received, both classically and in the recent computer vision literature, it remains a difficult task in practice. Perspective distortion, clutter, occlusion, non-uniform distribution of people, complex illumination, scale variation, and a host of other scene-incidental imaging conditions render person counting and crowd density estimation in unconstrained images an extremely daunting problem. Techniques for crowd counting have been recently improved using Convolutional Neural Networks (CNNs). These recent approaches We sample a decreasing sequence of sub-images I1, I2, and I3 from an unlabeled image. Though we do not know the exact person counts C(Ii), we use the fact that C(I1) ≥ C(I2) ≥ C(I3) as self-supervision to learn representations for person counting. include scale-aware regression models [25], multi-column CNNs [37], and switching networks [1]. As with most CNN architectures, however, these person counting and crowd density estimation techniques are highly data-driven. Even modestly deep architectures for visual recognition require massive amounts of labeled training data for learning. For person counting, the labeling burden is even more onerous than usual. Training data for person counting requires that each individual person be meticulously labeled in training images. It is for this reason that person counting and crowd density estimation datasets tend to have only a few hundred images available for training. As a consequence, the ability to train these sophisticated CNN-based models suffers. Recently, self-supervised learning has received more attention because it provides an alternative to collecting large hand-labeled datasets. Self-supervised learning is based on Methods Basic CNNs Scale-aware Context-aware Multi-task Fast inference [35] Shang et al. [28] Marsden et al. [23] Zhang et al. (2016) [37] Babu Sam et al. [1] Sindagi et al. [31] Ours Table 1. State-of-the-art crowd counting networks and their characteristics. the idea of using an auxiliary task (different, but related to the original supervised task) for which data is freely available and no annotation is required. As a consequence, selfsupervised learning can be much more scalable and flexible. A network trained to estimate the relative location of patches in images was shown to automatically learn features discriminative for semantic concepts in [10]. Other examples include methods that can generate color images from gray scale images and vice versa [17,36], recover a whole patch from the surrounding pixels by inpainting [26], and learn from equivalence relations [24]. In this paper, we propose a self-supervised task to improve the training of networks for crowd counting. Our approach leverages unlabeled crowd images at training time to significantly improve performance. Our key insight is that even though we do not have an exact count of the number of persons in a crowd image, we do know that crops sampled from a crowd image are guaranteed to contain the same or fewer persons than the original (see Figure 1). This gives a technique for generating a ranking of sub-images that can be used to train a network to estimate whether one image contains more persons than another image. The standard approach to exploiting self-supervised learning is to train the self-supervised task first, after which the resulting network is fine-tuned on the final task for which limited data is available. We show that this approach, which is used by the vast majority of self-supervised methods [10,20,24,26,36], does not produce satisfactory results for crowd counting. Our proposed self-supervision, however, yields significant improvement over the state-ofthe-art when added as a proxy task to supervised crowd counting in a multi-task network. The main contribution of this work is that we propose a method that can leverage unlabeled crowd imagery at training time. We propose two different approaches to automatically acquire this data from the Internet. In addition, we analyze three approaches to training using ranked image sets in combination with datasets of labeled crowd scenes. Finally, we demonstrate that our approach leads to state-ofthe-art results on two crowd counting datasets and obtains excellent results on a cross-dataset experiment. The rest of this paper is organized as follows. In the next section we briefly review the literature related to crowd counting. Then, in Section 3 we describe how to systematically generate ranked images from unlabeled crowd imagery. In Section 4 we introduce our approach to exploiting this ranked imagery at training time. We follow in Section 5 with an extensive experimental evaluation of our approach and a comparative analysis with the state-of-the-art. Related work We divide our discussion of related work into two main groups as in [32]: traditional approaches and CNN-based methods. We focus on crowd counting in still images, but we refer the interested reader to the following papers for examples of crowd counting in video [5,9,22]. Various traditional approaches have been proposed to deal with the crowd counting problem. The main strategies are divided into the various categories as in [21]. Most early work on crowd counting used detectors to detect the heads or full bodies of persons in the scene [11,19]. This information can then be used to count. However, detectionbased approaches fail in extremely dense crowded scenes due to occlusion and low resolution of persons. To address these issues, researchers proposed to map features learned from the crowded scene or patches to the number of people [4,6]. By counting using regression, the crowd counting problem is decomposed into two parts: feature extraction and a regression model. While regression-based approaches resulted in improvement, only global counting was considered without any spatial information (i.e. without estimating a density map). The authors of [18] proposed to learn a mapping from patches to corresponding density maps, which achieved great success on a variety of counting problems. As introduced in the review of [32], CNN-based approaches can be classified into different categories based on the properties of the CNN (see Table 1 for an overview of state-of-the-art networks and the properties they hold). Basic CNNs incorporate only basic CNN layers in their networks. The approaches in [12,34] use the AlexNet network [16] to map from crowd scene patches to global number of people by changing the output of AlexNet from 1000 to 1. The resulting network can be trained end-to-end. Due to the large variations of density in different images, recent methods have focused on scale-awareness. The method proposed in [37] trains a multi-column based architecture (MCNN) to capture the different densities by using different sizes of kernels in the network. Similarly, the authors of [25] propose the Hydra-CNN architecture that takes different resolutions of patches as inputs and has multiple output layers (heads) which are combined in the end. Most recently, in [1] the authors propose a switching CNN that can select an optimal head instead of combining the information from all network heads. Finally, context-aware models are networks that can learn from the context of images. In [12,31] the authors propose to classify images or patches into one of five classes: very high density, high density, medium density, low density and very low density. However, the definition of these five classes varies across datasets and must be carefully chosen using knowledge of the statistics of each dataset. Although CNN-based methods have achieved great success in crowd counting, due to lack of labeled data it is still challenging to train deep CNNs without over-fitting. The authors of [35] propose to learn density map and global counting in an alternating sequence to obtain better local optima. The method in [15] uses side information like groundtruth camera angle and height to help the network to learn. However, this side information is expensive to obtain and is not available in most existing crowd counting datasets. There are several works which have studied how to learn to rank, and they focus on learning a ranking function from ground-truth rankings [7,27]. However, these are very different from our approach in which we aim to learn from rankings. Most related to our work is a recent paper [20] proposing a method for image quality assessment using automatically generated rankings of distorted images. In that work the authors used ranking data to pre-train a network and then fine-tune on available labeled datasets. We will show that such an approach fails for crowd counting, and that only when posed as a multi-task learning problem is the network able to exploit the additional data from rankings. In another recent paper [24] a method is proposed where the self-supervised task is to learn to count. The authors propose two pretext tasks (scaling and tiling) which guide self-supervised training. The network aims to learn to count visual primitives in image regions. It is self-supervised by the fact that the number of visual primitives is not expected to change under scaling, and that the sum of all visual primitives in individual tiles should equal the total number of visual primitives in the whole image. Unlike our approach, they do not consider rankings of regions and their counts are typically very low (several image primitives). Also, their final tasks do not involve counting but rather unsupervised learning of features for object recognition. Generating ranked image sets for counting As discussed in the introduction, acquiring data for crowd counting is laborious because images often contain hundreds of persons which require precise annotation. Instead, we propose a self-supervised task for crowd-counting which exploits crowd images which are not hand-labeled with person counts during training. Rather than regressing to the absolute number of persons in the image, we train a network which compares images and ranks them according to the number of persons in the images. In this section, we show how to cheaply collect rank-labeled data which can be used to train these methods. The main idea is based on the observation that all patches contained within a larger patch must have a fewer or equal number of persons than the larger patch (see Figure 1). This observation allows us to collect large datasets of crowd images for which relative ranks exist. Rather than having to painstakingly annotate each person we are only required to verify if the image contains a crowd. Given a crowd image we extract ranked patches according to Algorithm 1. To collect a large dataset of crowd images from the Internet, we use two different approaches: • Keyword query: We collect a crowd scene dataset from Google Images by using different key words: Crowded, Demonstration, Train station, Mall, Studio, Beach, all of which have high likelihood of containing a crowd scene. Then we delete images not relevant to our problem. In the end, we collected a dataset containing 1180 high resolution crowd scene images, which is about 24x the size of the UCF CC 50 dataset, 2.5x the size of ShanghaiTech Part A, and 2x the size of ShanghaiTech Part B. Note that no other annotation of images is performed. Example images from this dataset are given in Figure 2 (top row). • Query-by-example image retrieval: For each specific existing crowd counting dataset, we collect a dataset by using the training images as queries with the visual image search engine Google Images. We choose the first ten similar images and remove irrelevant ones. For UCF CC 50 we collect 256 images, for ShanghaiTech Part A 2229 images, and for Shang-haiTech Part B 3819 images. An example of images returned for a specific query image is given in Figure 2 (bottom row). Learning from ranked image sets In this section we describe our approach to training a CNN to estimate the number of persons in dense crowd scenes. We use the ranked image set generation technique described in the previous section to generate data for the Algorithm 1 : Algorithm to generate ranked datasets. Input: A crowd scene image, number of patches k and scale factor s. Step 1: Choose an anchor point randomly from the anchor region. The anchor region is defined to be 1/r the size of the original image, centered at the original image center, and with the same aspect ratio as the original image. Step 2: Find the largest square patch centered at the anchor point and contained within the image boundaries. Step 3: Crop k − 1 additional square patches, reducing size iteratively by a scale factor s. Keep all patches centered at anchor point. Step 4: Resize all k patches to input size of network. Output: A list of patches ordered according to the number of persons in the patch. self-supervised task of ranking crowd images. We first introduce the network architectures used for counting and ranking, and then describe three different approaches to combining both losses. Crowd density estimation network Here we explain the network architecture which is trained on available crowd counting datasets with ground truth annotations. This network regresses to a crowd density image which indicates the number of persons per pixel (examples of such maps are given in Figure 5). A summation of all values in such a crowd density image gives an estimate of the number of people in the scene. In the experimental section we will consider this network as the baseline method to which we compare. Our baseline network is derived from the VGG-16 network [30], which consists of 13 convolutional layers followed by three fully connected layers. We adapt the network to regress to person density maps. We remove its two fully connected layers, and the max-pooling layer (pool5) to prevent further reduction of spatial resolution. In their place we add a single convolutional layer (a single 3 × 3 × 512 filter with stride 1 and zero padding to maintain same size) which directly regresses to the crowd density map. As the counting loss, L c , we use the Euclidean distance between the estimated and ground truth density maps: L c = 1 M M i=1 (y i −ŷ i ) 2(1) where M is the number of images in a training batch, y i is ground truth person density map of the i-th image in the batch, and the prediction from the network asŷ i . The network is indicated in orange in Figure 3. Ranking loss (over pairs) Counting loss (over images) Ground truth annotations for crowd counting typically consist of a set of coordinates which indicate the 'center' (typically head center of a person). To convert this data to crowd density maps, we place a Gaussian with standard deviation of 15 pixels and sum these for all persons in the scene to obtain y i . This is a standard procedure and is also used in [25,37]. The fact that we derive our architecture from the VGG-16 network has the advantage of being able to use pretrained features from ImageNet. Given the large success of pre-trained features in neural networks, it is somewhat surprising to note that the vast majority of deep learning methods for crowd counting train from scratch [37,1,31]. We found, however, that using pre-trained features significantly improves results. To further improve the performance of our baseline network, we introduce multi-scale sampling from the available labeled datasets. Instead of using the whole image as an input, we randomly sample square patches of varying size (from 56 to 448 pixels). In the experimental section we verify that this multi-scale sampling is important for good performance. Since we are processing patches rather than images, we will useŷ i to refer to the estimate of patch i from now on. The importance of multi-scale processing of crowd data was also noted in [2]. Crowd ranking network In the previous section we explained how to collect abundantly available data for crowd counting. This data does not have crowd density maps and only ranking data is avail-able via the sampling procedure described in Algorithm 1. This ranking indicates that an equal number or more persons are present in a patch when compared to another. Here we present the network which is trained based on this information. For this purpose, we replace the Euclidean loss by an average pooling layer followed by a ranking loss (network in blue in Figure 3). First, the average pooling layer converts the density map into an estimate of the number of persons per spatial unitĉ(I i ) according to: c (I i ) = 1 M jŷ i (x j ),(2) where x j are the spatial coordinates of the density map, and M = 14 × 14 is the number of spatial units in the density map. The ranking which is on the total number of persons in the patchĈ i also directly holds for its normalized version c i , sinceĈ (I i ) = M ×ĉ (I i ). To enforce the ranking, we apply the pairwise ranking hinge loss which for a single pair is defined as: L r = max (0,ĉ (I 2 ) −ĉ (I 1 ) + ε),(3) where ε is the margin, which is set to zero in our case. This loss increases with the difference between two count estimates when their order does not respect the correct ranking. Without loss of generality, we assume that the rank ofĉ(I 1 ) is higher thanĉ(I 2 ). The gradient with respect to the network parameters θ of the loss in Eq. 3 is given by: ∇ θ L r = 0 ifĉ (I 2 )−ĉ (I 1 ) + ε ≤ 0 ∇ θĉ (I 2 ) − ∇ θĉ (I 1 ) otherwise (4) When network outputs the correct ranking there is no backpropagated gradient. However, when the network estimates are not in accordance with the correct ranking the backpropagated gradient causes the network to increase its estimate for the patch with lower score and to decrease its estimate for the one with higher score (note that in backpropagation the gradient is subtracted). A typical implementation of the ranking loss would involve a Siamese network [8] where two images are sent through parallel branches of the network which share their parameters. However, in [20] the authors show that it is computationally advantageous (and sometimes leads to better minima) to send the images in a batch through a single branch and combine them when computing the ranking loss. The ranking loss is then computed with: L r = M i=1 j∈S(i) max (0,ĉ (I j ) −ĉ (I i ) + ε)(5) where S (i) is the set of patches containing fewer people than patch i. Note that this relation is only defined for patches which are contained by patch i. In practice we sample minibatches of 25 images which contain 5 sets of 5 images which can be compared among them resulting in a total of 5 × (4 + 3 + 2 + 1) = 50 pairs in one minibatch. Combining counting and ranking data Here we discuss three approaches to combining ground truth labeled crowd scenes with data for which only rank information is available. These three approaches are depicted in Figure 4. We shortly introduce them here: • Ranking plus fine-tuning: In this approach the network is first trained on the large dataset of ranking data, and is next fine-tuned on the smaller dataset for which density maps are available. To the best of our knowledge this is the approach which is used by all self-supervised methods in vision [10,26,36,24,20]. • Alternating-task training: While ranking plus finetuning works well when the two tasks are closely related, it might perform bad for crowd counting because no supervision is performed to indicate what the network is actually supposed to count. Therefore, we propose to alternate between the tasks of counting and ranking. In practice we perform train for 300 minibatches on a single task before switching to the other, then repeat. • Multi-task training: In the third approach, we add the self-supervised task as a proxy to the supervised counting task and train both simultaneously. In each minibatch we sample data from both ranking and labeled datasets and train both tasks simultaneously as shown in Figure 3. The loss function for multi-task training is: L = L c + λL r ,(6) where λ sets the relative weight between the counting and ranking loss. In the next section we compare these three approaches on several standard dataset for crowd counting. Experiments We report on several experiments to evaluate our approach with respect to baselines and the state-of-the-art methods. 1 Datasets and Experimental Protocol We use two standard benchmark crowd counting datasets. The UCF CC 50 dataset is a very challenging dataset introduced by [13]. It contains 50 annotated images of different resolutions, illuminations and scenes. The variation of densities is very large among images from 94 to 4543 persons with an average of 1280 persons per image. The ShanghaiTech dataset introduced by [37] is a largescale crowd counting dataset consisting of 1198 images with 330,165 annotated heads. This dataset includes two parts: 482 images in Part A which are randomly crawled from the Internet, and 716 images in Part B which are taken from busy streets. Both parts are further divided into training and evaluation sets. The training and test of Part A has 300 and 182 images, respectively, whereas that of Part B has 400 and 316 images, respectively. Following existing work, we use the mean absolute error (MAE) and the mean squared error (MSE) to evaluate different methods. These are defined as follows: M AE = 1 N N i=1 C (I i ) −Ĉ (I i ) , M SE = 1 N N i=1 C (I i ) −Ĉ (I i ) 2(7) where N is the number of test images, C (I i ) is the ground truth number of persons in the ith image andĈ (I i ) is the predicted number of persons in the ith image. We use the Caffe [14] framework and train using minibatch Stochastic Gradient Descent (SGD). The minibatch size for both ranking and counting is 25, and for multi-task training is 50. For the ranking plus fine-tuning method, the learning rate is 1e-6 for both ranking and fine-tuning. Similarly, for the alternating-task training method, the steps for training both tasks are 300 iterations. For the multi-task method, we found λ = 100 to provide good results on all datasets. Learning rates are decreased by a factor of 0.1 every 10K iterations for a total of 20K iterations. For both training phases we use 2 weight decay (set to 5e-4). During training we sample one sub-image from each training image per epoch. We perform down-sampling of three scales and up-sampling of one scale on the UCF CC 50 dataset and only up-sampling of one scale is used on the ShanghaiTech dataset. The number of ranked crops k = 5, the scale factor s = 0.75 and the anchor region r = 8. Method Ablation study We begin with an ablation study on the UCF CC 50 dataset. The aim is to evaluate the relative gain of the proposed improvements and to evaluate the use of a ranking loss against the baseline. The ranked images in this experiment are generated from the Keyword dataset. The results are summarized in Table 2. We can observe the benefit of using a pre-trained ImageNet model in crowd counting, with a significant drop in MAE of around 28% compared to the model trained from scratch. By using both multi-scale data augmentation and starting from a pre-trained model, another improvement of around 6% is obtained. Next, we compare the three methods we propose for combining the ranking and counting losses. The "Ranking plus fine-tuning" method, which is the approach applied by all self-supervised methods in the literature [10,26,36,24,20], performs worse than directly fine-tuning from a pretrained ImageNet model. This is probably caused by the poorly-defined nature of the self-supervised task. To optimize this task the network could decide to count anything, e.g. 'hats', 'trees', or 'people', all of which would agree with the ranking constraints that are imposed. By jointly learning both the self-supervised and crowd counting tasks, the self-supervised task is forced to focus on counting persons. As a result the "Alternating-task training" method improves the MAE by about 12% when compared to the direct fine-tuning method. Moreover, the "Multi-task training" approach reduces the MAE further to 279.6. Given its excellent results we consider only the "Multi-task training" approach for the remainder of the experiments. We also probe how performance scales with increasing training data. We ran an experiment in which we incrementally add supervised training data from Part A of the ShanghaiTech data. Our approach, using only 60% of the labeled data, yields about the same accuracy as training the counting objective alone on 100% of this data. Comparison with the state-of-the-art We start with the results on the UCF CC 50 dataset. A five-fold cross-validation was performed for evaluating [33,31], which means our methods works better in general but has more extreme outliers. Compared to training on the Keyword dataset, learning from the Query-by-example dataset is slightly worse, which might be because most images from UCF CC 50 are black and white with low resolution, which often does not lead to satisfactory query results. An example of prediction in UCF CC 50 using our network is shown in Figure 5. Next we compare with state-of-the-art on the two sets of the ShanghaiTech dataset. As shown in Table 4, similar conclusions as on UCF CC 50 can be drawn. We see that using the our approach further improves by about 2% on ShanghaiTech. For both Part A and Part B, our approach surpasses the state-of-the-art method [31]. An example of prediction by our network on ShanghaiTech is given in Figure 5. For comparison we also provide the results of our baseline method (including fine-tuning from a Part B our baseline already obtains state-of-the-art, with the best results for the multi-task approach obtaining around a 30% improvement when compared to the state-of-the-art. It should also be noted that the method of [31] is complementary to ours and an approach which combines both methods is expected to further improve results. Evaluation on transfer learning As proposed in [37], to demonstrate the generalization of the learned model, we test our method in the transfer learning setting by using Part A of the ShanghaiTech dataset as the source domain and using UCF CC 50 dataset as the target domain. The model trained on Part A of ShanghaiTech is used to predict the crowd scene images from UCF CC 50 dataset, and the results can be seen in Table 5. Using only counting information improves the MAE by 12% compared to reported results in [37]. By combining both ranking and counting datasets, the MAE decreases from 349.5 to 337.6, and MSE decreases from 475.7 to 434.3. In conclusion, these results show that our method significantly outperforms the only other work reporting results on the task of cross-dataset crowd counting. Conclusions In this work we proposed a method for crowd counting. The main novelty is based on the observation that a crop which is contained within a larger crop must contain fewer or an equal number of persons than the larger crop. This allows us to address one of the main problems for crowd counting, namely the lack of large training datasets. Our approach enables the exploitation of abundantly available training data from the Internet by automatically generating rankings from them. We showed how this additional data can be leveraged with available annotated data in a multitask network. Experiments show that the proposed self-supervised task improves results significantly when compared to a network which is only trained on the annotated data. We show that incorporating the self-supervised task in a multitask approach obtains optimal results. Furthermore, we obtain state-of-the-art results on two challenging datasets for crowd counting, namely the ShanghaiTech and the UCF CC 50 dataset. Finally, we show that the learned models generalize well to other datasets, significantly outperforming the only other crowd counting method which reports on this transfer learning task. Method MAE MSE Figure 1 . 1Using ranked sub-images for self-supervised training. Figure 2 . 2Example images from the retrieved crowd scene dataset. (top) Representative images using key words as query. (bottom) Representative images using training image as query image (the query image is depicted on the left). Figure 3 . 3The multi-task framework combining both counting and ranking information. This network can be trained end-to-end for crowd counting. VGG-conv refers to the convolutional layers of the VGG-16 network. See text for more details. Figure 4 . 4Three ways to combine ranking and counting datasets. Figure 5 . 5Examples of predicted density maps for the UCF CC 50 (Top row, true count: 3406 prediction: 3052) and ShanghaiTech datasets (Bottom row, true count: 361 prediction: 365). Left column: crowd image. Middle column: ground truth. Right column: prediction. pre-trained model and multi-scale data augmentation) on this dataset: M AE = 77.7 and M SE = 115.9 on Part A, and M AE = 14.7 and M SE = 24.7 on Part B. On Method MAE MSEIdrees et al.[13] 419.5 541.6 Zhang et al. (2015) [35] 467.0 498.5 Zhang et al. (2016) [37] 377.6 509.1 Onoro et al. [25] 333.7 425.2 Walach et al. [33] 364.4 341.4 Babu Sam et al. [1] 318.1 439.2 Sindagi et al. [31] 295.8 320.9 Ours: Multi-task (Query-by-example) 291.5 397.6 Ours: Multi-task (Keyword) 279.6 388.9 Table 3. MAE and MSE error on the UCF CC 50 dataset. Method MAE MSE MAE MSE Zhang et al. (2015) [35] 181.8 277.7Part A Part B 32.0 49.8 Zhang et al. (2016) [37] 110.2 173.2 26.4 41.3 Babu Sam et al. [1] 90.4 135.0 21.6 33.4 Sindagi et al. [31] 73.6 106.4 20.1 30.1 Ours: Multi-task (Query-by-example) 72.0 106.6 14.4 23.8 Ours: Multi-task (Keyword) 73.6 112.0 13.7 21.4 Table 4 . 4MAE and MSE error on the ShanghaiTech dataset. the methods. Results are shown inTable 3. Our multitask training method with the Keyword dataset reduces the MAE error from 295.8 to 279.6 compared to the state-ofthe-art. However the MSE of our method on UCF CC 50 dataset is worse then the state-of-the-art methods Zhang et al. (2016) [37] 397.6 624.1 Ours: Counting only 349.5 475.7 Ours: Multi-task 337.6 434.3 Table 5. Transfer learning across datasets. Models were trained on Part A of ShanghaiTech and tested on UCF CC 50. Contributions with respect to the state-of-the-art: Basic CNNs are simple and fast to train, but usually achieve lower accuracy. Combining different scale-aware models and context-aware models has been shown to significantly increase performance, but the complexity of these models is high. In addition, considering the scarcity of large annotated datasets, ranked patches are used as side information to decrease the effect of over-fitting. The model we propose in this paper is fast to train, uses no side information, supports fast inference, is scale-aware, is multi-task, and outperforms the state-of-the-art. Our key contribution is in showing how to effectively exploit unlabeled crowd imagery for pre-training CNNs. 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Springer, 2016. 7 Deep people counting in extremely dense crowds. C Wang, H Zhang, L Yang, S Liu, X Cao, Proceedings of ACM int. conf. on Multimedia. ACM int. conf. on MultimediaACMC. Wang, H. Zhang, L. Yang, S. Liu, and X. Cao. Deep people counting in extremely dense crowds. In Proceedings of ACM int. conf. on Multimedia. ACM, 2015. 2 Cross-scene crowd counting via deep convolutional neural networks. C Zhang, H Li, X Wang, X Yang, CVPR. 27C. Zhang, H. Li, X. Wang, and X. Yang. Cross-scene crowd counting via deep convolutional neural networks. In CVPR, 2015. 2, 3, 7 Colorful image colorization. R Zhang, P Isola, A A Efros, ECCV. 7R. Zhang, P. Isola, and A. A. Efros. Colorful image coloriza- tion. In ECCV, 2016. 2, 6, 7 Singleimage crowd counting via multi-column convolutional neural network. Y Zhang, D Zhou, S Chen, S Gao, Y Ma, CVPR. 7Y. Zhang, D. Zhou, S. Chen, S. Gao, and Y. Ma. Single- image crowd counting via multi-column convolutional neu- ral network. In CVPR, 2016. 1, 2, 3, 5, 6, 7, 8
[ "https://github.com/xialeiliu/CrowdCountingCVPR18" ]
[ "Implications of 2dFGRS results on cosmic structure", "Implications of 2dFGRS results on cosmic structure" ]
[ "J A Peacock \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nEH9 3HJEdinburghUK\n" ]
[ "Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nEH9 3HJEdinburghUK" ]
[ "* On behalf of the 2dF Galaxy Redshift Survey team" ]
The 2dF Galaxy Redshift Survey is the first to observe more than 100,000 redshifts, making possible precise measurements of many aspects of galaxy clustering. The spatial distribution of galaxies can be studied as a function of galaxy spectral type, and also of broad-band colour. Redshift-space distortions are detected with a high degree of significance, confirming the detailed Kaiser distortion from large-scale infall velocities, and measuring the distortion parameter β ≡ Ω 0.6 m /b = 0.49 ± 0.09. The power spectrum is measured to < ∼ 10% accuracy for k > 0.02 h Mpc −1 , and is well fitted by a CDM model with Ω m h = 0.18 ± 0.02 and a baryon fraction of 0.17 ± 0.06. A joint analysis with CMB data requires Ω m = 0.31 ± 0.05 and h = 0.67 ± 0.04, assuming scalar fluctuations. The fluctuation amplitude from the CMB is σ 8 = 0.76 ± 0.04, assuming reionization at z < ∼ 10, so that the general level of galaxy clustering is approximately unbiased, in agreement with an internal bispectrum analysis. Luminosity dependence of clustering is however detected at high significance, and is well described by a relative bias of b/b * = 0.85 + 0.15(L/L * ). This is consistent with the observation that L * in rich clusters is brighter than the global value by 0.28 ± 0.08 mag.The survey is designed around the 2dF multi-fibre spectrograph on the Anglo-Australian Telescope, which is capable of observing up to 400 objects simultaneously over a 2 degree diameter field of view. For details of the instrument and its performance see http://www.aao.gov.au/2df/, and alsoLewis et al. (2002).The source catalogue for the survey is a revised and extended version of the APM galaxy catalogue (Maddox et al. 1990a,b,c); this includes over 5 million galaxies down to b J = 20.5 in both north and south Galactic hemispheres over a region of almost 10 4 deg 2 (bounded approximately by declination δ ≤ +3 • and Galactic latitude b > ∼ 20 • ). This catalogue is based on Automated Plate Measuring machine (APM) scans of 390 plates from the UK Schmidt Telescope (UKST) Southern Sky Survey. The b J magnitude system for the Southern Sky Survey is defined by the response of Kodak IIIaJ emulsion in combination with a GG395 filter, and is related to the Johnson-Cousins system by b J = B − 0.304(B − V ), where the colour term is estimated from comparison with the SDSS Early Data Release(Stoughton et al. 2002)The photometry of the catalogue is calibrated with numerous CCD sequences and has a precision of approximately 0.15 mag for galaxies with b J = 17-19.5. The star-galaxy separation is as described inMaddox et al. (1990b), supplemented by visual validation of each galaxy image. FIGURE 7. The 2dFGRS redshift-space dimensionless power spectrum, ∆ 2 (k), estimated according to the FKP procedure. The solid points with error bars show the power estimate. The window function correlates the results at different k values, and also distorts the large-scale shape of the power spectrum An approximate correction for the latter effect has been applied. The solid and dashed lines show various CDM models, all assuming n = 1. For the case with non-negligible baryon content, a big-bang nucleosynthesis value of Ω b h 2 = 0.02 is assumed, together with h = 0.7. A good fit is clearly obtained for Ω m h ≃ 0.2. Note that the observed power at large k will be boosted by nonlinear effects, but damped by small-scale random peculiar velocities. It appears that these two effects very nearly cancel, but model fitting is generally performed only at k < 0.15 h Mpc −1 in order to avoid these complications.
10.1063/1.1581805
[ "https://arxiv.org/pdf/astro-ph/0301042v1.pdf" ]
16,820,080
astro-ph/0301042
9d0f84c4199715994089e5d3f4a9875dfa46e88f
Implications of 2dFGRS results on cosmic structure PederCopyright Peder3 Jan 2003 J A Peacock Institute for Astronomy University of Edinburgh Royal Observatory EH9 3HJEdinburghUK Implications of 2dFGRS results on cosmic structure * On behalf of the 2dF Galaxy Redshift Survey team St Andrews; George Efstathiou (IoA), Richard Ellis (Caltech), Carlos Frenk (Durham), Karl Glazebrook; Leeds), Steve Maddox; Nottingham; NorbergPeder3 Jan 2003 The 2dF Galaxy Redshift Survey is the first to observe more than 100,000 redshifts, making possible precise measurements of many aspects of galaxy clustering. The spatial distribution of galaxies can be studied as a function of galaxy spectral type, and also of broad-band colour. Redshift-space distortions are detected with a high degree of significance, confirming the detailed Kaiser distortion from large-scale infall velocities, and measuring the distortion parameter β ≡ Ω 0.6 m /b = 0.49 ± 0.09. The power spectrum is measured to < ∼ 10% accuracy for k > 0.02 h Mpc −1 , and is well fitted by a CDM model with Ω m h = 0.18 ± 0.02 and a baryon fraction of 0.17 ± 0.06. A joint analysis with CMB data requires Ω m = 0.31 ± 0.05 and h = 0.67 ± 0.04, assuming scalar fluctuations. The fluctuation amplitude from the CMB is σ 8 = 0.76 ± 0.04, assuming reionization at z < ∼ 10, so that the general level of galaxy clustering is approximately unbiased, in agreement with an internal bispectrum analysis. Luminosity dependence of clustering is however detected at high significance, and is well described by a relative bias of b/b * = 0.85 + 0.15(L/L * ). This is consistent with the observation that L * in rich clusters is brighter than the global value by 0.28 ± 0.08 mag.The survey is designed around the 2dF multi-fibre spectrograph on the Anglo-Australian Telescope, which is capable of observing up to 400 objects simultaneously over a 2 degree diameter field of view. For details of the instrument and its performance see http://www.aao.gov.au/2df/, and alsoLewis et al. (2002).The source catalogue for the survey is a revised and extended version of the APM galaxy catalogue (Maddox et al. 1990a,b,c); this includes over 5 million galaxies down to b J = 20.5 in both north and south Galactic hemispheres over a region of almost 10 4 deg 2 (bounded approximately by declination δ ≤ +3 • and Galactic latitude b > ∼ 20 • ). This catalogue is based on Automated Plate Measuring machine (APM) scans of 390 plates from the UK Schmidt Telescope (UKST) Southern Sky Survey. The b J magnitude system for the Southern Sky Survey is defined by the response of Kodak IIIaJ emulsion in combination with a GG395 filter, and is related to the Johnson-Cousins system by b J = B − 0.304(B − V ), where the colour term is estimated from comparison with the SDSS Early Data Release(Stoughton et al. 2002)The photometry of the catalogue is calibrated with numerous CCD sequences and has a precision of approximately 0.15 mag for galaxies with b J = 17-19.5. The star-galaxy separation is as described inMaddox et al. (1990b), supplemented by visual validation of each galaxy image. FIGURE 7. The 2dFGRS redshift-space dimensionless power spectrum, ∆ 2 (k), estimated according to the FKP procedure. The solid points with error bars show the power estimate. The window function correlates the results at different k values, and also distorts the large-scale shape of the power spectrum An approximate correction for the latter effect has been applied. The solid and dashed lines show various CDM models, all assuming n = 1. For the case with non-negligible baryon content, a big-bang nucleosynthesis value of Ω b h 2 = 0.02 is assumed, together with h = 0.7. A good fit is clearly obtained for Ω m h ≃ 0.2. Note that the observed power at large k will be boosted by nonlinear effects, but damped by small-scale random peculiar velocities. It appears that these two effects very nearly cancel, but model fitting is generally performed only at k < 0.15 h Mpc −1 in order to avoid these complications. AIMS AND DESIGN OF THE 2DFGRS The 2dF Galaxy Redshift Survey (2dFGRS) was designed to study the following key aspects of the large-scale structure in the galaxy distribution: (1) To measure the galaxy power spectrum P(k) on scales up to a few hundred Mpc, bridging the gap between the scales of nonlinear structure and measurements from the the cosmic microwave background (CMB). (2) To measure the redshift-space distortion of the large-scale clustering that results from the peculiar velocity field produced by the mass distribution. (3) To measure higher-order clustering statistics in order to understand biased galaxy formation, and to test whether the galaxy distribution on large scales is a Gaussian random field. FIGURE 1. The 2dFGRS fields (small circles) superimposed on the APM catalogue area (dotted outlines of Sky Survey plates). There are approximately 140,000 galaxies in the 75 • × 15 • southern strip centred on the SGP, 70,000 galaxies in the 75 • × 7.5 • equatorial strip, and 40,000 galaxies in the 100 randomly-distributed 2dF fields covering the whole area of the APM catalogue in the south. The survey geometry is shown in Figure 1, and consists of two contiguous declination strips, plus 100 random 2-degree fields. One strip is in the southern Galactic hemisphere and covers approximately 75 • ×15 • centred close to the SGP at (α, δ )=(01 h ,−30 • ); the other strip is in the northern Galactic hemisphere and covers 75 • × 7.5 • centred at (α, δ )=(12.5 h ,+0 • ). The 100 random fields are spread uniformly over the 7000 deg 2 region of the APM catalogue in the southern Galactic hemisphere. At the median redshift of the survey (z = 0.11), 100 h −1 Mpc subtends about 20 degrees, so the two strips are 375 h −1 Mpc long and have widths of 75 h −1 Mpc (south) and 37.5 h −1 Mpc (north). The sample is limited to be brighter than an extinction-corrected magnitude of b J = 19.45 (using the extinction maps of Schlegel et al. 1998). This limit gives a good match between the density on the sky of galaxies and 2dF fibres. Due to clustering, however, the number in a given field varies considerably. To make efficient use of 2dF, we employ an adaptive tiling algorithm to cover the survey area with the minimum number of 2dF fields. With this algorithm we are able to achieve a 93% sampling rate with on average fewer than 5% wasted fibres per field. Over the whole area of the survey there are in excess of 250,000 galaxies. SURVEY STATUS After an extensive period of commissioning of the 2dF instrument, 2dFGRS observing began in earnest in May 1997, andterminated in April 2002. In total, observations were made of 899 fields, yielding redshifts and identifications for 232,529 galaxies, 13976 stars and 172 QSOs, at an overall completeness of 93%. The galaxy redshifts are assigned a quality flag from 1 to 5, where the probability of error is highest at low Q. Most analyses are restricted to Q ≥ 3 galaxies, of which there are currently 221,496. An interim data release took place in July 2001, consisting of approximately 100,000 galaxies (see Colless et al. 2001 for details). A public release of the full photometric and spectroscopic database is scheduled for July 2003. The Colless et al. (2001) paper details the practical steps that are necessary in order to work with a survey of this sort. The 2dFGRS does not consist of a simple region sampled with 100% efficiency, and it is therefore necessary to use a number of masks in order to interpret the data. Two of these concern the input catalogue: the boundaries of this catalogue, including 'drilled' regions around bright stars where galaxies could not be detected; also, revisions to the photometric calibration mean that in practice the survey depth varies slightly with position on the sky. A futher mask arises from the way in which the sky is tessellated into 2dF tiles: near the survey edges and near internal holes, a lack of overlaps mean that the sampling fraction falls to about 50%. Finally, the spectroscopic success rate of each spectroscopic observation fluctuated according to the observing conditions. The median redshift yield was approximately 95%, but with a tail towards poorer data. The terminal stages of 2dFGRS observing were in fact devoted to re-observing these fields of low completeness; nevertheless, approximately 10% of fields have completeness lower than 80%. This variable sampling makes quantification of the large scale structure more difficult, particularly for any analysis requiring relatively uniform contiguous areas. However, the effective survey 'mask' can be measured precisely enough that it can be allowed for in analyses of the galaxy distribution. GALAXY SPECTRA AND COLOURS Beyond the basic data of positions, magnitudes and redshifts, it is important on physical grounds to be able to divide the 2dFGRS database into different categories of galaxies. This has been done in two different ways. Spectral classification of 2dFGRS galaxies was performed by Folkes et al. (1999) and Madgwick et al. (2002). Principal component analysis was used to split galaxies into a superposition of a small number of templates. Not all of these are robust, owing to uncalibrated spectral distortions in the 2dF instru-ment, but it was possible to derive a robust classification parameter (termed η) from the templates, which effectively measures the emission-line strength (closely related to the star-formation rate). Galaxies were divided into four spectral classes; their mean spectra and separate luminosity functions are shown in Figure 2. FIGURE 2. The type-dependent galaxy luminosity function according to Madgwick et al. (2002). Principal component analysis was used to split galaxies into a superposition of a small number of templates, and a categorization made based on the decomposition. Type 1 galaxies are generally E/S0, while later types range from Sa to Irr. This classification method has the drawback that it cannot be used beyond z = 0.15, where Hα is lost from the spectra. Also, the fibres do not cover the whole galaxy (although Madgwick et al. 2002 show that aperture corrections are not large in practice). More recently, we have been able to obtain total broad-band colours for the 2dFGRS galaxies, using the data from the SuperCOSMOS sky surveys (Hambly et al. 2001). These yield B J from the same UK Schmidt Plates as used in the original APM survey, but with improved linearity and smaller random errors (0.09 mag rms relative to the SDSS EDR data). The R F plates are of similar quality, so that we are able to divide galaxies by colour, with an rms in photographic B −R of about 0.13 mag. The systematic calibration uncertainties are negligible by comparison, and are at the level of 0.04 mag. rms in each band. Figure 3 shows that the colour information divides 'passive' galaxies with little active star formation cleanly from the remainder, uniformly over the whole redshift range of the 2dFGRS. As an immediate application, we can display the spatial distribution of 2dFGRS galaxies divided according to colour (Figure 4). The most striking aspect of this image is how closely both sets of galaxies follow the same structure. The passive subset display a more skeletal appearance, as expected owing to morphological segregation of ellipticals. A red-selected survey such as SDSS will appear more similar to the passive subset of the 2dFGRS, with relatively low sampling of the more active spectral type 2-4. The separation between 'passive' (red) and 'active' (blue) galaxies is very clear. Empirically, B − R − 2.8z defines a 'restframe' colour whose distribution is independent of redshift, and very clearly bimodal. This is strongly reminiscent of the distribution of the spectral type, η, and we assume that a division at (B − R) 0 = 0.85 achieves a separation of 'class 1' galaxies from classes 2-4, as was done using spectra by Madgwick et al. (2002). REDSHIFT-SPACE CORRELATIONS The simplest statistic for studying clustering in the galaxy distribution is the the twopoint correlation function, ξ (σ , π). This measures the excess probability over random of finding a pair of galaxies with a separation in the plane of the sky σ and a line-ofsight separation π. Because the radial separation in redshift space includes the peculiar velocity as well as the spatial separation, ξ (σ , π) will be anisotropic. On small scales the correlation function is extended in the radial direction due to the large peculiar velocities in non-linear structures such as groups and clusters -this is the well-known 'Finger-of-God' effect. On large scales it is compressed in the radial direction due to the coherent infall of galaxies onto mass concentrations -the Kaiser effect (Kaiser 1987). To estimate ξ (σ , π) we compare the observed count of galaxy pairs with the count estimated from a random distribution following the same selection function both on the sky and in redshift as the observed galaxies. We apply optimal weighting to minimise the uncertainties due to cosmic variance and Poisson noise. The redshift-space correlation function for the 2dFGRS computed in this way is shown in Figure 5. The correlationfunction results display very clearly two signatures of redshift-space distortions. The 'fingers of God' from small-scale random velocities are very clear, as indeed has been the case from the first redshift surveys (e.g. Davis & Peebles 1983). However, this is the first time that the large-scale flattening from coherent infall has been seen in detail. An initial analysis of this effect was performed in Peacock et al. (2001), and the final database was analysed by Hawkins et al. (2002). The degree of large-scale flattening is determined by the total mass density parameter, Ω m , and the biasing of the galaxy distribution. On large scales, it should be correct to assume a linear bias model, with correlation functions ξ g (r) = b 2 ξ (r), so that the redshift-space distortion on large scales depends on the combination β ≡ Ω 0.6 m /b. This is modified by the Finger-of-God effect, which is significant even at large scales and dom-FIGURE 4. The distribution of galaxies in part of the 2dFGRS, drawn from a total of 221,496 galaxies: slices 4 • thick, centred at declination −2.5 • in the NGP and −27.5 • in the SGP. The survey is divided at a rest-frame colour of photographic B − R = 0.85, into galaxies with and without active star formation. The This image reveals a wealth of detail, including linear supercluster features, often nearly perpendicular to the line of sight. It appears that these transverse features have been enhanced by infall velocities. inant at small scales. The effect can be modelled by introducing a parameter σ p , which represents the rms pairwise velocity dispersion of the galaxies in collapsed structures, σ p (see e.g. Ballinger et al. 1996). Considering both these effects, and marginalising over σ p , the best estimate of β and its 68% confidence interval according to Hawkins et β = 0.49 ± 0.09 (1) The quoted error is slightly larger than in Peacock et al. (2001): mainly, this reflects the decision of Hawkins et al. to concentrate on the better sampled volume at z < 0.2, although a more detailed comparison with mock data also indicates that the previous errors were too small by a factor of about 1.2. Our measurement of Ω 0.6 /b can only be used to determine Ω if the bias is known. We discuss below two methods by which the bias parameter may be inferred, which in fact favour a low degree of bias. Nevertheless, there are other uncertainties in converting a measurement of β to a figure for Ω. The 2dFGRS has a median redshift of 0.11; with weighting, the mean redshift in Hawkins et al. is 0.15, and our measurement should be interpreted as β at that epoch. The optimal weighting also means that our mean luminosity is high: it is approximately 1.4 times the characteristic luminosity, L * , of the overall galaxy population (Folkes et al. 1999;Madgwick et al. 2002). This means that we need to quantify the luminosity dependence of clustering. REAL-SPACE CLUSTERING AND ITS DEPENDENCE ON LUMINOSITY The dependence of galaxy clustering on luminosity is an effect that was controversial for a number of years. Using the APM-Stromlo redshift survey, Loveday et al. (1995) claimed that there was no trend of clustering amplitude with luminosity, except possibly at the very lowest luminosities. In contradiction, the SSRS study of Benoist et al. (1996) suggested that the strength of galaxy clustering increased monotonically with luminosity, with a particularly marked effect for galaxies above L * . The latter result was arguably more plausible, based on what we know of luminosity functions and morphological segregation. It has been clear for many years that elliptical galaxies display a higher correlation amplitude than spirals (Davis & Geller 1976). Since ellipticals are also more luminous on average, as shown above, some trend with luminosity is to be expected, but the challenge is to detect it. The difficulty with measuring the dependence of ξ (r) on luminosity is that cosmic variance can mask the signal of interest. It is therefore important to analyse volumelimited samples in which galaxies of different luminosities are compared in the same volume of space. This comparison was undertaken by Norberg et al. (2001), who measured real-space correlation functions via the projection Ξ(σ ) = ξ (σ , π) dπ, demonstrating that it was possible to obtain consistent results in both NGP and SGP. A very clear detection of luminosity-dependent clustering was achieved, as shown in Figure 6. The results can be described by a linear dependence of effective bias parameter on luminosity: b/b * = 0.85 + 0.15 (L/L * ),(2) and the scale-length of the real-space correlation function for L * galaxies is approximately r 0 = 4.8 h −1 Mpc. This trend is in qualitative agreement with the results of Benoist et al. (1996), but in fact these workers gave a stronger dependence on luminosity than is indicated by the 2dFGRS. Finally, with spectral classifications, it is possible to measure the dependence of clustering both on luminosity and on spectral type, to see to what extent morphological segregation is responsible for this result. Norberg et al. (2002) show that, in fact, the principal effect seems to be with luminosity: ξ (r) increases with L for all spectral types. This is reasonable from a theoretical point of view, in which the principal cause of different clustering amplitudes is the mass of halo that hosts a galaxy (e.g. Cole & Kaiser 1989;Mo & White 1996;Kauffman, Nusser & Steinmetz 1997). Finally, these results would lead us to infer that the LF must change in strongly clumped regions, shifting to higher luminosities. Such an effect has been sought for many years, but always yielded null results. However, De Propris et al. (2002) have shown that L * in rich clusters does obey a shift with respect to the global value, being brighter by 0.28 ± 0.08 mag. THE 2DFGRS POWER SPECTRUM Perhaps the key aim of the 2dFGRS was to perform an accurate measurement of the 3D clustering power spectrum, in order to improve on the APM result, which was deduced by deprojection of angular clustering (Baugh & Efstathiou 1993. The results of this direct estimation of the 3D power spectrum are shown in Figure 7. This powerspectrum estimate uses the FFT-based approach of Feldman, Kaiser & Peacock (1994), and needs to be interpreted with care. Firstly, it is a raw redshift-space estimate, so that the power beyond k ≃ 0.2 h Mpc −1 is severely damped by fingers of God. On large scales, the power is enhanced, both by the Kaiser effect and by the luminosity-dependent clustering discussed above. Finally, the FKP estimator yields the true power convolved with the window function. This modifies the power significantly on large scales (roughly a 20% correction). We have made an approximate correction for this in Figure 7. The precision of the power measurement appears to be encouragingly high, and the next task is to perform a detailed fit of physical power spectra, taking full account of the window effects. We summarize here results from this analysis (Percival et al. 2001). The fundamental assumption is that, on large scales, linear biasing applies, so that the nonlinear galaxy power spectrum in redshift space has a shape identical to that ow linear theory in real space. We believe that this assumption is valid for k < 0.15 h Mpc −1 ; the detailed justification comes from analyzing realistic mock data derived from N-body simulations (Cole et al 1998). The model free parameters are thus the primordial spectral index, n, the Hubble parameter, h, the total matter density, Ω m , and the baryon fraction, Ω b /Ω m . Note that the vacuum energy does not enter. Initially, we show results assuming n = 1; this assumption is relaxed later. In order to compare the 2dFGRS power spectrum to members of the CDM family of theoretical models, it is essential to have a proper understanding of the full covariance matrix of the data: the convolving effect of the window function causes the power at adjacent k values to be correlated. This covariance matrix was estimated by applying the survey window to a library of Gaussian realisations of linear density fields. Similar results were obtained using a covariance matrix estimated from a set of mock catalogues. It is now possible to explore the space of CDM models, and likelihood contours in Ω b /Ω m versus Ω m h are shown in Figure 8. At each point in this surface we have marginalized by integrating the likelihood surface over the two free parameters, h and the power spectrum amplitude. Assuming a uniform prior for h over a factor of 2 is arguably over-cautious, and we have therefore added a Gaussian prior h = 0.7 ± 10%. This corresponds to multiplying by the likelihood from external constraints such as the HST key project (Freedman et al. 2001); this has only a minor effect on the results. Figure 8 shows that there is a degeneracy between Ω m h and the baryonic fraction Ω b /Ω m . However, there are two local maxima in the likelihood, one with Ω m h ≃ 0.2 and ∼ 20% baryons, plus a secondary solution Ω m h ≃ 0.6 and ∼ 40% baryons. The high-density model can be rejected through a variety of arguments, and the preferred solution is Ω m h = 0.20 ± 0.03; Ω b /Ω m = 0.15 ± 0.07.(3) The 2dFGRS data are compared to the best-fit linear power spectra convolved with the window function in Figure 8. This shows where the two branches of solutions come from: the low-density model fits the overall shape of the spectrum with relatively small 'wiggles', while the solution at Ω m h ≃ 0.6 provides a better fit to the bump at k ≃ 0.065 h Mpc −1 , but fits the overall shape less well. A preliminary analysis of P(k) from the full final dataset shows that P(k) becomes smoother: the high-baryon solution becomes disfavoured, and the uncertainties narrow slightly around the lower-density solution: Ω m h = 0.18 ± 0.02; Ω b /Ω m = 0.17 ± 0.06. It is interesting to compare these conclusions with other constraints. These are shown on Figure 8, assuming h = 0.7 ± 10%. Latest estimates of the Deuterium to Hydrogen FIGURE 8. Likelihood contours for the best-fit linear power spectrum over the region 0.02 < k < 0.15. The normalization is a free parameter to account for the unknown large scale biasing. Contours are plotted at the usual positions for one-parameter confidence of 68%, and two-parameter confidence of 68%, 95% and 99% (i.e. −2 ln(L /L max ) = 1, 2.3, 6.0, 9.2). We have marginalized over the missing free parameters (h and the power spectrum amplitude). A prior on h of h = 0.7 ± 10% was assumed. This result is compared to estimates from X-ray cluster analysis (Evrard 1997), big-bang nucleosynthesis (Burles et al. 2001) and CMB results (Netterfield et al. 2001;Pryke et al. 2002). The CMB results assume that Ω b h 2 and Ω cdm h 2 were independently determined from the data. The second panel shows the 2dFGRS data compared with the two preferred models from the Maximum Likelihood fits convolved with the window function (solid lines). Error bars show the diagonal elements of the covariance matrix, for the fitted data that lie between the dotted vertical lines. The unconvolved models are also shown (dashed lines). The Ω m h ≃ 0.6, Ω b /Ω m = 0.42, h = 0.7 model has the higher bump at k ≃ 0.05 h Mpc −1 . The smoother Ω m h ≃ 0.20, Ω b /Ω m = 0.15, h = 0.7 model is a better fit to the data because of the overall shape. A preliminary analysis of the complete final 2dFGRS sample yields a slightly smoother spectrum than the results shown here (from Percival et al. 2001), so that the high-baryon solution becomes disfavoured. ratio in QSO spectra combined with big-bang nucleosynthesis theory predict Ω b h 2 = 0.020 ± 0.001 (Burles et al. 2001), which translates to the shown locus of f B vs Ω m h. X-ray cluster analysis predicts a baryon fraction Ω b /Ω m = 0.127 ± 0.017 (Evrard 1997) which is within 1σ of our value. These loci intersect very close to our preferred model. Moreover, these results are in good agreement with independent estimates of the total density and baryon content from data on CMB anisotropies (e.g. Netterfield et al. 2001;Pryke et al. 2002). Perhaps the main point to emphasise here is that the 2dFGRS results are not greatly sensitive to the assumed tilt of the primordial spectrum. We have used CMB results to motivate the choice of n = 1, as discussed below, but it is clear that very substantial tilts are required to alter the conclusions significantly: n ≃ 0.8 would be required to turn zero baryons into the preferred model. The main residual worry about accepting the above conclusions is probably whether the assumption of linear bias can really be valid. In general, concentration towards higher-density regions both raises the amplitude of clustering, but also steepens the correlations, so that bias is largest on small scales. One way in which this issue can be studied is to use the split by colour introduced above. Figure 9 shows the power spectra for the 2dFGRS divided in this way. The shapes are almost identical (perhaps not so surprising, since the cosmic variance effects are closely correlated in these co-spatial samples). However, what is impressive is that the relative bias is almost precisely independent of scale, even though the passive subset is rather strongly biased relative to the active subset (relative b ≃ 1.4). This provides some reassurance that the large-scale P(k) reflects the underlying properties of the dark matter, rather than depending on the particular class of galaxies used to measure it. FIGURE 9. The power spectra of passive galaxies (filled circles) and active galaxies (open circles). The shapes are strikingly similar. The square root of the ratio yields the right-hand panel: the relative bias in redshift space of passive and active galaxies. The error bars are obtained by a jack-knife analysis. The relative bias is consistent with a constant value of 1.4 over the range used for fitting of the power-spectrum data (0.015 < k < 0.15 h Mpc −1 ). COMBINATION WITH THE CMB AND COSMOLOGICAL PARAMETERS The 2dFGRS power spectrum contains important information about the key parameters of the cosmological model, but we have seen that additional assumptions are needed, in particular the values of n and h. Observations of CMB anisotropies can in principle measure most of the cosmological parameters, and combination with the 2dFGRS can lift most of the degeneracies inherent in the CMB-only analysis. It is therefore of interest to see what emerges from a joint analysis. These issues are discussed in Efstathiou et al. (2002). The CMB data alone contain two important degeneracies: the 'geometrical' and 'tensor' degeneracies. In the former case, one can evade the commonly-stated CMB conclusion that the universe is flat, by adjusting both Λ and h to extreme values. In the latter case, a model with a large tensor component can be made to resemble a zero-tensor model with large blue tilt (n > 1) and high baryon content. Efstathiou et al. (2002) show that adding the 2dFGRS data removes the first degeneracy, but not the second. This is reasonable: if we take the view that the CMB determines the physical density Ω m h 2 , then a measurement of Ω m h from 2dFGRS gives both Ω m and h separately in principle, removing one of the degrees of freedom on which the geometrical degeneracy depends. On the other hand, the 2dFGRS alone constrains the baryon content weakly, so this does not remove the scope for the tensor degeneracy. On the basis of this analysis, we can therefore be confident that the universe is very nearly flat (|Ω − 1| < 0.05), so it is defensible to assume hereafter that this is exactly true. The importance of tensors will of course be one of the key questions for cosmology over the next several years, but it is interesting to consider the limit in which these are negligible. In this case, the standard model for structure formation contains a vector of only 6 parameters: p = (n s , Ω m , Ω b , h, Q, τ).(4) Of these, the optical depth to last scattering, τ, is almost entirely degenerate with the normalization, Q -and indeed with the bias parameter; we discuss this below. The remaining four parameters are pinned down very precisely: using the latest CMB data plus the 2dFRGS power spectrum, we obtain (n s , Ω c h 2 , Ω b h 2 , h) = (0.963 ± 0.042, 0.115 ± 0.009, 0.021 ± 0.002, 0.665 ± 0.047), (5) or an overall density parameter of Ω m = 0.31 ± 0.05. It is remarkable how well these figures agree with completely independent determinations: h = 0.72 ± 0.08 from the HST key project (Mould et al. 2000;Freedman et al. 2001); Ω b h 2 = 0.020 ± 0.001 (Burles et al. 2001). This gives confidence that the tensor component must indeed be sub-dominant. For further details of this analysis, see Percival et al. (2002). MATTER FLUCTUATION AMPLITUDE AND BIAS The above conclusions were obtained by considering the shapes of the CMB and galaxy power spectra. However, it is also of great interest to consider the amplitude of mass fluctuations, since a comparison with the galaxy power spectrum allows us to infer the degree of bias directly. This analysis was performed by Lahav et al. (2002). Given assumed values for the cosmological parameters, the present-day linear normalization of the mass spectrum (e.g. σ 8 ) can be inferred. It is convenient to define a corresponding measure for the galaxies, σ 8g , such that we can express the bias parameter as b = σ 8g σ 8m .(6) In practice, we define σ 8g to be the value required to fit a CDM model to the powerspectrum data on linear scales (0.02 < k < 0.15 h Mpc −1 ). A final necessary complication of the notation is that we need to distinguish between the apparent values of σ 8g as measured in redshift space (σ S 8g ) and the real-space value that would be measured in the absence of redshift-space distortions (σ R 8g ). It is the latter value that is required in order to estimate the bias. A model grid covering the range 0.1 < Ω m h < 0.3, 0.0 < Ω b /Ω m < 0.4, 0.4 < h < 0.9 and 0.75 < σ S 8g < 1.14 was considered. The primordial index was assumed to be n = 1 initially, and the dependence on n studied separately. For fixed 'concordance model' parameters n = 1, k = 0, Ω m = 0.3, Ω b h 2 = 0.02 and a Hubble constant h = 0.70, we find that the amplitude of 2dFGRS galaxies in redshift space is σ S 8g (L s , z s ) = 0.94. Correcting for redshift-space distortions as detailed above reduces this to 0.86 in real space. Applying a correction for a mean luminosity of 1.9L * using the recipe of Norberg et al. (2001), we obtain an estimate of σ R 8g (L * , z s ) = 0.76, with a negligibly small random error. In order to obtain present-day bias figures, we need to know the evolution of galaxy clustering to z = 0. Existing data on clustering evolution reveals very slow changes: higher bias at early times largely cancels the evolution of the dark matter. We therefore assume no evolution in σ 8g . The value of σ 8 for the dark matter can be deduced from the CMB fits: σ 8 = (0.72 ± 0.04) exp τ,(7) where the error bar includes both data errors and theory uncertainty. The unsatisfactory feature is the degeneracy with the optical depth to last scattering. For reionization at redshift 8, we would have τ ≃ 0.05; it is unlikely that τ can be hugely larger (e.g. Loeb & Barkana 2001). Although direct removal of this theoretical prejudice is desirable (and will be possible with future CMB data), it seems reasonable to assume that the true value of σ 8 must be very close to 0.76. Within the errors, this agrees perfectly with our σ R 8g (L * , 0) = 0.76, implying that L * galaxies are very nearly exactly unbiased. As we have seen, there are large variations in the clustering amplitude with type, so that this outcome must be something of a coincidence. Finally, this conclusion of near-unity bias was reinforced in a completely independent way, by using the measurements of the bispectrum of galaxies in the 2dFGRS (Verde et al. 2002). As it is based on three-point correlations, this statistic is sensitive to the filamentary nature of the galaxy distribution -which is a signature of nonlinear evolution. One can therefore split the degeneracy between the amplitude of dark-matter fluctuations and the amount of bias. At the effective redshift and luminosity of their sample (z s = 0.17 and L = 1.9L * ), Verde et al. found b = 1.04 ± 0.11. Although the corrections to zero redshift and to luminosity L * are probably significant, this reinforces the point that on large scales there is no substantial difference in clustering between typical galaxies and the dark matter (small scales, of course, are another matter entirely). CONCLUSIONS The 2dFGRS is the first 3D survey of the local universe to achieve 100,000 redshifts, almost an order of magnitude improvement on previous work. The final database should yield definitive results on a number of key issues relating to galaxy clustering. For details of the current status of the 2dFGRS, see http://www.mso.anu.edu.au/2dFGRS. In particular, this site gives details of the 2dFGRS public release policy, in which approximately the first half of the survey data were made available in June 2001, with the complete survey database to be made public by mid-2003. Some key results of the survey to date may be summarized as follows: (1) The galaxy luminosity function has been measured precisely as a function of spectral type (Folkes et al. 1999;Madgwick et al. 2002). (2) The amplitude of galaxy clustering has been shown to depend on luminosity (Norberg et al. 2001). The relative bias is b/b * = 0.85 + 0.15 (L/L * ). (3) The redshift-space correlation function has been measured out to 30 h −1 Mpc. Redshift-space distortions imply β ≡ Ω 0.6 m /b = 0.49 ± 0.09, for galaxies with L ≃ 1.4L * . (4) The galaxy power spectrum has been measured to high accuracy (10-15% rms) over about a decade in scale at k < 0.15 h Mpc −1 . The results are very well matched by an n = 1 CDM model with Ω m h = 0.18 and 16% baryons. (5) Combining the power spectrum results with current CMB data, very tight constraints are obtained on cosmological parameters. For a scalar-dominated flat model, we obtain Ω m = 0.31 ± 0.05, and h = 0.68 ± 0.04, independent of external data. (6) Results from the CMB comparison imply a large-scale bias parameter consistent with unity. This conclusion is also reached in a completely independent way via the bispectrum analysis of Verde et al. (2002). Overall, these results provide precise support for a cosmological model that is flat, with (Ω b , Ω c , Ω v ) ≃ (0.04, 0.25, 0.71), to a tolerance of 10% in each figure. Although the ΛCDM model has been claimed to have problems in matching galaxy-scale observations, it clearly works extremely well on large scales, and any proposed replacement for CDM will have to maintain this agreement. So far, there has been no need to invoke either tilt of the scalar spectrum, or a tensor component in the CMB. If this situation is to change, the most likely route will be via new CMB data, combined with the key complementary information that the large-scale structure in the 2dFGRS can provide. FIGURE 3 . 3Photographic B − R colour versus redshift for the 2dFGRS. FIGURE 5 . 5The galaxy correlation function ξ (σ , π) as a function of transverse (σ ) and radial (π) pair separation is shown as a greyscale image. It was computed in 0.2 h −1 Mpc boxes and then smoothed with a Gaussian having an rms of 0.5 h −1 Mpc. The contours are for a model with β = 0.4 and σ p = 400 km s −1 , and are plotted at ξ = 10, FIGURE 6 . 6(a) The correlation length in real space as a function of absolute magnitude. The solid line shows the predictions of the semi-analytic model of Benson et al. (2001), computed in a series of overlapping bins, each 0.5 magnitudes wide. The dotted curves show an estimate of the errors on this prediction, including the relevant sample variance for the survey volume. (b) The real space correlation length estimated combining the NGP and SGP (filled circles). The open symbols show a selection of recent data from other studies. ACKNOWLEDGEMENTSThe 2dF Galaxy Redshift Survey was made possible by the dedicated efforts of the staff of the Anglo-Australian Observatory, both in creating the 2dF instrument, and in supporting it on the telescope. . W E Ballinger, J A Peacock, A F Heavens, MNRAS. 282877Ballinger W.E., Peacock J.A., Heavens A.F., 1996, MNRAS, 282, 877 . C M Baugh, G Efstathiou, MNRAS. 265145Baugh C.M., Efstathiou G., 1993, MNRAS, 265, 145 . C M Baugh, G Efstathiou, MNRAS. 267323Baugh C.M., Efstathiou G., 1994, MNRAS, 267, 323 . C Benoist, S Maurogordato, L N Da Costa, A Cappi, R Schaeffer, ApJ. 472452Benoist C., Maurogordato S., da Costa L.N., Cappi A., Schaeffer R., 1996, ApJ, 472, 452 . 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[ "The kinetic equation for filament density, formed during propagation of femtosecond laser radiation, in the approximation of self-consistent field", "The kinetic equation for filament density, formed during propagation of femtosecond laser radiation, in the approximation of self-consistent field" ]
[ "Bulygin A D V E Zuev \nInstitute of Atmospheric Optics\nSiberian Branch of the Russian Academy of Sciences\nAcademician Zuev Square\n634021TomskRussia\n" ]
[ "Institute of Atmospheric Optics\nSiberian Branch of the Russian Academy of Sciences\nAcademician Zuev Square\n634021TomskRussia" ]
[]
A general form of the kinetic equation (nonlinear Fokker-Planck equation) for filament number density considering the effects of their generation and decay is stated. It is consists of the phenomenological parameters, which have been determined from the direct numerical simulation of propagation of a high-power femtosecond laser pulse (HPFLP) based on the stationary nonlinear Schrödinger equation (NSE) for a series of the threshold particular cases. Possibility of theoretical forecast of HPFLP propagation in different regimes of multifilamentation is still unsolved problem in spite of high applied significance [1]. The cause is that the majority of the existing methods are reduced to numerical solution of NSE modeling HPFLP [2]. Such approach allows to study the beams with significantly less dimensions than those of the light beams applied in the atmospheric optics problem[3]. At the same time, the laws of large numbers become apparent for large multiterawatt HPFLPs [4] therefore it is natural to search the statistical regularities in multifilamentation generation[5][6][7][8]. This work is continuation of a series of the authors' papers[6,7], which suggests a solution of a key problem: construction of the equation for the field of filament number density ( , )
null
[ "https://export.arxiv.org/pdf/1306.5329v3.pdf" ]
117,137,871
1306.5329
a09df964f0ab33e417affce7809037718978fe96
The kinetic equation for filament density, formed during propagation of femtosecond laser radiation, in the approximation of self-consistent field Bulygin A D V E Zuev Institute of Atmospheric Optics Siberian Branch of the Russian Academy of Sciences Academician Zuev Square 634021TomskRussia The kinetic equation for filament density, formed during propagation of femtosecond laser radiation, in the approximation of self-consistent field 1 A general form of the kinetic equation (nonlinear Fokker-Planck equation) for filament number density considering the effects of their generation and decay is stated. It is consists of the phenomenological parameters, which have been determined from the direct numerical simulation of propagation of a high-power femtosecond laser pulse (HPFLP) based on the stationary nonlinear Schrödinger equation (NSE) for a series of the threshold particular cases. Possibility of theoretical forecast of HPFLP propagation in different regimes of multifilamentation is still unsolved problem in spite of high applied significance [1]. The cause is that the majority of the existing methods are reduced to numerical solution of NSE modeling HPFLP [2]. Such approach allows to study the beams with significantly less dimensions than those of the light beams applied in the atmospheric optics problem[3]. At the same time, the laws of large numbers become apparent for large multiterawatt HPFLPs [4] therefore it is natural to search the statistical regularities in multifilamentation generation[5][6][7][8]. This work is continuation of a series of the authors' papers[6,7], which suggests a solution of a key problem: construction of the equation for the field of filament number density ( , ) (2) /2 fil ll d l l l l ll Yh x x x                .(6) Here l l hh  . Then we finally obtain: 2 1 (2) 0 0 2 2 / 2 m ld m l l k s s k n I I z x x x                          .(7) Now pass to derivation of the equations for ( Then find from Eq. (8) neglecting the higher-order moments 2 0 0 2 4 ml l m k s s k n I z x x            . We finally obtain including point sources, i.e., filaments [10]: 4 0 22 0 0 2 22 0 0 2 2 2 2 0 0 2 0 2 2 / 2 / 2 42 eff f d eff f (2) d eff f k I n z k k n I n I z k k n I + n z k k n I k n I z                                                                                                       s r s s s r r r ss rr s s s s r r r / (2) eff f d +n                      .(12) Here 3/ 2 2 ( ) ( ) ( , ) f f f f f out n D n n C I n Bn F z z                     sr .(13) Here the diffusion coefficient D has simple relationship with diffraction divergence density: . 3 This circumstance is caused by the fact that conversions of amplitude perturbations into phase ones are implemented rapidly, i.e., at the distances considerably less not only than scales of variation of macroscopic characteristics of the system, but also the scales of filament length [14]. interference with the background field [11] (coefficient 1 CI ); and the coefficient B providing existence of the finite solutions of Eq. (13) for any I and corresponding to the saturation effects [11]; the inhomogeneous term ( , ) out Fz  r is responsible for external mechanisms of system excitation, such as initial conditions and medium turbulence. Let us demonstrate briefly our reasoning, on the basis of which the right part of the equation Here  , f I are the empirical parameters determined from our previous work [7], which gives rise to a clamped intensity of 13 2 5 10 W/cm  and the value eff  corresponded to a similar size in the complete model. The numerical simulation of solution of Eq. (15) was performed using the TSU SKIF Cyberia cluster. It should be noted that the approach of choice of the effective stationary model of nonlinearity is similar to those suggested in [11,15], but comparing with them, as our numerical experiments have shown, it allows to adjust more flexibly formation of angular divergence of the light beam to the complete model. Further it should be noted that the abovementioned characteristic of appearance of the critical parameter is typical for the nonlinear systems [16], phase portrait for which has the following form (Fig. 3). The following dynamic equation for n A corresponds to this phase portrait: 23 2 ( ) n n n n d A A C I A B A dz       .(16) Here / CC   ; / BB   . Physical meaning of the coefficients of n A has been explained above, however the values of () CI  and B  are not found yet. For extension of their definition we choose maximally simple type of dependence of these coefficients on mean intensity conforming this to the known fact that filament generation is connected with interference with the background field: [11]. Moreover, we note that the right part of Eq. (16) has two nontrivial roots besides trivial one; the largest from them is related to the stable equilibrium point and this is 7 nothing else but () eq n AI . Now we require that dependence of this maximal root of Eq. (16) on I satisfies empirically found dependence , then we obtain: Thus we have found a type of the self-action term responsible for generation and decay of filaments of Eq. (13) and suppose the procedure of their specification from the numerical simulation. (0)(1)() C I C C I     In conclusion we make a few remarks about the limits of applicability of the system of Eqs. (12)- (13) and form of function of external source ( , ) out Fz  r . The equations for filament number density have been obtained in the approximation of energy constancy in an area element, therefore they can be correct only under the conditions of relative smoothness of mean intensity, or at fulfillment of the following condition:   I I    s  . With respect to the source function ( , ) out Fz  r in the presence of constant external disturbances, such as medium turbulence, the independent investigation should be done, which we are going to perform in following works. In the case of absence of turbulence, the analytical construction of the function ( , ) out Fz  r is possible only for the simplest situation, when each disturbance on the intensity profile evolves rather independently; and the beginning of its filamentation can be found by the Marburger equation [2]. In general case, obtaining of this function is possible only by direct numerical modeling of NSE at the initial stage of propagation of HPFLP with subsequent joining with the equation system (12)- (13).  are the phenomenologically introduced coefficients responsible for the dissipation effects and the effects of increment of diffraction divergence density of the light field 2  during filamentation, which are determined by the model of medium nonlinearity [7,10,11] (see below on page 5). Derivation of this equation for f n is based on the approximation of the selfconsistent field [12,13] and also on the remark that the centers of the hot pointsfilamentsobey the second-order dynamic equations being in random velocity field      s = s + s . For extension of a definition of the equations, a series of the numerical simulations were required, where the empirical regularities of multifilamentation evolution in the closed system were established [7]. Write this nonlinear diffusion equation 2 : L and is caused first of all by random component of "velocity" is memory length (length of longitudinal correlation of amplitude-phase perturbations (rings) induced by filaments), which has been found on the basis of the numerical simulation under the stationary model [7] and is on the order of ten centimeters; m L  is regularized delta function with regularization parameter m L . Other quantities including in Eq. (13) have the following values:  is filaments decay "velocity" [12], i.e., the value inverse to filaments length being for air about 4 is the coefficient responsible for multiplication of the filaments [12] owing to interference of the rings of the filaments including 2 It should be noted that this equation can be rewritten in the form of the Fokker-Planck equation adding   f nD   to the right and left parts and introducing D Fig. 2 . 2establishment of equilibrium for the closed system that the amplitude of filament number density eq eq nf An  increases according to linear law (Fig. 2) from the mean intensity I after achievement threshold level of the mean intensity Dependence of filament number density eq eq nf An  in equilibrium state on the mean intensity of the light field I found in the numerical simulation (crosses) and extrapolation of this dependence after critical area cr II  (red line). Fig. 3 . 3Schematic phase portrait of the equation for the filament number density amplitude. Two phase curves are related to two values of the mean intensity parameter. Two additional equilibrium points (except trivial point at zero value of n A ) are formed, when mean intensity increases critical value: stable (red dotted line) and unstable (blue dotted line). Horizontal arrows show the directions of phase movement of the system. Differentiating z with respect to the Wigner function we obtain the equations of the type of the            sr r s r r r r r . continuity equation from Eq. (1) in the approximation of its smoothness of  r , so called the transfer equation: 2 0 0 2 20 m m mm k s k n I J z x x s                   . (2) Hence we derive the equations for the first two field moments J по  s (*). Thus for zero moment corresponding to light field intensity we obtain: 0 0 m m k s I zx       . (3) Here 22 1 2 1 2 /^/m m m s S I s Jds ds Jds ds     is the Poynting vector normalized to intensity. In turn, we find for the Poynting vector: 2 2 2 2 2 0 1 2 0 2 1 2 0 2 2m m m l l l l m m m l k S s s Jds ds k n s I Jds ds h k n I z x x s x x                          , (4) where the "velocity" correlation tensor 2 12 m m l l h s s Jds ds    is introduced that corresponds to the momentum flux density tensor in hydrodynamics. Let us consider in detail the value: m l m h x   ; introduce the following notation (center): m m m ll l h s s I   . (5) Further, to simplify we use reasonable physical approximations 11 22 ; ll ll h h h h   confirmed in a numerical simulation (see Fig. 1) for the case of multifilamentation corresponding to the situation of deterministic chaos [8]: In our current modeling scheme, the negative contribution of the plasma involved in a full threedimensional model is replaced by the effective negative refractive index of the following form: Physics and applications of atmospheric nonlinear optics and filamentation // Optics Express. J Kasparian, J.-P Wolf, 16Kasparian J., Wolf J.-P. Physics and applications of atmospheric nonlinear optics and filamentation // Optics Express. 2008. V. 16, No 1. P. 466-493. . Self-Focusing ; Past, Present, R W Boyd, S G Lukishova, Y R Shen, Topics in Applied Physics. V. 114SpringerSelf-focusing: Past and Present, Boyd R.W., Lukishova S.G., Shen Y.R., eds. // Topics in Applied Physics. V. 114, New York: Springer, 2009. 605 p. GPU accelerated fully space and time resolved numerical simulations of self-focusing laser beams in SBS-active media. S Mauger, De Verdiere Guillaume Colin, L Berge, S Skupin, Journal of Computational Physics. Mauger S., De Verdiere Guillaume Colin, Berge L., Skupin S. GPU accelerated fully space and time resolved numerical simulations of self-focusing laser beams in SBS-active media // Journal of Computational Physics 2013. V. 235. P. 606-625. Saturation of the filament density of ultrashort intense laser impulse in air. S Henin, Y Petit, J Kasparian, J.-P Wolf, A Jochmann, S D Kraft, S Bock, U Schramm, R Sauerbrey, W M Nakaema, K Stelmaszczyk, P Rohwetter, L Woste, C.-L Soulez, S Mauger, L Berge, S Skupin, 10.1007/s00340-010-3941-xAppl. Phys. B. Henin S., Petit Y., Kasparian J., Wolf J.-P., Jochmann A., Kraft S.D., Bock S., Schramm U., Sauerbrey R., Nakaema W.M., Stelmaszczyk K., Rohwetter P., Woste L., Soulez C.-L., Mauger S., Berge L., Skupin S. Saturation of the filament density of ultrashort intense laser impulse in air // Appl. Phys. B. 2010. doi: 10.1007/s00340-010-3941-x. Non-Gaussian statistics of multiple filamentation // Opt. Lett. P M Lushnikov, N Vladimirova, Lushnikov P.M., Vladimirova N. Non-Gaussian statistics of multiple filamentation // Opt. Lett. 2010. V. 35. P. 1965. Evolution of the effective characteristics of high-power femtosecond laser radiation in air under the optical turbulence conditions. The Gauss beam approximation // Atmospheric and oceanic optics. A A Zemlyanov, A D Bulygin, in RussianZemlyanov A.A., Bulygin A.D. Evolution of the effective characteristics of high-power femtosecond laser radiation in air under the optical turbulence conditions. The Gauss beam approximation // Atmospheric and oceanic optics. 2010. V. 23. No. 5. P. 378-382 [in Russian]. The density number of filaments in the state of the weak and optical turbulence. A D Bulygin, arXiv:1208.39168Bulygin A.D. The density number of filaments in the state of the weak and optical turbulence // arXiv:1208.39168. Pattern dynamics and filamentation of femtosecond terawatt laser pulses in air including the higher-order Kerr effects // PHYS. T W Huang, C T Zhou, X T He, REV. E. 2013. V. 87. P. 053103Huang T. W., Zhou C. T., and He X. T. Pattern dynamics and filamentation of femtosecond terawatt laser pulses in air including the higher-order Kerr effects // PHYS. REV. E. 2013. V. 87. P. 053103. . Statistical Mechanics (a Set, R P Of Lectures). Feynman, Institute, Technology, 371STATISTICAL MECHANICS (A SET OF LECTURES). Feynman R.P. //CALIFORNIA INSTITUTE OF TECHNOLOGY,1972., P.371. Evolution of the effective characteristics of high-power femtosecond laser radiation in air under the optical turbulence conditions. The Gauss beam approximation // J.Atmospheric and oceanic optics. A A Zemlyanov, A D Bulygin, 23Zemlyanov A.A., Bulygin A.D. Evolution of the effective characteristics of high-power femtosecond laser radiation in air under the optical turbulence conditions. The Gauss beam approximation // J.Atmospheric and oceanic optics. vol. 23, 2010, No.05, pp.378-382 Peculiarities of development of light field perturbations in Kerr medium with nonlinear absorption // Atmospheric and oceanic optics. A A Zemlyanov, A D Bulygin, in RussianZemlyanov A.A., Bulygin A.D. Peculiarities of development of light field perturbations in Kerr medium with nonlinear absorption // Atmospheric and oceanic optics. 2012. V. 25. No. 10. P. 852-856 [in Russian]. Critical phenomena in media with breeding, decay, and diffusion // Sov. Phys. Usp. A S Mikhailov, I V Uporov, 10.1070/PU1984v027n09ABEH00415627Mikhailov A S, Uporov I.V. Critical phenomena in media with breeding, decay, and diffusion // Sov. Phys. Usp. 27 695-714 (1984); DOI: 10.1070/PU1984v027n09ABEH004156. Bogolyubov The compensation principle and the self-consistent field method. N N , 2N.N. Bogolyubov The compensation principle and the self-consistent field method // 2 236-254 (1959) Optimum small-scale management of random beam perturbations in a femtosecond laser pulse. V P Kandidov, A E Dormidonov, O G Kosareva, N Akozbek, M Scalora, S L Chin, Appl. Phys. B. 87Kandidov V.P., Dormidonov A.E., Kosareva O.G., Akozbek N., Scalora M., Chin S.L. Optimum small-scale management of random beam perturbations in a femtosecond laser pulse // Appl. Phys. B. 2007. V. 87, iss. 1. P. 29-36. Abnormal wavelength dependence of the self-cleaning phenomenon during femtosecond-laser-pulse filamentation // PHYS. REV. A. Weiwei Liu, See Leang Chin, 13826Weiwei Liu, and See Leang Chin Abnormal wavelength dependence of the self-cleaning phenomenon during femtosecond-laser-pulse filamentation // PHYS. REV. A. 2007. V. 76. P. 013826. Dissipative optical solitons. N N Rozanov, Fizmatlit, 2011. 536 p. in RussianRozanov N.N. Dissipative optical solitons. From micro-to nano-and atto-. M.:Fizmatlit, 2011. 536 p. [in Russian].
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[ "Charged test-particle scattering and effective one-body metrics with spin", "Charged test-particle scattering and effective one-body metrics with spin", "Charged test-particle scattering and effective one-body metrics with spin", "Charged test-particle scattering and effective one-body metrics with spin", "Charged test-particle scattering and effective one-body metrics with spin", "Charged test-particle scattering and effective one-body metrics with spin" ]
[ "Jitze Hoogeveen \nNiels Bohr International Academy Niels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 17DK-2100Copenhagen ØDenmark\n", "Jitze Hoogeveen \nNiels Bohr International Academy Niels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 17DK-2100Copenhagen ØDenmark\n", "Jitze Hoogeveen \nNiels Bohr International Academy Niels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 17DK-2100Copenhagen ØDenmark\n" ]
[ "Niels Bohr International Academy Niels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 17DK-2100Copenhagen ØDenmark", "Niels Bohr International Academy Niels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 17DK-2100Copenhagen ØDenmark", "Niels Bohr International Academy Niels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 17DK-2100Copenhagen ØDenmark" ]
[]
Using recently developed techniques, we consider weak-field test-particle scattering angle calculations in two distinct settings: Charged test-particles in spacetimes of charged sources and Effective One-Body theory with spin. We present scattering angle calculations up to O(G 4 ) of charged particles in the Kerr-Newman metric, including electromagnetic interactions up to second order in charge. Coulomb scattering is also discussed, and the well-known Darwin scattering formula is rederived by resummation. An Effective One-Body metric for a Kerr-Schwarzschild binary is constructed in a post-Minkowskian framework up to O(G 2 ) and first order in spin. Facilitated by explicit scattering calculations, our approach is equivalent with existing literature through gauge-like transformations. Finally, we investigate if the Newman Janis Algorithm applied to an Effective One-Body metric of non-spinning binaries represents a binary system with spin.
null
[ "https://export.arxiv.org/pdf/2303.00317v2.pdf" ]
257,255,045
2303.00317
7f024a9c539e62f7aace44d5fe0c62c0e89e70f4
Charged test-particle scattering and effective one-body metrics with spin 10 Mar 2023 Jitze Hoogeveen Niels Bohr International Academy Niels Bohr Institute University of Copenhagen Blegdamsvej 17DK-2100Copenhagen ØDenmark Charged test-particle scattering and effective one-body metrics with spin 10 Mar 2023(Dated: March 13, 2023)arXiv:2303.00317v2 [hep-th] Using recently developed techniques, we consider weak-field test-particle scattering angle calculations in two distinct settings: Charged test-particles in spacetimes of charged sources and Effective One-Body theory with spin. We present scattering angle calculations up to O(G 4 ) of charged particles in the Kerr-Newman metric, including electromagnetic interactions up to second order in charge. Coulomb scattering is also discussed, and the well-known Darwin scattering formula is rederived by resummation. An Effective One-Body metric for a Kerr-Schwarzschild binary is constructed in a post-Minkowskian framework up to O(G 2 ) and first order in spin. Facilitated by explicit scattering calculations, our approach is equivalent with existing literature through gauge-like transformations. Finally, we investigate if the Newman Janis Algorithm applied to an Effective One-Body metric of non-spinning binaries represents a binary system with spin. Using recently developed techniques, we consider weak-field test-particle scattering angle calculations in two distinct settings: Charged test-particles in spacetimes of charged sources and Effective One-Body theory with spin. We present scattering angle calculations up to O(G 4 ) of charged particles in the Kerr-Newman metric, including electromagnetic interactions up to second order in charge. Coulomb scattering is also discussed, and the well-known Darwin scattering formula is rederived by resummation. An Effective One-Body metric for a Kerr-Schwarzschild binary is constructed in a post-Minkowskian framework up to O(G 2 ) and first order in spin. Facilitated by explicit scattering calculations, our approach is equivalent with existing literature through gauge-like transformations. Finally, we investigate if the Newman Janis Algorithm applied to an Effective One-Body metric of non-spinning binaries represents a binary system with spin. I. INTRODUCTION The breakthrough observation of Gravitational Waves [1] opens a new window to the universe, allowing for the first time detailed testing of General Relativity. Among these initial observations, signals from binary black holes intricately encode both dynamics of the binary, and single black hole properties. The prospect of gaining insight into these previously unprobed areas has in recent years catalysed a great theoretical effort, developing new analytical and numerical techniques solving the highly non-linear dynamics of General Relativity. One such area is the investigation of test-particle trajectories. This historically well-established subject provides both insight to single black hole properties, and facilitates a simple setting for developing new calculational tools, some of which have proven useful even to the full binary problem. Analytical expressions for geodesics of test-particles have been found for (off-)equatorial trajectories, expressed with elliptical functions (for a review, see ref. [2]). Calculations with a non-spinning Schwarzschild black hole were carried out in [3][4][5] followed by a charged non-spinning Reissner Nordström black hole [6,7]. Similar geodesics for a spinning Kerr black hole [8][9][10][11] including plunging orbits [12], notably introduced the Carter constant [8], and Mino time which is essential for integration [9]. A collection of results is presented e.g. in refs. [13,14]. Kerr-Newman geodesics are similarly expressible in terms of elliptical functions [15], considering even charge on the test-particle. Recently, test-particle scattering trajectories have seen increased attention. The associated calculations of the scattering angle are closely related to binary dynamics. For the purposes of this article, we will restrict ourselves to planar scattering. Examples of non-planar paths are given in e.g. refs. [15,16]. Although planar scattering angles are easily encapsulated within a Hamilton-Jacobi formalism, difficulties dealing with integration limits make actual calculations a non-trivial matter. Few closed form scattering angle calculations are possible (see e.g. ref. [17]). In a weak-field limit, where the scattering angle χ is expanded in Newtons gravitational constant G, these difficulties may be overcome. Hadamard regularization is the traditional approach to scattering angle evaluations in this limit [18,19]. It substitutes the difficult lower integration limit with something simple and manually removes emergent divergencies. A different method was recently developed in ref. [20]. Building on work in isotropic metrics [21,22], this technique provides a simple formula applicable to very general situations. It considers the scattering angle integral χ/2 + π/2 = ∞ rm dr dφ dr .(1) The lower integration limit, r m , is the distance of minimum approach, which may not always be explicitly obtained. This complicates integration. Ref. [20] showed, by writing the integral in a very general form and assuming a weakfield limit, that one may explicitly render the scattering angle a sum of easily calculable integrals independent of r m . Notably, this applies to test-particles in very general metrics (not necessarily restricting to black holes), for both scalar and spinning particles. In ref. [20], the formalism was specifically employed for scattering in the equatorial plane of a Kerr metric for spinning test-particles up to second order in spin. Test-particle scattering in a weak-field regime is linked to binary (black hole) dynamics in numerous ways. The extreme mass ratio limit of a two-body calculation naturally retrieves the test-particle regime. As such test-particle scattering in itself provides a useful tool for cross checking two-body calculations. However beyond this connection, dynamics of test-particles may encode dynamics of a two-body system through Effective One-Body (EOB) theory. Crucial for the EOB approach, to be introduced shortly, are the calculations of full binary Hamiltonians and scattering angles. Outside the test-particle limit, the work on binary dynamics in GR is rapidly evolving, due to its connection with Gravitational Wave observations. Precise knowledge of binary trajectories is crucial for constructing the waveform. Both numerical and analytical approaches have proven fruitful. On the analytical side, various classical methods yield the two-body Hamiltonian in an expanded form; a post-Newtonian (PN) approach expands around weak-field Newtonian gravity in velocities v 2 /c 2 ≪ 1 and Newton's gravitational constant GM/c 2 r ≪ 1, whereas a post-Minkowskian (PM) expansion considers weak-field interactions with arbitrary velocity as perturbations of Minkowski space in GM/c 2 r ≪ 1. This latter regime is equivalent to the weak-field approach discussed above in the test-particle limit. Partial expressions for the non-spinning binary black hole Hamiltonian are available up to 6PN, ie. O[(v/c) 12 ], in the post-Newtonian expansion [23][24][25][26][27][28] and 4PM, ie. O(G 4 ), in the post-Minkowskian expansion [29][30][31]. For spinning binaries, results up to 5PN including both spin-orbit [32][33][34] and spin-spin couplings [35][36][37][38][39][40][41][42][43] have been found. For the post-Minkowskian expansion all-order in spin expressions are available at 1PM [44], whereas second order in spin results are available at 2PM and 3PM [45][46][47][48]. Starting at O(G 4 ), radiative processes contribute to conservative dynamics. Dealing with these subtleties is still an open problem [49]. The post-Minkowskian (PM) expansion naturally lends itself to scattering trajectories. For reviews see e.g. refs. [49][50][51]. Importantly, the two-body Hamiltonian may be recovered from this regime [18,48,52,53]. Binary scattering angles, computed order by order in G, encode information about the Hamiltonian. These may be found by a plethora of methods. Early landmark calculations were performed by Westpfahl [54]. A linearised form of Einstein's equations have yielded exact results for aligned spinning binaries at first order in G [44]. Other manifestly classical approaches such as the world-line formalism [48,[55][56][57][58][59][60][61][62] and effective field theory [29][30][31][63][64][65][66][67][68][69][70] employ techniques lent from quantum field theory. Using these, scattering angles with both non-spinning and spinning binaries are available up to O(G 3 ) and second order in spin. Quantum calculations, e.g. amplitudes of massive particle scattering mediated by gravitons, provide further results up to O(G 5 ) [71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86] and various orders in spin [47,52,. Remarkably, the classical limit of such amplitude calculations is in correspondence with macroscopic black hole scattering [89,90,122], reproducing classical results at least up to 3PM and second order in spin [48,89,90]. Especially relevant in our current treatment is the scattering angle of aligned spinning particles from amplitudes, up to 2PM and fourth order in spin [89], χ (a1,a2) = 2GE −2(a 1 + a 2 )v + bv 2 + b v 2 (b 2 − (a 1 + a 2 ) 2 ) − πG 2 E ∂ ∂b [m 2 f (a 1 , a 2 ) + m 1 f (a 2 , a 1 )] + O(G 3 ) (2a) f (σ, a) = 1 2a 2 −b + (j + κ − 2a) 5 4vκ[(j + κ) 2 − (2va) 2 ] 3/2 + O(σ 5 ), j = vb + σ + a, κ = j 2 − 4va(b − vσ) (2b) evaluated in center of mass coordinates. This matches classical computations of aligned Kerr black holes at all orders in spin for O(G), up to linear order in spin at O(G 2 ), and conjecturally up to O(spin 4 ). Parameters (a 1 , a 2 ) are the binary spins with m 1 and m 2 their masses, E is the total (center of mass) energy of the system, v is the relative asymptotical (center of mass) velocity between objects, b is their impact parameter and γ = 1/ √ 1 − v 2 . As suggested in refs. [18,65,66,68,[123][124][125], post-Minkowskian angles encode binary dynamics even for bound orbits. This is crucial for the study of gravitational waves, typically emitted by inspiraling bound systems. One way to recover bound orbit dynamics from scattering data is with Effective One-Body (EOB) theory. Originally formulated in a Post Newtonian expansion of velocities [126][127][128][129], in 2016 it was naturally adopted to a post-Minkowskian, scattering-based approach [18,72]. The EOB formalism translates full binary motion (outside the scattering regime), to an effective test-particle moving in an EOB metric. This metric may be constructed by matching scattering angles of the full binary with those of the effective test-particle. This formalism is thus heavily reliant on scattering angle calculations of test-particles in complicated (EOB) metrics. The construction of such a metric has three major benefits, i) calculations of test-particle motion are much easier than directly solving Einstein's equations, ii) geodesics of the metric readily include bound orbits, and iii) the metric in effect resums post-Minkowskian data, widening the regime of applicability. EOB formalisms have been constructed up to O(G 2 ) without spin in refs. [18,19,53]. Including spin, current literature goes up to O(G 2 ) and O(spin 1 ) [45,46]. EOB mappings at higher orders in spin have been considered in ref. [52] at O(G 2 ), and an all order in spin result at O(G) was published by Justin Vines [44]. For recent developments in the post-Newtonian approach, see refs. [24,25]. Ref. [130] includes a discussion on non-conservative contributions. For older results see [126][127][128][129] of which a review is given in [131]. Notably, ref. [128] presents a post-geodesic Q term to the Hamiltonian which is reintroduced in post-Minkowskian theory in [53]. See also ref. [19] for discussions hereof, especially the rewriting to an effective potential W . Test-particle scattering calculations provide a way to both probe single black hole properties, and encode full black hole binary dynamics. The method of ref. [20] provides a novel tool for evaluating angles, enabling new streamlined analysis of these areas. In this paper, demonstrating the versatility of the method, we consider scattering in two distinct settings. First, the formalism of ref. [20] is applied to scattering of charged test-particles in the Kerr-Newman metric. Applications to other spacetimes and electric potentials is discussed. Among these is the treatment of relativistic Coulomb scattering. Second, the method of ref. [20], given its broad applicability, is considered with the EOB approach. A post-Minkowskian framework, such as that from refs. [18,132] is used. Particularly, an EOB metric for a Kerr-Schwarzschild black hole binary is constructed, up to second order in G and fourth order in spin. Comparisons with earlier approaches [19,44,132] are made. The formalism is restricted to orbits in the equatorial plane. As an accompanying study, the Newman-Janis Algorithm (NJA) is explored in context of the EOB formalism. Does the application of the NJA algorithm to non-spinning EOB metrics produce an EOB metric with spin? The success of this approach, based on a Schwarzschild-Schwarzschild binary EOB metric from ref. [132], will be explicitly checked by comparing post-Minkowskian scattering angles of the NJA-transformed metric with those of aligned Kerr black holes from amplitude methods [89]. Section II introduces the general formalism of ref. [20] with a view towards Kerr-Newman, but emphasises its general applicability. Sections III and IV then compute Kerr-Newman scattering angles of scalar and charged testparticles respectively. Sections V and VI turn to the EOB formalism based on a post-Minkowskian approach. In section V, a 2PM EOB metric describing Kerr-Schwarzschild binaries is constructed based on a deformed Kerr metric. Section VI treats the application of the NJA to the EOB metric from ref. [132]. Throughout we adopt natural units for the speed of light c = 1 and the Coulomb constant 1/(4πǫ 0 ) = 1. Newtons gravitational constant is denoted G, and the mostly plus sign convention (− + ++) is used. II. SCATTERING ANGLES IN POST-MINKOWSKIAN EXPANSION We introduce test-particle scattering in this section. Let us first consider a scalar (uncharged, non-spinning) testparticle. Adding electrodynamic behavior is covered in section IV. We start by reviewing the work of ref. [20], establishing a general method of evaluating test-particle scattering angles. The assumptions listed below are used implicitly throughout the article. We concern ourselves only with planar scattering in asymptotically flat metrics g µν = g µν (r) parametrised by polar coordinates {t, r, φ}. The test-particle, given mass m, follows a geodesic scattering trajectory r = {∞ → r m → ∞} with angular deflection φ = {0 → π/2 + χ/2 → π + χ}. The incident direction is chosen as φ = 0 without loss of generality and r m denotes the distance of minimum approach. Test-particle impact parameter is denoted b, asymptotical velocity v, asymptotical momentum p ∞ , energy E, and orbital angular momentum L. Energy and momenta E, L and p ∞ may be expressed in terms of b and v as E = γm, p ∞ = γmv, L = bp ∞ , where γ = 1/ 1 − v 2 .(3) Hamilton-Jacobi theory readily determines the test-particle trajectory, and therefore also the scattering angle. For a scalar test-particle, the Hamiltonian and associated Hamilton-Jacobi equations which normalise canonical momentum are H = 1 2 g µν p µ p ν , p µ = g µν dx ν dλ , −m 2 = g µν p µ p ν ⇒ẋ µ = ∂H ∂p µ ,(4) with affine parameter dλ = ds/m, given in terms of line element ds, parameterizing the particle path such that canonical momentum p µ matches test-particle four-momentum. Dots denote differentiation with respect to λ. The equations of motion forẋ µ may be readily found from the Hamiltonian. Translational symmetry of H in t and φ yields conservation of energy and orbital angular momentum, p t ≡ −E and p φ ≡ L. The radial component p r may be determined from the Hamilton-Jacobi equation as a function only of coordinate r. The scattering angle may readily be found from the Hamiltonian. It may generically be written as χ/2 + π/2 = ∞ rm dr dφ dr = − ∞ rm dr h(r) p r , h(r) ≡ −p r dφ dr(5) simply by integrating angular deflection over half a scattering trajectory. A conventional π/2 has been added to yield χ = 0 for straight-line motion. The lower integration limit r m may be found for Hamiltonians quadratic in p r by the requirementṙ = 0| r=rm ⇒ p r (r m ) = 0. Following ref. [20], we have rewritten the scattering angle integral simply by defining the function h(r) in terms of dφ dr and p r . However this form is suggestive: Note that the integral is naturally divergent in the lower integration limit p r (r m ) = 0. Factoring out this divergence, h(r) often takes very simple non-divergent forms. In fact from eq. (4) one identifies h(r) = − Lg φφ − Eg φt g rr (7) for a scalar test-particle in the general metric discussed above. This identification of h(r) is useful beyond scalar particles, and was also shown applicable to spinning test-particles in ref. [20]. Below, the same will be shown true also for charged test-particles. Exact calculation of the scattering angle, often does not yield a closed expression, instead returning an elliptical integral. However, in a weak-field expansion in G closed expressions may be found order by order in G. In ref. [20] a general calculation of the scattering angle in the weak-field regime was provided. It considers a scattering angle written in the form of eq. (5) with h(r) obeying the requirements h(r) is analytical on r ∈ [r m , ∞[ and falls off at least like 1/r 2 as r → ∞. One may readily confirm that eq. (7) indeed obeys these requirements. Furthermore, the metric is assumed written in what ref. [20] defines as a normal form. A metric is of normal form when it has the property g µν (r) → (Minkowski) =   −1 0 0 0 1 0 0 0 r 2   as G → 0(9) in the scattering plane. Metrics may readily be written in normal form by setting G → 0, and performing a coordinate transformation to recover the above structure. For asymptotically flat metrics, p r then takes the form p 2 r = p 2 ∞ − J 2 r 2 − U (r)(10) with some function U specified by the metric reminiscent of potentials in classical and isotropic amplitude calculations (see e.g. [22]). This potential may safely be assumed to drop off to zero as r → ∞, and may depend on any metric and test-particle quantities, e.g. angular momentum, energy etc. The above requirements are satisfied throughout this article. Under such requirements, ref. [20] provides the scattering angle in a summed form which readily yields arbitrary high orders in the weak-field expansion in G, χ + π = −2 ∞ n=0 ∞ 0 du d du 2 n h(r) r 2n U (r, b) n n!p 2n+1 ∞ , r 2 = u 2 + b 2 .(11) A very similar formula for isotropic metrics was previously obtained in ref. [22], and may readily be recovered by restricting U (r, b) → V ef f (r) independent of b and h(r) = −bp ∞ /r 2 . In this special case V ef f is independent of impact parameter b or, equivalently, angular momentum L. Restricting to scalar test-particles, eq. (11) obtains another useful form. Inspired by a derivative form of eq. (5), χ/2 + π/2 = − d dL ∞ rm dr p r , (scalar test-particle)(12) the h(r)-dependence may be converted to a derivative of the impact parameter, χ = ∞ n=0 ∞ 0 du d du 2 n d db r 2n U (r, b) n+1 (n + 1)!p 2(n+1) ∞ , r 2 = u 2 + b 2 (scalar test-particle),(13) removing the need to specify h(r) for each individual metric. In the next section, we demonstrate the use of eqs. (11) and (13) by explicitly calculating the scattering angle in the Kerr-Newman metric of a scalar test-particle, up to O(G 4 ). III. NON-CHARGED TEST-PARTICLES IN KERR-NEWMAN SPACETIME The Kerr-Newman metric describes a charged, spinning black hole, its mass denoted by M , charge Q and spin a. The Q → 0 and a → 0 limits are Kerr and Reissner-Nordström black holes respectively. Written in Boyer-Lindquist coordinates, it is rotationally symmetric only in the θ = π/2 plane. In this plane the metric reads g µν =      − 1 − 2GM/r + r 2 Q r 2 0 −a 2GM/r − r 2 Q r 2 0 r 2 a 2 −2GMr+r 2 Q +r 2 0 −a 2GM/r − r 2 Q r 2 0 a 2 (1 + 2GM/r − r 2 Q r 2 ) + r 2      ,(14) which is effectively a Kerr metric with 2GM/r replaced by 2GM/r − r 2 Q /r 2 . Parameter r Q = GQ 2 encodes the black hole electric charge. Notice that eq. (14) is not in normal form, which setting G → 0 confirms. One recovers a result identical to that of Kerr, g µν →   −1 0 0 0 r 2 r 2 +a 2 0 0 0 r 2 + a 2   as G → 0,(15) which may be brought to normal form by a coordinate transformation r 2 → ρ 2 = r 2 + a 2 . In these coordinates, the potential U may readily be found from eqs. (4) and (10), U (ρ, b) = G Q 2 − 2M ρ 2 − a 2 2a 2 b 2 p 2 ∞ − a 2 ρ 2 p 2 ∞ − 2abEρ 2 p ∞ − b 2 ρ 2 p 2 ∞ + 2ρ 4 p 2 ∞ + m 2 ρ 4 ρ 4 (ρ 2 − a 2 ) + O(G 2 ),(16) given here up to O(G) for brevity. Of course, equation (4) readily yields U to all orders in G. With this information, equation (13) straight forwardly gives the scalar test-particle scattering angle in such a spacetime. No explicit computation of h(r) is needed. Take the O(G 1 ) calculation as an example. The results at higher orders are listed in table II. Extracting only G 1 terms from eq. (13), one finds χ 1 = ∞ 0 du d db U (ρ, b) p 2 ∞ , ρ 2 = u 2 + b 2 = 1 p 2 ∞ ∞ 0 du d db   G Q 2 − 2M ρ 2 − a 2 2a 2 b 2 p 2 ∞ − a 2 ρ 2 p 2 ∞ − 2abEρ 2 p ∞ − b 2 ρ 2 p 2 ∞ + 2ρ 4 p 2 ∞ + m 2 ρ 4 ρ 4 (ρ 2 − a 2 )   = 2GM −2av + bv 2 + b v 2 (b 2 − a 2 ) + πGQ 2 2a 3 v − a 2 −v 2 √ b 2 − a 2 + 2bv 2 + b + b 2 v 2 b − √ b 2 − a 2 2a 2 v 2 (b 2 − a 2 ) 3/2 .(17) The Q → 0 limit is just the Kerr result. At any order O(G n ), Q only appears in even orders no greater than Q 2n . The scattering angle χ may thus be decomposed into orders of G and Q, χ = ∞ n=1 n k=0 χ (k) n , χ (k) n ∼ π (n+k+1) mod 2 Q 2k G n M n−k v 2n (b 2 − a 2 ) (3n+k−1)/2 f n,k (v).(18) These expressions are structurally similar to those found in ref. [20] for Schwarzschild and Kerr metrics. A prefactor π appears in χ (k) n only when n + k is even, much like the factors of π appearing at even order in ref. [20] for Kerr and Schwarzschild. Indeed, the Kerr-Newman potential U ∼ 1/r n+k at order O(G n ) and O(Q 2k ) resembles in its r dependence a Kerr potential at order n + k in G. Additional similarities may be found. Polynomials f n,k (v) of velocity v depend on fractional powers of b and a for even n + k, and integer powers of b and a for odd n + k. This mimics the identical behavior in Kerr, for n respectively even and odd. Interestingly, the v and (a, b) dependence of f n,k is partially factorized at orders (n, k) = (1, 0), (2, 1), (3,2), (4,3), ie. when n − k = 1. Here we observe f n,k = [f (v) − f (ab) −+ + f (v) + f (ab) +− ] n,k(19) where f (v) and f (ab) are polynomials exclusively dependent on either v or (a, b) respectively. Subscript +, − signs indicate whether v, a and b appear in even or odd powers, ie. f (v) + is even in v and f (ab) +− is even in a but odd in b. All polynomials differ in structure with varying n, k. We expect the structure of eq. (19) to continue to higher orders. IV. CHARGED TEST-PARTICLES IN KERR-NEWMAN SPACETIME Continuing our treatment of the Kerr-Newman metric, we next consider test-particles with charge e and mass m in a spacetime of a charged source. For completeness, as before, our initial analysis pertains to a general metric g µν as defined in the previous section. Hereafter, the specific case of the Kerr-Newman metric is treated. A short study of Coulomb scattering will also be discussed, as results are easily compared to the all-order exact Coulomb scattering angle [17] obtained by direct integration of eq. (5). Apart from gravitational effects, electromagnetic interactions with coupling constant e (charge of test-particle) and electromagnetic 4-potential A µ need to be accounted for. Charged bodies in curved spacetime may be treated with Einstein-Maxwell theory (for a review, see e.g. ref. [133], including [54] for early scattering calculations). Charged test-particle orbits around black holes are covered in ref. [13,15,134,135] and fully characterized for Kerr-Newman in ref. [15]. The treatment presented in this article is restricted to test-particle limits of e and m, neglecting self-force effects [54,133,136]. A µ is then entirely produced by the gravitational source. Subjected to an external potential A µ , the Hamiltonian and associated equations of motion of a test-particle with charge e and mass m in metric g µν are H = 1 2 g µν (p µ − eA µ )(p ν − eA ν ), p µ = g µν dx ν dλ + eA µ ⇒ẋ µ = ∂H ∂p µ(20) where p µ denotes canonical momentum. Affine parameter dλ = ds/m is defined in terms of the line element ds as in eq. (4), and we continue to denoteẋ µ = dx µ dλ . For generality, the discussion below will not assume any specific form of A µ , save require A µ → 0 as r → ∞, A µ = (A t (r), 0, 0, A φ (r)) θ=π/2(21) with θ = π/2 the equatorial plane of orbit. The scattering angle may still be calculated from eq. (5). Equation (24) yields dφ dr = (L − eA φ )g φφ + (−E − eA t )g φt g rr p r ,(22) and per identification h(r) = − (L − eA φ )g φφ + (−E − eA t )g φt g rr .(23) Asymptotically, h(r) ∼ 1/r 2 as required, provided A µ obeys eq. (21). The Hamilton-Jacobi equation from which p r may be determined, is found by normalizingẋ 2 = −m 2 with eq. (20) −m 2 = g µν (p µ − eA µ )(p ν − eA ν ).(24) Having found the radial momentum with g µν implicitly in normal form, the corresponding potential U is identified from eq. (10) p 2 r = T − U.(25) Crucially, T is now taken independent of both G and e, T ≡ p 2 r G=e=0(26) as we are dealing with two interactions. We now treat the Kerr-Newman metric, and subsequently relativistic Coulomb scattering in flat space. Both conform to the requirements set by eq. (11). The respective electrodynamic potentials read A µ = (− Q r , 0, 0, aQ r ) (Kerr-Newman), A µ = (− Q r , 0, 0, 0) (Coulomb).(27) where the Kerr-Newman potential, see ref. [137], has been evaluated at θ = π/2 in Boyer-Lindquist coordinates. The Coulomb potential may simply be considered the G = a = 0 limit of the Kerr-Newman solution. Let us therefore focus on Kerr-Newman. The normal form of the metric is recovered by the coordinate transformation of eq. (15), ρ 2 = r 2 + a 2 where r is the Boyer-Lindquist radial coordinate appearing in eq. (14). Equations (10) and (24) then yield U (ρ) = Qe −2ab ρ 2 − a 2 p ∞ + ρ 2 2E ρ 2 − a 2 − Qe + a 2 Qe ρ 4 − a 2 ρ 2 +G Q 2 − 2M ρ 2 − a 2 ρ 6 − a 2 ρ 4 p ∞ p ∞ −ρ 2 a 2 + b 2 + 2a 2 b 2 + 2ρ 4 + 4abQe ρ 2 − a 2 − 2abEρ 2 + 2Qρ 2 e Qe − 2E ρ 2 − a 2 − 2a 2 Q 2 e 2 + m 2 ρ 4 + O(G 2 ),(28) here truncated at O(G) for simplicity. The scattering angle is now found readily from eq. (11). It may be expanded simultaneously in G and e, ie. as a collective expansion in gravitational and electromagnetic interactions, χ = ∞ n=0 ∞ j=0 χ n,j , χ n,j ∼ G n e j .(29) As before, owing to the Q 2 -dependence of the Kerr-Newman metric, χ n,j may be decomposed as χ n,j = n k=0 χ (k) n,j , χ (k) n,j ∼ Q 2k ,(30) where k denotes the order of Q 2 coming from the metric. χ χ (k) n,j ∼ π (n+j+k+1)mod 2 G n e j Q 2k+j M n−k m j γ j v 2(j+n) (b 2 − a 2 ) (3(n+j)+k−1)/2 2(n+j)−1 p=0 d n,j,k a p b 2(n+j)−p f n,j,k (v),(31) involving only whole powers of a and b, contrary to fractional powers encountered with non-charged test-particles (table II). This behavior is structurally equivalent to odd powers of n+k for non-charged test-particles. Q appears exclusively as a prefactor Q 2k+j , containing a contribution from the metric (Q 2k ) and a contribution from the electromagnetic potential (Q j ). f n,j,k (v) are polynomials in v and d n,j,k are numerical constants. As observed with scalar test-particles in eq. (19), some angles factorise v and (a, b) dependence. Specifically, for (n, j, k) = (0, 1, 0), (1, 1, 1), (2, 1, 2) and (3, 1, 3), ie. when j = 1, the sum in eq. (31) may be written as two terms 2(n+j)−1 p=0 d n,j,k a p b 2(n+j)−p f n,j,k (v) = [f (v) − f (ab) −+ + f (v) + f (ab) +− ] n,j,k(32) where, again, f (v) and f (ab) are polynomials exclusively dependent on v or (a, b) respectively. Subscript +, − signs indicate whether v, a and b appear as even or odd powers, ie. f v and (a, b) terms. For instance, with (n, j, k) = (2, 1, 2), the coefficients {1, 5, 10} appear both in f (v) and f (ab) . We expect this behavior to continue to higher orders in n and k. It is similarly a straight forward matter to consider relativistic Coulomb scattering. The corresponding potential U is given by the G = a = 0 limit of eq. (28). It only involves one coupling constant, e, in terms of which the scattering angle is expanded and presented in table III, A). Order by order comparisons match the small Qe/J expansion of the well-known Coulomb scattering angle presented e.g. in ref. [17], We now apply the method of ref. [20] to the full binary problem by means of Effective One-Body (EOB) theory. Dynamics of aligned, spinning binary black holes are mapped to an effective system consisting of a test-particle in an EOB metric g χ + π = J J 2 − Q 2 e 2 π − 2 arctan Qe v J 2 − Q 2 e 2 (Coulomb(ef f ) µν . This is achieved by directly relating equatorial scattering angles of the binary to those of the effective test-particle. We stress that our approach is not unique -different EOB metrics may be constructed by similar methods. The EOB formalism is first presented in the post-Minkowskian regime by reviewing ref. [132]. Consider equatorial scattering of binary Kerr-Schwarzschild black holes with masses m 1 and m 2 , and spins a 1 = a and a 2 = 0. Center of mass coordinates may be used, in which E denotes the total energy of the system, v is the relative asymptotical velocity of the binary objects, p ∞ is the asymptotical momentum of a single body, L is the orbital angular momentum of the system, and b denotes the impact parameter. These quantities are related by v = |v 1 − v 2 |, p ∞ = (E 2 − (m 1 + m 2 ) 2 )(E 2 − (m 1 − m 2 ) 2 ) 4E 2 = m 1 m 2 E γv, L = bp ∞ ,(34) where v 1 and v 2 are velocities of the individual black holes. For convenience we have defined the reduced mass µ, total mass M , and asymptotical Lorentz contraction factor γ as µ ≡ m 1 m 2 m 1 + m 2 , M ≡ m 1 + m 2 , ν ≡ µ M , γ = 1 √ 1 − v 2 .(35) This binary is now described by an effective system consisting of a test-particle of mass µ scattering on a metric g (ef f ) µν . Two-body quantities are related to effective ones by an EOB map, which we now present. Label with subscript "eff" quantities of the effective system. From kinematic considerations of the effective test-particle, one may establish the following EOB map v ef f = v, E = M 1 + 2ν E ef f µ − 1 , p ef f = µγv = E M p ∞ , b ef f = b ⇒ L ef f = b ef f p ef f = L E M ,(36) where E ef f = µγ is the test-particle energy, and v ef f , p ef f , L ef f and b ef f are the asymptotical test-particle velocity, asymptotical momentum, angular momentum and impact parameter respectively. Furthermore, the effective formalism should have some notion of spin, call it a ef f . EOB maps between spin have been discussed in detail in e.g. refs. [44][45][46]129]. We shall here use a simple map, namely a ef f = a.(37) Last, a map between scattering angles is required. Denote by χ and χ ef f the full two-body and effective test-particle scattering angles. The most natural mapping between these, as discussed in ref. [18], is simply χ ef f = χ.(38) Both angles are treated perturbatively in G, χ = ∞ n=0 χ n G n . The calculation of χ ef f depends on g (ef f ) µν . With the full two-body system amplitude calculations give the individual components χ n . As stated before, ref. [89] provides the scattering angle of a Kerr-Schwarzschild binary with spin parameter a and respective masses m 1 and m 2 , χ (a,0) = 2GE −2av + bv 2 + b v 2 (b 2 − a 2 ) + πEG 2 32b 6 v 4 5a 4 m 1 35v 4 + 180v 2 + 24 + 24 4m 2 v 2 + m 2 − 96a 3 bv m 1 5v 2 + 4 + m 2 2v 2 + 3 + 6a 2 b 2 m 1 15v 4 + 72v 2 + 8 + 4m 2 2v 4 + 11v 2 + 2 − 16ab 3 v 3v 2 + 2 (4m 1 + 3m 2 ) + 24b 4 v 2 v 2 + 4 (m 1 + m 2 ) + O(a 5 ) + O(G 3 ).(39) The spin configuration is indicated by (a, 0). As mentioned, the connection between eq. (39) and Kerr black hole scattering has only been confirmed at O(G) for all orders in a, and at O(G 2 ) only up to O(a) [89]. We restrict to these orders in the text, and provide a conjectural Kerr binary EOB metric matching the full O(a 4 ) result in table I. The EOB metric g (ef f ) µν is constructed in such a way that eq. (38) is satisfied. An ansatz is provided, the parameters of which are constrained by eq. (38). As scattering is planar by construction, a rotationally symmetric, asymptotically flat ansatz is natural. We further demand that the metric reduces to Kerr in the test-particle limit m 2 → 0. One may thus naturally search for EOB metrics among generalizations of Kerr. It is instructive to review in short the results obtained by ref. [132]. Here binary Schwarzschild black holes were considered. The EOB metric had a generalized Schwarzschild form ds 2 = − 1 − α(r) 1 + α(r) 2 dt 2 + (1 + α(r)) 4 (dr 2 + r 2 dΩ 2 ), α(r) = ∞ n=1 α n G n r n ,(40) which simply replaces GM/(2r) → α(r) in the Schwarzschild spacetime written in isotropic coordinates. Adopting an isotropic calculation renders the scattering angle integrand, dφ/dr in a form comparable directly to amplitude calculations of refs. [22,132]. This allows specification of α n directly from comparing scattering angle integrands, as opposed to merely the scattering angles. One finds, comparing up to 2PM, α 1 = 1 2 E, α 2 = − 3(5γ 2 − 1) 8(2γ 2 − 1) 1 − M E E 2 .(41) The Schwarzschild metric in isotropic coordinates, meaning α → GM/(2r), is recovered in the test-particle limit. Now turn to Kerr-Schwarzschild binaries. We provide an EOB metric ansatz written in non-isotropic coordinates, meaning g rr = r 2 g φφ . Our method differs from that of ref. [132] by explicitly computing the effective scattering angle, instead of comparing integrands. Comparing our result with eq. (39) determines the EOB metric parameters. We choose an ansatz constructed from equatorial Kerr in Boyer-Lindquist coordinates by replacing GM/r → κ(r), g (ef f ) µν =   − (1 − κ(r)) 0 −aκ(r) 0 r 2 a 2 +r 2 (1−κ(r)) 0 −aκ(r) 0 a 2 (1 + κ(r)) + r 2   , κ(r) = ∞ n=1 κ n G n r n(42) parametrized with coordinates {t, r, φ}. This choice of EOB metric is entirely arbitrary. A different ansatz could be equally viable, producing a different final result. The metric above resums orders in a, in a structure similar to Kerr(-Newman) metrics. Note however that κ may also depend on a. Below we present κ to O(a 4 ) as eq. (39) naturally restricts hereto. Equation (13) readily yields the test-particle scattering angle. However, one should be careful about the parameterdependence of κ, which may possibly depend on all test-particle quantities, namely energy E ef f , impact parameter b ef f , and angular momentum L ef f , including also asymptotical velocity v and asymptotical momentum p ∞ . As an example, eq. (40) is dependent on E. However impact parameter, or equivalently, angular momentum dependence influences the application of d db in eq. (13). The result of calculating the post-Minkowskian scattering angle by eq. (13) (see footnote [138]), with arbitrary b-dependent κ, is therefore an expression with first-order derivatives of κ n in b. In our case, it is sufficient to assume κ 1 is independent of b, which will yield consistent solutions. Other κ n → κ n (b) will remain unspecified functions of b. Imposing the EOB map of eqs. (36)- (38), the effective test-particle scattering angle of the EOB metric from eq. (42) becomes χ ef f = Gκ 1 −2av + bv 2 + b v 2 (b 2 − a 2 ) + πG 2 128b 6 v 4 4v 2 κ 2 (b) 5a 4 5v 2 + 6 − 48a 3 bv + 6a 2 b 2 3v 2 + 4 − 32ab 3 v + 8b 4 v 2 + 2 − b a 4 5v 2 + 6 − 12a 3 bv + 2a 2 b 2 3v 2 + 4 − 16ab 3 v + 8b 4 v 2 + 2 dκ 2 (b) db + κ 2 1 5a 4 35v 4 + 180v 2 + 24 − 96a 3 bv 5v 2 + 4 + 6a 2 b 2 15v 4 + 72v 2 + 8 − 64ab 3 v 3v 2 + 2 + 24b 4 v 2 v 2 + 4 + O(a 5 ) + O(G 3 ),(43) presented here up to O(G 2 ) and truncated to O(a 4 ) to facilitate direct comparison with eq. (39). Of course, the angle could be evaluated to any order in G and a. At the current precision, equating eq. (39) with eq. (43), κ n may be determined up to O(a 4 ). This produces a first order differential equation. The solutions will therefore naturally involve a b-independent integration constant C. For brevity we will only present the O(a) result in the text, leaving the complete O(a 4 ) result to table I. One finds κ 1 =2E, κ 2 = 3 v 2 + 4 E(m 1 + m 2 − E) v 2 + 2 + b C v 2 + 2 + a E −2(m 1 − E) 3v 4 + 4v 2 + 8 − 3m 2 v 4 + 4 bv (v 2 + 2) 2 + 2Cv (v 2 + 2) 2 + O(a 2 ).(44) The integration constant C may be set to 0 by requiring g (ef f ) µν → Kerr in the test-particle limit of m 2 . Contrary, the limit m 1 → 0 describes a spinning test-particle in Schwarzschild. Here the EOB metric does not reduce to Schwarzschild, as it also encodes spin of the probe. We stress the simplicity of the EOB construction presented here. An EOB metric is readily found from an ansatz and EOB map, by directly matching scattering angles of the full two-body system with those of the effective test-particle. Non-metric, post-geodesic Finsler-type contributions of e.g. refs. [19,45] are not needed. Similar observations were made without spin in ref. [132]. We leave to future work the extension to higher Post Minkowskian orders, by the inclusion of κ n>2 terms. We emphasize that the EOB metric found above is by no means unique. Other solutions, based on a different ansatz, may exist. ) for O(a), with higher orders in a matched only by conjecture. Each row, labelled by n contains the O(a n ) contribution to κ2. Expressions are given in terms of full two-body quantities. These solutions follow from equating eq. (43) with eq. (39) and inserting the 1PM result κ1 = 2E from eq. (70). The arbitrary b-independent integration constant C associated with these solutions is set to C = 0. n [κ 2 at O(a n )] / a n E b n v n (v 2 +2) n+1 0 3 v 2 + 4 (m 1 + m 2 − E) 1 −2(m 1 − E) 3v 4 + 4v 2 + 8 − 3m 2 v 4 + 4 2 1 4 (m 1 − E) 6v 8 + 18v 6 + 148v 4 + 96v 2 + 32 + 1 4 m 2 −v 8 − 14v 6 + 72v 4 + 16v 2 + 32 3 − v 2 4 (m 1 − E) 12v 8 + 144v 6 + 176v 4 + 352v 2 + 192 + v 2 4 m 2 v 8 − 34v 6 + 60v 4 − 184v 2 − 128 4 v 2 16 (m 1 − E) 22v 12 + 230v 10 + 576v 8 + 3240v 6 + 3456v 4 + 1984v 2 + 512 + v 2 16 m 2 −27v 12 − 156v 10 − 484v 8 + 1264v 6 + 864v 4 + 832v 2 + 512 B. Comparison with earlier approaches without spin Our EOB metric is first compared to previous approaches without spin. Letting a = 0 in the previous section, below our treatment is shown equivalent with that of refs. [19,132]. Particularly, the above results are related to ref. [19] by a gauge transformation of the post-geodesic Q term and a coordinate-shift of the scattering angle integral to incorporate differing EOB maps. Ref. [19] presents both Schwarzschild-like and isotropic EOB metrics. We will compare exclusively with the Schwarzschild type. Ref. [19] uses an EOB map differing from eq. (36) and (38) purely by equating angular momenta and not impact parameters L = L ef f ⇒ b = b ef f E M . (EOB map of ref. [19])(45) We have expressly indicated above that these are relations used in ref. [19] and continue to denote L, L ef f and b as defined by eq. (36) in everything following below. This EOB map can readily be converted to eq. (36) as both maps employ χ = χ ef f . Equation (13) therefore implies d dL ∞ Rm dR p R = d dL ef f ∞ rm dr p r = M E d dL ∞ rm dr p r ,(46) where the integral in R is that of ref. [19], and the integral in r that of the current paper. p R and p r are the corresponding canonical radial momenta. This equality allows the natural identification, applicable when both formalisms use the same ansatz (Schwarzschild-like) metric, R = M E r, p R = p r ,(47) amounting to a coordinate-shift of the scattering angle integral. This is exactly the identification made by ref. [132], there interpreted as a canonical transformation between effective test-particle momentum of ref. [19] and center of mass momentum of the full two-body system. Particularly, eq. (54) and (51) in ref. [132] is exactly eq. (47) above. We show below that the coordinate transformation of eq. (47) yields a transformed momentum p R related to that found in ref. [19] by a gauge-transformation in Q. The formalisms are thus equivalent. To see this, consider the specific forms of p r and p R . One finds p r = p 2 ef f − L 2 ef f r 2 − U (r) = p 2 ef f − L 2 R 2 − U (r),(48a) with Canonical momentum p R is calculated very similarly in ref. [19], from a modified Hamilton-Jacobi equation, U (r) = 2GM R 3 L 2 + µ 2 R 2 v 2 + 1 v 2 − 1 + G 2 M 2 L 2 v 2 − 1 3M v 2 + 4 + v 2 − 4 E + µ 2 R 2 3M v 4 + 5v 2 + 4 + v 4 + v 2 + 4 E R 4 (v 2 − 1) (v 2 + 2) E + O(G 3 ),(48b)g µν D p µ p ν = −µ 2 − Q,(49) with post-geodesic correction Q. Subscript D indicates quantities from ref. [19]. It was shown in refs. [18,53] that, i) the EOB metric g D µν can be chosen to be a Schwarzschild metric with mass M = m 1 + m 2 , ii) Q starts at G 2 , and iii) the scattering angle is invariant under certain gauge-like transformations of Q. Furthermore, it was shown a suitable gauge could be chosen, such that Q only depends on R and quantities relating to energy. We therefore write Q = ∞ n=2 Q n (R)G n(50) without loss of generality. Truncating at O(G 2 ), one finds p R = p 2 ef f − L 2 R 2 − U (R),(51a)U (R) = 2GM L 2 + µ 2 R 2 (v 2 +1) v 2 −1 R 3 + G 2 Q 2 + G 2 M 2 4L 2 + R 2 4µ 2 (v 2 +2) v 2 −1 R 4 + O(G 3 ),(51b) where U (R) denotes the corresponding potential, suitably identified as g D µν is already in normal form. Note in particular that, in the potential, Q 2 term appears isolated from other metric-dependent terms. This becomes important in a moment. Q may be found from our EOB formalism by imposing eq. (47) and inserting eqs. (48) and (51). One finds U (r) = U (R)(52) to all orders in G. This equality is trivial up to 1PM as both EOB metrics are Schwarzschild-like. Q therefore starts at O(G 2 ) as expected. At 2PM the equality determines the lowest order coefficient of Q Q = Q M 2 G 2 + O(G 3 ) = 3 v 2 + 4 (M − E) L 2 v 2 − 1 + µ 2 R 2 v 2 + 1 R 2 (v 2 − 1) (v 2 + 2) E G 2 M 2 R 2 + O(G 3 ),(53) corresponding to translating the EOB metric of eq. (70) to a Finsler-type post-geodesic form directly comparable with ref. [18,19,53]. The obtained value of Q 2 is dependent on angular momentum, and thus definitely not in the gauge used in ref. [19]. Results for Q from refs. [18,19,53] instead yield, Q D 2 = 3µ 2 v 2 + 4 (M − E) 2 (v 2 − 1) E ,(54) which is different from eq. (53) by ∆Q ≡ Q D 2 − Q M 2 = − 3M 2 v 2 + 4 (M − E) 2L 2 v 2 − 1 + µ 2 R 2 v 2 2R 4 (v 2 − 1) (v 2 + 2) E .(55) However, Q M 2 may be related to Q D 2 by a gauge-transformation. After all, the scattering angle calculated with each is the same. Gauge-transformations may be introduced by considering, as in ref. [53], the scattering angle integral eq. (13). Plug in eq. (51) with unspecified Q 2 . Denoting by χ Q contributions to the scattering angle that come from Q, one finds up to O(G 2 ) χ Q = 1 p ef f d dL ∞ 0 du Q 2 (R)G 2 + O(G 3 ), R 2 = u 2 + L 2 /p 2 ef f = u 2 + b 2 M 2 E 2 ,(56) rewriting eq. (13) in terms of R, imposing in effect the EOB map of ref. [18]. Consider how the full scattering angle integral changes with Q M 2 compared to Q D 2 , keeping the Schwarzschild metric g D µν . By construction, the scattering angles calculated in either case are equal. One therefore concludes d dL ∞ 0 du Q D 2 (R) = d dL ∞ 0 du Q M 2 (R), R 2 = u 2 + b 2 M 2 E 2 .(57) Subtracting the LHS from the RHS yields d dL ∞ 0 du ∆Q = 0,(58) as confirmed by an explicit calculation. One may thus interpret Q M 2 as being related to Q D 2 by a gauge-like transformation which keeps the scattering angle invariant, Q D 2 = Q M 2 + d du G(R), R 2 = u 2 + b 2 M 2 E 2 ,(59) with contributions from G(R) vanishing in the integration limits of eq. (56). A very similar result was found in ref. [53], based on analogous considerations. Incidentally, an identical relation holds when imposing the EOB map of eq. (36), with the replacements b → b E M , R → r and u continuing to denote the integration parameter in eq. (13). For reference, G(R) inferred from eq. (55) is G(R) = 3G 2 µ 2 M 2 uv 2 v 2 + 4 (M − E) 2 (v 2 − 1) (v 2 + 2) E b 2 M 2 E 2 + u 2 , u 2 = R 2 − b 2 M 2 E 2 .(60) The gauge-transformation presented above is explicitly restricted to O(G). Similar arguments may be made at higher orders in G from a more complicated gauge-relation derived from multiple terms of eq. (13). Concluding, we may interpret our EOB metric without spin as a gauge-specific embedding of a Schwarzschild metric with post-geodesic Q contribution. We next compare our results with ref. [132], which is related to ref. [19] simply by the coordinate-shift of eq. (47). By means of the previous analysis, our result is therefore related to ref. [132] purely by the gauge-transformation in Q. Apparent from isotropic and Schwarzschild EOB constructions of ref. [19], this transformation cannot be interpreted as a simple coordinate-shift between Schwarzschild and isotropic metrics, eqs. (42) and (40). Such behavior is to be expected since the EOB metric is by no means unique -multiple distinct metrics, ie. not related by coordinate transformations, may encode full binary dynamics. To see this, try bringing the isotropic EOB metric of ref. [132] to a Schwarzschild form which satisfies g φφ = ρ 2 and g tt = −1/g ρρ . Denoting the transformed coordinates {t, ρ, φ}, and the transformed metricg µν , one does not recover eq. (42) with a = 0, nor even Schwarzschild structure. Instead g µν =   − (1 − A) 0 0 0 1 1− A 0 0 0 ρ 2   ,(61a) with A = 4α 1 G ρ + 4α 2 G 2 ρ 2 + O(G 3 ), A = 4α 1 G ρ + 8α 2 G 2 ρ 2 + O(G 3 ).(61b) A and A are unequal starting at O(G 2 ), breaking the Schwarzschild-like characteristics. α 1 and α 2 are given in eq. (41). We have used r = ρ − 2α 1 G − α 2 1 + 2α 2 G 2 /ρ + O(G 3 )(61c) and neglected O(G 3 ) terms in the metric. This is warranted as α(r) is only specified up to O(G 2 ) anyway. A resummation in G is presented merely to highlight deviations from the Schwarzschild form. As a consequence, the gauge-transformation in Q of eq. (59), does not correspond to a coordinate-transformation of the EOB metric. C. Comparison with earlier approaches with spin We now turn to consistency checks with earlier approaches including spin. Observations are similar to those without spin. At 1PM, our EOB metric is compared with that of Justin Vines ref. [44], section III,b. In ref. [44] the EOB map of ref. [19] is used, introducing a Kerr-like EOB metric with mass M = m 1 + m 2 and spinã = M E a. By an analysis identical to that without spin, one may propose the connection of eq. (46) yielding eq. (47) R = M E r, p R (R,ã, L) = p r (r, a, L ef f ).(62) This identification is indeed correct, as an explicit calculations of p r and p R shows. We remind the reader that L and L ef f are defined in eq. (36). The simplicity of this coordinate connection is purely due to the Kerr-like structure of both metrics. Our result is thus consistent with ref. [44]. VI. SPINNING BINARY EOB METRICS AND NEWMAN-JANIS ALGORITHM Finally, we consider the Newman-Janis Algorithm (NJA), a remarkable procedure for introducing spin to a nonspinning, so-called seed metric. Although presented first as an ad-hoc observation by Newman and Janis in 1965 [139], its uniqueness has subsequently been investigated [140]. Beyond useful only for its original application in obtaining the Kerr metric from a Schwarzschild spacetime, it successfully produces also the Kerr-Newman metric from a Reissner Nordström seed. As such it is interesting to explore the application of the NJA to EOB metrics. Namely, does one recover an EOB metric for aligned spinning Kerr black holes by applying the NJA to the EOB metric of two Schwarzschild black holes? We consider only equatorial, aligned spin scattering throughout. Adopting the EOB formalism above with a scalar effective test-particle of mass µ = m 1 m 2 M ,(63) and EOB maps given by eqs. (36) and (38), the requirement for such a metric is that it reproduces the aligned spinning binary scattering angles, eq. (2), order by order in G. We will concern ourselves with 1PM and 2PM scattering angles below. We choose to construct our NJA metric from the non-spinning binary Schwarzschild EOB metric of eqs. (40) and (41). We employ the NJA in the form discussed by refs. [139,140], and restrict ourselves purely to the equatorial plane (θ = π/2). Only the final result is presented in the main text, and specifics of the procedure are included in appendix A. A few remarks are worth noting. First, notice that eq. (40) is symmetric in binary masses m 1 and m 2 . The NJA-transformed metric will naturally preserve this symmetry, and we therefore expect the metric to describe some equal-in-spin binary, for which the scattering angle of eq. (2) is symmetric in masses. Furthermore, note we are only interested in scattering angles up to O(G 2 ), and therefore we take as the seed metric the O(G 2 ) accurate Schwarzschild-form of eq. (40) presented in eq. (61a). In the application of the NJA, the transformed tetrads defining the transformed metric are similarly truncated to O(G 2 ) (see step 4 in appendix A). The NJA transformed metric of eq. (61a), following the original procedure of refs. [139,140], becomes g (NJA) µν =      − r 2 (r(r−4α1G)−8α2G 2 ) (r 2 −2α2G 2 ) 2 0 − ar 2 (4α1Gr+6α2G 2 ) (r 2 −2α2G 2 ) 2 0 r 2 a 2 +r(r−4α1G)−8α2G 2 0 − ar 2 (4α1Gr+6α2G 2 ) (r 2 −2α2G 2 ) 2 0 r 2 a 2 (r(4α1G+r)+4α2G 2 ) (r 2 −2α2G 2 ) 2 + 1      ,(64) which has Boyer-Lidquist structure reminiscent of a Kerr-Newman metric. Specifically, eq. (64) recovers the equatorial Kerr-Newman metric by setting α 1 = M 2 − Q 2 4r and α 2 = 0, for which eq. (61a) becomes the Reissner-Nordström metric. We now discuss EOB interpretation of such a metric, based on scattering data. The equatorial scattering angle of a scalar (effective) test-particle in g (NJA) µν is readily calculated by eq. (13), χ NJA EOB = 2GE −2av + bv 2 + b v 2 (b 2 − a 2 ) + 3πG 2 M v 2 + 4 E 4b 2 v 2 − a πG 2 E 9M v 2 + 4 v 2 + 15v 4 + 4v 2 + 16 E 4 (b 3 v 3 (v 2 + 1)) + O(a 2 ) + O(G 3 ),(65) after coordinate transformation r 2 → ρ 2 = r 2 + a 2 retrieving the normal form. M = m 1 + m 2 denotes the total mass of the binary. To obtain EOB interpretation, this should reproduce a full binary scattering angle. We compare explicitly with the aligned Kerr binary from ref. [89], adopting the EOB map of eqs. (36) and (38). The 2PM angle above has been truncated at O(a) to make such comparisons. Further analysis could equally be carried out at higher orders in a, however eq. (2) is not yet confirmed to represent scattering of black holes at these orders. Consider all configurations (a 1 , a 2 ) of eq. (2) conceivably described by the NJA. The 1PM scattering angle matches exactly that of eq. (2) with configuration a 1 + a 2 = a. Spins a 1 and a 2 are further restricted by going to 2PM. Equation (65) is symmetric in masses m 1 and m 2 . This symmetry is only recovered in equation (2) if a 1 = a 2 = a/2. The corresponding full binary scattering angle from eq. (2) is χ (a/2,a/2) = 2GE −2av + bv 2 + b v 2 (b 2 − a 2 ) + 3πG 2 M v 2 + 4 E 4b 2 v 2 − 7a πG 2 M 3v 2 + 2 E 4b 3 v 3 + O(a 2 ) + O(G 3 ),(66) which is the only binary Kerr scattering angle conceivably matched by eq. (65). However comparing eq. (65) and eq. (66), they do not match up. In the present setting, based on the non-spinning EOB metric of eq. (40) and eq. (41), the NJA thus fails to produce an EOB metric for aligned spinning binaries. Although the present NJA interpretation fails, we have only considered scalar test-particles. One could equally consider scattering of a spinning test-particle on the metric of eq. (64). Whether this combination has EOB interpretation, and produces correct two-body scattering angles, is not pursued here. VII. CONCLUSION Using the method of ref. [20], eq. (11), scattering angles of systems involving both electromagnetic and gravitational interactions have been calculated. As a specific case-study, post-Minkowskian scattering angles in the Kerr-Newman metric have been computed for charged test-particles (table II and table III B). Purely electromagnetic situations are also treatable -the well known Coulomb scattering angle is readily obtained in resummed form in the weak field limit (table III A). Charged test-particle scattering angles in Kerr-Newman show definite structure. At order O(G n ) and O(e j ), χ = ∞ n=0 ∞ j=0 n k=0 χ (k) n,j , χ (k) n ∼ π (n+j+k+1)mod 2 G n e j Q 2k+j M n−k m j γ j v 2(j+n) (b 2 − a 2 ) (3(n+j)+k−1)/2 f n,j,k (v),(67) where f n,j,k (v) is a polynomial in v. For j = 0 and odd (even) n, f n,j,k (v) contains integer (fractional) orders of a and b. When j = 0 only integer orders of a and b are present, with the structure f n,j,k (v) ∼ 2(n+j) p=0 a p b 2(n+j)−pf n,j,k (v),(68) wheref n,j,k is some polynomial in v, independent of a and b. The flexibility of the scattering angle formula of ref. [20] is aptly suited for EOB formalisms with planar orbits. An EOB metric describing full binary motion may be constructed by explicitly matching scattering angles and adopting the EOB map of ref. [132] (eq. (36) of the present paper). An ansatz, g (ef f ) µν =   − (1 − κ(r)) 0 −aκ(r) 0 r 2 a 2 +r 2 (1−κ(r)) 0 −aκ(r) 0 a 2 (1 + κ(r)) + r 2   , κ(r) = ∞ n=1 κ n G n r n ,(69) for the EOB metric may be constructed from the Kerr metric in Boyer-Lindquist coordinates by replacing 2GM/r → κ(r). The result is a resummed metric in spin parameter a and Newtons gravitational constant G. κ n may be determined by matching the resulting test-particle scattering angle with that of a Kerr-Schwarzschild binary [89], order by order in G and a. Our current treatment extends to 2PM and O(a), yielding κ 1 =2E, κ 2 = 3 v 2 + 4 E(m 1 + m 2 − E) v 2 + 2 + a − E 6m 1 v 4 + 8m 1 v 2 + 16m 1 + 3m 2 v 4 + 12m 2 − 6v 4 E − 8v 2 E − 16E bv (v 2 + 2) 2 + O(a 2 )(70) by requiring that the metric reduces to Kerr in the m 2 → 0 limit. By identical comparisons with higher orders of a in ref. [89], κ 2 is presented up to O(a 4 ) in table I. We see no obstruction in continuing the treatment to further orders in G and a. In the non-spinning limit, our EOB formalism is related to previous approaches of refs. [19,53,132] through a gauge-transformation of the post-geodesic Finsler-type Q term. Including spin, at 1PM, our formalism is equivalent to that of ref. [44] after a coordinate-reparametrisation identical to the connection of ref. [132] with ref. [19]. Furthermore, for spinning binaries, a subtlety with b dependence of the metric was noted. Corresponding to angular momentum dependence, the EOB metric may readily depend on it, however requires extra care when evaluating test-particle scattering angles. In particular with the current ansatz metric, the scattering angle becomes a function of linear derivatives of κ n with respect to b. Supplementary to our EOB analysis, the Newman-Janis Algorithm was explored in its application to the nonspinning EOB metric of ref. [132], see eq. (40). The complexification technique and coordinate-transformations involved are those originally introduced by Newman and Janis [139], and the EOB map of eqs. (36) and (38) is assumed. Applying the NJA to eq. (40), the result is g (NJA) µν =      − r 2 (r(r−4α1G)−8α2G 2 ) (r 2 −2α2G 2 ) 2 0 − ar 2 (4α1Gr+6α2G 2 ) (r 2 −2α2G 2 ) 2 0 r 2 a 2 +r(r−4α1G)−8α2G 2 0 − ar 2 (4α1Gr+6α2G 2 ) (r 2 −2α2G 2 ) 2 0 r 2 a 2 (r(4α1G+r)+4α2G 2 ) (r 2 −2α2G 2 ) 2 + 1      ,(71) derived with O(G 2 )-accurate manipulations. Scalar test-particle scattering angles in this metric are computed, and compared to different combinations of aligned spin (a 1 , a 2 ) of the full binary result [89]. No spin map is a priori assumed. At 1PM we find that angles indeed match, due to natural Kerr-like structure in both cases. At 2PM, however, no spin configurations are possible for which the angles are equal. It is therefore, within the confines of the current construction, not possible to interpret the NJA transformed metric above as an EOB metric of aligned binary Kerr black holes. This analysis is based on a scalar effective test-particle. Alternatively, the same analysis including spin on the test-particle might conceivably have EOB interpretation. This is left to future work. (n, k) χ (k) n / G n Q 2k M n−k v 2n (b 2 −a 2 ) (3n+k−1)/2 (1,0) 2 −2av + bv 2 + b (1,1) π/(2a 2 ) 2a 3 v − a 2 −v 2 √ b 2 − a 2 + 2bv 2 + b + b 2 v 2 b − √ b 2 − a 2 (2,0) π/(2a 2 ) − 4a 5 v − 4a 3 b 2 v 3v 2 + 2 + 2a 2 b 2 v 2 b 2v 2 + 3 − v 2 √ b 2 − a 2 + b 4 v 4 √ b 2 − a 2 − b + a 4 v 4 √ b 2 − a 2 + 3b 4v 2 + 1 (2,1) (8a 3 + 24ab 2 )(1 + v 2 )v + (−6a 2 b − 2b 3 )(1 + 6v 2 + v 4 ) (2,2) −(3π)/(16a 4 ) 4a 7 2v 3 + v + 4a 5 b 2 v 3v 2 + 4 + 2b 6 v 4 b − √ b 2 − a 2 + a 2 b 4 v 4 6 √ b 2 − a 2 − 7b − a 6 b 8 v 2 + 3 v 2 + 3 − 2v 4 √ b 2 − a 2 + 2a 4 b 2 b 4v 4 − 3v 2 − 1 − 3v 4 √ b 2 − a 2 (3,0) 1 3 4a 5 v 3v 4 − 10v 2 − 9 + 6a 4 b −v 6 + 15v 4 + 45v 2 + 5 − 8a 3 b 2 v 15v 4 + 70v 2 + 27 + 4a 2 b 3 11v 6 + 135v 4 + 105v 2 + 5 − 36ab 4 v 5 v 2 + 2 v 2 + 1 + 2b 5 5v 2 v 4 + 9v 2 + 3 − 1 (3,1) (3π)/(8a 4 ) 2a 9 v 20v 2 + 9 + 12a 7 b 2 v 10v 4 + 35v 2 + 12 + 6a 5 b 4 v 15v 4 + 40v 2 + 8 + 2b 8 v 6 b − √ b 2 − a 2 + a 2 b 6 v 6 8 √ b 2 − a 2 − 9b − a 8 2v 6 √ b 2 − a 2 + 15b 8v 4 + 12v 2 + 1 − a 6 b 2 5b 8v 6 + 72v 4 + 63v 2 + 4 − 8v 6 √ b 2 − a 2 + 3a 4 b 4 v 2 b 4v 4 − 15v 2 − 10 − 4v 4 √ b 2 − a 2 (3,2) (−4a 5 − 40a 3 b 2 − 20ab 4 )(3 + 10v 2 + 3v 4 )v + (10a 4 b + 20a 2 b 3 + 2b 5 )(1 + 15v 2 + 15v 4 + v 6 ) (3,3) (5π)/(128a 6 ) 2a 11 v 24v 4 + 40v 2 + 9 + 4a 9 b 2 v 60v 4 + 205v 2 + 54 + 18a 7 b 4 v 5v 4 + 20v 2 + 8 + 8b 10 v 6 b − √ b 2 − a 2 + 4a 2 b 8 v 6 10 √ b 2 − a 2 − 11b − a 10 3b 16v 6 + 120v 4 + 90v 2 + 5 − 8v 6 √ b 2 − a 2 + 5a 8 b 2 b 8v 6 − 108v 4 − 123v 2 − 8 − 8v 6 √ b 2 − a 2 − a 6 b 4 b 118v 6 + 45v 4 + 60v 2 + 8 − 80v 6 √ b 2 − a 2 + a 4 b 6 v 6 99b − 80 √ b 2 − a 2 (4,0) (3π)/(16a 4 ) − 8a 11 v 14v 2 + 5 − 8a 9 b 2 v 140v 4 + 273v 2 + 60 − 8a 7 b 4 v 70v 6 + 455v 4 + 392v 2 + 40 − 56a 5 b 6 v 3 5 v 2 + 4 v 2 + 8 + 2b 10 v 8 √ b 2 − a 2 − b + a 2 b 8 v 8 11b − 10 √ b 2 − a 2 + a 10 35b 16 v 4 + v 2 + 1 − 2v 8 √ b 2 − a 2 + 10a 8 b 2 v 8 √ b 2 − a 2 + 7b 16v 6 + 56v 4 + 26v 2 + 1 + 2a 6 b 4 v 2 7b 8v 6 + 120v 4 + 195v 2 + 40 − 10v 6 √ b 2 − a 2 + 4a 4 b 6 v 4 5v 4 √ b 2 − a 2 + b −4v 4 + 35v 2 + 35 (4,1) 4 3 8a 7 v −v 6 + 7v 4 + 21v 2 + 5 + 5a 6 b v 8 − 28v 6 − 210v 4 − 140v 2 − 7 + 24a 5 b 2 v 7 v 4 + 13v 2 + 19 v 2 + 25 − 5a 4 b 3 13v 8 + 420v 6 + 1190v 4 + 532v 2 + 21 + 40a 3 b 4 v 7 3v 4 + 19v 2 + 17 v 2 + 15 − 3a 2 b 5 31v 8 + 700v 6 + 1330v 4 + 364v 2 + 7 + 40ab 6 v 7 v 4 + 5v 2 + 3 v 2 + 1 + b 7 1 − 7v 2 v 6 + 20v 4 + 30v 2 + 4 (4,2) (15π)/(128a 6 ) − 8a 13 v 56v 4 + 84v 2 + 15 − 8a 11 b 2 v 7 24v 4 + 200v 2 + 237 v 2 + 270 − 8a 9 b 4 v 7 60v 4 + 395v 2 + 372 v 2 + 360 − 24a 7 b 6 v 7 5 v 2 + 6 v 2 + 24 v 2 + 16 + 8b 12 v 8 √ b 2 − a 2 − b + 4a 2 b 10 v 8 13b − 12 √ b 2 − a 2 + a 12 8v 8 √ b 2 − a 2 + 21b 8 8v 4 + 30v 2 + 15 v 2 + 5 + 4a 10 b 2 7b 16v 8 + 360v 6 + 930v 4 + 395v 2 + 15 − 12v 8 √ b 2 − a 2 + 2a 8 b 4 60v 8 √ b 2 − a 2 + 7b 8v 8 + 540v 6 + 1185v 4 + 400v 2 + 12 + 4a 6 b 6 v 2 b 58v 6 + 105v 4 + 210v 2 + 56 − 40v 6 is the O(G n ) contribution to the full scattering angle, of which the Q 2k -proportional part is taken. k ranges from 0 to n owing to the GQ 2 charge-dependence of the metric, eq. (14). √ b 2 − a 2 + a 4 b 8 v 8 120 √ b 2 − a 2 − 143b Null coordinates may readily be identified from eq. (61a), yielding the line element (step 1 of the NJA) ds 2 = −(1 − A)du 2 − 2 1 − A 1 − A dudρ + ρ 2 dΩ 2 ,(79a) where A = 4α 1 G ρ + 4α 2 G 2 ρ 2 + O(G 3 ) and A = 4α 1 G ρ + 8α 2 G 2 ρ 2 + O(G 3 ).(79b) As mentioned in ref. [140], the same analysis with a Reissner-Nordström metric yields instead A = A = 2GM/ρ − GQ 2 /ρ 2 , (Reissner Nordström metric)(80) simplifying the dudρ component. This difference follows from the observation of eq. (61a); eq. (40) cannot be brought to exact Schwarzschild form, meaning g ρρ = −1/g tt . Were this the case, the NJA-transformed metric would simply have Kerr-Newman structure. The line-element of eq. (79a) is written in terms of null-tetrads (step 2 of the NJA), which may easily by identified by comparing eq. (79a) with eq. (72) e Φ(ρ) = (1 − A) 1/2 , e λ(ρ) = (1 − A) −1/2 , (81a) l µ = δ µ 1 , n µ = 1 − A 1 − A δ µ 0 − 1 2 (1 − A)δ µ 1 , m µ = 1 √ 2ρ (δ µ 2 + i sin θ δ µ 3 ),(81b) and coordinates are complexified (step 3 of the NJA) by replacing 1/ρ → 1 2 (1/r ′ + 1/r ′ ) and 1/ρ 2 → 1/(r ′r′ ). A coordinate transformation x µ = x ′µ − ia cos θ(δ µ 0 − δ µ 1 ) is now performed (step 4 of the NJA). Reuse notation by denoting x µ = {u, r, θ, φ}. The tetrads transform according to eq. (77). In particular, expressions are reduced by expandingZ µ a to O(G 2 ). The transformed metric in the equatorial (θ = π/2) plane reads g (NJA) µν = −l µñν −ñ µlν +m µmν +m µmν =        −(1 − A) − √ 1−A √ 1− A −a √ 1−A √ 1− A + A − 1 − √ 1−A √ 1− A 0 a √ 1−A √ 1− A −a √ 1−A √ 1− A + A − 1 a √ 1−A √ 1− A r 2 − a 2 − 2 √ 1−A √ 1− A − A + 1        (83) =       − r 2 (r(r−4α1G)−8α2G 2 ) (r 2 −2α2G 2 ) 2 − r 2 r 2 −2α2G 2 − aGr 2 (4α1r+6α2G) (r 2 −2α2G 2 ) 2 − r 2 r 2 −2α2G 2 0 − ar 2 2α2G 2 −r 2 − aGr 2 (4α1r+6α2G) (r 2 −2α2G 2 ) 2 − ar 2 2α2G 2 −r 2 −r 2 − a 2 (r(4α1G+r)+4α2G 2 ) (r 2 −2α2G 2 ) 2 − 1       , where the first expression uses all-order expressions ofZ µ a and the second uses their expansion to 2PM. Last (step 5 of the NJA), g tr terms may be removed by the coordinate transformation u → t and φ → ψ dt = du + D 01 (r)dr, dψ = dφ + D 31 (r)dr, D 01 = a 2 − 2α 2 G 2 + r 2 a 2 − 4α 1 Gr − 8α 2 G 2 + r 2 , D 31 = a a 2 − 4α 1 Gr − 8α 2 G 2 + r 2 (84) producing the final result g (NJA) µν =       − r 2 (r(r−4α1G)−8α2G 2 ) (r 2 −2α2G 2 ) 2 0 − ar 2 (4α1Gr+6α2G 2 ) (r 2 −2α2G 2 ) 2 0 r 2 a 2 +r(r−4α1G)−8α2G 2 0 − ar 2 (4α1Gr+6α2G 2 ) (r 2 −2α2G 2 ) 2 0 r 2 a 2 (r(4α1G+r)+4α2G 2 ) (r 2 −2α2G 2 ) 2 + 1       ,(85) which is the NJA metric from the seed of eq. (61a), presented in eq. (64) of the main text. j are tabulated in table III, B) up to O(G 3 ) and O(e 3 ).We again find definite structure is even in a but odd in b. The exact structure of polynomials f depends on n, j and k. Furthermore, f (v) and f (ab) share numerical coefficients of by inserting eq. (42) in eq. (4), subsequently rewriting in terms of R = M E r and L ef f = E M L. (4, 3 ) 3(16a 7 + 336a 5 b 2 + 560a 3 b 4 + 112ab 6 )(1 + 7v 2 + 7v 4 + v 6 )v + (−14a 6 b − 70a 4 b 3 − 42a 2 b 5 − 2b 7 )(1 + 28v 2 + 70v 4 + 28v 6 + v 8 ) (4,4) (35π)/(2048a 8 ) − 8a 15 v 2v 2 + 5 8 v 4 + v 2 + 1 − 24a 13 b 2 v 7 8v 4 + 52v 2 + 45 v 2 + 40 − 120a 11 b 4 v 7 2v 4 + 15v 2 + 16 v 2 + 16 − 8a 9 b 6 v 7 5 v 2 + 8 v 2 + 48 v 2 + 64 + 16b 14 v 8 √ b 2 − a 2 − b + 8a 2 b 12 v 8 15b − 14 √ b 2 − a 2 + a 14 b 32 4v 6 + 56v 4 + 105v 2 + 35 v 2 + 35 − 16v 8 √ b 2 − a 2 + 14a 12 b 2 8v 8 √ b 2 − a 2 + 15b 32v 6 + 80v 4 + 30v 2 + 1 + 42a 10 b 4 b 16v 8 + 80v 6 + 225v 4 + 104v 2 + 4 − 8v 8 √ b 2 − a 2 + 4a 8 b 6 140v 8 √ b 2 − a 2 + b −200v 8 + 35v 6 + 105v 4 + 56v 2 + 4 + 5a 6 b 8 v 8 143b − 112 √ b 2 − a 2 + 6a 4 b 10 v 8 56 √ b 2 − a 2 − 65b TABLE I . IParameter κ2 from the EOB metric of eq. (42) computed up to O(a 4 ) by matching the scattering angle of the amplitude calculations from [89], provided in eq. (39) above. Equation (39) only matches black hole scattering at O(G 2 TABLE II . IIScattering angle of a scalar test-particle in the Kerr-Newman metric, orbiting in the equatorial plane. χ(k) n A) Coulomb scatteringQe v √ J 2 −Q 2 e 2 − π B) Kerr-Newman equatorial scattering of charged test-particle (n, j, k) χ(k)n,j / G n e j Q 2k+j M n−k m j v 2(j+n) (b 2 −a 2 ) (3(n+j)+k−1)/2 For completeness, we first present the Newman-Janis algorithm with a review of ref.[140], before discussing its specific application to eq. (61a). The Newman-Janis Algorithm is a five-step procedure, proceeding as follows;1. Consider a given spherically symmetric seed metric to which spin should be endowed. Write it in advanced null coordinates, genericallyspecifying functions Φ(r) and λ(r) on a case-to-case basis.2. Invert the metric and express it in terms of a null tetrad of vectors {l µ , n µ , m µ },where m µ can be complex valued, andm µ is its complex conjugate, such that the metric is real-valued. Furthermore l, m, n obey l µ l µ = m µ m µ = n µ n µ = 0, l µ n µ = −m µm µ = 1, l µ m µ = n µ m µ = 0.One may abbreviate notation by introducing Z µ a = (l µ , n µ , m µ ,m µ ). The metric of eq. (72) has3. Complexify coordinates by letting x ρ → x ′ρ = x ρ + iy ρ ∈ C and Z µ a → Z ′µ a . The transformation only requires g ′ µν ∈ R and Z ′ = Z when x ′ =x ′ , a bar denoting complex conjugation. Multiple complexification transformations obey this requirement, and as such the Newman-Janis Algorithm is ambiguous. The transformation adopted here is that originally introduced by Newman and Janis,Note in particular that 1/r 2 terms transform differently from 1/r terms. Justification of this fact was given in ref.[140]. However, as will be demonstrated, this ambiguity vanishes in the equatorial plane.4. Perform a complex coordinate transformation x ′µ = x µ + iy µ (x) whereby Z ′µ a transforms asThe new coordinates x µ ∈ {u, r, θ, φ} are real-valued. 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[ "Farook Rahaman ", "P K F Kuhfittig ", "B C Bhui ", "Masiur Rahaman ", "Saibal Ray ", "U F Mondal ", "\nDepartment of Mathematics\nDepartment of Mathematics\nMilwaukee School of Engineering\nDepartment of Mathematics\nJadavpur University\n700 032, 53202-3109Kolkata, MilwaukeeWest Bengal, WisconsinIndia, USA\n", "\nDepartment of Physics\nGovernment College of Engineering and Ceramic Technology\nDepartment of Mathematics\nMeghnad Saha Institute of Technology\n700, 010Kolkata-700150, KolkataWest BengalIndia, India\n", "\nBehala College\n700060Parnasree, KolkataIndia\n" ]
[ "Department of Mathematics\nDepartment of Mathematics\nMilwaukee School of Engineering\nDepartment of Mathematics\nJadavpur University\n700 032, 53202-3109Kolkata, MilwaukeeWest Bengal, WisconsinIndia, USA", "Department of Physics\nGovernment College of Engineering and Ceramic Technology\nDepartment of Mathematics\nMeghnad Saha Institute of Technology\n700, 010Kolkata-700150, KolkataWest BengalIndia, India", "Behala College\n700060Parnasree, KolkataIndia" ]
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In this paper a Bañados, Teitelboim and Zanelli (BTZ) [1] black hole is constructed from an exact solution of the Einstein field equations in a (2+1)-dimensional anti-de Sitter spacetime in the context of noncommutative geometry. The BTZ black hole turns out to have two horizons, no horizon or a single horizon corresponding to a minimal mass. Certain thermodynamical properties are investigated, including Hawking temperature, entropy and heat capacity. Also discussed is the geodesic structure of BTZ black holes for both massless and massive particles. In particular, it is shown that bound orbits for test particles are possible.
10.1103/physrevd.87.084014
[ "https://arxiv.org/pdf/1301.4217v3.pdf" ]
119,295,242
1301.4217
a0786400bf7d2de3c293cf60990317c073e5fc7b
13 Mar 2013 (Dated: December 11, 2013) Farook Rahaman P K F Kuhfittig B C Bhui Masiur Rahaman Saibal Ray U F Mondal Department of Mathematics Department of Mathematics Milwaukee School of Engineering Department of Mathematics Jadavpur University 700 032, 53202-3109Kolkata, MilwaukeeWest Bengal, WisconsinIndia, USA Department of Physics Government College of Engineering and Ceramic Technology Department of Mathematics Meghnad Saha Institute of Technology 700, 010Kolkata-700150, KolkataWest BengalIndia, India Behala College 700060Parnasree, KolkataIndia 13 Mar 2013 (Dated: December 11, 2013)arXiv:1301.4217v3 [gr-qc] BTZ black holes inspired by noncommutative geometry PACS numbers: 04.40.Nr, 04.20.Jb, 04.20.Dw In this paper a Bañados, Teitelboim and Zanelli (BTZ) [1] black hole is constructed from an exact solution of the Einstein field equations in a (2+1)-dimensional anti-de Sitter spacetime in the context of noncommutative geometry. The BTZ black hole turns out to have two horizons, no horizon or a single horizon corresponding to a minimal mass. Certain thermodynamical properties are investigated, including Hawking temperature, entropy and heat capacity. Also discussed is the geodesic structure of BTZ black holes for both massless and massive particles. In particular, it is shown that bound orbits for test particles are possible. Recent years have seen rapid advances in the applications of noncommutative geometry, an outgrowth of string theory. The approach is based on the realization that coordinates may become noncommuting operators on a Dbrane [2][3][4][5]. The result is a discretization of spacetime due to the commutator [x µ , x ν ] = iθ µν , where θ µν is an antisymmetric matrix. It is similar to the way that the Planck constant discretizes phase space [2]. The noncommutativity eliminates point-like structures and replaces them with smeared objects. The noncommutative geometry is an intrinsic property of spacetime and does not depend on particular features such as curvature. A number of studies inspired by noncommutative geometry can be found in the literature. In one of the earlier studies, Garattini and Lobo [6] obtained exact wormhole solutions that were then analyzed in semiclassical gravity. In a subsequent study [7] they found an exact gravastar solution and explored its physical characteristics. Rahaman et al. [8], discussing galactic rotation curves, concluded that a noncommutativegeometry background is sufficient for producing stable * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: bikas˙[email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] * * Electronic address: [email protected] circular orbits without the need for dark matter. Kuhfittig [9] found that a special class of thin-shell wormholes that are unstable in classical general relativity exhibit small regions of stability in noncommutative geometry. In another study by Rahaman et al. [10] on higherdimensional wormholes, it is shown that wormhole solutions exist in the usual four, as well as in five dimensions, but they do not exist in higher-dimensional spacetimes. In a more recent study, Radinschi et al. [11] calculated the energy-momentum for a noncommuting radiating Schwarzschild black hole in order to obtain the expression for energy. Common to all these studies is that the effect of the smearing is mathematically implemented by using a Gaussian distribution of minimal length √ θ instead of the Dirac delta function. Interest in (2 + 1)-dimensional gravity has increased in recent years due to the discovery of various aspects of black-hole solutions. Some general works in this line are: quasinormal modes of charged dilaton black holes in (2 + 1)-dimensional solutions in low-energy string theory with asymptotically anti-de Sitter spacetimes [12], branes with naked singularities, analogous to linear or planar defects in crystals and showing that zero-branes in AdS spacetimes are "negative mass black holes" [13], Hawking radiation from covariant anomalies in (2 + 1)dimensional black holes [14] and so on. On the other hand, specialized investigations have been carried out by Rahaman et al. [15] on gravastars in (2 + 1) anti-de Sitter spacetimes with charge as an alternative to charged black holes. Also, nonstatic charged BTZ-like black holes in (N + 1)-dimensions have been considered by Ghosh [16], which in the static limit, for N = 2, reduces to (2 + 1) BTZ black hole solutions. There are also charged, regular extensions of the BTZ black hole solutions available in the literature by employing nonlinear Born-Infeld electrodynamics to eliminate the inner singularity [17]. There are several proposals in the literature for constructing noncommutative black holes [18][19][20][21]. However, following Nicolini et al [3], we construct a BTZ black-hole solution from the Einstein field equations in (2 + 1)-dimensional anti-de Sitter spacetime, given a noncommutative-geometry background. This is followed by a discussion of the black hole's thermodynamical properties such as Hawking temperature, entropy and heat capacity, as well as the geodesic structure for both massless and massive particles. Some of the thermodynamical properties are similar to those obtained by Liang and Liu [22] who used a Lorentzian smeared mass distribution instead of a Gaussian one in AdS 3 spacetime. One consequence of this is that in the limit, r/ √ θ → ∞, the solution reduces to a rotating BTZ black hole. Returning to a Gaussian framework, rotating black holes in (2+1) dimensions are also discussed in Ref. [23] by Tejeiro and Larranaga. They [24] also describe charged black holes and compare them to charged BTZ black holes. II. THE INTERIOR SPACETIME Let us write the line element describing the interior spacetime of a static spherically symmetric distribution of matter in (2 + 1) dimensions in the following form: ds 2 = −f (r)dt 2 + [f (r)] −1 dr 2 + r 2 dφ 2 ,(1) where f (r) is denoted by e 2ν(r) and [f (r)] −1 by e 2µ(r) . We take the matter distribution to be anisotropic in nature and therefore choose the most general energymomentum tensor in the form T ij = (ρ + p t )u i u j + p t g ij + (p r − p t )χ i χ j ,(2) where ρ, p r and p t represent the energy density, radial pressure and tangential pressure, respectively. Also, χ i = e −µ(r) δ i r is a unit four vector along the radial direction and u i is the 4-velocity of the fluid. The Einstein field equations with cosmological constant (Λ < 0), together with the general energymomentum tensor given in Eq. (2), yield (letting G = c = 1) 2πρ + Λ = µ ′ e −2µ r ,(3)2πp r − Λ = ν ′ e −2µ r ,(4)2πp t − Λ = e −2µ ν ′2 + ν ′′ − ν ′ µ ′ .(5) We have, in addition, the conservation equation in (2+1) dimensions: (ρ + p r ) ν ′ + p ′ r + 1 r (p r − p t ) = 0.(6) In (2+1) dimensions, the maximally localized source of energy of the static and spherically symmetric distributions having a minimal spread Gaussian profile is taken as [4] ρ = M 4πθ exp − r 2 4θ .(7) Here M is the total mass of the source. Due to the uncertainty, it is diffused throughout a region of linear dimension √ θ. The vacuum Einstein field equations in (2 + 1) spacetime dimensions, with a negative cosmological constant, admit a black hole solution known as a BTZ solution [1]. For a BTZ black hole, we have g rr = g −1 tt . So to retain the structure, we require that p r = −ρ.(8) This ansatz is known in the literature as a 'ρ-vacuum' or 'vacuum equation of state' in connection with the 'zero point energy of quantum fluctuation' [25][26][27][28] where pressure is of a repulsive nature. With this equation, one can solve Eq. (6) to yield p t = M r 2 2θ − 1 4πθ exp − r 2 4θ .(9) Using the field equations, we get the following solution for the metric coefficients: e −2µ = e 2ν = −A + 2M exp − r 2 4θ − Λr 2 ,(10) where A is an integration constant. In the limit, r √ θ → ∞, so that Eq. (10) reduces to a BTZ black hole, where the integration constant A plays the role of the mass of the BTZ black hole, i.e., A = M . Observe that asymptotically far away, ρ = p r = p t = 0. To determine the mass distribution from Eq. (7), we use an approach similar to that in Refs. [4] and [6]: m(r) = M π (m−2)/2 γ m 2 , α 2 r 2M 2 ,(11) where α 2 = M 2 /θ and γ is the lower incomplete gamma function γ a b , x = x 0 u a/b e −u du u .(12) For a BTZ black hole,m = 2, and we obtain from α 2 r 2 /4M 2 = r 2 /4θ the expression for mass as m(r) = M r 2 /4θ 0 e −t dt = M 1 − exp − r 2 4θ .(13) The parameter α plays a critical role in determining the horizons, as we will see later on. At the origin, m(0) = 0, which is consistent with Eq. (11). Near the origin, the geometry is given by e −2µ = e 2ν = −A + 2M − 2M 4θ + Λ r 2 + O(r 4 ). One can identify this result with a BTZ black hole spacetime, where the total mass M and the noncommutative parameter θ combine to modify the cosmological constant, a point also made in Ref. [4]. This indicates that different mass particles experience different cosmological constants. We therefore conclude that our line element describes the geometry of a noncommutative-geometry inspired BTZ black hole. III. FEATURES OF THE BLACK HOLE In this section we study some of the effects of the noncommutative geometry on BTZ black holes. Let A = M in Eq. (10). Then the equation g tt (r h ) = 0 gives the event horizon(s): r 2 h = M Λ 2 exp − r 2 h 4θ − 1 .(14) Even though we cannot obtain a closed-form solution for r h in Eq. (14), we can readily write the mass M as a function of r h : M = Λr 2 h 2exp − r 2 h 4θ − 1 .(15) The existence of horizons and their radii can be found at the points where g tt cuts the r-axis, as shown in (iii) no horizon for α < 0.214. Fig. 1 shows that a noncommutative-geometry inspired BTZ black hole has two horizons and that the distance between the horizons will increase with an increasing black-hole mass. Fig. 1 also indicates that there is a minimal mass M 0 below which no black hole exists. Moreover, at the minimal mass M = M 0 , the two horizons coincide at the minimal horizon radius r 0 , which lies between the horizons. This r 0 is therefore the horizon radius of the extremal black hole. It can also be determined from the conditions f = 0 and df dr = 0, leading to the equation 2 exp r 2 0 4θ + r 2 0 /θ 1 − 2exp − r 2 0 4θ = 0.(16) Using the condition Λ √ θ = −0.02 in Fig. 1, we can obtain r 0 / √ θ = 2.59, showing that the two approaches are consistent. The minimal mass of the extremal black hole can be written in terms of the minimal radius r 0 , so that M 0 √ θ = r 2 0 θ (Λ √ θ) 2 exp − r 2 0 4θ − 1 = 0.214.(17) The variation of this factor with respect to the horizon radius is shown in Fig. 2. At this point let us also plot ρ √ θ from Eq. (7) for various values of M/ √ θ (Fig. 3). Similarly, Fig. 4 shows p t √ θ plotted against r/ √ θ from Eq. (9). It is worth noting that the radius of the extremal black hole is always less than the radius of the outer horizon. The result significant: if the initial mass of the black hole is M > M 0 , then it can radiate until the value M 0 is reached. It follows that evaporation of the black hole may indeed be occurring. Next, consider the Hawking temperature, which is given by T H = 1 4π dg tt dr −g tt g rr | r=r h = − r h 4π 2Λ + M θ exp − r 2 h 4θ .(18) The plot, Fig. 5, is obtained from T H = − r h / √ θ 4π 2Λ √ θ + M √ θ exp − r 2 h 4θ .(19) Eq. (19) and Fig. 5 show that the noncommutative geometry leads to the minimal horizon radius r 0 , since T H cannot be negative. This is exactly where dgtt dr = 0 in Eq. (10). Observe that the temperature sinks to absolute zero at r 0 . For r h √ θ >> 1, the Hawking temperature assumes the value T H ≈ − r h Λ 2π .(20) For completeness, let us also state the closely related surface gravity κ = 1 2 dg tt dr | r=r h = − r h 2 2Λ + M θ exp − r 2 h 4θ ,(21) as well as the Bekenstein-Hawking entropy (S) of the black hole. It is twice the perimeter L of the event horizon: As a final comment, the noncommutative geometry inspired BTZ black hole is stable if the heat capacity C is positive [22], where S = 2L = 4πr h .(22C = ∂M (r h ) ∂T (r h ) = ∂M (r h ) ∂r h 1 ∂T (r h ) ∂r h . The nature of the heat capacity is shown in Fig. 6. The plot shows that C vanishes at the extremal event horizon r 0 and becomes negative for r h √ θ < r0 √ θ , just as in the case of the Hawking temperature. So this region is definitely unphysical. On the other hand, for r h √ θ > r0 √ θ , C is positive, which implies that the BTZ black hole is stable. At this point we would like to comment on how the physical quantities such as temperature, entropy, etc., could be affected by the noncommutativity for small θ. We calculate the different physical quantities of the standard BTZ black hole plus perturbative terms. The BTZ black hole has the horizon located at r h = √ M l,(23) where Λ = − 1 l 2 to emphasize that Λ is negative. Equation (14) can be solved by iteration. The result is r h = √ M l 1 − 2 exp − M l 2 4θ 1 2 .(24) For small θ and r h √ θ >> 1, the above equation can be written as r h ≈ √ M l 1 − exp − M l 2 4θ .(25) Here the first term is the BTZ black hole horizon radius, while the second term is the θ correction. It now becomes apparent that the effect of noncommutativity is small, as expected, because spacetime should be a smooth classical manifold at large distances. The next step is to find the θ corrections of the Bekenstein-Hawking entropy (S) of the BTZ black hole: r 2 dφ dτ = p,(32) and dt dτ = E f (r) ,(33) where f (r) = −M + 2M exp − r 2 4θ − Λr 2 and the constants E and p are identified as the energy per unit mass and angular momentum, respectively. Here τ is the affine parameter and L is the Lagrangian having values 0 and −1, respectively, for massless and massive particles. From the geodesic Eq. (31) we can write 1 2 dr dτ 2 = 1 2 E 2 + f (r) L − p 2 r 2 .(34) Now, comparing Eq. (34) withṙ 2 2 + V ef f = 0, the effective potential can be written V ef f = − 1 2 E 2 + f (r) L − p 2 r 2 .(35) A. Null geodesics For massless particles, i.e., for photons, we have L = 0, and the corresponding effective potential is V ef f = − E 2 2 + p 2 r 2 −M + 2M exp − r 2 4θ − Λr 2 . (36) As r → 0, the effective potential V ef f (r) becomes very large, but it approaches − E 2 2 − Λp 2 as r → ∞. At the horizons, V ef f = − E 2 2 . The shape of V ef f (r), shown in Fig. 7, indicates that a photon will fall into a black hole [29]. Taking various values for the masses does not alter the nature of the geodesics. B. Time-like geodesics For massive particles, L = −1, and the corresponding effective potential is V ef f = − E 2 2 + 1 + p 2 r 2 −M + 2M exp − r 2 4θ − Λr 2 ,(37) shown in Fig. 8. The effective potential becomes very large as r → 0, as well as when r → ∞. At the minimal horizon r 0 , it assumes the constant value V ef f = − E 2 2 , while Fig. 9 shows that the roots of the V ef f coincide with the horizons. Also, the shape of the effective potential indicates that the particle can move only inside the The effective potential has a minimum between two horizons, i.e., stable circular orbits do exist. black hole. Since the effective potential assumes negative values between the horizons, the particle is confined to the region between the two horizons, and, as a result, cannot hit the singularity. Finally, observe that the minimum of V ef f (r) occurs between the horizons, so that stable circular orbits are going to exist. V. TEST PARTICLES Let us consider a test particle having mass m 0 and moving in the gravitational field of the BTZ black hole inspired by noncommutative geometry and described by the metric ansatz (1). The Hamilton-Jacobi [H-J] equa- The effective potential has a minimum between two horizons, i.e., stable circular orbits do exist. tion for the test particle is [30,31] − 1 f ∂S ∂t 2 + f ∂S ∂r 2 + 1 r 2 ∂S ∂φ 2 + m 2 0 = 0. (38) As there is no explicit dependence on t and φ, let us choose the H-J function S as [30,31] S(t, r, θ, φ) = −E.t + S 1 (r) + p.φ, where E and p are identified as the energy and angular momentum of the particle. The radial velocity of the particle is given by dr dt = f 3 2 E E 2 f − m 2 0 − p 2 r 2 .(39) For detailed calculations, see Refs. [30,31]. The turning points of the trajectory are determined from dr dt = 0 and, as a consequence, the potential curve is given by V (r) ≡ E m 0 = f 1 + p 2 m 2 0 r 2 1/2 .(40) The extremals of the potential curve are the solutions of the equation dV dr = 0 and are found to be dV dr = − 2p 2 m 0 r 3 −M + 2M exp − r 2 4θ − Λr 2 + 1 + p 2 m 0 r 2 − M r θ exp − r 2 4θ − 2Λr = 0. While difficult to tell from the equation, the plot of dV dr , given in Fig. 10 VI. CONCLUSION This paper investigates the properties of a BTZ black hole constructed from the exact solution of the Einstein field equations in a (2 + 1)-dimensional anti-de Sitter spacetime in the context of noncommutative geometry. It was found that a BTZ black hole has two horizons, no horizons or a single horizon r = r 0 corresponding to a minimal mass M = M 0 . In this connection we note the comments by Mazharimousavi et al. [17]: "It is well-known that unlike its chargeless version the charged Banados-Teitelboim-Zanelli (BTZ) black hole solution in (2 + 1)-dimensional spacetime is singular". Thus they construct a charged, regular extension of the BTZ black hole solution by employing nonlinear Born-Infeld electrodynamics, supplemented with the Hoffmann term and gluing different spacetimes. However, our observation is that even the noncommutative geometry inspired BTZ black hole is not free from any singularity. Beside this, in the present paper we continue our investigation with a discussion of Hawking temperature, entropy and heat capacity. We observe that the noncommutativity leads to the same minimal radius r 0 at which the black hole cools down to absolute zero. A discussion of the geodesic structure leads to the effective potential for both massless and massive particles. It is shown that photons will fall into the black hole, while massive particles are trapped between the two horizons. The use of the Hamilton-Jacobi equation confirms that bound orbits are possible for test particles. Fig. 1 , 1using Λ √ θ = −0.02. Here three possibilities present themselves graphically in terms of the approximate value of α: (i) two horizons when α > 0.214, or M > M 0 = 0.214 √ θ; (ii) one horizon corresponding to the extremal black hole with M = M 0 , i.e., α = 0.214; FIG. 1 : 1The singularities occur where gtt cuts r-axis. Representation: the solid curve for M = 1.2 √ θ and dotted curve for M = 0.8 √ θ indicate two horizons. The dashed curve for M = 0.214 √ θ represents one degenerate horizon r0 ≈ 2.59 √ θ, i.e., an extremal black hole. For M = 0.1 √ θ, no horizon exists (chain curve). The intercepts on the r-axis give the radii of the event horizons. FIG. 2 : 2The the solid curve for M = 1.2 √ θ, dotted curve for M = 0.8 √ θ, dashed curve for M = 0.214 √ θ, and chain curve for M = 0.1 √ θ. FIG. 6 : 6Plot for C vs. r h √ θ . FIG. 7 : 7Plot for V ef f √ θ vs. r √ θ for different values of M √ θ . Representation: the solid curve for M = 1.2 √ θ, dotted curve for M = 0.8 √ θ, dashed curve for M = 0.214 √ θ, and chain curve for M = 0.1 √ θ. The shape of the V ef f √ θ indicates that a photon will fall into black hole. FIG. 8: Plot for V ef f √ θ vs. r √ θ for different values of M √ θ . Representation: the solid curve for M = 1.2 √ θ, dotted curve for M = 0.8 √ θ, dashed curve for M = 0.214 √ θ, and chain curve for M = 0.1 √ θ. FIG. 9 : 9Plot for V ef f √ θ vs. r √ θ for different values of M √ θ . Representation: the solid curve for M = 0.8 √ θ, dashed curve for M = 0.214 √ θ, and chain curve for M = 0.1 √ θ. ) for an extremal black hole, pt assumes negative values and beyond that positive values. For large values of r, i.e., for r √ θ >> 1, it dies out. FIG. 5: Plot for Hawking temperature vs. rFIG. 4: Plot for pt √ θ vs. r √ θ for different values of M √ θ . Representation: the solid curve for M = 1.2 √ θ, dotted curve for M = 0.8 √ θ, dashed curve for M = 0.214 √ θ, and chain curve for M = 0.1 √ θ. Note that up to the degenerate horizon r0 ≈ 2.59 √ θ √ θ for different values of M √ θ . Representation: the solid curve for M = 1.2 √ θ, dotted curve for M = 0.8 √ θ, dashed curve for M = 0.214 √ θ, and chain curve for M = 0.1 √ θ. , shows that real positive solutions existFIG. 10: Plot for dV√θ -axis. Hence bound orbits for the test particles exist. In other words, the test particles can be trapped by BTZ black holes inspired by noncommutative geometry.dr √ θ vs. r √ θ . Note that dV dr √ θ is zero for certain value of r √ θ . This implies that V (r) has at least one extremal. wherever 1 √ θ dV dr cuts the r AcknowledgmentsFR and SR are thankful to the authority of Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing them Visiting Associateship under which a part of this work was carried out. FR is also thankful UGC, Govt. of India under research award scheme, for providing financial support. We are very grateful to an anonymous referee for his/her insightful comments that have led to significant improvements, particularly on the interpretational aspects.Here the first term is the Bekenstein-Hawking entropy (S) of the BTZ black hole, while the second term is the θ correction. Our final task is to find the θ corrections for the Hawking temperature and surface gravity of the BTZ black hole.From eq. (18), we have,Keeping the first order of exp − Ml 2 4θ , the above expres-sion yieldsNote that the first term is the Hawking temperature of the BTZ black hole, while the second term is the θ cor-rection. The correction term of surface gravity is given byIV. THE GEODESICS From the geodesics equationwe obtain[29]1 f (r) . M Bañados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Bañados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 1849 (1992). . A Smailagic, E Spalluci, J. 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Obregn, M. Sabido and C. Ramrez. Phys. Rev. D 74, 084024 (2006) . O Obregn, M Sabido, E Mena, Mod. Phys. Lett. A. 241907O. Obregn, M. Sabido and E. Mena. Mod. Phys. Lett. A 24 , 1907(2009) . M Rinaldi, Class. Quant. Gravit. 28105022M. Rinaldi, Class. Quant. Gravit. 28, 105022 (2011). . Y S Myung, M Yoon, arXiv:0810.0078Y.S. Myung and M. Yoon, arXiv: 0810.0078. . J Liang, B Liu, EPL. 10030001J. Liang and B. Liu, EPL 100, 30001 (2012). . J M Tejeiro, A Larranaga, Pramana -J. Phys. 781J.M. Tejeiro and A. Larranaga, Pramana -J. Phys. 78, 1 (2012). . A Larranaga, J M Tejeiro, Abra. Zel. J. 428A. Larranaga and J.M. Tejeiro, Abra. Zel. J., 4, 28 (2011). . J J Blome, W Priester, Naturwissen. 71528J.J. Blome and W Priester, Naturwissen. 71, 528 (1984). . P C W Davies, Phys. Rev. D. 30737P.C.W. Davies, Phys. Rev. D 30, 737 (1984). . C Hogan, Nat. 310365C. Hogan, Nat. 310, 365 (1984). . N Kaiser, A Stebbins, Nat. 310391N. Kaiser and A. Stebbins, Nat. 310, 391 (1984). . 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[ "Spatial dependence of quantum friction amplitudes in a scalar model", "Spatial dependence of quantum friction amplitudes in a scalar model" ]
[ "Aitor Fernández \nCentro Atómico Bariloche\nInstituto Balseiro Comisión Nacional de Energía Atómica R8402AGP S. C. de Bariloche\nArgentina\n", "C D Fosco \nCentro Atómico Bariloche\nInstituto Balseiro Comisión Nacional de Energía Atómica R8402AGP S. C. de Bariloche\nArgentina\n" ]
[ "Centro Atómico Bariloche\nInstituto Balseiro Comisión Nacional de Energía Atómica R8402AGP S. C. de Bariloche\nArgentina", "Centro Atómico Bariloche\nInstituto Balseiro Comisión Nacional de Energía Atómica R8402AGP S. C. de Bariloche\nArgentina" ]
[]
We study the spatial dependence of the quantum friction effect for an atom moving at a constant velocity, in a parallel direction to a material plane. In particular, we determine the probability per unit time and unit area, for exciting degrees of freedom on the plane, as a function of their position, for a given trajectory of the atom. We also show that the result of integrating out the probability density agrees with previous results for the same system. arXiv:2209.12986v2 [quant-ph] 28 Sep 2022
null
[ "https://export.arxiv.org/pdf/2209.12986v2.pdf" ]
252,545,028
2209.12986
0fa4f427f9ec557bd641f311795d6ae61f1d4237
Spatial dependence of quantum friction amplitudes in a scalar model September 29, 2022 Aitor Fernández Centro Atómico Bariloche Instituto Balseiro Comisión Nacional de Energía Atómica R8402AGP S. C. de Bariloche Argentina C D Fosco Centro Atómico Bariloche Instituto Balseiro Comisión Nacional de Energía Atómica R8402AGP S. C. de Bariloche Argentina Spatial dependence of quantum friction amplitudes in a scalar model September 29, 2022 We study the spatial dependence of the quantum friction effect for an atom moving at a constant velocity, in a parallel direction to a material plane. In particular, we determine the probability per unit time and unit area, for exciting degrees of freedom on the plane, as a function of their position, for a given trajectory of the atom. We also show that the result of integrating out the probability density agrees with previous results for the same system. arXiv:2209.12986v2 [quant-ph] 28 Sep 2022 Introduction The intrinsically quantum nature of the elementary constituents of matter and their interactions can sometimes manifest itself macroscopically, in a rather straightforward way. Indeed, among the most distinctive features of quantum systems are their vacuum fluctuations, which produce observable effects when subjected to non-trivial boundary conditions. This is the case in the celebrated Casimir effect [1], where material media imposes boundary conditions on the electromagnetic (EM) field fluctuations. A different kind of phenomenon, where quantum fluctuations are also responsible of observable effects, is the so-called "non-contact friction" or "Casimir friction", whereby a frictional force appears on lossy media in nonaccelerated relative motion. It is a somewhat complementary situation to the Casimir effect case, since the zero point fluctuations of the EM field are not directly relevant; rather, its role is to mediate the interaction between the microscopic degrees of freedom on the two media. The frictional effect does not happen for perfects mirrors [2] but if may appear in non-dispersive media [3] when their relative speed overcomes the threshold posed by the speed of light in the media. The dissipative force may also appear on a single atom moving with constant velocity, parallel to a plate [5]. There are also thresholds related to the speed of the modes on the material media; for instance, in a recent paper [4], for an atom in the proximity of a graphene plate the atom must move faster than v F , the Fermi speed of the electrons in graphene, for dissipation to occur. In this paper, we use a quantum field theory model to study quantum friction between an atom, moving at a constant parallel speed with respect to a planar medium, in an approach which allows us to study the spatial distribution of the media excitations which play a role for the existence of the frictional force. The model we use is essentially the same we had used in [6], which is based on [7], namely, a vacuum scalar field linearly coupled to a set of uncoupled quantum harmonic oscillators which are the microscopic "matter" degrees of freedom on the mirror. Note that when considering the quantum friction effect between two planes, the spatial details we want to study are lost because of the very geometry of the system. Here, we use a perturbative quantum field theory approach to calculate transition amplitudes, and those amplitudes account for processes whereby modes on the medium are excited, what allows us to study the spatial distribution of the phenomenon. Besides, by integrating out the transition probabilities, we also provide an indirect verification of the result obtained in [6], where the total probability of vacuum decay was otained from the imaginary part of the in-out effective action to the frictional force on the plates. A similar approach has been used in [8] and [9] while in [10] a CTP in-in formulation [11] has been applied to evaluate the frictional force between two plates in relative motion at a constant speed. The system The system we deal with here is, regarding both its dynamical variables and the interactions between them, essentially the same as the one considered in [6]. It contains a scalar variable q, associated with the "electron": a scalar degree of freedom sitting on a moving atom, while the atom's center of mass trajectory, r(t), is externally driven. The variable q is coupled to a vacuum real scalar field ϕ, which also interacts with a medium, represented by microscopic independent scalar degrees of freedom Q, uniformly distributed on a plane. Regarding conventions, in this paper we shall use natural units, so that c = 1 and = 1; space-time coordinates are denoted by x = (x µ ) 3 µ=0 , x 0 = t, and we use the Minkowski metric (g µν ) ≡ diag{1, −1, −1, −1}. Our choice of coordinates is such that the spacetime occupied by the medium is x 3 = 0. Correspondingly, space-time coordinates relevant to the degrees of freedom on the plane, shall be denoted by x = (x α ) 2 α=0 = (t, x ). Here, x ≡ (x 1 , x 2 ) are two Cartesian coordinates on the spatial plane. The real-time action S for the whole system will thus be conveniently defined as follows: S(q, Q, ϕ; r(t)) = S (0) (q, Q, ϕ) + S (q, Q, ϕ; r(t)) ,(1) where S (0) determines the free evolution, while S does so for the interactions. The free part will consist of three terms: S S (0) (q, Q, ϕ) = S (0) e (q) + S (0) m (Q) + S (0) v (ϕ) ,(2) where: S (0) e (q) = dt m 2 q 2 (t) − Ω 2 e q 2 (t) ,(3)S (0) v (ϕ) = d 4 x 1 2 ∂ µ ϕ(x)∂ µ ϕ(x) ,(4) and S (0) m (Q) = d 3 x 1 2 ∂ t Q(x )∂ t Q(x ) − Ω 2 m Q 2 (x ) ,(5) where m is the mass of the electron's degree of freedom. It is assumed that its free dynamics is the one of a harmonic oscillator with the frequency Ω e determining its energy levels. As we shall see, at the lowest non-trivial order, the physics of the quantum friction process is determined by the ground state and the first excited state. Thus, results should be in this respect quite universal regarding the potential for q(t), except for a redefinition of the parameters; for example, the energy gap. On the other hand, note that the medium may be thought of as a continuous distribution of decoupled oscillators with frequency Ω m . It corresponds to taking the u → 0 limit in a more general model, namely, one whose elementary excitations have speed u: S (0) m (Q) = lim u→0 d 3 x 1 2 ∂ t Q(x ) 2 − u 2 ∇ Q(x ) 2 − Ω 2 m Q 2 (x ) . (6) The interaction term S may, on the other hand, be conveniently written as follows: S (q, Q, ϕ; r(t)) = d 4 x J(x) ϕ(x) , J(x) ≡ J e (x) + J m (x)(7) where J e and J m are, respectively, concentrated on the atom and on the medium. They are given by: J e (x) = g q(t) δ 3 (x − r(t)) , J m (x) = λ Q(x ) δ(x 3 ) ,(8) where g and λ are coupling constants. Transition amplitudes In order to study the transition amplitudes and probabilities which are responsible for the quantum friction phenomenon, we adopt the interaction picture, based on our choice for the free and interaction actions. In this situation, we have the following expression for the time evolution of the operators corresponding to the dynamical variables: q(t) = 1 √ 2mΩ e âe −iΩet +â † e iΩet Q(x ) = 1 √ 2Ω m α(x ) e −iΩmt +α † (x ) e iΩmt ϕ(x) = d 3 k (2π) 3/2 1 2|k| â(k) e −ik·x +â † (k) e ik·x .(9) Here, the creation and annihilation operators satisfy the standard commutation relations; namely, the only non-vanishing commutators are: [â ,â † ] = 1 , [â(k) ,â † (p)] = δ 3 (k − p) ,(10) and, taking into account the independence of the degrees of freedom for different spatial points, [α(x ) ,α † (x )] = δ 2 (x − x ) .(11) In order to study quantum friction, we consider the usual situation of the atom moving at a constant velocity, which is assumed to be parallel to the plane. Without any loss of generality, we use coordinates such that the velocity points towards the x 2 direction. Analogously, also by a proper choice of origin for space and time coordinates, the atom will pass just above the origin of the plane at t = 0. Denoting by a the (constant) distance between the atom and the plane, we then have: r(t) = (0, vt, a) .(12) The transition amplitudes T f i = f | T |i shall be determined from the scattering matrix S =Î + i T , namely, from the evolution operator in the interaction representation, U (t f , t i ), for t i → −∞ and t f → +∞: S = U (+∞, −∞) = T exp iS (q, Q, ϕ; r(t)) ,(13) where T denotes time-ordering. The initial quantum state |i of the full system is assumed to be the vacuum for all the modes, namely, for the electron, the medium, and the vacuum field. In a self explanatory notation, |i = |0 e ⊗ |0 m ⊗ |0 v .(14) Regarding the final state, |f , in quantum friction there is no production of vacuum-field particles (photons); indeed, that would require a non-vanishing acceleration. Therefore, in quantum friction, only even terms in the expansion of the exponential in (13) can intervene. The lowest order contribution to the transition amplitude is, therefore, the second order one, which yields: T f i = i d 4 x d 4 x ( f e | ⊗ f m |) J m (x) J e (x ) (|0 e ⊗ |0 m ) G(x − x ) ,(15) where we introduced the final states for the electron and the medium, and the scalar field propagator G: G(x − x ) = d 4 k (2π) 4 e −ik·(x−x ) i k 2 + iε ,(16) and we have assumed the initial and final states to be normalized. It is rather straightforward to see that the only contribution to the transition amplitude (to this order) contains a quantum for both e and m, namely: |f e =â † |0 e , |f m = d 2 x f (x )α † (x )|0 m ,(17) where d 2 x |f (x )| 2 = 1. Inserting this into (15), and integrating out x and x , we obtain: T f i = − 2πgλ √ 2mΩ e 2Ω m d 3 k (2π) 3 δ(Ω e + Ω m + k 2 v) e ik 3 af * (k ) Ω 2 m − k 2 + iε (18) wheref (k ) = d 2 x e −ik ·x f (x ) . Thus, integrating out k 2 and k 3 , and denoting by k x ≡ k 1 the only remaining component to integrate, we have that T f i = gλ 4v √ mΩ e Ω m +∞ −∞ dk x 2πf * k x , − Ω e + Ω m v e −a √ k 2 x +Ω 2 k 2 x + Ω 2 ,(19) where we have defined Ω 2 = Ω e + Ω m v 2 − Ω 2 m .(20) In what follows, for the sake of notational clarity, we use denote by x and y the two Cartesian coordinates x 1 and x 2 , respectively. We want to study the spatial properties of the transition probabilities, we consider a spatially localized function f , centered about a point with coordinates (ξ, η) (see Fig. 1) on the plane, and with size (σ x , σ y ): f (x, y) ≡ φ σx (x − ξ) φ σy (y − η) (21) where φ σ (x) ≡ e − x 2 4σ 2 (2π) 1/4 √ σ , +∞ −∞ dx |φ σ (x)| 2 = 1 .(22) Then: T f i = gλ v πσ x σ y 2mΩ e Ω m e −σ 2 y ( Ωe+Ωm v ) 2 e iη( Ωe+Ωm v ) × +∞ −∞ dk x 2π e −σ 2 x k 2 x e −ikxξ e −a √ k 2 x +Ω 2 k 2 x + Ω 2 .(23) Note that ρ(ξ) ≡ |T f i | 2 /σ x σ y is a probability per unit area, and making σ x , σ y → 0 would give the probability (per area) of having an oscillator of the medium excited at (ξ, η) 1 ρ(ξ) ≡ lim σx,σy→0 |T f i | 2 σ x σ y = g 2 λ 2 8πmΩ m Ω e v 2 ∞ −∞ dk x e −ikxξ e −a √ k 2 x +Ω 2 k 2 x + Ω 2 2 ,(24) which does not depend on η. With Ω as defined in (20), and giving different values to the adimensional combination Ωa for fixed Ω e , we can see in the Fig.2 how this distribution varies with the distance between the medium and the atom, and with the quotient f ≡ Ω m /Ω e . Integrating ρ(ξ) for all ξ and multiplying by a characteristic length in the direction of movement of the atom, i.e. vT , where we can think of T as the time the atom has been moving with constant speed v, gives the probability of this process to happen. Then, dividing by T , would give the probability per unit time P ≡ 1 T vT ∞ −∞ dξ ρ(ξ) = g 2 λ 2 2mvΩ e Ω m ∞ 0 dk x e −2a √ k 2 x +( Ωe+Ωm v ) 2 −Ω 2 m k 2 x + ( Ωe+Ωm v ) 2 − Ω 2 m ,(25) which matches with the result obtained in equation (69) of [6]. Case of u = 0 Now we consider the case where waves can be propagated through the medium at speed u = 0, and see that results match with the previous model when u → 0. The action for the medium is now S (0) m (Q) = d 3 x 1 2 ∂ t Q(x ) 2 − u 2 ∇ Q(x ) 2 − Ω 2 m Q 2 (x ) ,(26) and the second expression of (9) turns intô Q(x ) = d 2 k 2π 1 √ 2k 0 α(k ) e −ik ·x +α † (k ) e ik ·x ,(27) where k 0 = u 2 |k | 2 + Ω 2 m and [α(k ) ,α † (p )] = δ 2 (k − p ). The (normalized) final state that we consider now is |f m = 2π α † (p ) |0 m , i.e. an T f i = 2π gλ e −a √ |p | 2 (1−u 2 )−Ω 2 m mΩ e |p | 2 (1 − u 2 ) − Ω 2 m u 2 |p | 2 + Ω 2 m × δ Ω e + u 2 |p | 2 + Ω 2 m − vp y .(28) The arising Dirac delta gives us some information about the process. First, p y has to be positive, so the momentum of the excitation in the medium has a positive component along the velocity of the atom. Then, it shows that there is a threshold for this process to occur: the speed of the atom should be greater than the speed of the waves propagating in the medium v > u. up y < u|p | < u 2 |p | 2 + Ω 2 m = vp y(29) Dividing |T f i | 2 by T with u → 0 and integrating all possible momentums for the excitation of the medium gives the probability per unit time of having this process. P = 1 T 2π 2 d 2 p lim u→0 |T f i | 2 = g 2 λ 2 πmvΩ e Ω m ∞ 0 dp x e −2a √ p 2 x +( ΩeΩm v ) 2 −Ω 2 m p 2 x + ( Ωe+Ωm v ) 2 − Ω 2 m ,(30) which is essentially the same as (25). Conclusions In this paper, we have evaluated the transition amplitudes corresponding to the elementary processes which lead to the phenomenon of quantum friction between a moving atom and a material plane. Our choice of system, and model, allows for a determination of the spatial dependence of the Casimir friction phenomenon, by providing the funcional form of the amplitude as a function of the distance, on the plane, to the projection of the atom's trajectory. The dependence of the previous results on the distance between the atom and the plane is modulated by a precise combination of the model's parameters and the velocity of the atom. Finally, by integrating out the probabilities to this order, we have found agreement with the total vacuum decay probability obtained, for the same model, by an evaluation of the imaginary part of the effective action, in a functional integral approach. Figure 1 : 1Sketch of the system function of a (the distance between the atom and the medium), and the ratio between frequencies. The four curves are for v = 0.1, increasing the speed has a similar effect than increasing f . excitation with momentum p . 2 This gives a transition amplitude This is equivalent to taking f (x ) ∝ δ(x − ξ)δ(y − η) in (17). When u = 0, this is equivalent to taking f (x ) ∝ e −ip ·x in (17). AcknowledgementsThe authors thank ANPCyT, CONICET, CNEA and UNCuyo for financial support. P W Milonni, The Quantum Vacuum. San DiegoAcademic PressP.W. Milonni, The Quantum Vacuum (Academic Press, San Diego, 1994); . M Bordag, U Mohideen, V M Mostepanenko, Phys. Rep. 3531M. Bordag, U. Mohideen, and V.M. Mostepanenko, Phys. Rep. 353, 1 (2001); The Casimir Effect: Physical Manifestations of the Zero-Point Energy. K A Milton, World ScientificSingaporeK. A. Milton, The Casimir Effect: Physical Manifesta- tions of the Zero-Point Energy (World Scientific, Singapore, 2001); . S Reynaud, A Lambrecht, C Genet, M T Jaekel, C. R. Acad. Sci. Paris Ser. IV. 21287S. Reynaud, A. Lambrecht, C. Genet, and M.T. 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[ "Discrete Laguerre-Sobolev expansions: A Cohen type inequality", "Discrete Laguerre-Sobolev expansions: A Cohen type inequality" ]
[ "A Peña ", "M L Rezola ", "Ana Peña [email protected] ", "\nDepartamento de Matemáticas and IUMA\nDepartamento de Matemáticas\nUniversidad de Zaragoza\nSpain\n", "\nUniversidad de Zaragoza\n50009-Zaragoza (Spain\n" ]
[ "Departamento de Matemáticas and IUMA\nDepartamento de Matemáticas\nUniversidad de Zaragoza\nSpain", "Universidad de Zaragoza\n50009-Zaragoza (Spain" ]
[]
C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre-Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán.2000MSC: 42C05
10.1016/j.jmaa.2011.06.036
[ "https://arxiv.org/pdf/1107.0832v1.pdf" ]
54,694,655
1107.0832
8d596ed397ee91c267503c813e020486c2734352
Discrete Laguerre-Sobolev expansions: A Cohen type inequality 5 Jul 2011 A Peña M L Rezola Ana Peña [email protected] Departamento de Matemáticas and IUMA Departamento de Matemáticas Universidad de Zaragoza Spain Universidad de Zaragoza 50009-Zaragoza (Spain Discrete Laguerre-Sobolev expansions: A Cohen type inequality 5 Jul 2011Corresponding author:Laguerre polynomialsLaguerre-Sobolev type polynomialsCohen type inequalityBessel functions C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre-Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán.2000MSC: 42C05 Introduction and notations Littlewood conjectured in 1948 that for any trigonometric polynomial F N (x) = N k=1 a k e in k x where 0 < n 1 < n 2 < · · · < n N , N ≥ 2, and |a k | ≥ 1 for 1 ≤ k ≤ N , there holds the estimate from below 2π 0 |F N (x)dx| ≥ C logN where C is an absolute constant (see [6]). Cohen's inequality [3] was the first result on the way to the solution of this conjecture. Later, inequalities of this type have been established in various other contexts, e.g., on compact group (see [5]). In [12] Markett proved such inequalities for classical orthogonal polynomial expansions in the appropriate weighted L p spaces, here in terms of the highest coefficient. The main purpose of this paper is to extend these results to discrete Laguerre-Sobolev expansions. More precisely, we obtain such inequalities, in the appropriate weighted L p spaces, for Fourier expansions in terms of orthonormal polynomials with respect to an inner product of the form p, q S = 1 Γ(α + 1) ∞ 0 p(x)q(x) x α e −x dx + N j=0 M j p (j) (0)q (j) (0),(1) where α > −1 and M j ≥ 0, j = 0, . . . , N. Such inner products are called of discrete Sobolev type. Recently in [4], the authors Fejzullahu and Marcellán obtained Cohen type inequalities for orthonormal expansions with respect to the above inner product in the case N = 1, i.e. at most two masses in the discrete part. In this particular case, the authors benefit from the fact that there are explicit formulas for the connection coefficients which appear in the representation of discrete Laguerre-Sobolev type polynomials in terms of three standard Laguerre polynomials (see [9]). For a general discrete Laguerre-Sobolev inner product, we only know that these coefficients are a nontrivial solution of a system of N + 1 equations on N + 2 unknowns (see [8]). If the system is solved, we get an intricate expression with which it is difficult to work. Our contribution in this paper is that we can assure that there exists limit of these connection coefficients and this is enough for our purpose. Let {L α n (x)} n≥0 be the sequence of Laguerre polynomials, orthogonal on [0, ∞) with respect to the probability measure dµ(x) = 1 Γ(α + 1) x α e −x dx where α > −1 and normalized by L α n (0) = n + α n . We denote the orthonormal Laguerre polynomial of degree n by l α n (x) = L α n (x) L α n where L α n 2 = ∞ 0 L α n (x) 2 dµ(x). Let {Q α n } n≥0 be the sequence of discrete Laguerre-Sobolev orthogonal polynomials with respect to the inner product (1) and such that Q α n (x) and L α n (x) have the same leading coefficient. We denote by q α n (x) = Q α n , Q α n −1/2 S Q α n (x) the orthonormal discrete Laguerre-Sobolev polynomials. From now on, for simplicity we write Q n (x) = Q α n (x) and q n (x) = q α n (x). Laguerre expansions have been investigated mainly in the following two sets of weighted Lebesgue spaces, namely in the classical spaces ( [2], [10]) L p (x α dx) = {f ; ∞ 0 |f (x)e −x/2 | p x α dx < ∞}, if 1 ≤ p < ∞; {f ; ess sup 0<x<∞ |f (x)e −x/2 | < ∞}, if p = ∞, for α > −1 as well as in the spaces L p (x αp/2 dx) = {f ; ∞ 0 |f (x)e −x/2 x α/2 | p dx < ∞}, if 1 ≤ p < ∞; {f ; ess sup 0<x<∞ |f (x)e −x/2 x α/2 | < ∞}, if p = ∞, for α > − 2 p if 1 ≤ p < ∞ and α ≥ 0 if p = ∞. In order to unify the two results we are going to prove, we introduce an auxiliary parameter β which means either α or αp/2. We consider the class S β p , 1 ≤ p ≤ ∞, defined as the space of measurable functions f defined on [0, ∞), such that there exits f (k) (0) for k = 0, . . . , N and if 1 ≤ p < ∞ f p S β p = f p Lp(x β dx) + N i=0 M j |f j (0)| p < ∞, where f p Lp(x β dx) = ∞ 0 |f (x)e −x/2 | p x β dx, 1 ≤ p < ∞, and if p = ∞ f S β ∞ = max{ f L∞(x β dx) , |f (0)|, . . . , |f (N ) (0)|} < ∞, where f L∞(x β dx) =    ess sup 0<x<∞ |f (x)e −x/2 |, if β = α; ess sup 0<x<∞ |f (x)e −x/2 x α/2 |, if β = αp/2. (If some M j = 0 the corresponding derivative does not appear in the maximum.) Let f ∈ S β p , 1 ≤ p ≤ ∞, then the Fourier expansion in terms of orthonormal discrete Laguerre-Sobolev polynomials {q n } n≥0 , is ∞ k=0f (k) q k (x) wheref (k) = f, q k S . In the following, [S β p ] denotes the space of all bounded linear operators T from the space S β p into itself, endowed with the usual operator norm, T [S β p ] = sup 0 =f ∈S β p T f S β p f S β p . Let 1 ≤ p ≤ ∞. For a family of complex numbers {c k,n } n k=0 , n ∈ N ∪ {0}, with |c n,n | > 0 we define the operators T α,S n : S β p → S β p by T α,S n (f ) = n k=0 c k,nf (k) q k . Let us denote q 0 = 4α+4 2α+1 for β = α and q 0 = 4 for β = pα/2, and let p 0 be the conjugate of q 0 , i.e. 1/p 0 + 1/q 0 = 1. Now, we can state our main theorem, which extends the ones given in [12] and [4]. Theorem 1 Let 1 ≤ p ≤ ∞. There exists a positive constant C, independent of n, such that: For α > −1/2 T α,S n [S α p ] ≥ C |c n,n |      n 2α+2 p − 2α+3 2 , if 1 ≤ p < p 0 ; (log(n + 1)) 2α+1 4α+4 , if p = p 0 , p = q 0 ; n 2α+1 2 − 2α+2 p , if q 0 < p ≤ ∞. For α > −2/p if 1 ≤ p < ∞ and α ≥ 0 if p = ∞ T α,S n [S pα/2 p ] ≥ C |c n,n |      n 2 p − 3 2 , if 1 ≤ p < p 0 ; (log(n + 1)) 1 4 , if p = p 0 , p = q 0 ; n 1 2 − 2 p , if q 0 < p ≤ ∞. This theorem will be proved in Section 3. In Section 2, we obtain some new results for discrete Laguerre-Sobolev polynomials, which we will use to establish Theorem 1. More concretely, we prove a technical lemma that will be used to deduce a Mehler-Heine type formula for Laguerre-Sobolev polynomials and a sharp estimation for their norm in the appropriate weighted L p spaces. In the sequel we use the following notation, a n ∼ b n means that there exist positive constants c 1 and c 2 , such that c 1 a n ≤ b n ≤ c 2 a n for n large enough, while a n ∼ = b n means that the sequence an bn converges to 1. Throughout the paper, the values of the constants may change from line to line. Estimates for discrete Laguerre-Sobolev polynomials Consider the standard Laguerre polynomials L α n and the Laguerre-Sobolev polynomials Q n with the same leading coefficient. Let us recall some properties of Laguerre polynomials for α > −1 (see [14]). The evaluation at x = 0 of the polynomials L α n and its successive derivatives are given by (L α n ) (k) (0) = (−1) k Γ(n + α + 1) (n − k)! Γ(α + k + 1) , k ∈ N ∪ {0}, and their L 2 -norm is L α n 2 = 1 Γ(α + 1) ∞ 0 (L α n (x)) 2 x α e −x dx = Γ(n + α + 1) n! Γ(α + 1) .(2) As usual, we denote the derivatives of the nth kernels of Laguerre polynomials by K (k,h) n (x, y) = ∂ k+h ∂x k ∂y h K n (x, y) = n i=0 (L α i ) (k) (x)(L α i ) (h) (y) L α i 2 with k, h ∈ N ∪ {0} and the convention K (0,0) n (x, y) = K n (x, y). In the next lemma, we obtain an asymptotic estimate for Q (k) n (0), that will play an important role along this paper. Lemma 1 Let Q n be the polynomials orthogonal with respect to the inner product (1). Then the following statements hold: (a) Q (k) n (0) (L α n ) (k) (0) ∼ = C k n α+2k+1 , for k such that M k > 0; C k , otherwise, where C k is a nonzero constant independent of n. (b) Q n , Q n S ∼ = L α n 2 . Proof. If all the masses in the inner product (1) are zero the result is trivial because Q n = L α n . We will prove the result by induction concerning the number of positive masses in the inner product (1). We take the first mass which is positive, namely M j 1 (j 1 ≥ 0), and consider the sequence of polynomials {Q n,1 } n≥0 orthogonal with respect to the inner product (p, q) 1 = 1 Γ(α + 1) ∞ 0 p(x)q(x) x α e −x dx + M j 1 p (j 1 ) (0)q (j 1 ) (0). The Fourier expansion of the polynomial Q n,1 in the orthogonal basis {L α n } n≥0 leads to Q n,1 (x) = L α n (x) − M j 1 Q (j 1 ) n,1 (0)K (0,j 1 ) n−1 (x, 0) . Therefore Q n,1 (x) = L α n (x) − M j 1 (L α n ) (j 1 ) (0) 1 + M j 1 K (j 1 ,j 1 ) n−1 (0, 0) K (0,j 1 ) n−1 (x, 0) ,(3) and (Q n,1 , Q n,1 ) 1 = L α n 2 + M j 1 ((L α n ) (j 1 ) (0)) 2 1 + M j 1 K (j 1 ,j 1 ) n−1 (0, 0) .(4) These relationships are very well known in the literature of discrete Sobolev type orthogonal polynomials. Taking derivatives k times in (3) and evaluating at x = 0, we obtain Q (k) n,1 (0) (L α n ) (k) (0) = 1 − M j 1 K (k,j 1 ) n−1 (0, 0) 1 + M j 1 K (j 1 ,j 1 ) n−1 (0, 0) (L α n ) (j 1 ) (0) (L α n ) (k) (0) .(5) Applying the Stolz criterion (see, e.g. [7]), we have lim n K (k,j 1 ) n−1 (0, 0) n α+k+j 1 +1 = lim n (L α n−1 ) (k) (0)(L α n−1 ) (j 1 ) (0) L α n−1 2 (α + k + j 1 + 1)n α+k+j 1 = 0,(6) and therefore K (k,j 1 ) n−1 (0, 0) K (j 1 ,j 1 ) n−1 (0, 0) (L α n ) (j 1 ) (0) (L α n ) (k) (0) ∼ = (α + 2j 1 + 1) (α + k + j 1 + 1) (L α n−1 ) (k) (0) (L α n−1 ) (j 1 ) (0) (L α n ) (j 1 ) (0) (L α n ) (k) (0) ∼ = α + 2j 1 + 1 α + k + j 1 + 1 .(7) Thus, from (5), (6) and (2), we have Q (j 1 ) n,1 (0) (L α n ) (j 1 ) (0) = 1 1 + M j 1 K (j 1 ,j 1 ) n−1 (0, 0) ∼ = C j 1 n α+2j 1 +1 and for k = j 1 Q (k) n,1 (0) (L α n ) (k) (0) ∼ = 1 − α + 2j 1 + 1 α + k + j 1 + 1 = 0. So, we achieve (a) for Q n,1 . Besides, taking limits in (4) and using again the size of derivatives of Laguerre polynomials, we get (b) for the polynomials Q n,1 . If there are no more positive masses, since Q n,1 = Q n we have concluded the proof. Otherwise, suppose that the results (a) and (b) hold for the sequence of polynomials {Q n,s−1 } n≥0 orthogonal with respect to the inner product (p, q) s−1 = 1 Γ(α + 1) ∞ 0 p(x)q(x) x α e −x dx + M j 1 p (j 1 ) (0)q (j 1 ) (0) + · · · + M j s−1 p (j s−1 ) (0)q (j s−1 ) (0), where j 1 < j 2 < · · · < j s−1 and all these masses are positive. Now, we have to prove the result for the polynomials Q n,s orthogonal with respect to (p, q) s = 1 Γ(α + 1) ∞ 0 p(x)q(x) x α e −x dx + M j 1 p (j 1 ) (0)q (j 1 ) (0) + · · · + M js p (js) (0)q (js) (0), where M js > 0. Since (p, q) s = (p, q) s−1 + M js p (js) (0)q (js) (0) we can work as before. Then the Fourier expansion of the polynomial Q n,s in the orthogonal basis {Q n,s−1 } n≥0 leads to Q n,s (x) = Q n,s−1 (x) − M js Q (js) n,s (0)K (0,js) n−1,s−1 (x, 0) , where K n,s−1 denotes the corresponding nth kernel for the sequence {Q n,s−1 } and K (k,h) n,s−1 (x, y) = n i=0 Q (k) i,s−1 (x)Q (h) i,s−1 (y) (Q i,s−1 , Q i,s−1 ) s−1 , k, h ∈ N ∪ {0}. Therefore, in the same way as in (3) and (4), we get . Taking derivatives k times in (8) and evaluating at x = 0, we obtain Q (k) n,s (0) (L α n ) (k) (0) = Q (k) n,s−1 (0) (L α n ) (k) (0) 1 − M js K (k, where C k is a nonzero constant. Indeed, for k = j 1 , . . . , j s−1 , lim n K (k,js) n−1,s−1 (0, 0) n α+k+js+1 = lim n Q (k) n−1,s−1 (0)Q (js) n−1,s−1 (0) (Q n−1,s−1 , Q n−1,s−1 ) s−1 (α + k + j s + 1) n α+k+js = lim n Q (k) n−1,s−1 (0) (L α n−1 ) (k) (0) lim n Q (js) n−1,s−1 (0) (L α n−1 ) (js) (0) lim n (L α n−1 ) (k) (0)(L α n−1 ) (js) (0) L α n−1 2 (α + k + j s + 1) n α+k+js ,(12) and, for k = j 1 , . . . , j s−1 , lim n K (k,js) n−1,s−1 (0, 0) n js−k = lim n Q (k) n−1,s−1 (0)Q (js) n−1,s−1 (0) (Q n−1,s−1 , Q n−1,s−1 ) s−1 (j s − k) n js−k−1 (13) = lim n (L α n−1 ) (k) (0)(L α n−1 ) (js) (0) L α n−1 2 (j s − k) n α+k+js lim n n α+2k+1 Q (k) n−1,s−1 (0) (L α n−1 ) (k) (0) lim n Q (js) n−1,s−1 (0) (L α n−1 ) (js) (0) . Then, from (10), (11) and the hypothesis for Q n,s−1 , we have Q (js) n,s (0) (L α n ) (js) (0) = Q (js) n,s−1 (0) (L α n ) (js) (0) 1 1 + M js K (js,js) n−1,s−1 (0, 0) ∼ = C js n α+2js+1 , with C js a nonzero constant. Moreover, for k = j s , taking into account (12), (13) and the hypothesis for Q n,s−1 , we can deduce K (k,js) n−1,s−1 (0, 0) K (js,js) n−1,s−1 (0, 0) Q (js) n,s−1 (0) Q (k) n,s−1 (0) = K (k,js) n−1,s−1 (0, 0) K (js,js) n−1,s−1 (0, 0) Q (js) n,s−1 (0) (L α n ) (js) (0) (L α n ) (k) (0) Q (k) n,s−1 (0) (L α n ) (js) (0) (L α n ) (k) (0) ∼ = (L α n ) (js) (0) (L α n−1 ) (js) (0) (L α n−1 ) (k) (0) (L α n ) (k) (0) α+2js+1 α+k+js+1 , if k = j 1 , . . . , j s−1 ; α+2js+1 js−k , if k = j 1 , . . . , j s−1 , ∼ = α+2js+1 α+k+js+1 , if k = j 1 , . . . , j s−1 ; α+2js+1 js−k , if k = j 1 , . . . , j s−1 . Thus, taking limits in (10) and (9), we get (a) and (b) for the polynomials Q n,s , i.e. Q (k) n,s (0) (L α n ) (k) (0) ∼ = C k n α+2k+1 , if k = j 1 , . . . , j s ; C k , otherwise, and (Q n,s , Q n,s ) s ∼ = L α n 2 . Hence the result follows. ✷ Observe that the part (a) of Lemma 1 is also true for the ratio of the corresponding orthonormal polynomials, and therefore there exists lim n q (k) n (0) (l α n ) (k) (0) = 0, for k such that M k > 0; C k = 0, otherwise.(14) Consider the following representation of the orthonormal polynomials q n in terms of the orthonormal Laguerre polynomials l α n (see [8,Section 9] ) q n (x) = N +1 j=0 b j (n)x j l α+2j n−j (x).(15) For the inner product (1) with N = 1, the coefficients b j (n) was explicitly obtained in [9], and their estimation was essential to obtain the result in [4]. Now in the general case, using Lemma 1, we can prove that there is always limit of the connection coefficients b j (n) for an arbitrary N . Moreover, the first index j such that b j = 0 corresponds with the first j such that M j = 0 in the inner product (1). (We understand that if all the masses are positive, then the unique coefficient b j different from zero is the last one). Proof. Taking derivatives k times in (15) and evaluating at x = 0, we deduce q (k) n (0) (l α n ) (k) (0) = k j=0 b j (n) k j j! A j (k, n), k ∈ {0, . . . , N + 1},(16) where A 0 (k, n) = 1 and A j (k, n) = (l α+2j n−j ) (k−j) (0) (l α n ) (k) (0) ∼ = (−1) j Γ(α + k + 1) Γ(α + k + j + 1) Γ(α + 2j + 1) Γ(α + 1) 1/2(17) Since there exists lim n A j (k, n) = 0 , applying recursively (14) and (16) we can assure there exists lim n b j (n) = b j , j ∈ {0, . . . , N +1}. More precisely, for k = 0 we have lim n b 0 (n) = lim n q n (0) l α n (0) = b 0 = 0, if M 0 > 0; C = 0, if M 0 = 0. Now, from (16) for k = 1, (14) and (17) we get lim n b 1 (n) = lim n 1 A 1 (1, n) q ′ n (0) (l α n ) ′ (0) − b 0 (n) = b 1 Observe that b 1 = 0, if M 0 > 0 and M 1 > 0 ; C = 0, if M 0 > 0 and M 1 = 0 . In this way, recursively, if M 0 M 1 . . . M i > 0 and M i+1 = 0 we can assure that b j = 0, if 0 ≤ j ≤ i; C = 0, if j = i + 1, and we obtain the result. ✷ As a consequence of the above lemma, we can establish a Mehler-Heine type formula for general discrete Laguerre-Sobolev orthonormal polynomials. This formula shows how the presence of the masses in the discrete part of the inner product changes the asymptotic behavior around the origin. Moreover, it supplies information on the location and asymptotic distribution of the zeros of the polynomials in terms of the zeros of known special functions. We recall the corresponding formula for orthonormal Laguerre polynomials (see [14]) lim n l α n (x/(n + k)) n α/2 = Γ(α + 1) x −α/2 J α (2 √ x)(18) uniformly on compact subsets of C and uniformly for k ∈ N ∪ {0}, where J α is the Bessel function of the first kind. Proposition 1 The polynomials q n satisfy the following Mehler-Heine type formula: lim n q n (x/n) n α/2 = Γ(α + 1) N +1 j=0 b j x −α/2 J α+2j (2 √ x)(19) uniformly on compact subsets of C. Proof. The proof is a straightforward consequence of formula (15), Lemma 2 and (18). ✷ Remark. According to Lemma 2, the first Bessel function which appears in (19) corresponds with the first index j such that M j = 0, in the inner product (1). We want to highlight that this result generalizes the one obtained in [1,Theorem 3], where the authors only deal with inner products with a unique "gap" in the discrete part. The above proposition allows us to deduce a lower estimate of q n Lp(x β dx) , for β = α and β = αp/2, that will play an important role in the proof of Theorem 1. Proposition 2 Let 1 ≤ p ≤ ∞. Then, the following statements hold: For α > −1/2 q n Lp(x α dx) ≥ C n −1/4 (log(n + 1)) 1/p , if p = 4α+4 2α+1 ; n α/2−(α+1)/p , if 4α+4 2α+1 < p ≤ ∞, and for α > −2/p if 1 ≤ p < ∞ and α ≥ 0 if p = ∞ q n Lp(x αp/2 dx) ≥ C n −1/4 (log(n + 1)) 1/p , if p = 4; n −1/p , if 4 < p ≤ ∞, where C is an absolute positive constant. Proof. Assume 1 ≤ p < ∞. Then, q n p Lp(x β dx) = ∞ 0 |q n (x)e −x/2 | p x β dx > 1/ √ n 0 |q n (x)e −x/2 | p x β dx ≥ Cn −β−1 √ n 0 |q n (t/n)| p t β dt According to formula (19), ∃ n 0 ∈ N such that ∀n ≥ n 0 √ n 0 |q n (t/n)| p t β dt ≥ Cn pα/2 √ n 0 | N +1 j=0 b j t −α/2 J α+2j (2 √ t)| p t β dt and therefore ∀n ≥ n 0 q n p Lp(x β dx) ≥ Cn pα/2−β−1 2n 1/4 0 u 2β−pα+1 | N +1 j=0 b j J α+2j (u)| p du. Working as Stempak in [13, Lemma 2.1], we can prove that for α > −1, and λ > −1 − αp 2n 1/4 0 u λ | N +1 j=0 b j J α+2j (u)| p du ∼ 1, if λ < p/2 − 1; log(n + 1), if λ = p/2 − 1. Thus, if 1 ≤ p < ∞, we obtain the first and the second result for β = α and β = pα/2 respectively. The results for p = ∞ can be deduced from the previous one by passing to the limit when p goes to ∞. ✷ It is worth to noticing that these lower bounds are sharp in the following sense. Proposition 3 Let 1 ≤ p ≤ ∞. Then: For α ≥ 0, q n Lp(x α dx) ∼ n −1/4 (log(n + 1)) 1/p , if p = 4α+4 2α+1 ; n α/2−(α+1)/p , if 4α+4 2α+1 < p ≤ ∞, and for α > −2/p if 1 ≤ p < ∞ and α ≥ 0 if p = ∞, q n Lp(x αp/2 dx) ∼ n −1/4 (log(n + 1)) 1/p , if p = 4; n −1/p , if 4 < p ≤ ∞. Proof. From Lemma 1 of [11] it can be deduced that for α ≥ 0 ∞ 0 |x j l α+2j n (x)e −x/2 | p x α dx ∼ n −p/4 log(n + 1), if p = 4α+4 2α+1 ; n αp/2−(α+1) , if 4α+4 2α+1 < p ≤ ∞, and for α > −2/p if 1 ≤ p < ∞ and α ≥ 0 if p = ∞ ∞ 0 |x j l α+2j n (x)e −x/2 x α/2 | p dx ∼ n −p/4 log(n + 1), if p = 4; n −1 , if 4 < p ≤ ∞. Thus, using the representation formula for the polynomials q n (see (15)), and the fact that the connection coefficients are bounded (see Lemma 2), we get one of the two inequalities. The other one has been proved in Proposition 2 and therefore the result follows. ✷ A Cohen type inequality In this section we prove a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre-Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So we extend the result due to Fejzullahu and Marcellán which deals with a discrete Laguerre-Sobolev inner product with at most two masses in the discrete part (see [4]). Proof of Theorem 1. Let us consider the following test functions which were already used in [12] and later in [4] g α,j n (x) = x j L α+j n (x) − (n + 1)(n + 2) (n + α + j + 1)(n + α + j + 2) L α+j n+2 (x) , with j ∈ N \ {1, . . . , N }. Notice that (g α,j n ) (i) (0) = 0, i = 0, . . . , N. These functions can be written as (see formula (2.15) in [12]) g α,j n (x) = j+2 m=0 a m,j (α, n)L α n+m (x) with a 0,j (α, n) = Γ(n + α + j + 1) Γ(n + α + 1) ∼ = n j . From (20), (21), and 0 ≤ k ≤ n, we have g α,j n (k) = g α,j n , q k S = 1 Γ(α + 1) By the orthogonality of Laguerre polynomials, we obtain g α,j n (k) = 0, if 0 ≤ k ≤ n − 1; 1 Γ(α+1) a 0,j (α, n) ∞ 0 L α n (x)q n (x)e −x x α dx, if k = n. Thus, from Lemma 1 (b), the estimate of a 0,j (α, n) and the value of the norm of Laguerre polynomials (see (2)), we can deduce g α,j n (n) = 1 Γ(α + 1) a 0,j (α, n) Observe that Q n and L α n have always equivalent norms, and, therefore this estimation does not depend neither on the number of positive masses, nor on the existence or non-existence of any gap in the inner product. Applying the operator T α,S n,s , Q n,s ) s = (Q n,s−1 , Q n,s−1 ) s−1 + M js (Q n, Stolz criterion and the hypotheses (a) and (b) for {Q n,s−1 } n≥0 , n α+k+js+1 , if k = j 1 , . . . , j s−1 ; C k n js−k , if k = j 1 , . . . , j s−1 , Lemma 2 2Let {b j (n)} N +1 0 be the coefficients in formula (15). Then, there exists lim n b j (n) = b j ∈ R, j ∈ {0, . . . , N + 1}. n+m (x)q k (x)e −x x α dx . n to the functions g α,j n , we get T α,S n (g α,j n ) = c n,n g α,j n (n)q n , and thereforeOn the other hand, for j > α − 1/2 − 2(α + 1)/p we have Hence, by duality the theorem follows. ✷ Remark. In particular, for M i = 0, i = 0, . . . , N , the above theorem extends Theorem 1 in[12]to negative values of α.In the particular case of c k,n = 1, k = 0, . . . , n, the operator T α,S n is the nth partial sum of the Fourier expansion, so, we can assure the following result.Corollary 1 If p is outside the Pollard interval (p 0 , q 0 ), we have S n [S β p ] → ∞, n → ∞ where S n denotes the nth partial sum of the Fourier expansion. A new approach to the asymptotics of Sobolev type orthogonal polynomials. M Alfaro, J J Moreno-Balcázar, A Peña, M L Rezola, J. Approx. Theory. 163M. Alfaro, J.J. Moreno-Balcázar, A. Peña, M.L. Rezola, A new ap- proach to the asymptotics of Sobolev type orthogonal polynomials, J. Approx. Theory 163 (2011) 460-480. Mean convergence of expansions in Laguerre and Hermite series. R Askey, S Wainger, Amer. J. Math. 87R. Askey, S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965) 695-708. On a conjecture of Littlewood and idempotent measures. P J Cohen, Amer. J. Math. 82P. J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960) 191-212. A Cohen type inequality for Laguerre-Sobolev expansions. B Xh, F Fejzullahu, Marcellán, J. Math. Anal. Appl. 352B. Xh. Fejzullahu, F. Marcellán, A Cohen type inequality for Laguerre- Sobolev expansions, J. Math. Anal. Appl. 352 (2009) 880-889. A Cohen type inequality for compact Lie groups. S Giulini, P M Soardi, G Travaglini, Proc. Amer. Math. Soc. 77S. Giulini, P. M. Soardi, G. Travaglini, A Cohen type inequality for compact Lie groups, Proc. Amer. Math. Soc. 77 (1979) 359-364. A new proof of a theorem on rearrangements. G H Hardy, J E Littlewood, J. London Math. Soc. 23G. H. Hardy, J. E. Littlewood, A new proof of a theorem on rearrange- ments, J. London Math. Soc., 23 (1948) 163-168. Aspects of Calculus. G Klambauer, Springer-VerlagNew YorkG. Klambauer, Aspects of Calculus, Springer-Verlag, New York 1986. Generalizations of Laguerre polynomials. R Koekoek, J. Math. Anal. Appl. 153R. Koekoek, Generalizations of Laguerre polynomials, J. Math. Anal. Appl. 153 (1990) 576-590. A generalization of Laguerre polynomials. R Koekoek, H G Meijer, SIAM J. Math. Anal. 24R. Koekoek, H.G. Meijer, A generalization of Laguerre polynomials, SIAM J. Math. Anal. 24 (1993) 768-782. Mean convergence of Hermite and Laguerre series II. B Muckenhoupt, Trans Amer. Math. Soc. 147B. Muckenhoupt, Mean convergence of Hermite and Laguerre series II, Trans Amer. Math. Soc. 147 (1970) 433-460. Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter. C Markett, Anal. Math. 8C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982) 19-37. Cohen type inequalities for Jacobi, Laguerre and Hermite expansions. C Markett, Siam J. Math. Anal. 14C. Markett, Cohen type inequalities for Jacobi, Laguerre and Hermite expansions, Siam J. Math. Anal. 14 (1983) 819-833. On convergence and divergence of Fourier-Bessel series. K Stempak, Electron. Trans. Numer. Anal. 14K. Stempak, On convergence and divergence of Fourier-Bessel series, Electron. Trans. Numer. Anal. 14 (2002) 223-235. . G Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23Amer. Math. Soc., Providence R.IFourth EditionG. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. vol. 23, Amer. Math. Soc., Providence R.I., 1975. Fourth Edition.
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[ "Stochastic Robustness Interval for Motion Planning with Signal Temporal Logic", "Stochastic Robustness Interval for Motion Planning with Signal Temporal Logic" ]
[ "Roland B Ilyes ", "Qi Heng Ho ", "Morteza Lahijanian " ]
[]
[]
In this work, we present a novel robustness measure for continuous-time stochastic trajectories with respect to Signal Temporal Logic (STL) specifications. We show the soundness of the measure and develop a monitor for reasoning about partial trajectories. Using this monitor, we introduce an STL sampling-based motion planning algorithm for robots under uncertainty. Given a minimum robustness requirement, this algorithm finds satisfying motion plans; alternatively, the algorithm also optimizes for the measure. We prove probabilistic completeness and asymptotic optimality of the motion planner with respect to the measure, and demonstrate the effectiveness of our approach on several case studies.
10.48550/arxiv.2210.04813
[ "https://export.arxiv.org/pdf/2210.04813v2.pdf" ]
252,780,090
2210.04813
1a9690ed9a22942e66e2f6800e8e4baf397ac9dc
Stochastic Robustness Interval for Motion Planning with Signal Temporal Logic 28 May 2023 Roland B Ilyes Qi Heng Ho Morteza Lahijanian Stochastic Robustness Interval for Motion Planning with Signal Temporal Logic 28 May 2023 In this work, we present a novel robustness measure for continuous-time stochastic trajectories with respect to Signal Temporal Logic (STL) specifications. We show the soundness of the measure and develop a monitor for reasoning about partial trajectories. Using this monitor, we introduce an STL sampling-based motion planning algorithm for robots under uncertainty. Given a minimum robustness requirement, this algorithm finds satisfying motion plans; alternatively, the algorithm also optimizes for the measure. We prove probabilistic completeness and asymptotic optimality of the motion planner with respect to the measure, and demonstrate the effectiveness of our approach on several case studies. I. INTRODUCTION In recent years, Temporal Logics (TLs) [1] have been increasingly employed to formalize complex robotic tasks. These logics allow precise description of properties over time by combining Boolean logic with temporal operators. A popular choice is Linear TL [2], where time is treated as linear and discrete. However, robotic systems operate in continuous time, and their tasks often include properties in dense time. Signal TL (STL) [3] is a variant of TL that provides the means for expressing such tasks with evaluations over continuous signals (trajectories). For motion planning, however, STL introduces both computational and algorithmic challenges precisely due to the same reason that makes it powerful, i.e., reasoning in continuous time. This challenge is exacerbated in real-world robotics, where uncertainty cannot be avoided. Hence, motion planners must reason about the robustness of plans by accounting for uncertainty. This requires a proper notion of a robustness measure. STL does in fact admit such a measure but only for deterministic trajectories [4]. No such a measure is known for stochastic systems. This paper takes on this challenge and aims to develop a stochastic robustness measure for STL with the purpose of using it for efficient motion planning with robustness guarantees. Consider the following mission for the robot in Fig. 1: Go to the charger in the next 10 minutes. But, if you traverse a puddle of water, stay away from the charger until you dry off on the carpet within 3 minutes. Such a specification is easily captured in STL (ϕ 3 in (14)). Two sample trajectories of an uncertain robot are shown in Fig. 1, where the ellipses represent the 90% confidence contours around the nominal trajectories. Here, both trajectories appear to satisfy the temporal aspects of the specification. However, their spatial robustness is different with regards to their uncertainty. Note (14). Trajectory 1 has a StoRM of 0.96 and Trajectory 2 has a StoRM of 0.71. that, even though the nominal component of Trajectory 2 gets to the charger without passing through the water puddle, its uncertainty ellipses overlap with the puddle and walls, making it less robust than Trajectory 1 with respect to the task. A capable motion planner must be able to reason about the robustness of these trajectories with respect to both temporal and spatial aspects of the STL task algorithmically. Much of the literature for control synthesis for STL specifications has been focused on deterministic systems. Success has been found using optimization-based techniques [5]- [8], control barrier function approaches [9], and samplingbased methods [10], [11]. The sampling-based approaches are particularly suitable for robotics applications given their efficiency and scalability. None of those methods, however, account for stochasticity in the robot's dynamics. Reasoning about systems under uncertainty with continuous-time TL specifications is a rapidly developing topic. Recent works [12]- [17] introduce new TL as extensions to STL that incorporate uncertainty in the logic itself. Nevertheless, they do not define a robustness measure for STL. Other works [13], [18] use STL and reason about the distribution of the STL robustness measure over the realizations of stochastic trajectories. This measure, however, is defined with respect to deterministic trajectories and ignores the knowledge of the system's uncertainty. In this work, we develop a novel measure of the robustness of continuous-time stochastic trajectories with respect to STL specifications. We refer to it simply as the Stochastic Robustness Measure (StoRM). StoRM allows us to quantify how well a stochastic trajectory satisfies a specification. The measure is based on the recursive evaluation of the Stochastic Robustness Interval (StoRI) according to the semantics of STL over continuous stochastic trajectories and the probability measure. We also propose a technique to compute the StoRI of partial trajectories, which allows us to reason about (monitor) the satisfaction robustness of the STL properties on an evolving stochastic trajectory. Next, we present a sampling-based algorithm that produces motion plans that can (i) satisfy a user-defined bound on StoRM, or (ii) asympotically optimize for StoRM. We prove the theoretical properties of the algorithm under StoRM constraints, and demonstrate the effectiveness of our measure and planner on a variety of STL formulas in several case studies. In summary, the contributions of this work are four-fold: (i) a novel measure (StoRM) to quantify the robustness of stochastic trajectories against STL specifications, (ii) a new monitor to compute the robustness of partial trajectories, (iii) a probabilistically-complete planning algorithm that satisfies STL specifications with StoRM constraints for systems under uncertainty and asymptotically optimizes for StoRM, and (iv) a series of case studies and benchmarks that reveal properties of StoRM and performance of the planner. II. SYSTEM SETUP Consider a robotic system whose dynamics are described by a linear stochastic differential equation (SDE): dx(t) = (Ax(t) + Bu(t))dt + Gdw(t),(1) where x ∈ X ⊂ R n is the state, u ∈ U ⊂ R m is the control input, w(·) is an r-dimensional Wiener process (Brownian motion) with diffusion matrix Q ∈ R r×r representing noise, A ∈ R n×n , B ∈ R n×m , and G ∈ R n×r . The initial state of the robot is x(0) = x 0 . The solution to the SDE in (1) is a continuous-time Gaussian process [19], i.e., x(t) ∼ b t = N (x(t), P (t)), where b t , called the belief of x(t), is a normal distribution with meanx(t) ∈ X and covariance P (t) ∈ R n×n . The evolution of b t is governed by: x(t) = Ax(t) + Bu(t),(2)P (t) = AP (t) + P (t)A T + GQG T(3) with initial conditionsx(0) = x 0 and P (0) = 0. Note that if, instead of a deterministic initial state, the robot has initial uncertainty described by a Gaussian distribution, only the initial conditionsx(0) and P (0) change. Let T ∈ R ≥0 be a time duration. Then, given a controller u : [0, T ] → U , a belief (stochastic) trajectory b over time window [0, T ] for the robotic system (1) can be computed (predicted) using (2) and (3). An execution of this controller on the robotic system, called a realization or sample of b, is a state trajectory x over time duration [0, T ]. We are interested in properties of System (1) with respect to a set of linear predicates defined in state space X. Let H = {h 1 , h 2 , ..., h l } be a given set of functions where h i : R n → R is a linear function for every 1 ≤ i ≤ l. Then, the set of predicates M = {µ 1 , ..., µ l } is defined on H such that ∀i ∈ {1, ..., l}, the Boolean values of µ i : X → {⊤, ⊥} is determined by the sign of function h i as: µ i (x) = ⊤ if h i (x) ≥ 0 ⊥ if h i (x) < 0.(4) To express desired properties of the robot with respect to the set of predicates M , we use signal temporal logic (STL) [3]. STL is a logic that allows specification of realtime temporal properties, and is therefore well-suited for continuous-time systems such as the one in (1). Definition 1 (STL Syntax). The STL Syntax is recursively defined by : φ := ⊤ | µ | ¬φ | φ ∧ φ | φ U I φ where µ ∈ M , and I = a, b is a time interval with ∈ {(, [}, ∈ {), ]}, a, b ∈ R ≥0 and a < b < ∞. Notations ⊤, ¬, and ∧ are the Boolean "true," "negation," and "conjunction," respectively, and U denotes the temporal "until" operator. The temporal operator eventually (♦) is defined as ♦ I φ ≡ ⊤U I φ and the operator globally is defined as I φ ≡ ¬♦ I ¬φ. Definition 2 (STL Semantics). The semantics of STL is defined over a state trajectory realization x at time t as: (x, t) |= ⊤ ⇐⇒ ⊤ (x, t) |= µ ⇐⇒ h(x(t)) ≥ 0 (x, t) |= ¬φ ⇐⇒ (x, t) |= φ (x, t) |= φ 1 ∧ φ 2 ⇐⇒ (x, t) |= φ 1 ∧ (x, t) |= φ 2 (x, t) |= φ 1 U a,b φ 2 ⇐⇒ ∃t ′ ∈ t + a, t + b s.t. (x, t ′ ) |= φ 2 ∧ ∀t ′′ ∈ [t, t ′ ], (x, t ′′ ) |= φ 1 where |= denotes satisfaction. A state trajectory x satisfies an STL formula φ if (x, 0) |= φ. In addition to the Boolean semantics, STL admits quantitative semantics [1]. This is traditionally defined as a robustness metric [4] based on Euclidian distance. This metric is well-defined for deterministic systems. For a stochastic process such as System (1), however, evaluation of the satisfaction of a belief trajectory is not straightforward. In this work, we aim to develop an appropriate measure of robustness to evaluate the satisfaction of STL formulas by a belief trajectory. Specifically, we are interested in using this measure for robust motion planning for System (1). Therefore, the robustness measure must be appropriate for planning, namely sampling-based motion planning algorithms. This entails that the measure must be able to provide useful information for complete as well as partial trajectories. In the next section, we introduce such a measure. Then, we present a motion planning algorithm that asymptotically optimizes for this measure. III. STOCHASTIC ROBUSTNESS MEASURE A popular method of measuring robustness of belief trajectories in motion planning is based on the notion of chance constraints [20]- [22]. Chance constraints require that the probability of constraint violation (e.g., avoiding obstacles) not exceed some prescribed value. Motion planners typically enforce this chance constraint at each time step. Such an approach is difficult to extend to STL formulas. This is primarily due to temporal operators and their time intervals, e.g., the probability of violating ♦ I φ only needs to be below a prescribed value at one time step in I, but it is not clear at which time step to enforce this. Some approaches find success by generating constraints over time windows, and allocating risk of violating the constraints among time steps [12], [13]. However, those methods are limited to reasoning about discrete-time trajectories. This is in conflict with the fundamental idea of STL, which is expressing properties over real-valued continuous-time intervals. Furthermore, those approaches only provide a qualitative (boolean) judgement of a trajectory's satisfaction with respect to a chance constraint. In contrast, we seek to define a measure that provides a quantitative judgement of a trajectory's satisfaction. This is analogous to robustness for deterministic STL that provides a quantitative measure beyond the qualitative Boolean semantics. Such a measure can then be extended to reason about partial trajectories, and hence, is advantageous for iterative methods for planning such as sampling-based algorithms. To illustrate what such a measure might convey, consider the following example. Example 1. Consider the formulas φ 1 = I µ and φ 2 = ♦ I µ. In the case of the operator, which states that a property must hold for all time t ∈ I, a quantitative measure of robustness could be characterized by the point with the lowest probability of satisfying µ in I, i.e., min t∈I P (h(x(t)) ≥ 0), where h is the linear function that µ is defined on. If the probability of violation at that point is below a certain threshold, then it is also below that threshold at every other point. Similarly, for the ♦ operator, which states that a property must hold for a time point t ∈ I, we could look at the point with the highest probability of satisfying µ, i.e., max t∈ a,b P (h(x(t)) ≥ 0). If the probability of violation at that point is above a certain threshold, then formula φ 2 is satisfied. A quantitative measure of stochastic robustness must incorporate this conflicting treatment of temporal operators. This intuition guides the development of the Stochastic Robustness Interval (StoRI) as defined below. Definition 3 (Stochastic Robustness Interval). The Stochastic Robustness Interval (StoRI) of a belief trajectory b over time window [0, T ] with respect to an STL formula φ is a functional f (φ, b): f (φ, b) = [f ↓ (φ, b), f ↑ (φ, b)] such that 0 ≤ f ↓ (φ, b) ≤ f ↑ (φ, b) ≤ 1. For a t ∈ [0, T ], let b t be the time-shifted suffix of trajectory b such that for all t ′ ∈ [0, T − t], b t (t ′ ) = b(t + t ′ ). Then, lower bound f ↓ (φ, b) and upper bound f ↑ (φ, b) are recursively defined by: f ↓ (⊤, b) = f ↑ (⊤, b) = 1, f ↓ (µ, b) = f ↑ (µ, b) = P (h(x(0)) ≥ 0), f ↓ (¬φ, b) = 1 − f ↑ (φ, b), f ↑ (¬φ, b) = 1 − f ↓ (φ, b), f ↓ (φ 1 ∧ φ 2 , b) = max f ↓ (φ 1 , b) + f ↓ (φ 2 , b) − 1, 0 , f ↑ (φ 1 ∧ φ 2 , b) = min f ↑ (φ 1 , b), f ↑ (φ 2 , b)}, f ↓ (φ 1 U a,b φ 2 , b) = max t∈ a,b max f ↓ (φ 2 , b t ) + min t ′ ∈[0,t] f ↓ (φ 1 , b t ′ ) − 1, 0 , f ↑ (φ 1 U a,b φ 2 , b) = max t∈ a,b min f ↑ (φ 2 , b t ), min t ′ ∈[0,t] f ↑ (φ 1 , b t ′ ) . We define the Stochastic Robustness Measure (StoRM) to be the lower bound of StoRI. Definition 4 (Stochastic Robustness Measure). The Stochas- tic Robustness Measure (StoRM) of a belief trajectory b with respect to STL formula φ is the lower bound of the StoRI, i.e, f ↓ (φ, b). The StoRI is an interval that aims to quantify how robustly a belief trajectory b satisfies an STL formula φ. The StoRM of a belief trajectory is the lower bound of its StoRI. A trajectory always satisfies a Boolean ⊤, so the StoRI for φ = ⊤ is [1, 1] for every trajectory. If φ = µ is a linear predicate, both bounds of the StoRI are the probability that the state at time zero x(0) satisfies the linear predicate. Works [20] [23] outline an efficient way to calculate this probability for Gaussian distributions. However, we stress that the StoRI is defined for general belief distributions, and also applies to non-Gaussian beliefs. The StoRI of the negation of a formula ¬φ derives from the Unit Measure Axiom of Probability [24]. Note that the lower bound of the StoRI of ¬φ depends on the upper bound of the StoRI of φ, and vice-versa. The StoRI for a conjunction of two formulas φ 1 ∧ φ 2 is inspired by the lower and upper bounds on the probability of a conjunction of two events. These are the Boole-Fréchet inequalities for logical conjunction [25], which make no assumptions on the independence on the events, and hence are general. Note that the measure seeks to quantify both upper and lower bounds because the upper bound is conservative when the specification seeks to enforce 'avoidance' of the conjunction, but the lower bound is conservative when it seeks to enforce the 'occupancy' of the conjunction. The StoRI of φ 1 U a,b φ 2 is less straightforward. Intuitively, it aims to report the StoRM at the time t ∈ a, b that best balances the point-wise probabilities that φ 1 hold for all t ′ ∈ [0, t] and the probability that φ 2 hold at t. Note that by following Definition 3, the obtained StoRI for eventually and globally are: f (♦ a,b φ, b) = max t∈ a,b f ↓ (φ, b t ), max t∈ a,b f ↑ (φ, b t ) (5) f ( a,b φ, b) = min t∈ a,b f ↓ (φ, b t ), min t∈ a,b f ↑ (φ, b t ) (6) These intervals are fully inline with the intuition of the measure as discussed in Example 1. (7) Example 2. Consider the environment in Fig. 2 and formula An important property of the StoRI is that a trajectory which has a StoRM of 1 satisfies STL formula φ with probability 1, as stated in the following theorem, which illustrates soundness of StoRM. Proof. Consider any realization x of b. If f ↓ (φ, b) = 1 =⇒ (x, 0) |= φ and f ↑ (φ, b) = 0 =⇒ (x, 0) |= φ, the StoRI follows STL semantics as defined in Definition 2, which defines the satisfaction of a deterministic trajectory (realizations). This can be shown recursively as follows. ϕ 1 = ¬(x ≥ 1 ∧ y ≥ 2 ∧ x ≤ 2)U I (x ≥ 3 ∧ y ≥ 2). (7) For boolean ⊤, the StoRI of b trivially follows STL semantics: f ↓ (⊤, b) = 1 =⇒ (x, 0) |= ⊤, For predicate µ, the StoRI of b follows STL semantics: f ↓ (µ, b) = 1 ⇐⇒ P (h(x(0)) ≥ 0) = 1 ⇐⇒ h(x(0)) ≥ 0 =⇒ (x, 0) |= µ, For boolean negation ¬φ, the StoRI for b follows STL semantics: f ↓ (¬φ, b) = 1 ⇐⇒ 1 − f ↑ (φ, b) = 1 ⇐⇒ f ↑ (φ, b) = 0 =⇒ (x, 0) |= φ =⇒ (x, 0) |= ¬φ, For boolean conjunction φ 1 ∧ φ 2 , the StoRI follows STL semantics: f ↓ (φ 1 ∧ φ 2 , b) = 1 =⇒ f ↑ (φ 1 ∧ φ 2 , b) = 1 ⇐⇒ min f ↓ (φ 1 , b), f ↓ (φ 2 , b) = 1 ⇐⇒ f ↓ (φ 1 , b) = 1 ∧ f ↓ (φ 2 , b) = 1 =⇒ (x, 0) |= φ 1 ∧ (x, 0) |= φ 2 =⇒ (x, 0) |= (φ 1 ∧ φ 2 ), and finally, for temporal Until φ 1 U I φ 2 , the StoRI for b reduces to STL semantics: f ↓ (φ 1 U I φ 2 , b) = 1 =⇒ f ↑ (φ 1 U I φ 2 , b) = 1 ⇐⇒ max t∈I min{f ↑ (φ 2 , b t ), min t ′ ∈[0,t] f ↑ (φ 1 , b t ′ )} = 1 =⇒ ∃t ∈ I s.t. min f ↑ (φ 2 , b t ), min t ′ ∈[0,t] f ↑ (φ 1 , b t ′ ) = 1 ⇐⇒ ∃t ∈ I s.t. f ↑ (φ 2 , b t ) = 1 ∧ min t ′ ∈[0,t] f ↑ (φ 1 , b t ′ ) = 1 =⇒ ∃t ∈ I s.t.(x, t) |= φ 2 ∧ ∀t ′ ∈ [0, t]f ↑ (φ 1 , b t ′ ) = 1 =⇒ ∃t ∈ I s.t.(x, t) |= φ 2 ∧ ∀t ′ ∈ [0, t](x, t ′ ) |= φ 1 =⇒ (x, 0) |= (φ 1 U I φ 2 ) Therefore, a belief trajectory b having a StoRM of 1 with respect to STL formula φ implies that every non-zero measure realization x of b satisfies the STL specification. A. StoRI Monitor StoRI is defined for a given belief trajectory. It does not account for what could happen if we extend the trajectory. However, in some cases (such as planning), we are interested in extending trajectories to achieve a higher StoRM. Hence, we need a monitor for StoRI, which assumes the given trajectory is to be extended. The monitor must account for all the possible suffixes of the trajectory and how they might change the StoRM. In this section, we present a monitor for the StoRI with respect to a partial belief trajectory, that acts to bound the achievable StoRI of any extensions of it. Definition 5 (StoRI Monitor). The Stochastic Robustness Interval (StoRI) Monitor of a partial belief trajectory b t over time window [0, t] with respect to an STL formula φ is a functionalf (φ, b t ): f (φ, b t ) = [f ↓ (φ, b t ),f ↑ (φ, b t )] such that 0 ≤f ↓ (φ, b t ) ≤f ↑ (φ, b t ) ≤ 1, wherẽ f (⊤, b t ) = f (⊤, b t ) f (µ, b t ) = f (µ, b t ) f ↓ (¬φ, b t ) = 1 −f ↑ (φ, b t ), f ↑ (¬φ, b t ) = 1 −f ↓ (φ, b t ), f ↓ (φ 1 ∧ φ 2 , b t , t) = max f ↓ (φ 1 , b t ) +f ↓ (φ 2 , b t ) − 1, 0 , f ↑ (φ 1 ∧ φ 2 , b t , t) = min f ↑ (φ 1 , b t ),f ↑ (φ 2 , b t )}, f ↓ (φ 1 U a,b φ 2 , b t ) = f ↓ (φ 1 U a,b φ 2 , b t ) if t ≥ a 0 otherwise, f ↑ (φ 1 U a,b φ 2 , b t ) =        f ↑ (φ 1 U a,b φ 2 , b t ) if t ≥ b max f ↑ (φ 1 U a,b φ 2 , b t ), min t ′ ∈[0,b] f ↑ (φ 1 , b t ′ t ) if t ∈ [0, b) 1 otherwise. The differences between the StoRI and StoRI Monitor arise in the temporal operators. These operators seek to bound possible future robustness, and also quantify the robustness of the behavior already seen. This is apparent in the StoRI Monitor with respect to the ♦ and operators: f (♦ a,b φ, b t ) =        f (♦ a,b φ, b t ) if t ≥ b max t∈ a,b f ↓ (φ, b t t ), 1 if a ≤ t < b 0, 1 if t < ã f ( a,b φ, b t ) =        f ( a,b φ, b t ) if t ≥ b 0, min t∈ a,b f ↑ (φ, b t t ) if a ≤ t < b 0, 1 if t < a The case when a ≤ t < b is of particular interest. In the case of the operator, the StoRI is upper bounded by the best point in the partial trajectory, but is lower bounded by zero to account for possible future violation. In the case of the ♦ operator, the StoRI is lower bounded by the best point in the partial trajectory, but is upper bounded by one to account for possible future "perfect" satisfaction. The following theorem proves the correctness of StoRI Monitor by showing that it always subsumes StoRI. Proof. Without loss of generality, consider φ = ψ 1 U [a,b] ψ 2 . From Definition 5, the lower bound of the monitor of φ is: f ↓ (φ 1 U a,b φ 2 , b t ) = = f ↓ (φ 1 U a,b φ 2 , b t ) = max t∈ a,b max f ↓ (φ 2 , b t t ) + min t ′ ∈[0,t] f ↓ (φ 1 , b t ′ t ) − 1, 0 , for t ≥ a, which monotonically increases with respect to t. For t ≥ a, the monitor takes on a value of 0. Because 0 is the lowest value the StoRI Monitor can take, it will never decrease with respect to t. Therefore, the lower bound of the monitor monotonically increases with respect to t. Proof. We show that for any belief trajectory b, the value of the StoRI Monitor's upper bound is greater than or equal to the value of the StoRI's upper bound, i.e., f ↑ (φ 1 U a,b φ 2 , b) = = max f ↑ (φ 1 U a,b φ 2 , b), min t ′ ∈[0,b] f ↑ (φ 1 , b t ′ ) ≥ f ↑ (φ 1 U a,b φ 2 , b) for t ≥ 0 andf ↑ (φ 1 U a,b φ 2 , b) = 1 ≥ f ↑ (φ 1 U a,b φ 2 , b) for t ≥ 0, and the value of the StoRI Monitor's lower bound is less than or equal to the value of the StoRI's lower bound, i.e., IV. MOTION PLANNING ALGORITHM This section presents two sampling-based motion planners that utilize StoRI and the StoRI Monitor. The first planner finds solutions that satisfy a given StoRM constraint, and the second finds solutions that directly optimize for the StoRM. f ↓ (φ 1 U a,b φ 2 , b) = f ↓ (φ 1 U a,b φ 2 , b) for when t ≥ ã f ↓ (φ 1 U a,b φ 2 , b) = 0 ≤ f ↓ (φ 1 U a,b φ 2 , b) for when t ≥ a A. StoRI-A A kinodynamic sampling-based tree planner A grows a motion tree in the state space X according to the robot dynamics through sampling and extension procedures. We generalize planner A to StoRI-A (Alg. 1) to generate plans for System (1) that are guaranteed to satisfy a given lower bound κ ∈ [0, 1] on StoRM with respect to STL formula φ. Algorithm 1: StoRI-A (X, U, φ,x 0 , N, κ) 1 P 0 ← 0 n×n ; 2 V ← {(x 0 , P 0 )}, E ← ∅; 3 G ← {V, E}; 4 for N iterations do 5x rand , t rand ,x near , P near ← sample(X, V, T ); 6x new , P new ← Extend(x near , P near , U ); 7 iff ↑ (φ, (x 0 , P 0 )...(x new , P new )) > κ then 8 V ← V ∪ {(x new , P new )} ; 9 E ← E ∪ {[(x near , P near ), (x new , P new )]}; 10 if f ↓ (φ, (x 0 , P 0 )...(x new , P new )) > κ then 11 return (x 0 , P 0 )...(x new , P new ); 12 return ∅ The algorithm first initializes the tree G with the belief of x(0). Each node in this tree is a tuple (x, P ) of the mean and covariance of the distribution that describes the state. At every iteration, the algorithm samples a random statex rand and time t rand and computes the nearest existing node (x near , P near ). t rand is sampled from the timehorizon of the STL formula as defined in [26], and the nearest node is selected using a distance metric that accounts for both state distance x rand −x near and time distance |t rand − t near |. Second, a random control input u ∈ U and time duration t is sampled and propagate the system from (x near , P near ) to generate a new belief node (x new , P new ). Third, the StoRI Monitor of the partial belief trajectory b t = (x 0 , P 0 )(x 1 , P 1 ) · · · (x new , P new ) is computed for the formula φ. If the StoRI Monitor has an upper bound f ↑ (φ, b t ) ≤ κ, b t has already violated the STL Specification and the new node is discarded. Otherwise, we add (x new , P new ) and the edge (x near , P near ), ( x new , P new ) to G. Finally, b t is a solution if the StoRM f ↓ (φ, b t ) > κ. This process repeats until a solution is found or for a maximum of N iterations. Theorem 3 (Probabilistic Completeness). Planner StoRI- A in Alg. 1 is probabilistically complete if the underlying planner A is probabilistically complete. Proof. StoRI-A mimics the behavior of planner A, only modifying its validity check. From Theorem 2, this modification only rejects nodes that are guaranteed to violate the STL Specification φ. Therefore, if a solution exists, StoRI-A will find it with probability 1 as N → ∞. B. Asymptotically Optimal StoRI-A (AO-StoRI-A) Since the StoRM allows us to compute a quantitative value of robustness, we can also optimize for the StoRM of a belief trajectory in a sampling-based motion planner. This is enabled through the AO-A meta-algorithm in [27], by repeatedly calling StoRI-A with increasing κ bounds by setting κ as the StoRM of the previous solution at each iteration. Algorithm 2: AO-StoRI-A (X, U, φ,x 0 , N )) 1 κ ← 0; 2 while N = 0 do 3 b, n ←StoRI-A (X, U, φ,x 0 , N, κ)); 4 κ ← f ↓ (φ, b); 5 N ← N − n; 6 return b if solution found, No Solution otherwise V. EVALUATIONS We evaluate efficacy and efficiency of the proposed measure and planners subject to a variety of STL specifications. The algorithms are implemented with A = RRT [28], i.e., StoRI-RRT. All algorithms are implemented in the Open Motion Planning Library (OMPL) [29], and computations were performed on 3.9 GHz CPU and 64 GB of RAM. The implementation is readily available on GitHub [30]. We considered a noisy second-order unicycle system whose stochastic dynamics are given by: dx = v cos(θ)dt + dw, dy = v sin(θ)dt + dw, dθ = u ω dt + dw, dv = u a dt+dw where u a and u ω are the acceleration and steering angle inputs. We linearized the dynamics according to the feedback linearization in [31]. We evaluate the StoRM and StoRI Monitor using sequences of points sampled from the continuous trajectory with dt = 0.15 seconds, and propagate dynamics and uncertainty according to the process in [32]. The beliefs at the sampled times are described by: x k+1 = A ′x k + B ′ u k ,(8)P k+1 = A ′ P k A ′T + Q ′(9) where A ′ =     Our sample procedure uses the distance metric x rand − x near + 0.25 × |t rand − t near |. We considered environments in Figs. 2, 3a, and 1 respectively with STL formulae ϕ 1 in (7), ϕ 2 = ¬OU [0,10] A ∧ ¬OU [0,10] B,(13)O = x ≥ 1.5 ∧ x ≤ 3.5 ∧ y ≥ −0.5 ∧ y ≤ 0.5, A = x ≥ 2 ∧ x ≤ 3 ∧ y ≥ 1, B = x ≥ 2 ∧ x ≤ 3 ∧ y ≤ −1, ϕ 3 = (Puddle → ¬ChargeU [0,3] Carpet)U [0,10] Charge, (14) Puddle = y ≤ 2.5 ∧ x ≥ 2 ∧ x ≤ 3, Charge = y ≥ 2 ∧ x ≥ 4, Carpet = y ≤ 1 ∧ x ≥ 4. Here, ϕ 2 requires avoiding (black) region O until both regions A and B are visited within 10 minutes in Fig. 3a. ϕ 3 requires the robot to go to the charger within 10 minutes and also that, if it visits the puddle, it must avoid the charger until it visits the carpet within 3 minutes of visiting the puddle in Fig. 1. For all formulas, we also require the robot to remain in the workspace for the duration of the mission. A. Case Study 1 -Computation Time vs StoRI Threshold This case study seeks to analyze the relationship between computation time and the StoRM constraint κ. Specifically, we study how StoRI-RRT performs for formula ϕ 2 , in the environment in Fig. 3a. We ran 100 trials, with a maximum computation time of 300 seconds for each trial. Table I reports the computation time and success rate for different κ values. We see that computation time increases and success rate decreases as the κ threshold increases. This is due to the increased difficulty of satisfying the robustness constraint. Fig. 3a shows two sample trajectories for this specification, where the ellipses represent the 90% confidence bounds of their uncertainty. Trajectory 1 uses κ = 0.9 and goes through the wider opening on the left. In contrast, Trajectory 2 uses κ = 0.5 and finds less robust paths that go through the narrow opening on the right. B. Case Study 2 -Computation Time for Different Formulas This case study compares computation time for different STL formulas. Table II shows the average computation time of StoRI-RRT for the three formulas and environments over 100 trials with a maximum timeout of 300s. We used StoRM bound κ = 0.9. We see that, even with a tight StoRM threshold κ, the algorithm finds solutions with a good success rate and computation time. This shows that the algorithm is generally applicable to all STL formulas. C. Case Study 3 -Asymptotic Optimality In this case study, we analyze the relationship between given computation time and the StoRM of the resulting trajectory. Here, we study how AO-StoRI-RRT performs when planning for formula ϕ 3 , in the environment in Fig. 1. Fig. 3b presents the results of 100 trials. It clearly shows that the solutions asymptotically approach an optimal StoRM. Fig. 1 gives two sample trajectories with different StoRMs. Trajectory 2 is found earlier in the optimization, and makes it through the gap. When the planner further optimizes for the StoRM, however, it favors trajectories like Trajectory 1, that enter the puddle and dry off before going to the carpet. D. Case Study 4 -Simulated Performance This case study seeks to analyze how the StoRM relates to the statistical satisfaction rate (SSR) defined as #satisfying runs total # runs . We do this by simulating the robot's motion plans from StoRI-RRT. Table III compares the StoRI of the belief trajectories against the SSR of the simulated realizations. Each motion plan is simulated 1000 times. We use the Breach Toolbox [26] to evaluate the realizations' satisfaction. These results show correlation between the StoRM and probability of satisfaction. VI. CONCLUSION AND FUTURE WORK This paper proposes a measure, StoRM, for quantifying the robustness of stochastic systems' trajectories with re-spect to STL specifications. We develop a monitor for this measure that reasons about partial trajectories, and use it in a sampling-based motion planner. We show desirable properties of this measure, that the algorithm is probabilistically complete, and that the algorithm asymptotically optimizes for the StoRM. Emprical evaluation demonstrates the measure and algorithm's effectiveness and utility. For future work, we plan to investigate guiding the growth of the motion tree and incorporating measurement uncertainty in the planner. Authors are with the department of Aerospace Engineering Sciences at the University of Colorado Boulder, CO, USA {firstname.lastname}@colorado.edu Fig. 1 : 1Two trajectories for ϕ 3 in Fig. 2 : 2Sample trajectories that have a StoRI of (left-to-right) [0.72, 0.91], [0.52, 0.847], [0, 0] with respect to ϕ 1 in Fig. 2 2shows three belief trajectories, where the ellipses represent the 90% confidence bounds of their uncertainty. The first two trajectories (left and middle) have the same expectation but are subject to different amounts of process noise. When evaluating against ϕ 1 with time interval I = [0, 6], we see both bounds of the StoRI decrease from [0.72, 0.91] for the first trajectory (left) to [0.52, 0.85] for the the second trajectory that has more uncertainty (middle). This trajectory is less likely to clear the obstacle, and also less likely to arrive in the goal region. The third plot (right) shows the first trajectory but evaluated against ϕ 1 with smaller interval I = [0, 4]. The robot does not arrive to goal until after 5 seconds. Hence, the robot fails to satisfy the specification and its StoRI is [0, 0]. Theorem 1 . 1Given belief trajectory b and STL formula φ, if the Stochastic Robustness Measure f ↓ (φ, b) = 1, then every realization x of b satisfies the specification, i.e., x |= φ, except realizations on the set of measure zero. Lemma 1 . 1Let b be a belief trajectory over time window [0, T ] and b t be a prefix of b where t ∈ [0, T ]. Then, given STL formula φ, the upper bound of the StoRI Monitor for b t is monotonically decreasing with respect to t.Proof. The proof is provided in the Appendix.Lemma 2. Let b be a belief trajectory over time window[0, T ] and b t be a prefix of b where t ∈ [0, T ]. Then, given STL formula φ, the lower bound of the StoRI Monitor for b t is monotonically increasing with respect to t. Lemma 3 . 3Let b be a belief trajectory over time window [0, T ]. Then, given STL formula φ, the StoRI Monitor of b subsumes the StoRI of b. Therefore, StoRI Monitor of b subsumes StoRI of b. Theorem 2 . 2Let b be a belief trajectory over time window [0, T ] and b t be a prefix of b where t ∈ [0, T ]. Then, given STL formula φ, the StoRI Monitor for b t subsumes the StoRI of b for allt ∈ [0, T ], i.e., ∀t ∈ [0, T ], f (φ, b) ⊆f (φ, b t ).Proof. From Lemma 3, when t = T , the StoRI Monitor of b t subsumes StoRI of b. From Lemmas 1 and 2, for every t < T , the StoRI Monitor of b t subsumes the StoRI Monitor of b T . Then it follows that the the StoRI monitor of b t subsumes the StoRI of b for all t ≤ T .Example 3. Recall the STL formula in Example 2 and sample trajectories in Fig. 2. The StoRI Monitor of the trajectories at different points in their evolution are shown in the plots in Fig. 2. Note that they subsume the StoRI of the final trajectory and the interval gets smaller with time. Fig. 3 : 3Experiment Results. (a) shows two trajectories where f ↓ (ϕ 2 , b 1 ) = 0.98, f ↓ (ϕ 2 , b 2 ) = 0.67, and (b) shows benchmarking results for Case Study 3. TABLE I : IBenchmarking Results for Case Study 1StoRM Threshold κ Computation Time (s) Success Rate 0.50 60.10 ± 64.87 97% 0.70 71.37 ± 67.48 97% 0.90 83.30 ± 69.35 90% 0.95 92.48 ± 81.56 86% TABLE II : IIBenchmarking Results for Case Study 2, κ = 0.9.Formula Computation Time (s) Success Rate ϕ 1 1.98 ± 2.56 100% ϕ 2 85.95 ± 82.56 90% ϕ 3 44.55 ± 53.89 99% TABLE III : IIIComparison of StoRI against SSRFormula f ↓ (φ, b) (StoRM) f ↑ (φ, b) Satisfaction Rate ϕ 1 0.568 0.568 0.367 ϕ 1 0.755 0.781 0.767 ϕ 1 0.986 0.986 0.985 ϕ 2 0.572 0.589 0.582 ϕ 2 0.873 0.998 0.754 ϕ 2 0.912 0.913 0.889 ϕ 3 0.625 0.677 0.669 ϕ 3 0.876 0.878 0.872 ϕ 3 0.989 0.994 0.985 t ′ ∈[0,t] f ↑ (φ 1 , b t ′ t1 )(22) APPENDIXProof of Lemma 1. The proof is as follows: Consider the value of the StoRI Monitor upper bound of two prefixes b t1 and b t2 , where t 2 ≥ t 1 . By definition 5, the value of the upper bound of the StoRI Monitor is one of 4 terms. We show that, for each case, the value of the upper bound of the StoRI Monitor of b t2 is always less than or equal to the value of the upper bound of the StoRI Monitor of b t1 . It follows then, that the upper bound of the StoRI Monitor monotonically decreases with respect to t. Without loss of generality, consider ψ = φ 1 U [a,b] φ 2 . From Definition 5, the upper bound of the monitor of b t1 with respect to ψ when t 1 ∈ [0, b) is:We now examine the case where the second term is greater than the first, the case where the first term is greater than the second, the case where there is not yet any information relevant to this formula, and the case where the time interval is closed (t 1 ≥ b). Case 1: We first examine the case where the second term is greater, i.e.,The upper bound of the monitor of b t2 with respect to ψ is:We now examine how both of these terms relate to equation 15.Recall, by our assumptions for case 1, term S 1 is less than or equal to the value in equation 15, i.e.,The term S 2 is upper bounded by:From equations 17 and 18, we conclude that:And this value is always less than the bound in equation 15, i.e., minEquations 19 and 20 show that, for case 1, the StoRI Monitor of b t2 with respect to ψ is less than or equal to the StoRI Monitor of b t1 :Case 2: We next examine the case where the first term of the StoRI Monitor of b t1 with respect to ψ is greater, i.e., max t∈ a,t1We now examine how this value relates to both terms in the upper bound of the monitor of b t2 with respect to ψ (equation 16). Case 2.1: We first look at how the first term in equation 16 relates to the bound in 22. This term is again split into S 1 and S 2 . For case 2, we see that S 1 is the same value as equation 22:Futhermore, equation 18 still holds. We know by our assumptions for case 2 that this value is less than or equal to the value in equation 22, i.e.,From equations 23 and 24, we conclude that:≤ max t∈ a,t1Equations 25 and 27 show that, for case 2, the StoRI Monitor of b t2 is less than or equal to the StoRI Monitor of b t1 :Case 3: The third case is when t < 0. This could happen when the U I operator is nested within another temporal operator, resulting in the time-shifted trajectory having a negative time. It means that there are no points in the trajectory relevant to the operator yet. Definition 5 defines the upper bound as 1 for this case. The Monitor itself is defined as having an upper bound less than or equal to 1; the value of the monitor cannot increase past 1. Therefore, for case 3, the StoRI Monitor of b t2 is less than or equal to the StoRI Monitor of b t1 :Case 4: The fourth case is when t 1 ≥ b. 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[ "THE QUANTIZATION OF PROCA FIELDS ON GLOBALLY HYPERBOLIC SPACETIMES: HADAMARD STATES AND MØLLER OPERATORS", "THE QUANTIZATION OF PROCA FIELDS ON GLOBALLY HYPERBOLIC SPACETIMES: HADAMARD STATES AND MØLLER OPERATORS" ]
[ "Valter Moretti s:[email protected] \nDipartimento di Matematica\nUniversità di Trento & INdAM& INFN-TIFPA\nVia Sommarive 14I-38123PovoItaly\n", "Simone Murro [email protected] \nDipartimento di Matematica\nUniversità di Genova & INdAM& INFN\nVia Dodecaneso 3516146GenovaItaly\n", "Daniele Volpe [email protected] \nDipartimento di Matematica\nUniversità di Trento & INdAM& INFN-TIFPA\nVia Sommarive 14I-38123PovoItaly\n" ]
[ "Dipartimento di Matematica\nUniversità di Trento & INdAM& INFN-TIFPA\nVia Sommarive 14I-38123PovoItaly", "Dipartimento di Matematica\nUniversità di Genova & INdAM& INFN\nVia Dodecaneso 3516146GenovaItaly", "Dipartimento di Matematica\nUniversità di Trento & INdAM& INFN-TIFPA\nVia Sommarive 14I-38123PovoItaly" ]
[]
This paper deals with several issues concerning the algebraic quantization of the real Proca field in a globally hyperbolic spacetime and the definition and existence of Hadamard states for that field. In particular, extending previous work, we construct the so-called Møller˚isomorphism between the algebras of Proca observables on paracausally related spacetimes, proving that the pullback of these isomorphisms preserves the Hadamard property of corresponding quasifree states defined on the two spacetimes. Then, we pull-back a natural Hadamard state constructed on ultrastatic spacetimes of bounded geometry, along this˚isomorphism, to obtain an Hadamard state on a general globally hyperbolic spacetime. We conclude the paper, by comparing the definition of an Hadamard state, here given in terms of wavefront set, with the one proposed by Fewster and Pfenning, which makes use of a supplementary Klein-Gordon Hadamard form. We establish an (almost) complete equivalence of the two definitions.
10.1007/s00023-023-01326-w
[ "https://export.arxiv.org/pdf/2210.09278v3.pdf" ]
252,918,072
2210.09278
519a31b71bf8cced8fc390fd778a5410f62c64ce
THE QUANTIZATION OF PROCA FIELDS ON GLOBALLY HYPERBOLIC SPACETIMES: HADAMARD STATES AND MØLLER OPERATORS 1 Jun 2023 Valter Moretti s:[email protected] Dipartimento di Matematica Università di Trento & INdAM& INFN-TIFPA Via Sommarive 14I-38123PovoItaly Simone Murro [email protected] Dipartimento di Matematica Università di Genova & INdAM& INFN Via Dodecaneso 3516146GenovaItaly Daniele Volpe [email protected] Dipartimento di Matematica Università di Trento & INdAM& INFN-TIFPA Via Sommarive 14I-38123PovoItaly THE QUANTIZATION OF PROCA FIELDS ON GLOBALLY HYPERBOLIC SPACETIMES: HADAMARD STATES AND MØLLER OPERATORS 1 Jun 2023Hadamard statesMøller operatorsProca operatorsalgebraic quantum field the- oryglobally hyperbolic manifoldsparacausal deformation MSC 2020: Primary: 81T0581T20; Secondary: 58J4058J4558J47 This paper deals with several issues concerning the algebraic quantization of the real Proca field in a globally hyperbolic spacetime and the definition and existence of Hadamard states for that field. In particular, extending previous work, we construct the so-called Møller˚isomorphism between the algebras of Proca observables on paracausally related spacetimes, proving that the pullback of these isomorphisms preserves the Hadamard property of corresponding quasifree states defined on the two spacetimes. Then, we pull-back a natural Hadamard state constructed on ultrastatic spacetimes of bounded geometry, along this˚isomorphism, to obtain an Hadamard state on a general globally hyperbolic spacetime. We conclude the paper, by comparing the definition of an Hadamard state, here given in terms of wavefront set, with the one proposed by Fewster and Pfenning, which makes use of a supplementary Klein-Gordon Hadamard form. We establish an (almost) complete equivalence of the two definitions. Introduction The (algebraic) quantization of a quantum field propagating in a globally hyperbolic curved spacetime pM, gq [7,56] and the definition of meaningful quantum states has been and continues to be at the forefront of scientific research. Linearized theories are the first step of all perturbative procedures, so the definition of physically meaningful states for linearized field equations is an important task. Gaussian, also known as quasifree, states ω : A Ñ C on the relevant CCR or CAR unital˚algebra A of observables of a given quantum field are an important family of (algebraic) states [40]. They are completely defined by assigning the two-point function, a bi-distribution ω 2 px, yq on the sections used to smear the field operator. A crucial physical requirement on ω is the so-called Hadamard condition, which is needed, in particular, for defining locally-covariant renormalization procedures of Wick polynomials [18,40] and for the mathematical formulation of locally covariant perturbative renormalization in quantum field theory [52]. Generalized Klein-Gordon vector fields All the notations and conventions used in this section to briefly summarize our results will be defined precisely later. For a charged (i.e. complex) Klein-Gordon field A, possibly vector-valued, the construction of Hadamard states amounts to finding distributional bi-solutions Λ2 px, yq of the Klein-Gordon equation NA " 0 describing the two-point functions 1 ωpâpfqâ˚pf 1 qq " ż MˆM Λ2 px, yq cd γ ca pxqγ db pyqf a pxqf 1 b pyqvol g b vol g ": Λ2 pf, f 1 q , and ωpâ˚pf 1 qâpfqq " ż MˆM Λ2 px, yq cd γ ca pxqγ db pyqf 1 a pxqf b pyqvol g b vol g ": Λ2 pf,f 1 q . Above, the generators of the CCR˚-algebra of the Proca fieldâpfq andâ˚pf 1 q "âpfq˚are the (algebraic) field operators smeared with smooth compactly supported complex sections f, f 1 of the relevant complex vector bundle E Ñ M. That bundle is equipped with a non-degenerate Hermitian 2 fiberwise scalar product (not necessarily positive) γ. In case of the standard complex vector Klein-Gordon field over pM, gq constructed out the 1-form Hodge D'Alembertian or the Levi-Civita vector D'Alembertian, the vector bundle E is the one of smooth 1-forms T˚M C :" T˚M`iT˚M and the Hermitian scalar product γ is the indefinite one induced by the metric g in T˚M C , i.e., γ " g 7 . In the general case, a Klein-Gordon operator N is by definition a second-order operator on the smooth sections of E which is normally hyperbolic [3,4]: its principal symbol σ N satisfies σ N pξq "´g 7 pξ, ξq Id E for all ξ P T˚M, where Id E is the identity automorphism of E. N is also required to be formally selfadjoint with respect to the Hermitian scalar product (generally non-positive!) induced on the space of complex sections f by γ and the volume form vol g , pf|gq : " ż M f a pxqγ ab pxqg b pxqvol g pxq . The scalar complex Klein-Gordon field is encompassed by simply taking C as canonical fiber of E and using the trivial positive scalar product. The requirements on the bi-distributions Λ2 are, where G N is the causal propagator of N, p1q N x Λ2 px, yq " Λ2 px, yqN y " 0 and Λ2´Λ2 "´iG N ; p2q Λ2 pf, fq ě 0 , where Λ˘pf, fq " 0 implies f " Ng for a compactly supported section g; p3q W F pΛ2 q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u . The second part of (1) corresponds to the canonical commutations relations, the first part is the "on-shell" condition, while condition (2) is the positivity requirement on two-functions. Then, the Gelfand-Naimark-Segal construction gives rise to a˚-representation of A g in terms of densely defined operators in a Hilbert space which, as a consequence of the above requirements (1) and (2) and the Wick rule, is a Fock space. Here ω is the expectation value referred to vacuum state and the action of the image of the representation on the vacuum state produces the dense invariant domain of the representation itself. Requirement (3) is the celebrated Hadamard condition (also known as the microlocal spectrum condition) which ensures the correct short-distance behavior of the n-point functions of the field. This condition has a long history which can be traced back to [22], passing to [41] and [50,51] (see [40] for a review). It plays a crucial role in various contexts of quantum field theory in curved spacetime. In particular, but not only, in perturbative renormalization and semiclassical quantum gravity. More recently, Gérard and Wrochna in [25,27], proved that condition (1)- (3) can be controlled at the same time by using methods of pseudodifferential calculus in spacetimes of bounded geometry (see also the subsequent papers [28][29][30][31][32]). When dealing with real quantum fields, as in this work, for instance the Klein-Gordon real vector field A, a single bidistribution ω 2 px, yq is sufficient to define a quasifree state ω: ωpâpfqâpf 1 qq " ż MˆM ω 2 px, yq cd γ ca pxqγ db pyqf a pxqf 1 b pyqvol g b vol g whereâpfq "âpfq˚is the (algebraic) field operators smeared with smooth real compactly supported sections f of a relevant real vector bundle E Ñ M, equipped with a fiberwise real symmetric nondegenerate (but not necessarily positive) scalar product γ. As before, a Klein-Gordon operator N is by definition a second-order differential operator on the smooth sections of E which is normally hyperbolic (same definition as for the complex case) and formally selfadjoint with respect to the real symmetric scalar product (not necessarily positive) pf|gq :" ż M f a pxqγ ab pxqg b pxqvol g pxq . In the case of the standard real vector Klein-Gordon field (constructed out of the Hodge D'Alembertian or the Levi-Civita D'Alembertian) the bundle is exactly T˚M, equipped with a real symmetric non-degenerate but indefinite fiberwise scalar product induced by the metric g on T˚M, namely γ " g 7 . The theory of the scalar real Klein-Gordon field is encompassed simply by taking R as canonical fiber of E with trivial positive scalar product. In the real case, defining the symmetric bilinear form µpf, f 1 q :" 1 2 pω 2 pf, f 1 q`ω 2 pf 1 , fqq, conditions (1)-(3) are replaced by p1q 1 N x ω 2 px, yq " ω 2 px, yqN y " 0 and ω 2 pf, f 1 q´ω 2 pf 1 , fq " iG N pf, f 1 q ; p2q 1 µ 2 pf, fq ě 0 where µpf, fq " 0 implies f " Ng for a compactly supported section g; p3q 1 |G N pf, f 1 q| 2 ď 4µpf, fq µpf 1 , f 1 q ; p4q 1 W F pω2 q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u . The apparently new continuity condition (3)' for the real case is actually embodied in the positivity condition (2) for the complex case [23]. As a matter of fact (2)' and (3)' together give rise to positivity of the whole state ω on A g induced by ω 2 in the real case. Once again, the GNS construction gives rise to a representation of the (complex) unital˚-algebra A g generated by the field operatorsâpfq exactly as in the complex case. Since the Klein-Gordon equations are normally hyperbolic, not only they are Green hyperbolic so that the Green operators GP and the causal propagator G P " GP´GṔ can be therefore defined, but the Cauchy problem is also automatically well posed [3,4]. An important implication of this fact is that the two-point function of a quasifree state can be defined as a Hermitian or real bilinear form -in the complex and real case respectively -on the Cauchy data of solutions of the Klein-Gordon equation (e.g., see [48]). We follow this route in the present paper and, to this end, we will translate (1)-(3) and (1)'-(4)' in the language of Cauchy data. Issues with the quantization of the Proca field Most of the quantum theories are described by Green hyperbolic operators [3,4], as Klein-Gordon operators N discussed above or the Proca operator [16,55], studied in this work, P " δd`m 2 acting on smooth 1-forms A P Ω 1 pMq and where m 2 ą 0 is a constant. These operators are usually formally self-adjoint w.r.t. a (Hermitian or real symmetric) scalar product induced by the analog γ on the fibers of the relevant vector bundle. In general γ is not positive definite. Very common and physical examples are: the standard vector Klein-Gordon field, the Proca field, the Maxwell field, more generally, the Yang-Mills field and also the linearized gravity. Referring to the Proca, and in general all 1-form fields, we have that γ " g 7 is the inverse (indefinite!) Lorentzian metric of the spacetime pM, gq. Unfortunately, in those situations, the Hadamard condition (4) and (5)' are in conflict with the positivity of states, respectively, (3) and (2)'-(3)'. It is known that for a vectorial Klein-Gordon operator that is formally self-adjoint w.r.t. an indefinite Hermitian/real symmetric scalar product, the existence of quasifree Hadamard states is forbidden (see the comment after [53,Proposition 5.6] and [29,Section 6.3]). The case of a (real) Proca field seems to be even more complicated at first glance. In fact, on the one hand differently from the Klein-Gordon operator, the Proca operator is not even normally hyperbolic and this makes more difficult (but not impossible) the proof of the well-posedness of the Cauchy problem, in particular. On the other hand, similarly to the case of the vectorial Klein-Gordon theory, the Proca theory deals with an indefinite fiberwise scalar product. Actually, as we shall see in the rest of the work, these two apparent drawbacks cooperate to permit the existence of quasifree Hadamard states. Positivity of the two-point function ω 2 is restored when dealing with a constrained space of Cauchy conditions that make well-posed the Cauchy problem. In the present paper, we study the existence of quasifree Hadamard states for the real Proca field on a general globally hyperbolic spacetime. A definition of Hadamard states for the Proca field was introduced by Fewster and Pfenning in [16], to study quantum energy inequalities, with a definition more involved than the one based on conditions (3) and (4)' above. They also managed to prove that such states exist in globally hyperbolic spacetimes whose Cauchy surfaces are compact. Differently from Fewster-Pfenning's definition, here we adopt a definition of Hadamard state which directly relies on conditions (3) and (4)' above and we consider a generic globally hyperbolic spacetime. At the end of the work, we actually prove that the two definitions of Hadamard states are substantially equivalent. Before establishing that equivalence, using the technology of the Møller operators we introduced in [48] for normally hyperbolic operators, and here extended to the Proca field, we prove the existence of quasifree Hadamard states in every globally hyperbolic spacetime, also in the case in which their Cauchy hypersurfaces are not compact. As a matter of fact, it is enough to focus our attention on ultrastatic spacetimes of bounded geometry. In this class of spacetimes, we directly work at the level of initial data for the Proca equation and we establish the following, also by taking advantage of some technical results of spectral theory applied to elliptic Hilbert complexes [5]. 1. The initial data of the Proca equations are a subspace C Σ of the initial data of a couple of Klein-Gordon equations, one scalar and the other vectorial, however both defined on bundles with fiberwise positive real symmetric scalar product; 2. The difference of a pair of certain Hadamard two-point functions for two above-mentioned Klein-Gordon fields becomes positive once that its arguments are restricted to C Σ . There, it defines a two-point function ω 2 for a quasifree state ω of the Proca field; 3. ω is also Hadamard since it is the difference of two two-point functions of Klein-Gordon fields which are Hadamard. They are Hadamard in view of known results of microlocal analysis of pseudodifferential operators on Cauchy surfaces of bounded geometry, for more details the interested reader can refer to [23]. Every field theory defined on a globally hyperbolic spacetime pM, gq is connected to one defined on an ultrastatic spacetime of bounded geometry pRˆΣ,´dt 2`h q through a Møller operator: the associated Møller˚-isomorphism between the algebras of Proca observables preserves the Hadamard condition. We therefore conclude that every globally hyperbolic spacetime pM, gq admits a Hadamard state for the Proca field. This state is nothing but the Hadamard state on pRˆΣ,´dt 2`h q pulled back to pM, gq by the Møller˚-isomorphism. One novelty of this paper is in particular a direct control of the positivity of the two-point functions, obtained by spectral calculus of elliptic Hilbert complexes. Some microlocal property of the Møller operators then guarantees the validity of the Hadamard condition without exploiting the classical so called deformation argument, or better, by re-formulating it into a new form in terms of Møller operators. Main results We explicitly state here the principal results established in this paper referring, for the former, to the notions introduced in the previous section. Below, GP denote the retarded and advanced Green operators of the Proca equation (2.3), we shall discuss in Section 3. The symbol κ g 1 g denotes a linear fiber-preserving isometry from the spaces of smooth sections ΓpV g q to ΓpV g 1 q constructed in Section 3. Here, V g indicates the vector bundle of real 1-forms over the spacetime pM, gq whose sections are the argument of the Proca operator P. Theorem 1 (Theorems 3.2 and 3.7). Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, with associated real Proca bundles V g and V g 1 and Proca operators P, P 1 . If the metric are paracausally related g » g 1 , then there exists a R-vector space isomorphism R : ΓpV g q Ñ ΓpV g 1 q, called Møller operator of g, g 1 (with this order), such that the following facts are true. (1) The restriction, called Møller map S 0 :" R| KerscpPq : Ker sc pPq Ñ Ker sc pP 1 q is well-defined vector space isomorphism with inverse given by pS 0 q´1 :" R´1| KerscpP 1 q : Ker sc pP 1 q Ñ Ker sc pPq . (2) It holds κ gg 1 P 1 R " P. (3) The causal propagators G P :" GP´GṔ and G P 1 :" GP 1´GṔ1 , respectively of P and P 1 , satisfy RG P R : gg 1 " G P 1 . (4) It holds R : gg 1 P 1 κ g 1 g | ΓcpVgq " P| ΓcpVg q , where the adjoint : gg 1 is defined in Definition 3.3. The next result (Theorem 2) permits us to promote R to a˚-isomorphism R of the algebras of field operators A, A 1 respectively associated to the paracausally related metrics g and g 1 , with the associated P, P 1 , and generated by respective Hermitian field operators apfq and a 1 pf 1 q with f, f 1 compactly supported smooth real sections of V. We will introduce these notions in Section 4. These field operators satisfy respective CCRs rapfq, aphqs " iG P pf, hqI , ra 1 pf 1 q, a 1 ph 1 qs " iG P 1 pf 1 , h 1 qI 1 and the said unital˚-algebra isomorphism R : A 1 Ñ A is uniquely determined by the requirement Rpa 1 pfqq " apR : gg 1 fq , f P Γ c pV g 1 q . The final important result regards the properties of R for the algebras of a pair of paracausally related metrics g, g 1 when it acts on the states ω : A Ñ C, ω 1 : A 1 Ñ C of the algebras in terms of pull-back. ω 1 " ω˝R . As is known, the most relevant (quasifree) states in algebraic QFT are Hadamard states characterized by the microlocal spectrum condition valid for the wavefront set of their two-point functions or, equivalently, an universal short distance structure of the distribution defining the two-point function. A definition of Hadamard state for the Proca field was first stated by Fewster and Pfenning in [16] and corresponds to Definition 6.1 in this paper. That definition requires the existence of a bisolution of the Klein Gordon field satisfying the microlocal spectrum condition. This bisolution is next used to construct the two-point function of the Proca field. Differently, in this work we adopt a direct definition (Definition 4.5) which only requires the validity of the microlocal spectrum condition directly for the two-point function of the Proca two-point function. We also prove that our definition, exactly as it happens for Fewster and Pfenning's definition, satisfies some physically relevant properties. In addition to these general results, we also prove that the Hadamard property is preserved by the Møller operators as one of main results of this work. Theorem 2 (Theorem 4.9). Let g, g 1 be paracausally related metric and consider the corresponding Proca operators P, P 1 . Finally refer to the associated on-shell CCR-algebras A and A 1 . Let ω : A Ñ C be a quasifree Hadamard state. The pull-back state ω 1 : A 1 Ñ C by ω 1 " ω˝R satisfies, 1. ω 1 is a well-defined state; 2. ω 1 is quasifree; 3. ω 1 is a Hadamard state. Attention is next focused on the existence problem of quasifree Hadamard states for the real Proca field in a generic globally hyperbolic spacetime. The technology of Møller operators allows us to reduce the construction of Hadamard states for the real Proca field to the special case of an ultrastatic spacetime pRˆΣ,´dt 2`h q. In this class of spacetimes, if assuming the further geometric hypothesis of bounded geometry, we provide a direct construction of a Hadamard state just working on the space of initial data C Σ for the Proca equation PA " 0 where A P ΓpV g q has compact Cauchy data. Here, A decomposes as A " A p0q dt`A p1q , where A p0q and A p1q and are, respectively, a 0-form and a 1-form on ttuˆΣ. As we shall prove, this space of initial data is actually constrained in order to satisfy the existence and uniqueness theorem for the Cauchy problem: C Σ :" ! pa p0q , π p0q , a p1q , π p1q q P Ω 0 c pΣq 2ˆΩ1 c pΣq 2ˇπp0q "´δ p1q h a p1q , p∆ p0q h`m 2 qa p0q " δ p1q h π p1q ) , where pa p0q , π p0q q :" pA p0q , B t A p0q q| t"0 and pa p1q , π p1q q :" pA p1q , B t A p1q q| t"0 . Theorem 3 (Propositions 5.8 and 5.10). Consider the˚-algebra A g of the real Proca field on the ultrastatic spacetime pM, gq " pRˆΣ,´dt b dt`hq, with pΣ, hq a smooth complete Riemannian manifold. Let η 0 :"´1, η 1 :" 1 and h 7 pjq denote the standard inner product of j-forms on Σ induced by h. Then the two-point function ω µ papfqapf 1 qq " ω µ2 pf, f 1 q :" µpA, A 1 q`i 2 σ pP q pA, A 1 q defines a quasifree state ω µ on A g where f, f 1 P Γ c pV g q. Above A " G P f , A 1 " G P f 1 , σ pP q pA, A 1 q " ż M g 7 pf, G P f 1 q vol g µpA, A 1 q :" 1 ÿ j"0 η j 2 ż Σ h 7 pjq pπ pjq , p∆ pjq`m2 q´1 {2 π pjq 1 q`h 7 pjq pa pjq , p∆ pjq`m2 q 1{2 a pjq 1 q vol h where ∆ pjq is the Hodge Laplacian for compactly supported real smooth j-forms on pΣ, hq. Finally, ω µ is Hadamard if pΣ, hq is of bounded geometry. Above the bar denotes the closure of the considered operators defined in suitable L 2 -spaces of forms according to the theory of elliptic Hilbert complexes. Using the fact that every globally hyperbolic spacetime is paracausally related to an ultrastatic spacetime with bounded geometry and combining the two previous Theorems, we can conclude that Proca fields can be quantized in any globally hyperbolic spacetime and admit Hadamard states. Theorem 4. Let pM, gq be a globally hyperbolic spacetime and refer to the associated CCRalgebras A g of the real Proca field. Then there exists a quasifree Hadamard state on A g . Eventually, coming back to the alternative definition of Hadamard states proposed by Fewster and Pfenning in [16], we prove an almost complete equivalence theorem, which is the last main achievement of this work. Theorem 5 (Theorem 6.6). Consider the globally hyperbolic spacetime pM, gq and a quasifree state ω : A g Ñ C for the Proca algebra of observables on pM, gq with two-point function ω P Γ 1 c pV g b V g q. The following facts are true. (a) If ω is Hadamard according to Fewster and Pfenning, then it is also Hadamard according to Definition 4.5. (b) If pM, gq admits a Hadamard state according to Fewster and Pfenning and ω is Hadamard according to Definition 4.5, then ω is Hadamard in the sense of Fewster-Pfenning's definition. The existence of Hadamard states according to Fewster-Pfenning's definition was proved in [16] for spacetimes with compact Cauchy surfaces. For these spacetimes the equivalence of the two definitions is complete. Structure of the paper The paper is structured as follows. In Section 3 we will provide a detailed analysis of the Møller maps and the Møller operator for classical Proca fields. In particular, we will analyze the relation between the Møller operators and the causal propagators of Proca operators on paracausally related globally hyperbolic spacetimes. In Section 4 we will extend the action of the Møller operators to a˚-isomorphism of the CCR-algebras of free Proca fields. This will allow us to pullback quasifree Hadamard states preserving the microlocal spectrum condition. In this section we also analyze the general properties of Hadamard states including their existence. The explicit construction of Hadamard states in an ultrastatic spacetime is performed in Section 5. In Section 6 we show that the microlocal spectrum condition is essentially equivalent to the definition of Hadamard states proposed by Fewster and Pfenning. Finally, we conclude our paper with Section 7, where open issues and future prospects are presented. Acknowledgments We are grateful to Nicolò Drago, Chris Fewster, Christian Gérard, and Igor Khavkine for helpful discussions related to the topic of this paper. This work was written within the activities of the INdAM-GNFM Conventions and notation of geometric tools in spacetimes Throughout all the paper the symbols Ă and Ą allow the case ". We explicitly adopt the signature p´,`,¨¨¨,`q for Lorentzian metrics. Throughout pM, gq denotes a spacetime, i.e., a paracompact, connected, oriented, timeoriented, smooth, Lorentzian manifold M, whose Lorentzian metric is g. As in [48], the Lorentzian metrics g of spacetimes are hereafter supposed to be equipped with their own temporal orientation. All considered spacetimes pM, gq are also globally hyperbolic. In other words, a (smooth) Cauchy temporal function t : M Ñ R exists. By definition dt is timelike, past directed and pM, gq is isometric to pRˆΣ, g 1 q with metric g 1 "´βdt b dt`h t , where β : RˆΣ Ñ R is a smooth positive function, h t is a Riemannian metric on each slice Σ t :" ttuˆΣ varying smoothly with t, and these slices are smooth spacelike Cauchy hypersurfaces. By definition they are achronal sets intersected exactly once by every inextensible timelike curve (see [47] for a recent up-to-date survey on the subject). According to [48], given two globally hyperbolic metrics g and g 1 on M , g ĺ g 1 means that V gp Ă V g 1p for all p P M, where V gp Ă T p M is the open cone of future directed timelike vectors at p in pM, gq. Two globally hyperbolic metrics g and g 1 on M are paracausally related , written g » g 1 , if there exists a finite sequence of globally hyperbolic metrics g 1 " g, g 2 . . . , g n " g 1 on M such that for each pair of consecutive metrics either g k ĺ g k`1 or g k`1 ĺ g k . For a discussion on this notion, its properties, and examples we refer to [48,Section 2]. We henceforth denote by ΓpEq the real vector space of smooth sections of any real vector bundle E Ñ M. More precisely, as in [48], we denote with Γ c pEq, Γ f c pEq, Γ pc pEq, Γ sc pEq the space of sections respectively with compact support, future-compact (i.e. whose support stays before a smooth spacelike Cauchy surface), past-compact (i.e. whose support stays after a smooth spacelike Cauchy surface), and spatially-compact support (i.e. whose support on every smooth spacelike Cauchy surface is compact). If E Ñ M and E 1 Ñ M 1 are two vector bundles, E b E 1 denotes the external tensor product of these vector bundles. This vector bundle has base MˆM 1 and fiber at pp, p 1 q given by the tensor products of the respective fibers at p P M and p 1 P M 1 respectively. If f P ΓpEq and f 1 P ΓpE 1 q, the section f b f 1 P ΓpE b E 1 q is defined by f b f 1 pp, p 1 q :" fppq b f 1 pp 1 q. The tensor product of linear operators acting on sections of an external product bundle are denoted by b. Smooth forms, Hodge operators, and the Proca equation In this work we frequently deal with real smooth k-forms f, g P Ω k pMq, where k " 0, . . . , n " dim M (and one usually adds Ω n`1 pMq " Ω´1pMq " t0u). The Hodge real inner product can be computed by integrating the fiberwise product with respect to the volume form induced by g: pf|gq g,k :" ż M f^˚g " ż M g 7 pkq pf, gq vol g , where at least one of the two forms has compact support and g 7 pkq is the natural inner product of k-forms induced by g. This symmetric real scalar product p¨|¨q g,k is always non-degenerate but it is not positive when g is Lorentzian as in the considered case. It is positive when g is Riemannian. If k " 1, we simply write pf|gq g " ż M g 7 pf, gq vol g . (2.1) In this context, d pkq : Ω k pMq Ñ Ω k`1 pMq is the exterior derivative and δ pkq g : Ω k pMq Ñ Ω k´1 pMq is the codifferential operator acting on the relevant spaces of smooth k-forms Ω k pMq on M depending on the metric g on M. d pkq and δ pk`1q g are the formal adjoint of one another with respect to the Hodge product (2.1) in the sense that pd pkq f|gq g,k`1 " pf|δ pk`1q g gq g,k , @f P Ω k pMq , @g P Ω k`1 pMq if f or g is compactly supported. In the rest of the paper we will often omit the indices g,k and pkq referring to the metric and the order of the used forms, when the choice of the used metric and order will be obvious from the context. If pM, gq is globally hyperbolic, we call Proca bundle the real vector bundle V g :" pT˚M, g 7 q obtained by endowing the cotangent bundle with the fiber metric given by the dual metric g 7 (also appearing in (2.1)) defined by g 7 pω p , ω 1 p q :" gp7ω p , 7ω 1 p q for every ω, ω 1 P ΓpT˚Mq and p P M, where 7 : ΓpT˚Mq Ñ ΓpTMq is the standard musical isomorphism. By construction ΓpV g q " Ω 1 pMq and Γ c pV g q " Ω 1 c pMq. Here and henceforth Ω k c pMq Ă Ω k pMq is the subspace of compactly supported real smooth k-forms on M. The formally selfadjoint Proca operator P on pM, gq is defined by choosing a (mass) constant m ą 0, the same for all globally hyperbolic metrics we will consider on M in this work, P " δd`m 2 : ΓpV g q Ñ ΓpV g q, (2.2) where d :" d p1q , δ :" δ p2q g . Actually P depends also on g, but we shall not indicate those dependencies in the notation for the sake of shortness. The Proca equation we shall consider in this paper reads PA " 0 for A P Γ sc pV g q ,(2.3) where, as said above, Γ sc pV g q is the space of real smooth 1-forms which have compact support on the Cauchy surfaces of the globally hyperbolic spacetime pM, gq. Møller Maps and Møller Operators The construction of the so-called Møller operator for hyperbolic PDEs (coming from the realm of quantum field theories on curved spacetimes) has been studied extensively in various contexts in Quantum Field Theory, see e.g. [9,10,12,14,48,49]. The key idea was to inspired by the scattering theory: Starting with two "free theories" described by the space of solutions of normally hyperbolic operators (see (3.3) below) N 0 and N 1 in corresponding spacetimes pM, g 0 q and pM, g 1 q, respectively, we connected them through an "interaction spacetime" pM, g χ q with a "temporally localized" interaction defined by interpolating the two metrics by means of a smoothing function χ. Here we need two Møller maps: Ω`connecting pM, g 0 q and pM, g χ q -which reduces to the identity in the past when χ is switched off -and a second Møller map connecting pM, g χ q to pM, g 1 q -which reduces to the identity in the future when χ constantly takes the value 1. The "S-matrix" given by the composition S :" Ω´Ω`will be the Møller map connecting N 0 and N 1 . As remarked in [48,Section 6], all the results concern vector-valued normally hyperbolic operators acting on real vector bundles whose fiber metric does not depend on the globally hyperbolic metrics g chosen on M. These operators are also assumed to be formally selfadjoint with respect to the associated real symmetric scalar product on the sections of the bundle. As already pointed out in the introduction, to quantize the theory defining quantum states on an associated˚-algebra of observables, the fiberwise metric on E should be assumed to be positive. This section aims to extend the construction of the Møller operators to Proca fields. The main difficulties we have to face with respect to the case of the Klein-Gordon equation are the following: • the fiber metric of the Proca bundle depends on the underlying globally hyperbolic metrics g chosen on M (and it is not positive definite); • Proca operators are not normally hyperbolic. The next two sections are devoted to tackle these technical issues before starting with the construction of the Møller maps. Linear fiber-preserving isometry As said above, to construct Møller maps for the Proca field we should be able to compare different fiberwise metrics on T˚M when we change the metric g on M. This will be done by defining suitable fiber preserving isometries. If g and g 1 are globally hyperbolic on M and g » g 1 , it is possible to define a linear fiberpreserving isometry from ΓpV g q to ΓpV g 1 q we denote with κ g 1 g and we shall take advantage of it very frequently in the rest of this work. In other words, if f P ΓpV g q, then κ g 1 g f P ΓpV g 1 q, the map κ g 1 g : ΓpV g q Ñ ΓpV g 1 q is R linear, and g 17 ppκ g 1 g fqppq, pκ g 1 g gqppqq " g 7 pfppq, gppqq @p P M . Let us describe the (highly non-unique) construction of κ gg 1 . If χ P C 8 pM; r0, 1sq and g 0 ĺ g 1 , then g χ :" p1´χqg 0`χ g 1 (3.1) is a Lorentzian metric globally hyperbolic on M (see [48,Section 2] for details) and satisfies g 0 ĺ g χ ĺ g 1 . Now consider the product manifold N :" RˆM, equipped with the indefinite non-degenerate metric h :"´dt b dt`g t , where g t " p1´f ptqqg 0`f ptqg χ and f : R Ñ r0, 1s is smooth and f ptq " 0 for t ď 0, f ptq " 1 for t ě 1. Notice that g t is Lorentzian according to [48] because g 0 ĺ g χ and h is indefinite nondegenerate by construction. At this point r κ χ0 : TM Ñ TM is the fiber preserving diffeomorphism such that r κ χ0 px, vq is the parallel transport form p0, xq to p1, xq of v P T x M Ă T p0,xq N along the complete h-geodesic R Q t Þ Ñ pt, xq P N. Standard theorems on joint smoothness of the flow of ODEs depending on parameters assure that r κ χ0 : TM Ñ TM is smooth. Notice that r κ χ0 | TxM : T x M Ñ T x M is also a h-isometry from known properties of the parallel transport and thus it is a g 0 , g χ -isometry by construction because h pt,xq pv, vq " g t pv, vq if v P T x M Ă T pt,xq N. Taking advantage of the musical isomorphisms, r κ χ0 induces a fiber-bundle map κ χ0 : T˚M Ñ T˚M which can be seen as a map on the sections of ΓpV g 0 q and producing sections of ΓpV gχ q, preserving the metrics g 7 0 , g 7 χ . Then the required Proca bundle isomorphism κ g 1 g " κ g 1 g 0 is defined by composition: κ 1,0 " κ 1χ κ χ0 . where κ 1χ from ΓpV gχ q to ΓpV g 1 q is defined analogously to κ χ0 . The general case g » g 1 can be defined by composing the fiber preserving linear isometries κ g k`1 g k or κ g k ,g k`1 . Klein-Gordon operator associated to a Proca operator and Green operators We pass to tackle the issue of normal hyperbolicity of P. As we shall see here, it is not really necessary to construct the Møller maps, and the weaker requirement of Green hyperbolicity is sufficient. Let N be the Klein-Gordon operator associated to the Proca operator P (2.2) acting on 1-forms N :" δd`dδ`m 2 : ΓpV g q Ñ ΓpV g q . Notice that this operator is normally hyperbolic: its principal symbol σ N satisfies σ N pξq "´g 7 pξ, ξq Id Vg for all ξ P T˚M, where Id Vg is the identity automorphism of V g . (3.3) Therefore the Cauchy problem for N is well-posed [3,4]. Both N and P are formally selfadjoint with respect to the Hodge scalar product (2.1) on Ω 1 c pMq " Γ c pV g q. Since m 2 ą 0 and δ p1q g δ p2q g " 0, it is easy to prove that the Proca equation (2.3) is equivalent to the pair of equations NA " 0 , for A P Γ sc pV g q , (3.4) δA " 0 . (3.5) As already noticed, differently from N, the Proca operator is not normally hyperbolic. However, it is Green hyperbolic [3,4,6] as N, in particular there exist linear maps, dubbed advanced Green operator GP : Γ pc pV g q Ñ ΓpV g q and retarded Green operator GṔ : Γ f c pV g q Ñ ΓpV g q uniquely defined by the requirements (i.a) GP˝P f " P˝GP f " f for all f P Γ pc pV g q , (ii.a) supp pGP fq Ă J`psupp fq for all f P Γ pc pV g q; (i.b) GṔ˝P f " P˝GṔ f " f for all f P Γ f c pV g q, (ii.b) supp pGṔ fq Ă J´psupp fq for all f P Γ f c pV g q; The causal propagator of P is defined as G P :" GP´GṔ : Γ c pV g q Ñ Γ sc pV g q . (3.6) All these maps are also continuous with respect to the natural topologies of the definition spaces [6]. As a matter of fact (see [6,Proposition 3.19] and also [4]), the advanced and retarded Green operator GP : Γ pc{f c pV g q Ñ Γ pc{f c pV g q can be written as GP :"ˆId`d δ m 2˙GN " GNˆId`d δ m 2ẇ here GN are the analogous Green operators for the Klein-Gordon operator N. Therefore G P :"ˆId`d δ m 2˙G N " G NˆI d`d δ m 2˙. (3.7) The fact that P is normally hyperbolic can be proved just by checking that the operators above satisfy the requirements which define the Green operators as stated above, using the analogous properties for GN . Eq. (3.7) and the analogous properties for G N entail G P pΓ c pV g qq " tA P Γ sc pV g q | PA " 0u . (3.8) Indeed, if PA " 0 then NA " 0 and δA " 0. If A P Γ sc pV g q, [48, Theorem 3.8] implies A " G N f for some f P Γ c pV g q, so that A "`Id`d δ m 2˘A " G P f as said. Furthermore, KerG P " tPg | g P Γ c pV g qu . (3.9) Indeed, if PA " 0 then m 2`I d`d δ m 2˘P A " NA " 0. If A P Γ sc pV g q, again [48, Theorem 3.8] implies that A " Nf for some f P Γ c pV g q. Since we also know that δA " 0, the form (3.3) of N yields A " Pf. On the other hand, if A " Pf for some f P Γ c pV g q, then G P A " GP f´GṔ f " f´f " 0. On account of [48, Proposition 3.6], for any smooth function ρ : M Ñ p0,`8q also ρP is Green hyperbolic and Gρ P " GP ρ´1. Proca Møller maps A smooth Cauchy time function in a globally hyperbolic spacetime pM, gq relaxes the notion of temporal Cauchy function, it is a smooth map t : M Ñ R such that dt is everywhere timelike and past directed, the level surfaces of t are smooth spacelike Cauchy surfaces and pM, gq is isometric to pRˆΣ, hq. Here, t identifies with the natural coordinate on R and the Cauchy surfaces of pM, gq identify with the sets ttuˆΣ. From now on we indicate by N 0 , N 1 , N χ the Klein-Gordon operators (3.2) on M constructed out of g 0 , g 1 and g χ respectively, where the globally hyperbolic metric g χ is defined as in (3.1) (and thus g 0 ĺ g χ ĺ g 1 [48, Theorem 2.18]) and depends on the choice of a function χ P C 8 0 pM, r0, 1sq. Similarly, P 0 , P 1 , P χ denote the Proca operators (2.2) on M constructed out of g 0 , g 1 and g χ respectively. We can state the first technical result. Proposition 3.1. Let g 0 , g 1 be globally hyperbolic metrics satisfying g 0 ĺ g 1 and let be χ P C 8 pM; r0, 1sq. Choose (a) a smooth Cauchy time g 1 -function t : M Ñ R and χ P C 8 pM; r0, 1sq such that χppq " 0 if tppq ă t 0 and χppq " 1 if tppq ą t 1 for given t 0 ă t 1 ; (b) a pair of smooth functions ρ, ρ 1 : M Ñ p0,`8q such that ρppq " 1 for tppq ă t 0 and ρ 1 ppq " ρppq " 1 if tppq ą t 1 . (Notice that ρ " ρ 1 " 1 constantly is allowed.) Then the following facts are true where g χ is defined as in (3.1). (1) The operators R`: ΓpV g 0 q Ñ ΓpV gχ q R`:" κ χ0´Gρ Pχ pρP χ κ χ0´κχ0 P 0 q , R´: ΓpV gχ q Ñ ΓpV g 1 q R´:" κ 1χ´Gρ P 1`ρ 1 P 1 κ 1χ´ρ κ 1χ P χȃ re linear space isomorphisms, whose inverses are given by R´1 : ΓpV gχ q Ñ ΓpV g 0 q R´1 " κ 0χ`GP 0 pρκ 0χ P χ´P0 ρκ 0χ q, R´1 : ΓpV g 1 q Ñ ΓpV gχ q R´1 :" κ χ1`Gρ Pχ`ρ 1 κ χ1 P 1´ρ κ 1χ P χ˘. By composition we define the Møller operator: R : ΓpV g 0 q Ñ ΓpV g 1 q R :" R´˝R`, which is also a linear space isomorphism. (2) It holds ρκ 0χ P χ R`" P 0 and ρ 1 κ χ1 P 1 R´" ρP χ . and also ρκ 0χ P χ " P 0 R´1 and ρ 1 κ χ1 P 1 " P χ R´1 . (3) If f P ΓpV g 0 q or ΓpV gχ q respectively, then pR`fqppq " fppq for tppq ă t 0 , (3.10) pR´fqppq " fppq for tppq ą t 1 . (3.11) Proof. First of all, we notice that the operator R`is well defined on the whole space ΓpV g 0 q since for all sections f P ΓpV g 0 q we have that pP χ κ χ0 ρ´κ χ0 ρ P 0 qf P Γ pc pV g 1 q: indeed by definition, there exists a t 0 P R such that on t´1p´8, t 0 q and we have that P χ " P 0 , κ χ0 " Id and t is a smooth g 1 -Cauchy time function. Moreover, since g χ ĺ g 1 it follows that Γ pc pV g 1 q Ă Γ pc pV gχ q " DompG Pχ q. To prove (1), we can first notice that R´1˝R`"´κ 0χ`GP 0 pρκ 0χ P χ´P0 κ 0χ q¯˝´κ χ0´Gρ Pχ pρP χ κ χ0´κχ0 P 0 q" Id´κ 0χ Gρ Pχ pρP χ κ χ0´κχ0 P 0 q`GP 0 pρκ 0χ P χ´P0 κ 0χ qκ χ0 GP 0 pρκ 0χ P χ´P0 ρκ 0χ qGρ Pχ pρP χ κ χ0´κχ0 P 0 q . To conclude it is enough to show that everything cancels out except the identity operator, but that just follows by using basic properties of Green operators and straightforward algebraic steps. We easily see that the last addend can be recast as: GP 0 pρκ 0χ P χ´P0 κ 0χ q Gρ Pχ pρP χ κ χ0´κχ0 P 0 q " GP 0 ρκ 0χ P χ Gρ Pχ pρP χ κ χ0´κχ0 P 0 q´GP 0 P 0 κ 0χ Gρ Pχ pρP χ κ χ0´κχ0 P 0 q " GP 0 κ 0χ pρP χ κ χ0´κχ0 P 0 q´κ 0χ Gρ Pχ pρP χ κ χ0´κχ0 P 0 q, which fulfills its purpose. A specular computation proves that R´1 is also a right inverse. Almost identical reasonings prove that R´1 is a two sided inverse of R´which is also well defined, then bijectivity of R is obvious. (2) follows by the following direct computation ρκ 0χ P χ R`" ρκ 0χ P χ´κχ0´Gρ Pχ pρP χ κ χ0´κχ0 P 0 q" κ 0χ κ χ0 P 0 " P 0 . (3) Let us prove (3.10). In the following P˚denotes the formal dual operator of P acting on the sections of the dual bundle Γ c pVg q. If f 1 P Γ c pVg q and f P Γ pc pV g q or f P Γ f c pV g q respectively, ż M xGṔ˚f 1 , fy vol g " ż M xf 1 , GP fy vol g , ż M xGP˚f 1 , fy vol g " ż M xf 1 , GṔ fy vol g ,(3.12) where GP indicate the Green operators of P and GP˚indicate the Green operators of P˚. Consider now a compactly supported smooth section h whose support is included in the set t´1pp´8, t 0 qq. Taking advantage of the Equation (3.12), we obtain ż M xh, Gρ Pχ pρP χ´P0 qfy vol gχ " ż M xGṕ ρPχq˚h , pρP χ´P0 qfy vol gχ " 0 since supppGṕ ρPχq˚h q Ă J gχ psuppphqq and thus that support does not meet suppppρP χ´P0 qfq because ppρP χ´P0 qfqppq vanishes if tppq ă t 0 . As h is an arbitrary smooth section compactly supported in t´1pp´8, t 0 qq, ż M xh, Gρ Pχ pρP χ´P0 qfy vol gχ " 0 entails that Gρ Pχ pρP χ´P0 qf " 0 on t´1pp´8, t 0 qq. The proof of (3.11) is strictly analogous, so we leave it to the reader. Using Proposition 3.1, we can pass to the generic case g » g 1 . Theorem 3.2. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, with associated Proca bundles V g and V g 1 and Proca operators P, P 1 . If g » g 1 , then there exist (infinitely many) vector space isomorphisms, (1) referring to the said domains, µκ gg 1 P 1 R " P for some smooth µ : M Ñ p0,`8q (which can always be chosen µ " 1 constantly in particular), and a smooth fiberwise isometry κ gg 1 : ΓpV g 1 q Ñ ΓpV g q. (2) The restriction, called Møller map S 0 :" R| KerscpPq : Ker sc pPq Ñ Ker sc pP 1 q is well-defined vector space isomorphism with inverse given by pS 0 q´1 :" R´1| KerscpP 1 q : Ker sc pP 1 q Ñ Ker sc pPq . Proof. Since g » g 1 , there exists a finite sequence of globally hyperbolic metrics g 0 " g, g 1 , .., g N " g such that at each step g k ĺ g k`1 or g k`1 ĺ g k . For all k P t0, .., N u we can associate to the metric a Proca operator P k . At each step the hypotheses of Proposition 3.1 are verified, in fact we can choose functions ρ k and ρ 1 k and the Møller map is given by R k " R k´˝Rk`. The general map is then built straightforwardly by composing the N maps constructed step by step: R " R N˝. ..˝R 1 . Regarding (1), by direct calculation we get that µ " ś N k"1 ρ 1 k , while κ gg 1 " κ g 0 g 1˝. ..˝κ g N´1 g N . The proof of (2) is trivial. Causal propagator and Møller operator The rest of this section is devoted to study the relation between Møller maps and the causal propagator of Proca operators. To this end, we use a natural extension of the notion of adjoint operator introduced in [48, Section 4.5]. Let g and g 1 (possibly g ‰ g 1 ) globally hyperbolic metric and let V g and V g 1 be a Proca bundle on the manifold M. Consider a R-linear operator T : DompTq Ñ ΓpV g 1 q , where DompTq Ă ΓpV g q is a R-linear subspace and DompTq Ą Γ c pV g q. Definition 3.3. An operator T : gg 1 : Γ c pV g 1 q Ñ Γ c pV g q is said to be the adjoint of T with respect to g, g 1 (with the said order) if it satisfies ż M g 17 ph, Tfq pxq vol g 1 pxq " ż M g 7´T: gg 1 h, f¯pxq vol g pxq @f P DompTq , @h P Γ c pEq. When g " g 1 , we use the simplified notation T : :" T :gg . As in [48], the adjoint operator satisfies a lot of useful properties which we summarize in the following proposition. Since the proof is analogous to the one of [48,Proposition 4.11], we leave it to the reader. Though the rest of this paper deal with the real case only, we state the theorem encompassing the case where the sections are complex and the fiber scalar product is made Hermitian by adding a complex conjugation of the left entry in the usual fiberwise real g 7 inner product, which becomes g 7 pf, gq, where the bar denotes the complex conjugation. Definition 3.3 extends accordingly. For this reason K will denote either R or C, and the complex conjugate c reduces to c itself when K " R. We keep the notation V g for indicating either the real or complex vector bundle T˚M or T˚M`iT˚M corresponding to two possible choices of K. (1) If the adjoint T : gg 1 of T exists, then it is unique. (2) If T : ΓpV g q Ñ ΓpV g 1 q is a differential operator and g " g 1 , then T :gg exists and is the restriction of the formal adjoint to Γ c pEq. (In turn, the formal adjoint of T is the unique extension to ΓpEq of the differential operator T : as a differential operator.) (3) Consider a pair of K-linear operators T : DompTq Ñ ΓpV g 1 q, T 1 : DompT 1 q Ñ ΓpV g 1 q with DompTq, DompT 1 q Ă ΓpV g q and a, b P K. Then paT`bT 1 q : gg 1 " aT : gg 1`b T 1: gg 1 provided T : gg 1 and T 1: gg 1 exist. (4) Consider a pair of K-linear operators T : DompTq Ñ ΓpV g 1 q, T 1 : DompT 1 q Ñ ΓpV g 2 q with DompTq Ă ΓpV g q and DompT 1 q Ă ΓpV g 1 q such that (i) DompT 1˝T q Ą Γ c pV g q, (ii) T : gg 1 and T 1: g 1 g 2 exist, then pT 1˝T q : gg 2 exists and pT 1˝T q : gg 2 " T : gg 1˝T1: g 1 g 2 . (5) If T : gg 1 exists, then pT : gg 1 q : g 1 g " T| ΓcpVg q . (6) If T : DompT q " ΓpV g q Ñ ΓpV g 1 q is bijective, admits T : gg 1 , and T´1 admits pT´1q : g 1 g , then T : gg 1 is bijective and pT´1q : g 1 g " pT : gg 1 q´1. Now we are ready to prove that the operators R admit adjoints and we explicitly compute them. Proposition 3.5. Let g 0 , g 1 be globally hyperbolic metrics satisfying g 0 ĺ g 1 . Let R`, R´and R be the operators defined in Proposition 3.1 and fix, once and for all, ρ " c χ 0 and ρ 1 " c 1 0 where c χ 0 , c 1 0 are the unique smooth functions on M such that: vol gχ " c χ 0 vol g 0 vol g 1 " c 1 0 vol g 0 . (3.13) Then we have: (1) R :g 0 gχ : Γ c pV gχ q Ñ Γ c pV g 0 q satisfies: R :g 0 gχ "´c χ 0 κ 0χ´p c χ 0 κ 0χ P χ´P0 κ 0χ q GṔ χ¯| ΓcpVχq and can be recast in the form R :g 0 gχ " P 0 κ 0χ GṔ χ | ΓcpVχq . (2) R :g χg1 : Γ c pV g 1 q Ñ Γ c pV gχ q satisfies R :g χg1 "´c χ 1 κ χ1´p c χ 1 κ χ1 P 1´Pχ κ χ1 q GP 1¯| ΓcpV 1 q , and can be recast in the form R :g χg1 " P χ κ χ1 GP 1 | ΓcpV 1 q . (3) The map R :g 0 g 1 : Γ c pV g 1 q Ñ Γ c pV g 0 q defined by R :g 0 g 1 :" R :g 0 gχ˝R:gχg 1 is invertible and pR :g 0 g 1 q´1 " pR´1q :g 1 g 0 : Γ c pV g 1 q Ñ Γ c pV g 0 q . We call it adjoint Møller operator. Moreover R :g 0 g 1 is a homeomorphism with respect to the natural (test section) topologies of the domain and of the co-domain. Proof. We start by proving points (1) and (2). For any f P DompR`q " ΓpV g 0 q and h P Γ c pV gχ q we have ż M g 7 χ ph, R`fq vol gχ " ż M g 7 χ´h ,`κ χ0´Gc χ 0 Pχ pc χ 0 P χ κ χ0´κχ0 P 0 q˘f¯vol gχ " ż M g 7 χ ph, κ χ0 fq vol gχ´ż M g 7 χ´h ,`Gcχ 0 Pχ pc χ 0 P χ κ χ0´κχ0 P 0 q˘f¯vol gχ . We now split the problem and compute the adjoint of the two summands separately. The adjoint of the first one follows immediately by exploiting the properties of the existing isometry and Equations (3 .13) ż M g 7 χ ph, κ χ0 fq vol gχ " ż M g 7 0 pc χ 0 κ 0χ h, fq vol g 0 . For the second summand the situation is trickier and we cannot split the calculation in two more summands since it is crucial that the whole difference pc χ 0 P χ κ χ0´κχ0 P 0 q acts on a general f P ΓpV gχ q before we apply the Green operator whose domain is Γ pc pV gχ q. So we first rewrite Gcχ 0 Pχ " GP χ 1 c χ 0 use the properties of standard adjoints of Green operators for formally self-adjoint Green hyperbolic differential operators to get ż M g 7 χ´h ,`Gcχ 0 Pχ pc χ 0 P χ κ χ0´κχ0 P 0 q˘f¯vol gχ " ż M g 7 χˆGṔ χ h,`P χ κ χ0´κ χ0 c χ 0 P 0˘f˙v ol gχ . Now we are tempted to exploit the linearity of the integral and of the fiber product, but first, to ensure that the two integrals individually converge, we need to introduce a cutoff function: • We notice again that there is a Cauchy surface of the foliation Σ t 0 such that for all leaves with t ă t 0 the operator´P χ κ χ0´κ χ0 c χ 0 P 0¯" 0; • So take a t 1 ă t 0 and define a cutoff smooth function s : M Ñ r0, 1s such that s " 0 on all leaves with t ă t 1 . In this way we are allowed to rewrite our last integral and split it in two convergent summands without modifying its numerical value. ż M g 7 χ´GṔ χ h,`P χ κ χ0´κ χ0 c χ 0 P 0˘s f¯vol gχ " " ż M g 7 χ´GṔ χ h, P χ κ χ0 sf¯vol gχ´ż M g 7 χˆGṔ χ h, κ χ0 c χ 0 P 0 sf˙vol gχ " ż M g 7 0´c χ 0 κ 0χ P χ GṔ χ h, sf¯vol g 0´ż M g 7 0 pP 0 κ 0χ GṔ χ h, sfqvol g 0 " ż M g 7 0´`c χ 0 κ 0χ P χ´P0 κ 0χ˘GṔ χ h, sf¯vol g 0 " ż M g 7 0´`c χ 0 κ 0χ P χ´P0 κ 0χ˘GṔ χ h, f¯vol g 0 . where in the last identities we have used properties of the standard adjoints of the formally selfadjoint operators, of the isometries and of the cutoff function. Since the domain of the operator is just made up of compactly supported sections, we may exploit the inverse property of the Green operators to immediately obtain that c χ 0 κ 0χ´p c χ 0 κ 0χ P χ´P0 κ 0χ q GṔ χ | ΓcpVχq " P 0 κ 0χ GṔ χ | ΓcpVχq . To see that the image of the operators is indeed compactly supported we can focus on R :g 0 gχ , the rest follows straightforwardly. The first summand c χ 0 κ 0χ does not modify the support of the sections, whereas the second does. Let us fix f P Γ c pV gχ q, then supp pGṔ χ fq Ă Jǵ χ psupp fq which means that GṔ χ f P Γ sf c , i.e it is space-like and future compact. The thesis follows by again observing that there is a Cauchy surface such that in its past´P χ κ χ0´κ χ0 c χ 0 P 0¯GṔ χ f " 0. The computation of the adjoint of R´is almost identical to the one just performed. The first part of (3) is an immediate consequence of property (4) in Proposition 3.4, while the invertibility of the adjoint can be proved by explicitly showing that the operator pR :g 0 gχ q´1 "ˆκ χ0 c χ 0`ˆP χ κ χ0´κ χ0 c χ 0 GṔ 0˙˙ˇΓ c pVg 0 q serves as a left and right inverse of R :g 0 gχ . An analogous argument can be used for R :g χg1 . The continuity of both the adjoint and its inverse comes by the same arguments used in the proof of [48,Theorem 4.12] (with the only immaterial difference that this time the smooth isometry κ χ0 is included in the definition of the Møller operator.) Remark 3.6. An interesting fact to remark is that having defined the adjoints over compactly supported sections makes the dependence on the auxiliary volume fixing functions disappear. We conclude the section, by proving the second part of Theorem 1. Theorem 3.7. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, with associated Proca bundles V g and V g 1 and Proca operators P, P 1 . If g » g 1 , it is possible to specialize the R-vector space isomorphism R : ΓpV g q Ñ ΓpV g 1 q of Proposition 3.2 such that the following further facts are true. (1) The causal propagators G P and G P 1 (3.6), respectively of P and P 1 , satisfy RG P R : gg 1 " G P 1 . (2) It holds R : gg 1 P 1 κ g 1 g | ΓcpVgq " P| ΓcpVg q . R as above is called Møller operator of g, g 1 (with this order). Proof. Since g » g 1 and the Møller map is defined as the composition R " R N˝. ..˝R 1 , we can use properties (4) in Proposition 3.4 and reduce to the case where g " g 0 ĺ g 1 " g 1 . With this assumption, (2) can be obtained following the proof of Proposition 3.1 and (3) is identical to [48,Theorem 4.12 (5)]. So we leave it to the reader. It remains to prove (1). Decomposing R as above, we define the maps R g 0 gχ , R gχg 1 by choosing the various arbitrary functions as in Proposition 3.5. We first notice R`GP 0 R :g 0 gχ "´κ χ0´Gc χ 0 Pχ pc χ 0 P χ κ χ0´κχ0 P 0 q¯GP 0´P 0 κ 0χ GṔ χ¯| ΓcpVχq " Gcχ 0 Pχ κ χ0´P0 κ 0χ GṔ χ¯| ΓcpVχq " GP χ´GPχˆP χ´κ χ0 c χ 0 P 0 κ 0χ˙GṔ χ . where the first equality follows by definition, in the second one we have used the properties of Green operators, while in the third one we have just equated the two expressions for the adjoint operator according to (1) in Proposition 3.5 and performed some trivial algebraic manipulations. Another analogous computation can be performed for the retarded Green operator yielding R`GP 0 R :g 0 gχ " GṔ χ´GPχˆP χ´κ χ0 c χ 0 P 0 κ 0χ˙GṔ χ . Therefore, subtracting the two terms we get R`G P 0 R :g 0 gχ " R`pGP 0´GṔ0 qR :g 0 gχ " G Pχ . Applying now R´and its adjoint we get the claimed result. Møller˚-Isomorphisms and Hadamard States The CCR algebra of observables of the Proca field We now pass to introduce the algebraic formalism to quantize the Proca field [16,55]. Let pM, gq be a globally hyperbolic spacetime, V g be a Proca bundle and denote by P : ΓpV g q Ñ ΓpV g q the Proca operator. Following [40], we call on-shell Proca CCR˚-algebra, the˚-algebra defined as A g " A g {I g where: -A g is the free complex unital algebra finitely generated by the set of abstract elements I (the unit element), apfq and apfq˚for all f P Γ c pV g q, and endowed with the unique (antilinear) -involution which associates apfq to apfq˚and satisfies I˚" I and pabq˚" b˚a˚. -I g is the two-sided˚-ideal generated by the following elements of A f : 1. apaf`bhq´aapfq´baphq , @a, b P R @f, h P Γ c pV g q; 2. apfq˚´apfq , @f P Γ c pV g q; 3. apfqaphq´aphqapfq´iG P pf, hqI , @f, h P Γ c pV g q; 4. apPfq , @f P Γ c pV g q. The four entries of the list respectively implement linearity, hermiticity of the generators, canonical commutation relations and the equations of motion for the quantum field. Remark 4.1. As in [16], we adopt in this paper the interpretation of apfq is pa|fq, where the pairing is the Hodge inner product of 1-forms (2.1). An equivalence class in A g is denoted by rapfqs "âpfq, the equivalence class corresponding to the identity is denoted by rIs " Id. The hermitian elements of the algebra A g are called observables. Remark 4.2. Requirement 4, when we pass to the quotient algebra corresponds to the distributional relation Pâ " 0, when we take Remark 4.1 into account and the fact that P is formally selfadjoint. Since every solution of the Proca equation is a co-closed solution of the Klein-Gordon equation and vice versa, we conclude that δâ " 0, i.e.âpdfq " 0 for every f P Γ c pV g q, must be valid. If, moreover, we deprive the ideal I g of the generators in 4, the quotient algebra is said to be off-shell, however it would still be convenient to assume the-closedness constraint when defining the off-shell algebra. That is when defining the relevant ideal of the free off-shell algebra, we should keep 1-3, we should drop 4, and we should replace it with the weaker condition 4'.âpdfq , @f P Γ c pV g q. This work however deals with the on-shell algebra only, we shall indicate by A g throughout. A study of the off-shell algebra, which may result in some relevance in perturbative renormalization procedure will be done elsewhere. Møller˚-isomorphism and Hadamard states From now on let X be a topological vector space, we indicate by X 1 its topological dual. For example Γ 1 c pV g q represents the space of distributions acting on compactly supported test functions, and shall not be confused with the space of compactly supported distributions. Having built the CCR-algebra, the subsequent step in quantization consists in finding a way to associate numbers to the abstract operators in A g by identifying a distinguished state. For sake of completeness, let us recall that a state over the Proca algebra A g a C-linear functional ω : A g Ñ C which is (i) Positive: ωpa˚aq ě 0 @a P A g , (ii) Normalized: ωpIq " 1 A generic element of the CCR-algebras A g of a quantum field can be written as a finite polynomial of the generatorsâpf q, where the zero grade term is proportional to I. To specify the action of a state it is sufficient to know its action on the monomials, i.e its n-point functions: ω n pf 1 , . . . , f n q :" ωpâpf 1 q . . .âpf n qq with f 1 , . . . , f n P Γ c pV g q. If we impose continuity with respect to the usual topology on the space of compactly supported test sections we can uniquely extend the n-point functions to distributions in Γ 1 c pV nb g q we shall hereafter indicate by the symbol r ω n . Among all possible states the physical ones are the so-called quasifree (or Gaussian) Hadamard states. Quasifree means that the n-point distributions of the state have a structure resembling the one of a free theory, i.e they all can be recovered just by knowing the two-point distribution. Definition 4.3. Consider the globally hyperbolic spacetime pM, gq and a state ω : A g Ñ C for the Proca algebra of observables on pM, gq. ω is called quasifree, if for all choices of f i P Γ c pV g q (i) ω n pf 1 , . . . , f n q " 0, if n P N is odd, (ii) ω 2n pf 1 , . . . , f 2n q " ř Π ω 2 pf i 1 , f i 2 q¨¨¨ω 2 pf i n´1 , f in q, if n P N is even, where Π refers to the class of all possible decompositions of the set t1, 2, . . . , 2nu into n pairwise disjoint subsets of 2 elements ti 1 , i 2 u, ti 3 , i 4 u, . . ., ti n´1 , i n u with i 2k´1 ă i 2k for k " 1, 2, . . . , n. Regarding the notion of Hadamard state for the Proca field, which is a vector field, we adopt the notions of microlocal analysis for vector-valued distributions introduced in [53]. Remark 4.4. The interpretation of the action of a distribution on test sections is formalized in the sense of the Hodge product (2.1). This interpretation is necessary in order to agree with the interpretation of the symbolâpfq stated in Remark 4.1, since some of the distributions we shall consider in the rest of the paper arise from field operators, as the two-point functions ω 2 pf, gq :" ωpâpfqâpgqq. If Γ c pV g q Q g Þ Ñ ω 2 p¨, gq P Γ 1 c pV g q is well-defined and continuous, ω 2 actually defines a distribution of Γ 1 c pV g b V g q and vice versa, as a consequence of the Schwartz kernel theorem as clarified below. From now on, if F P Γ 1 c pV g q and f P Γ c pV g q, the action of the former on the latter is therefore interpreted as the Hodge product (2.1) F pfq " pF |fq " pf|F q " ż M g 7 pF, fqvol g . With a straightforward extension of the Definition 3.3, operators working on a generic space of k test-forms T : Ω k c pMq Ñ Ω k c pMq can be extended to the topological duals, i.e the associated distributions, in terms of the action T : on the argument of the distribution: pTF qpfq :" F pT : fq . For instance, if F P Ω 2 1 c pMq and H P Ω 0 1 c pMq, pδF qpfq :" F pdfq , pdHqpfq :" Hpδfq , f P Ω 1 c pMq . If S : Γ c pV g q Ñ Γ 1 c pV g q is continuous (in particular if S : Γ c pV g q Ñ Γ c pV g q is continuous), the standard Schwartz kernel theorem permits to introduce the distribution indicated with the same symbol S P Γ 1 c pV g b V g q, which is the unique distribution such that Spf b gq :" Spf, gq :" pSgqpfq " " pf|Sgq 2 . Conversely, a distribution of Γ 1 c pV g b V g q defines a unique map Γ c pV g q Ñ Γ 1 c pV g q that fulfills the identity above. In the rest of the work we shall take advantage of these facts and notations above. Furthermore, we adopt the notion of wavefront set of a distribution on test sections of a vector bundle on M as defined in [53]. Definition 4.5. Consider the globally hyperbolic spacetime pM, gq and a state ω : A g Ñ C for the Proca algebra of observables on pM, gq. ω is called Hadamard if it is quasifree and its two-point function ω 2 P Γ 1 c pV g b V g q satisfies the microlocal spectrum condition 3 , i.e. W F pω 2 q " H :" tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u . Above, px, k x q " py, k y q means that x and y are connected by a lightlike geodesic and k y is the co-parallel transport of k x from x to y along said geodesic, whereas k x Ź0 means that the covector k x is future pointing. As for Klein-Gordon scalar field theory, Hadamard states for Proca fields have two important properties which were also established in [16] for the notion of Hadamard state adopted there. We present here independent proofs only based on Definition 4.5. Indeed, [16] uses a definition of Hadamard states which is apparently different from our definition. A comparison of the two definitions and an equivalence result appear in Section 6. The first property of Hadamard states is the fact that the difference between the two-point functions of a pair of Hadamard states is a smooth function. This fact plays a crucial role in the point-splitting renormalization procedure (for instance of Wick polynomials and time-ordered polynomials [36][37][38][39] and of the stress-energy tensor [35,45,56]) and is, in fact, one of the reasons for assuming that Hadamard states are the physically relevant ones. Proposition 4.6. Let ω, ω 1 P Γ 1 c pV g b V g q be a pair of Hadamard states on the algebra A g of the Proca field according to Definition 4.5. Then, ω´ω 1 P ΓpV g b V g q, i.e., ω´ω 1 is smooth. More generally, ω´ω 1 is smooth if ω, ω 1 are distributions satisfying (4.1) such that their antisymmetric parts coincide mod. C 8 . Proof. Let us first prove the second statement. Let us define ω2 pf, gq :" ω 2 pf, gq and ω2 pf, gq :" ω 2 pg, fq, N`:" tpx, kq P T˚Mzt0u | k a k a " 0 , k Ź 0u , N´:" tpx, kq P T˚Mzt0u | k a k a " 0 , k Ÿ 0u , Γ 1 :" tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x ; y, k y q P Γu . for every Γ Ă T˚M 2 zt0u. If both distributions satisfy (4.1), then W F pω2 q 1 Ă N˘ˆN˘. (4.3) With the hypotheses of the proposition define R˘:" ω2´ω 12 . Since ω2´ω2 " ω 12´ω12`F where F is a smooth function, we have that R`"´R´mod. C 8 . At this juncture, (4.3) yields W F pR`q 1 X W F pR´q 1 " H because N`X N´" H. Since W F pR`q " W F p´R´`F q " W F p´R´q " W F pR´q, we conclude that the wavefront set of the distributions R˘is empty and thus they are smooth functions. This is the thesis of the second statement. The latter statement implies the former because, since both ω and ω 1 are states on the Proca˚-algebra, their antisymmetric part must be identical and it amounts to iG P , furthermore ω and ω 1 satisfy (4.1) in view of Definition 4.5. The second property regards the so called propagation property of the Hadamard singularity or also the local-global feature of Hadamard states. It has a long history which can be traced back to [22] passing through [41], [50,51] and [53] (and the recent [44]) at least. Proposition 4.7. Consider a globally hyperbolic spacetime pM, gq and a globally hyperbolic neighborhood N of a smooth spacelike Cauchy surface Σ of pM, gq. Finally, let ω N be a quasifree state for the on-shell algebra of the Proca field in pN , g| N q. The following facts are valid. (a) There exists a unique a quasifree state ω : A g Ñ C for the Proca field on the whole pM, gq which restricts to ω N on the Proca algebra on N . (b) If ω N is Hadamard according to Definition 4.5, then ω is. Proof. (a) According to (3.9), G P f " 0 for f P Γ c pV g q if and only if f " Pg for some g P Γ c pV g q. We will use this fact to construct ω out of ω N . Consider two other smooth spacelike surfaces (for both M and N ) Σ`in the future of Σ and Σ´in the past of Σ. Let χ`, χ´: M Ñ r0, 1s be smooth maps such that χ`ppq " 0 if p stays in the past of Σ´and χ`ppq " 1 if p stays in the future of Σ`and χ``χ´" 1. Then, defining Tf :" Pχ`G P f , f P Γ c pV g q (4.4) we have that Tf P Γ c pV g | N q (more precisely supppTfq stays between Σ´and Σ`), and Tf´f " Pg for some g P Γ c pV g q , (4.5) because by standard properties of Green operators: G P Tf " GP Tf´GṔ Tf "`GP P˘χ`G P f´GṔ Pp1´χ´qG P f " χ`G P f´GṔ pPG P fq`GṔ Pχ´G P f " χ`G P f`χ´G P f " G P f. Therefore we can apply (3.9) obtaining (4.5). With these results, let us define ω 2 pf, gq :" ω N 2 pTf, Tgq , f, g P Γ c pV g q . Taking the continuity properties of G P into account, we leave to the reader the elementary proof of the fact that there is a unique distribution Γ 1 c pV g b V g q such that its value on f b g coincides with 4 ω 2 pf, gq. (We will indicate that distribution by ω 2 with the usual misuse of language.) Furthermore, in view of the definition of T, it is obvious that ω 2 is also a bisolution of the Proca equation, since G P P " PG P " 0. Using Definition 4.3 to construct a candidate quasifree state ω on A g out of its two-point function ω 2 , it is clear that the positivity requirement is guaranteed because ω N satisfies it. We conclude that there is a quasifree state ω on A g , whose two point function is (4.6), and this two point function is a distribution which is also bisolution of the Proca equation. Finally, observe that ω extends to the whole A g the state ω N since the states are quasifree and the two-point function of the former extends the two point function of the latter. Indeed, ω 2 pf, gq " ω N 2 pTf, Tgq " ω N 2 pf, gq if f, g P Γ c pV g | N q . This is because, specializing (3.9) and (4.4)-(4.5) to the globally hyperbolic spacetime pN , g| N q since f P Γ c pV g | N q, we have that Tf´f " Pg with g P Γ c pV g | N q and ω N 2 vanishes when one argument has the form Pg, because it is a bisolution of the Proca equation in N . There is only one such quasifree state which is an extension of ω N to the whole algebra A g , and such that its two-point function is a bisolution of the Proca equation. In fact, another such extension ω 1 would satisfy ω 1 2 pf, gq " ω 1 2 pTf, Tgq " ω N pTf, Tgq " ω 2 pTf, Tgq " ω 2 pf, gq , for all f, g P Γ c pV g q. (b) We pass to the proof that ω is Hadamard if ω N is. We have to prove that (4.1) is valid if it is valid for ω N in pN , g| N q. Interpreting the two-point functions as distributions of Γ 1 c pV g b V g q, ω 2 " ω N 2˝P χ`G P b Pχ`G P . (4.7) The wavefront sets of G P and Pχ`G P can be computed as follows. First of all, from (3.7), G P " QG N " G N Q (4.8) where Q " I`m´2dδ g . It is known that W F pG N q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y qu Notice that, in particular k x ‰ 0 and k y ‰ 0 nor simultaneously by definition, nor separately since they are connected by a coparallel transport. So, since Q is a differential operator we immediatly deduce by 4.8 that W F pG P q Ă W F pG N q. Then we associate to the two operator their distributional kernels G P px, yq and G N px, yq and recast equation 4.8 in the form: G P px, yq " pId x b Q y q G N px, yq, which, by standard microlocal analysis results, implies that W F pG N q Ă CharpId x b Q y q Y W F pG P q. However explicit computations give that CharpId x b Q y q " tpx, k x ; y, 0q|px, k x q P T˚M, y P Mu which does not intersect W F pG N q at any point, implying W F pG N q Ă W F pG P q Ă W F pG N q. So G P and G Q have the same wavefront set. Therefore, since Pχ`is a smooth differential operator, W F pPχG N q Ă tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y qu A this point, a standard estimate of composition of wavefront sets in (4.7) yields (see, e.g., [40]) W F pω 2 q Ă H where the Hadamard wavefront set H is the one in (4.1). To conclude the proof, it is sufficient to establish the converse inclusion. To this end, observe that, since the antisymmetric part of ω 2 is ω2´ω2 " iG P , W F pG P q Ă W F pω2 q Y W F pω2 q , where we adopted the same notation as at the beginning of the proof of Proposition 4.6: ω2 " ω 2 , ω2 pf, gq " ω 2 pg, fq. If, according to that notation, the prime applied to wavefront sets is defined as in (4.2), the above inclusion can be re-phrased to tpx, k x ; y, k y q P T˚M 2 zt0u | px, k x q " py, k y qu " W F pG P q 1 Ă W F pω2 q 1 Y W F pω2 q 1 (4.9) Above W F pω2 q 1 Ă H 1 " tpx, k x ; y, k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u and, with a trivial computation, W F pω2 q 1 Ă tpx,´k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k y Ź 0u , Now suppose that px, k x ; y, k y q P H 1 does not belong to W F pω2 q 1 . According to (4.9), px, k x ; y, k y q R W F pG P q 1 (notice that H 1 Q px, k x ; y, k y q R W F pω2 q 1 since the two sets are disjoint). This is impossible because every px, k x ; y, k y q P H 1 belongs to W F pG P q 1 as it immediately arises by comparing the explicit expressions of these two sets written above. In summary H 1 Ă W F pω 2 q 1 , that is H Ă W F pω 2 q, concluding the proof. Hadamard states turned also out to be relevant in the study of quantum energy conditions [16,17,19] and in black hole physics [13,24,42,46,54] (see references in [44] for a summary) We are finally ready to extend the Møller operator to the quantum algebras, proving that they are indeed isomorphic. To this end, for any paracausally related metric g » g 1 , we define an isomorphism of the free algebras R gg 1 : A g 1 Ñ A g as the unique unital˚-algebra isomorphism between the said free unital˚-algebras such that R gg 1 pa 1 pfqq " apR : gg 1 fq @f P Γ c pV g 1 q , where R is a Møller operator of g, g 1 and the adjoint R : gg 1 is defined as in Proposition 3.5. Møller˚-isomorphism and the pullback of Hadamard states When we pass to the quotient algebras, the preservation of the causal propagators discussed in the previous sections, immediately implies that the induced map on the quotient algebras is an isomorphism, that we call Møller˚-isomorphism. Proposition 4.8. Let now R gg 1 : A g 1 " A g 1 {I g 1 Ñ A g " A g {I g be the quotient morphism constructed out of R gg 1 . Then R is well defined and is indeed a˚-algebra isomorphism. Proof. The proof of this statement is identical to the one performed in [48,Proposition 5.6]. Indeed it just relies on the preservation of the causal propagators proved in Theorem 3.7, which implies that the associated CCR-ideals are˚-isomorphic. The final step in our construction is to define a pullback of the Møller˚-isomorphism to the states and then to prove that the Hadamard condition is preserved, as done in [48,Theorem 5.14] for normally hyperbolic field theories. Theorem 4.9. Let R gg 1 be the Møller *-isomorphism and let ω : A g Ñ C be a quasifree Hadamard state, we define the pull-back state ω 1 : A g 1 Ñ C by ω 1 " ω˝R gg 1 . The following facts are true: 1 ω 1 is a well-defined state; 2 ω 1 is quasifree; 3 ω 1 is a Hadamard state. Proof. The proof of 1-2 is trivial and discussed in [48,Proposition 5.11]. The proof of 3 follows from the Hadamard propagation property stated in Proposition 4.7. To prove the statement we can just focus on the case in which the Møller operator is constructed out of two spacetimes such that g ĺ g 1 , the reasoning can then be iterated at each step of the paracausal chain. The two-point function of the pullback state can be written as ω 1 2 pf, hq " ω 1 pâ 1 pfqâ 1 phqq " ωpR gg 1 pâ 1 pfqâ 1 phqqq " ωpâpR : gg 1 fqâpR : gg 1 hqq " ω 2 pR : gg 1 f, R : gg 1 hq. We recall that the operator is the composition of two pieces R : gg 1 " R :gg χ˝R : gχg 1 and split the proof in two steps. First we focus on the bidistribution ω χ 2 pf, hq :" ω 2 pR :gg χ f, R :gg χ hq on pM, g χ q defining a quasifree state therein. By the property 3.10, in the region in which g χ " g, there is a t 0 a Cauchy surface Σ t 0 in common for g and g χ , a common globally hyperbolic neighborhood N of that Cauchy surface such that ω χ 2 pf, hq " ω 2 pf, hq when the supports of f and g are chosen in N and thus the corresponding state is Hadamard in pN , g χ q. Now Proposition 6.3 implies that ω χ 2 is Hadamard in the whole pM, g χ q. Secondly, the same argument can be used once again for the operator R : gχg 1 on the Hadamard state ω χ on pM, g χ q, proving that the state induced by ω 2 pR : gg 1¨, R : gg 1¨q " ω χ 2 pR : gχg 1¨, R : gχg 1¨q is Hadamard as well on pM, g 1 q. In other words the full Møller operator preserves the Hadamard form. Existence of Proca Hadamard states in globally hyperbolic spacetimes This section is devoted to the construction of Hadamard states for the real Proca field in a generic globally hyperbolic spacetime. Actually, the technology of Møller operators, in particular Theorem 4.9, allows us to reduce the construction of Hadamard states for the Proca equation to the special case of an ultrastatic spacetime with Cauchy hypersufaces of bounded geometry. Indeed, as shown in [48,Corollary 2.23], for any globally hyperbolic spacetime pM, gq, there exists a paracausally related globally hyperbolic spacetime pM, g 0 q which is ultrastatic. In other words, first of all pM, g 0 q is isometric to RˆΣ where pΣ, h 0 q is a t-independent complete Riemannian manifold and g 0 "´dt b dt`h 0 , where t is the natural coordinate on R and dt is past directed. We also denote by B t the tangent vector to the submanifold R of RˆΣ. In view of the completeness of h, these spacetimes are globally hyperbolic (see e.g. [20]) and Σ is a Cauchy surface of this spacetime. In turn, it is possible to change the metric on Σ in order that the final metric, indicated by h is both complete and of bounded geometry [34]. By construction, the final ultrastatic spacetime pM,´dt b dt`hq is paracausally related to pM, g 0 q because the intersection of the corresponding open cones is non-empty as it always contains B t . By transitivity pM, gq is paracausally related with pRˆΣ,´dt b dt`hq. Hence, we assume without loss of generalities, that pM, gq " pRˆΣ,´dt b dt`hq is a globally hyperbolic ultrastatic spacetime, with dt past directed, whose spatial metric h is complete. When dealing with the construction of Hadamard states we also assume that the spatial manifold pΣ, hq is also of bounded geometry. In the final part of the section, we will come back to consider a generic globally hyperbolic spacetime pM, gq The Cauchy problem in ultrastatic spacetimes We study here the Cauchy problem for the Proca (real and complex) field in ultrastatic spacetimes pM, gq " pRˆΣ,´dt b dt`hq, where pΣ, hq is complete. A more general treatise appears in [55] where the Cauchy problem is studied, also in the presence of a source of the Proca field, in a generic globally hyperbolic spacetime and the continuity of the solutions with respect to the initial data is focused. Let us consider the Proca equation (2.3) (where m 2 ą 0) on the above ultrastatic spacetime. As observed in [16], every smooth 1-form A P Ω 1 pMq naturally uniquely decomposes as Apt, pq " A p0q pt, pqdt`A p1q pt, pq (5.1) where A piq pt,¨q P Ω i pΣq for i " 0, 1 and t P R. By direct inspection and taking the equivalence of (2.3) and (3.4)-(3.5) into account, one sees that Eq. (2.3) is equivalent to the constrained double Klein-Gordon system B 2 t A p0q "´p∆ p0q h`m 2 qA p0q , (5.2) B 2 t A p1q "´p∆ p1q h`m 2 qA p1q , (5.3) B t A p0q "´δ p1q h A p1q . (5.4) Above, ∆ pkq h :" δ pk`1q h d pkq`dpk´1q δ pkq h is the Hodge Laplacian on pΣ, hq for k-forms and the last condition (5.4) is nothing but the constraint δ p1q g A " 0 arising from (2.3). The theory for the fields A p1q and A p0q is a special case of the theory of normally hyperbolic equations on corresponding vector bundles with positive inner product over a globally hyperbolic spacetime [3,4]. In our case, (1) there is a real vector bundle V p1q g with basis M, canonical fiber isomorphic to Tq Σ, and equipped with a fiberwise real symmetric scalar product induced by h 7 q . A p1q P ΓpV p1q g q; (2) there is another real vector bundle V p0q g with basis M, canonical fiber isomorphic to R, and equipped with a positive fiberwise real symmetric scalar product given by the natural product in R. A p0q P ΓpV p0q g q. Evidently V g " V p0q g ' V p1q g . (5.5) Equations (5.2) and (5.3) admit existence and uniqueness theorems for smooth compactly supported Cauchy data and corresponding smooth spacelike compact solutions in Γ sc pV p0q g q and Γ sc pV p1q g q respectively, as a consequence of very well-known results in the theory of normally hyperbolic equations [3,4,33]. However, when viewing A p0q and A p1q as parts of the Proca field A, we have also to deal with the additional constraint (5.4). Notice that (5.4) imposes two constraints on the Cauchy data of A p0q and A p1q on Σ: B t A p0q p0, pq "´δ p1q h A p1q p0, pq B 2 t A p0q p0, pq "´δ p1q h B t A p1q p0, pq . The second constraint is only apparently of the second order. Indeed, taking (5.2) into account, it can be re-written as a condition of the Cauchy data p∆ p0q h`m 2 qA p0q p0, pq " δ p1q h B t A p1q p0, pq . At this juncture we observe that, with some elementary computation (use ∆ pB 2 t`∆ p0q h´m 2 qpB t A p0q`δ p1q h A p1q q " 0 which, in turn, implies Equation (5.4) , if the initial condition of that scalar Klein-Gordon equation for pB t A p0q`δ p1q h A p1q q are the zero initial conditions. This exactly amounts to B t A p0q p0, pq "´δ p1q h A p1q p0, pq and p∆ p0q h`m 2 qA p0q " δ p1q h B t A p1q p0, pq . In summary, we are naturally led to focus on this Cauchy problem B 2 t A p0q`p ∆ p0q h`m 2 qA p0q " 0 , (5.6) B 2 t A p1q`p ∆ p1q h`m 2 qA p1q " 0 , (5.7) pB 2 t`∆ p0q h´m 2 qpB t A p0q`δ p1q h A p1q q " 0 , (5.8) with initial data A p0q p0,¨q " a p0q p¨q, B t A p0q p0,¨q " π p0q p¨q, A p1q p0,¨q " a p1q p¨q, B t A p1q p0,¨q " π p1q p¨q (5.9) where a p0q , π p0q , a p1q , π p1q are pairs of smooth compactly supported, respectively 0 and 1, forms on Σ, and the constraints are valid 4), for the fields A p0q P Γ sc pV p0q g q and A p1q P Γ sc pV p1q g q, with the same initial data (5.9) and constraints (5.10). As a consequence, (1) every smooth spacelike compact solution of the Proca equation A P Γ sc pV g q (2.3) defines compactly supported smooth Cauchy data on Σ which satisfy the constraints (5.10); π p0q "´δ p1q h a p1q , p∆ p0q h`m 2 qa p0q " δ p1q h π p1q . (2) if the Cauchy data are smooth, compactly supported and satisfy (5.10), then there is a unique smooth spacelike compact solution of the Proca equation A P Γ sc pV g q (2.3) associated to them; (3) the support of a solution A P Γ sc pV g q with smooth compactly supported initial data satisfies supppAq Ă J`pSqYJ´pSq, where S Ă Σ is the union of the supports of the Cauchy data. Remark 5.2. (1) All the discussion above, and Proposition 5.1 in particular, extends to the case of a complex Proca field and corresponding associated complex Klein Gordon fields. The stated results can be extended easily to the case of the non-homogeneous Proca equation and also considering continuity properties of the solutions with respect to the source and the initial data referring to natural topologies. (See [55] for a general discussion.) (2) A naive idea may be that we can freely fix smooth compactly supported Cauchy data for A p1q and then define associated Cauchy conditions for A p0q by solving the constraints (5.10). In this case the true degrees of freedom of the Proca field would be the vector part A p1q , whereas A p0q would be a constrained degree of freedom. This viewpoint is incorrect, if we decide to deal with spacelike compact solutions, because the second constraint in Equation (5.10) in general does not produce a compactly supported function a p0q when the source δ p1q h π p1q is smooth compactly supported (the smoothness of a p0q is however guaranteed by elliptic regularity from the smoothness of δ p1q h π p1q ). a p0q is compactly supported only for some smooth compactly supported initial conditions π p1q . Therefore the linear subspace of initial data (5.9) compatible with the constraints (5.10) does not include all possible compactly supported initial conditions π p1q which, therefore, cannot be freely chosen. (3) However this space of constrained Cauchy data is non-trivial, i.e., it does not contain only zero initial conditions and in particular there are couples pa p0q , π p1q q such that both elements do not vanish. This is because, for every smooth compactly supported 1-form f p1q (with δ p1q f p1q ‰ 0 in particular) and for every smooth compactly supported 2-form f p2q , a p0q :" δ p1q h f p1q π p1q :"´∆ p1q h`m 2¯f p1q`δ p2q h f p2q are smooth, and compactly supported, they solve the nontrivial constraint in (5.10) δ p1q h π p1q " p∆ p0q`m 2 qa p0q and f p1q , f p2q can be chosen in order that neither of a p0q and π p1q vanishes. The easier constraint π p0q "´δ p1q h a p1q is solved by every smooth compactly supported 1-form a p1q by defining the smooth compactly supported 0-form π p0q correspondingly. The Proca symplectic form in ultrastatic spacetimes Consider two solutions A, A 1 P Γ sc pV g q X KerP of the Proca equation in our ultrastatic spacetime, choose t P R and consider the bilinear form σ pPq t pA, A 1 q :" ż Σ h 7 pa p1q t , π p1q 1 t´d a p0q 1 t q´h 7 pa p1q 1 t , π p1q t´d a p0q t q vol h ,(5.11) where we are referring to the Cauchy data on Σ of the smooth spacelike compact solutions of the Proca equation. Σ is viewed as the time slice at time t. As is well known, it is possible to define a natural symplectic form for the Proca field in general globally hyperbolic specetimes [6] with properties analogous to the ones we are going to discuss here. In this section we however stick to the ultrastatic spacetime case which is enough for our ends. According to [6] (with an argument very similar to the proof of Propositions 3.12 and 3.13 in [48]) we have immediately that σ pPq t pA, A 1 q " σ pPq t 1 pA, A 1 q @t, t 1 P R , and, omitting the index t as the symplectic form is independent of it, σ pPq pA, A 1 q " ż M g 7`f , G P f 1˘v ol g (5.12) where A, f (resp. A, f 1 ) are related by A :" G P f (resp. A 1 :" G P f 1 ). Remark 5.3. The important identity (5.12) is also valid in a generic globally hyperbolic spacetime when σ pP q is interpreted as the general symplectic form of the Proca field according to [6]. Let us suppose to deal with the Cauchy data of the real vector space C Σ Ă Ω 0 c pΣq 2Ω 1 c pΣq 2 of smooth compactly supported Cauchy data pa 0 , π 0 , a 1 , π 1 q subjected to the linear constraints (5.10), C Σ :" ! pa p0q , π p0q , a p1q , π p1q q P Ω 0 c pΣq 2ˆΩ1 c pΣq 2ˇπp0q "´δ p1q h a p1q , p∆ p0q h`m 2 qa p0q " δ p1q h π p1q ) . (5.13) Not only the Cauchy problem is well behaved in that space as a consequence of Proposition 5.1, but we also have the following result which, in particular, implies that the Weyl algebra of the real Proca field has trivial center. Proposition 5.4. The bilinear antisymmetric map σ pP q : C ΣˆCΣ Ñ R defined in (5.11) is non-degenerate and therefore it is a symplectic form on C Σ . where the smooth compactly supported sections are complex. We have used the same symbols as for the real case for the causal propagators since the associated operators commute with the complex conjugation. As a consequence, a standard argument about the uniqueness of Green operators implies that the causal propagators for the real case are nothing but the restriction of the causal propagator of the complex case which, in turn, are the trivial complexification of the real ones. The Proca energy density in ultrastatic spacetimes Starting from the Proca Lagrangian in every curved spacetime (see, e.g, [15]) L "´1 4 F µν F µν´m 2 2 A µ A µ with F µν :" B µ A ν´Bν A µ and referring to local coordinates px 0 , . . . , x n´1 q adapted to the split M " RˆΣ of our ultrastatic spacetime, where x 0 " t runs along the whole R and x 1 , . . . , x n´1 are local coordinates on Σ, the energy density reads in terms of initial conditions on Σ of the considered Proca field T 00 " 1 2 h 7 pπ p1q´d a p0q , π p1q´d a p0q q`1 2 h 7 p2q pda p1q , da p1q q m 2 2´h 7 pa p1q , a p1q q`a p0q a p0q¯ě 0 . (5.15) Above h 7 p2q is the natural scalar product for the 2-forms on Σ induced by the metric tensor. It is evident that the energy density is non-negative since the metric h and its inverse h 7 are positive by hypothesis. The total energy at time t is the integral of T 00 on Σ, using the natural volume form, when replacing A p0q and A p1q for the respective Cauchy data. As B t is a Killing vector and the solution is spacelike compact, the total energy is finite and constant in time. E pP q " 1 2 ż Σ´h 7 pπ p1q´d a p0q , π p1q´d a p0q q`h 7 p2q pda p1q , da p1q q m 2`h7 pa p1q , a p1q q`a p0q a p0q˘¯v ol h . (5.16) Using Hodge duality of d and δ and the definition of the Hodge Laplacian, the expression of the total energy can be re-arranged to E pP q " 1 2 ż Σ´h 7 pπ p1q , π p1q q`h 7 pda p0q , da p0q q´2h 7 pπ p1q , da p0q q´δ p1q h a p1q δ p1q h a p1q h 7`ap1q , ∆ p1q h a p1q q`m 2 pa p0q a p0q`h7 pa p1q , a p1q q˘¯vol h . Using again the Hodge duality of d and δ the third term in the integral can be rearranged tó ż Σ h 7 pπ p1q , da p0q qvol h "´ż Σ a p0q δ p1q h π p1q vol h . The term δ p1q π p1q above and the term δ p1q h a p1q δ p1q h a p1q appearing in the expression for the total energy can be worked out exploiting the constraints (5.10). Inserting the results in the found formula for the total energy, we finally find, with the notation already used for the symplectic form, E pP q " 1 ÿ i"0 η i 1 2 ż Σ h 7 piq pπ piq , π piq q`h 7 piq pa piq , p∆ piq h`m 2 Iqa piq q vol h ,(5.17) when the used Cauchy data belong to the constrained space C Σ . It is now clear that the total energy of the Proca field is the difference between the total energies of the two Klein-Gordon fields composing it exactly as it happened for the symplectic form. This difference is however positive when working on smooth compactly supported initial conditions satisfying the constraints (5.10), because the found expression of the energy is the same as the one computed with the density (5.15). Remark 5.6. We notice that the negative energy component of the field can be interpreted as a ghost, in this case however no issues arise since dynamical constraints covariantly remove such a state. A different approach to the problem by generalizing to curved spacetime the Stuckelberg lagrangian, can be found in [2], where it is appearently argued the no Hadamard states exist for the Proca field, contrarily to the results of [16] and of this work. Remark 5.7. With the same argument, the found result immediately generalizes to the case of complex k-forms and one finds 1 ÿ i"0 η i 1 2 ż Σ h 7 piq pπ piq , π piq q`h 7 piq pa piq , p∆ piq h`m 2 Iqa piq qvol h " " 1 2 ż Σ´h 7 pπ p1q´d a p0q , π p1q´d a p0q q`h 7 p2q pda p1q , da p1q q m 2`h7 pa p1q , a p1q q`a p0q a p0q˘v ol h¯ě 0 (5.18) where the bar over the forms denotes the complex conjugation and pa p0q , π p0q , a p1q , π p1q q are complex forms of C Σ`i C Σ . Elliptic Hilbert complexes and Proca quantum states in ultrastatic spacetimes We can proceed to the construction of quasifree states. As we shall see shortly, this construction for the Proca field uses some consequences of the spectral theory applied to the theory of elliptic Hilbert complexes [5] defined in terms of the closure of Hodge operators in natural L 2 spaces of forms. Some of the following ideas were inspired by [16]. However we now work in the space of Cauchy data instead of in the space of smooth supportly compacted forms and/or modes. Furthermore we do not assume restrictions on the topology of the Cauchy surfaces used in [16] to impose a pure point spectrum to the Hodge Laplacians. To define quasifree states for the Proca field we observe that, as P is Green hyperbolic, the CCR algebra A g is isomorphic to the analogous unital˚-algebra A psympq g generated by the solution-smeared field operators σ pPq pâ, Aq, for A P Ker sc pP q, which are R-linear in A, Hermitian, and satisfy the commutation relations 5 " σ pPq pâ, Aq, σ pPq pâ, A 1 q ı " iσ pPq pA, A 1 qI . (5.19) The said unital˚-algebra isomorphism F : A g Ñ A psympq g is completely defined as the unique homomorphism of unital˚-algebras that satisfies F :âpfq Þ Ñ σ pPq pâ, G P fq with A " G P f, f P Γ c pV g q . The definition is well-posed in view of (5.12), (3.8), (3.9), and the definition of A g . Within this framework, the two point function ω 2 is interpreted as the integral kernel of ω´σ pPq pâ, Aqσ pPq pâ, A 1 q¯. In particular, its antisymmetric part is universally given by i 2 σ pP q pA, A 1 q due to (5.19). The specific part of the two point function is therefore completely embodied in its symmetric part µpA, A 1 q. According to this observation, a general recipe for real (bosonic) CCR in generic globally hyperbolic spacetimes to define a quasifree state on the˚-algebra A g (e.g., see [40,41,56] for the scalar case and [23, Chapter 4, Proposition 4.9] for the generic case of real bosonic CCRs) is to assign a real scalar product on the space of spacelike compact solutions µ : Ker sc pPqˆKer sc pPq Ñ R satisfying (a) the strict positivity requirement µpA, Aq ě 0 where µpA, Aq " 0 implies A " 0; (b) the continuity requirement with respect to the relevant symplectic form σ pP q (see, e.g., [23,Proposition 4.9]), σ pPq pA, A 1 q 2 ď 4µpA, AqµpA 1 , A 1 q . (5.20) The continuity requirement directly arises form the fact that the quasifree state induced by µ on the whole˚-algebra A g " A symp g according to Definition 4.3 is a positive functional. The converse implication, though true, is less trivial [23,41]. The two mentioned requirements are nothing but the direct translation of (2)' and (3)' stated in the introduction. (Regarding the latter, observe that σ pP q corresponds to the causal propagator at the level of solutions -Eq. (5.12) in our case -as discussed in Section 5.2.) At this point, it should be clear that the quasifree state defined by µ has two-point function, viewed as bilinear map on Γ c pV g qˆΓ c pV g q, ω µ papfqapf 1 qq " ω µ2 pf, f 1 q :" µpG P f, G P f 1 q`i 2 σ pP q pG P f, G P f 1 q . However, since the Cauchy problem is well posed on the time slices Σ of an ultrastatic spacetime pRˆΣ,´dt b dt`hq, as proved in Proposition 5.1, we can directly define µ (and σ pPq ) in the space of Cauchy data C Σ on Σ, for smooth spacelike compact solutions, viewed as the time slice at t " 0, µ : C ΣˆCΣ Ñ R . In view of the peculiarity of the Cauchy problem for the Proca field as discussed in Section 5.1, the real vector space of the Cauchy data C Σ is constrained. We underline that working at the level of constrained initial data does not affect the construction of quasifree states. Indeed, it is sufficient that the space of constrained initial conditions is a real (or complex) vector space and that the constrained Cauchy problem is well posed. With this in mind, referring to the canonical decomposition A " A p0q dt`A p1q of a real smooth spacelike compact solution A of the Proca equation, we remember that C Σ :" ! pa p0q , π p0q , a p1q , π p1q q P Ω 0 c pΣq 2ˆΩ1 c pΣq 2ˇπp0q "´δ p1q h a p1q , p∆ p0q h`m 2 qa p0q " δ p1q h π p1q ) . Above pa p0q , π p0q q :" pA p0q , B t A p0q q| t"0 and pa p1q , π p1q q :" pA p1q , B t A p1q q| t"0 . With the said definitions and where A denotes both a solution of Proca equation and its Cauchy data on Σ, we have the first result. Proposition 5.8. Consider the˚-algebra A g of the real Proca field on the ultrastatic spacetime pM, gq " pRˆΣ,´dt b dt`hq, with dt past directed and pΣ, hq a smooth complete Riemannian manifold. Let η 0 :"´1, η 1 :" 1 and h 7 pjq denote the standard inner product of j-forms on Σ induced by h. The bilinear map on the space C Σ of real smooth compactly supported Cauchy data (5.13) µpA, A 1 q :" 1 ÿ j"0 η j 2 ż Σ h 7 pjq pπ pjq , p∆ pjq`m2 q´1 {2 π pjq 1 q`h 7 pjq pa pjq , p∆ pjq`m2 q 1{2 a pjq 1 q vol h (5.21) is a well defined symmetric positive inner product which satisfies (5.20) and thus it defines a quasifree state ω µ on A g completely defined by its two-point function ω µ`a pfqapf 1 q˘" ω µ2 pf, f 1 q :" µ`G P f, G P f 1˘`i 2 σ pP q`G P f, G P f 1˘( 5.22) where f, f 1 P Γ c pV g q satisfy σ pP q`G P f, G P f 1˘" ż M g 7 pf, G P f 1 q vol g . The bar over the operators in (5.21) denotes the closure in suitable Hilbert spaces of the operators originally defined on domains of compactly supported smooth functions. To explain this formalism, before starting with the proof we have to permit some technical facts about the properties of the Hodge operators at the level of L 2 spaces. Given the complete Riemannian manifold pΣ, hq, with n :" dimpΣq consider the Hilbert space H h :" À n k"0 L 2 k pΣ, vol h q, where the sum is orthogonal and L 2 k pΣ, vol h q is the complex Hilbert space of the square-integrable k-forms with respect to the relevant Hermitian Hodge inner product: pa|bq k :" ż Σ h 7 pkq pa, bq vol h , a, b P L 2 k pΣ, vol h q , where a denotes the pointwise complex conjugation of the complex form a. The overall inner product on H h will be indicated by p¨|¨q and the Hilbert space adjoint of a densely-defined operator A : DpAq Ñ H h , with DpAq Ă H h , will be denoted by A˚: DpA˚q Ñ H h . The closure of A will be denoted by the bar: A : DpAq Ñ H h . If Ω c pΣq C :" À n k"0 Ω k c pΣq C denotes the dense subspace of complex complactly supported smooth forms Ω k c pΣq C :" Ω k c pΣq`iΩ k c pΣq, define the two operators (we omit the index h for shortness) d :" ' n k"0 d pkq : Ω c pΣq C Ñ Ω c pΣq C , δ :" ' n k"0 δ pkq : Ω c pΣq C Ñ Ω c pΣq C with d pnq :" 0 and δ p0q :" 0. Finally, introduce the Hodge Laplacian as ∆ :" n ÿ k"0 ∆ pkq : Ω c pΣq C Ñ Ω c pΣq C with ∆ pkq :" δ pk`1q d pkq`dpk´1q δ pkq . Since pΣ, hq is complete, ∆ can be proved to be essentially selfadjoint, for instance exploiting the well-known argument by Chernoff [8] (or directly referring to [1]). Since ∆ is essentially selfadjoint, if c P R, also ∆`cI is essentially selfadjoint. In particular, its unique selfadjoint extension is its closure ∆`cI. Referring to the theory of elliptic Hilbert complexes developed in [5,Section 3] and focusing in particular on [5,Lemma 3.3] based on previous achievements established in [1], we can conclude that the following couple of facts are true. (The compositions of operators are henceforth defined with their natural domains: DpA`Bq :" DpAq X DpBq, DpABq " tx P DpBq | Bx P DpAqu, DpaAq :" DpAq for a ‰ 0, Dp0Aq :" H h , and A Ă B means DpAq Ă DpBq with B| DpAq " A.) ∆ pkq with ∆ pkq :" δ pk`1q d pkq`dk´1 δ pkq . (5.24) A trivial generalization of the decomposition as in (5.24) holds for ∆`cI " ∆`cI with c P R. We are now prompt to prove a preparatory technical lemma -necessary to establish Proposition 5.8 -that will be fundamental for showing that the bilinear map µ is positive on the space C Σ . Lemma 5.9. For every given k " 0, 1, . . . , n, c ą 0, and α P R, the identites hold p∆ pk`1q`c Iq α d pkq x " d pkq p∆ pkq`c Iq α x , @x P Dpp∆ pkq`c Iq α q X Dpp∆ pk`1q`c Iq α d pkq q p∆ pk´1q`c Iq α δ pkq y " δ pk´1q p∆ pkq`c Iq α y , @y P Dpp∆ pkq`c Iq α q X Dpp∆ pk´1q`c Iq α δ pkq q . Proof. Since dd " 0 and δδ " 0, from (5.23), we also have d dx " 0 if x P Dpdq and δ δy " 0 if y P Dpδq, and thus (5.24) yields 6 d ∆ Ą d δ d " ∆ d . However, if Dpd ∆q Ľ Dpd δ dq, we would have x P Dp∆q " Dpδ dq X Dpd δq such that ∆x " δdx`dδx P Dpdq, but x R Dpdδdq, namely δdx R Dpdq. This is impossible since δdx`dδx P Dpdq, Dpdq is a subspace and d δx P Dpdq (and more precisely d d δx " 0 as stated above). Therefore d ∆ " d δ d " ∆ d and the same result is valid with δ in place of d. Evidently, in both cases ∆ can be replaced by the selfadjoint operator ∆`cI = ∆`cI for every c P R: d ∆`cI " ∆`cI d , δ ∆`cI " ∆`cI δ . (5.25) We henceforth assume c ą 0. In that case, as ∆ is already positive on its domain, the spectrum of the selfadjoint operator ∆`cI is strictly positive and thus ∆`cI´1 : H h Ñ Dp∆`cIq is well defined, selfadjoint and bounded. The former identity in (5.25) also implies that Dpd ∆`cIq " Dp∆`cI dq, so that ∆`cI´1d ∆`cI| Dpd ∆`cIq x " d| Dpd ∆`cIq x . By construction, we can choose x " ∆`cI´1y with y P Dpdq in view of the definition of the natural domain of the composition d ∆`cIq. In summary ∆`cI´1dy " d ∆`cI´1y , @y P Dpdq . Since the argument is also valid for δ, we have established that ∆`cI´1d Ă d ∆`cI´1 , ∆`cI´1δ Ă δ ∆`cI´1 Iterating the argument, for every n " 0, 1, . . ., p∆`cI´1q n d Ă d p∆`cI´1q n , p∆`cI´1q n δ Ă δ p∆`cI´1q n . This result extends to complex polynomials of ∆`cI´1 in place of powers by linearity. Using the spectral calculus (see e.g. [43]) where µ xy pEq " px|P E yq and P is the projector-valued spectral measure of ∆`cI´1, the found result for d can be written ż r0,bs ppλqdµ x,dy pλq " ż r0,bs ppλqdµ δx,y pλq (5.26) for every complex polynomial p, where r0, bs is a sufficiently large interval to include the (bounded positive) spectrum of ∆`cI´1, x P Dpδq, y P Dpdq, and where we have used δ " d˚. Since the considered regular Borel complex measures are finite and supported on the compact r0, bs, we can pass in (5.26) from polynomials p to generic continuous functions f in view of the Stone-Weierstrass theorem. At this juncture, PE " P E and the uniqueness part of Riesz' representation theorem for regular complex Borel measures, implies that pP E δy|xq " pP E y|dxq for all x P Dpδq, y P Dpdq, and every Borel set E Ă R. which means P E δ Ă d˚P E , namely P E δ Ă δP E . Analogously, we also have P E d Ă dP E . If f : R Ñ C is measurable and bounded, the standard spectral calculus and (5.23), with a procedure similar to the one used to prove P E δ Ă δP E and taking into account the fact that Dpf p∆`cI´1qq " H h , yields f p∆`cI´1qδ Ă δf p∆`cI´1q , f p∆`cI´1qd Ă df p∆`cI´1q (5.27) If f is unbounded, we can choose a sequence of bounded measurable functions f n such that f n Ñ f pointwise. It is easy to prove that (see, e.g. [43]) x P Dp ş R f dP q entails ş R f n dP x Ñ ş R f dP x. This is the case for instance for f pλq " λ β with β ă 0 restricted to r0, bs. Referring to this function and the pointed out result for some sequence of bounded functions with f n Ñ f pointwise, the latter of (5.27) implies that 7 , p∆`cIq α dx " dp∆`cIq α x if x P Dpp∆`cIq α q X Dpdq and dx P Dpp∆`cIq α q, where we used also the fact that d is closed. The case of δ can be worked out similarly. Summing up, we have proved that, if α P R, p∆`cIq α dx " dp∆`cIq α x , @x P Dpp∆`cIq α q X Dpp∆`cIq α dq p∆`cIq α δy " δp∆`cIq α y , @y P Dpp∆`cIq α q X Dpp∆`cIq α δq . Let us remark that for α ď 0 it is sufficient to choose x P Dpdq and y P Dpδq. For every given k " 0, 1, . . . , n, c ą 0, and α P R, taking the decomposition of H h into account the above formulae imply p∆ pk`1q`c Iq α d pkq x " d pkq p∆ pkq`c Iq α x , @x P Dpp∆ pkq`c Iq α q X Dpp∆ pk`1q`c Iq α d pkq q p∆ pk´1q`c Iq α δ pkq y " δ pk´1q p∆ pkq`c Iq α y , @y P Dpp∆ pkq`c Iq α q X Dpp∆ pk´1q`c Iq α δ pkq q . That is the thesis. We are now prompted to prove that the bilinear map defined by Equation (5.21) defines a quasifree state defined by the two-point function given by (5.22) establishing the thesis of Proposition 5.8. Proof of Proposition 5.8. To continue with the proof of the proposition, we now demonstrate that µ is well-defined and positive. That bilinear form is well-defined because Ω pjq c pΣq Ă Dp∆ pjq`m2 I α q for α ď 1 as one immediately proves from spectral calculus. Furthermore, the integrand in the right-hand side of Equation (5.21) is the linear combination of products of L 2 functions (of which one of the two has also compact support). Let us pass to the positivity issue. Our strategy is to re-write µpA, Aq, where A " pa p0q , π p0q , a p1q , π p1q q P C Σ , as the quadratic form of the energy µpA, Aq " E pP q pA o q, where the right-hand side is defined in Equation (5.16), for a new set of initial data A o which are not necessarily smooth and compactly supported but such that E pP q pA o q is well defined. If A P C Σ , define for j " 0, 1 A o " pa p0q o , π p0q o , a p1q o , π p1q o q a pjq o :" p∆ pjq`m2 Iq´1 {4 a pjq π pjq o :" p∆ pjq`m2 Iq´1 {4 π pjq (5.28) Notice that the definition is well posed and the forms a pjq o and π pjq o belong to the respective Hilbert spaces of j-forms, because Ω pjq c pΣq Ă Dp∆ pjq`m2 I α q for α ď 1 as said above. Furthermore the new forms are real since the initial ones are real and ∆ pjq`m2 I α commutes with the complex conjugation 8 . At this juncture, we have from (5.21) µpA, Aq " 1 ÿ j"0 η j ż Σ h 7 pjq pπ pjq o , π pjq o q`h 7 pjq pa pjq o , p∆ pjq`m2 Iqa pjq o qvol h (5.29) Furthermore, the new Cauchy data, though they stay outside C Σ in general, they however satisfy the natural generalization of the constraints defining C Σ in view of Lemma 5.9: π p0q o "´δ p1q h a p1q o , p∆ p0q h`m 2 qa p0q o " δ p1q h π p1q o . (5.30) These identities arise immediately from Definitions (5.28), the constraints (5.10), and by applying Lemma 5.9 and paying attention to the fact that Ω pjq c pΣq Ă Dpp∆ pj´1q`c Iq α δ pjq q for every α ď 1 and also using p∆ pjq`m2 Iqp∆ pjq`m2 Iq´1 {4 " p∆ pjq`m2 Iq´1 {4 ∆ pjq`m2 I (for, e.g., [43, (f) in Proposition 3.60 ]). Using (5.23) and (5.30) in the right-hand side of (5.29), we can proceed backwardly as in the proof that (5.16) is equivalent to (5.17). Indeed, the only ingredients we used in that proof were the constraint equations which are valid also for A o and the duality of δ and d with respect to the Hodge inner product, which extends to δ and d. In summary, µpA, Aq " 1 2 ż Σ´h 7 p1q pπ p1q o´d p0q a p0q o , π p1q o´d p0q a p0q o q`h 7 p2q pd p1q a p1q o , d p1q a p1q o q m 2`h7 p1q pa p1q o , a p1q o q`a p0q o a p0q o˘¯v ol h . From that identity, it is clear that µpA, Aq ě 0 and µpA, Aq " 0 implies A o " 0, which in turn yields A " 0 because the operators ∆ pjq`m2 I 1{4 are injective. We have established that µ : C ΣˆCΣ Ñ R is a positive real symmetric inner product. Let us pass to prove (5.20). First of all, we change the notation concerning the scalar product µ making explicit the decomposition of A, and we work with complex valued forms. We use A " pa, πq " pa p0q , π p0q , a p1q , π p1q q , a :" pa p0q , a p1q q , π :" pπ p0q , π p1q q so that, if pa, πq, pa 1 , π 1 q P pL 2 0 pΣ, vol h q ' L 2 1 pΣ, vol h qqˆpL 2 0 pΣ, vol h q ' L 2 1 pΣ, vol h qq are such that the right-hand side below is defined, we can write µppa, πq, pa 1 , π 1 qq :" 1 ÿ j"0 η j 2 ż Σ h 7 pjq pπ pjq , H´1 pjq π pjq 1 q`h 7 pjq pa pjq , H pjq a pjq 1 qvol h where H pjq :" ∆ pjq`m2 I 1{2 , and the bar on forms denotes the complex conjugation. Finally, for α "˘1, we defined H α a :" pH α p0q a p0q , H α p1q a p1q q , H α π :" pH α p0q π p0q , H α p1q π p1q q . By direct inspection one sees that, if pa, πq, pa 1 , π 1 q P C Σ`i C Σ , then the right-hand side of the first identity below is well-defined and Λppa, πq, pa 1 , π 1 qq :" 1 2 µ`pπ`iH´1a, a´iHπq, pπ 1´i H´1a 1 , a 1`i Hπ 1 q" µppa, πq, pa 1 , π 1 qq`i 2 σ pPq ppa, πq, pa 1 , π 1 qq where σ pP q is the right-hand side of (5.14), which however coincides with the original symplectic form (5.11) evaluated on complex Cauchy data because pa, πq, pa 1 , π 1 q P C Σ`i C Σ and Remark 5.5 holds. Finally notice that if pa, πq P C Σ`i C Σ then a o :" π´iHa and π o :" a`iH´1π satisfy the constraints (though they do not belong to C Σ`i C Σ in general) π p0q o "´δ p1q h a p1q o , H p0q a p0q o " δ p1q h π p1q o . The proof is direct, using Lemma 5.9 once more. As a consequence, exploiting the same argument to prove (5.18) and observing that H α commutes with the complex conjugation -so that it holds π´iH´1a " π`iH´1a for instance -we have that 2Λppa, πq, pa 1 π 1 qq " µ`pπ`iH´1a, a´iHπq, pπ´iH´1a, a`iHπq" µ´pπ´iH´1a, a`iHπq, pπ´iH´1a, a`iHπq¯ě 0 . The final inequality is due to the fact that µ is (the complexification of) a real positive bilinear symmetric form. All that means in particular that the Hermitian form Λ on pC Σ`i C Σ qp C Σ`i C Σ q is (semi)positively defined and thus it satisfies the Cauchy-Schwartz inequality. In particular, pImΛppa, πq, pa 1 , π 1 qqq 2 ď |Λppa, πq, pa 1 , π 1 qq| 2 ď Λppa, πq, pa, πqq Λppa 1 , π 1 q, pa 1 , π 1 qq . If choosing pa, πq, pa 1 , π 1 q P C Σ (thus real forms), the above inequality specialises to σ pPq ppa, πq, pa 1 , π 1 qq 2 ď 4µppa, πq, pa, πqq µppa 1 , π 1 q, pa 1 , π 1 qq which is the inequality (5.20) we wanted to prove. Hadamard states in ultrastatic and generic globally hyperbolic spacetimes With the next proposition, we show that the quasifree states defined in Proposition 5.8 is a Hadamard state when pΣ, hq is of bounded geometry. To prove the assertion we will take advantage of the general formalism developed in [23] and [26]. An alternative proof, which does not assume that the manifold is of bounded geometry (however we here take advantage of [34]), could be constructed along the procedure developed in [21] and extending it to the vectorial Klein-Gordon field. where H pjq :" ∆ pjq`m2 1{2 , σ pjq are the symplectic forms of the corresponding Klein-Gordon fields taking place in the right-hand side of (5.14), now evaluated on complex fields. Above, a pjq , π pjq P Ω j c pΣq C are the Cauchy data on Σ of A pjq respectively and a pjq 1 , π pjq 1 P Ω j c pΣq C are the Cauchy data on Σ of A pjq 1 respectively. Notice that we are not imposing constraints on these initial data since we are dealing with independent Klein-Gordon fields. λp jq are evidently positive because, if all involved forms in the right-hand side are smooth and compactly supported, then the right-hand side of the identity above is well-defined and λp jq pA pjq , A pjq 1 q :" 1 2 ż Σ h 7 pjq pH 1{2 a pjq`i H´1 {2 π pjq , H 1{2 pjq a pjq 1`i H´1 {2 π pjq 1 q vol h . The case of λṕ jq is strictly analogous. Furthermore λp jq pA pjq , A pjq 1 q´λṕ jq pA pjq , A pjq 1 q " iσ pjq pA pjq , A pjq 1 q . Therefore λp jq satisfy the hypotheses of [23, Proposition 4.14] 9 so that they define a pair, for j " 0, 1, of gauge-invariant quasifree states for the complex Klein-Gordon fields respectively associated to Equations (5.2) and (5.3). We pass to prove that both states are Hadamard exploiting the fact that pΣ, hq is of bounded geometry. By rewriting the covariances λp jq as λp jq "˘qcp jq (q " iσ pjq ) a quick computation shows that cp jq " 1 2 « I˘H´1 pjq H pjq I ff . We can immediately realize that the operator cp jq is the same Hadamard projector obtained in [26, Section 5.2] 10 -see also [23,Section 11] for a more introductory explanation for the scalar case. This operator belongs to the necessary class of pseudodifferential operators C 8 b pR; Ψ 1 b pΣqq because pΣ, hq is of bounded geometry. Therefore, on account of [26,Proposition 5.4], the two quasifree states associated to λp jq , for A pjq and j " 0, 1, are Hadamard. In other words, the Schwartz kernels provided by the two-point functions λp jq pG pjq¨, G pjq¨q , viewed as distributions of ΓpV pjq g b V pjq g q 1 , satisfy W F pλp jq pG pjq¨, G pjq¨q q " H , where H is defined in (4.1) and G piq , i " 0, 1 are the causal propagators for the normally hyperbolic operators N piq :" B 2 t`∆ piq h`m 2 I : Γ sc pV piq g q Ñ Γ sc pV piq g q i " 0, 1 . Above and from now on we use the same notation to indicate a bidistribution and the associated Schwartz kernel. Notice that we have used the same symbol G pjq of the causal propagator we used for the real vector field case. This is because the causal propagators for the complex fields are the direct complexification of the scalar case (see Remark 5.5). We pass now to focus on the expression of ω µ2 provided in (5.22) taking the usual decomposition Ω 1 c pMq C Q f " f p0q dt`f p1q into account. It can be written ω µ2 pf, f 1 q " ω p1q µ2 pf p1q , f p1q 1 q´ω p0q µ2 pf p0q , f p0q 1 q where, comparing (5.21) and (5.22) with (5.31) for real arguments f, f 1 P ΓpV g q, we find ω pjq µ2 pf pjq , f pjq 1 q " λp jq pG pjq f pjq , G pjq f pjq 1 q . We have W F p˘ω pjq µ2 q " W F p˘λp jq pG pjq¨, G pjq¨q q " W F pλp jq pG pjq¨, G pjq¨q q " H for j " 0, 1. 9 The reader should pay attention to the fact that the Cauchy data used in [23], in the complex case, are defined as pf0, f1q :" pa,´iπq instead of our pa, πq! This is evident by comparing (2.4) and (2.20) in [23]. With the choice of [23], ipf0, f1q t¨q pf 1 0 , f 1 1 q " ş f0f 1 1`f1 f 1 0 vol h " iσppa, πq, pa 1 , π 1 qq, where¨q " σ1 (the Pauli matrix) according to [23]. 10 It follows immediately since b`ptq "´b´ptq " H :" ∆ pjq`m2 I 1{2 . Taking (5.5) into account, we now observe that ω µ2 P ΓpV g bV g q 1 " ΓppV p0q g 'V p1q g qbpV p0q g 'V p1q g qq 1 . As a matter of fact, however, ω µ2 does not have mixed components acting on sections of V g . These are respectively represented by´ω p0q µ2 and ω p1q µ2 whose wavefront set is H in both cases. The remaining two components have empty wavefront set since they are the zero distributions. Applying the definition of wavefront set of a vector-valued distribution [53], we conclude that W F pω µ2 q " W F p´ω p0q µ2 q Y W F pω p1q µ2 q Y H Y H " H Y H Y H Y H " H , concluding the proof. Combining the results obtained so far, we get the main result of this paper. Theorem 5.11. Let pM, gq be a globally hyperbolic spacetime and refer to the CCR-algebra A g of the real Proca field. Then there exists a quasifree Hadamard state on A g . Proof of Theorem 4. As already explained in the beginning of Section 5, for any globally hyperbolic spacetime pM, gq, there exists a paracausally related globally hyperbolic spacetime pM, g 0 q which is ultrastatic and whose spatial metric is of bounded geometry. In particular, in this class of spacetimes, the quasifree states defined in Proposition 5.8 satisfy the microlocal spectrum condition, as proved in Proposition 5.10. Therefore, since the pull-back along a Møller˚-isomorphism preserves the Hadamard condition on account of Theorem 4.9, we can conclude. Comparison with Fewster-Pfenning's definition of Hadamard states Though the paper [16] by Fewster and Pfenning concerns quantum energy inequalities, it also offers a general theoretical discussion about the algebraic quantization of the Proca and the Maxwell fields in curved spacetime. In particular, the authors propose a definition of a Hadamard state which appears to be technically different from ours at first glance, even if it shares a number of important features with ours. This section is devoted to a comparison of the two definitions for the Proca field. Proca Hadamard states according to Fewster and Pfenning The definition of Hadamard state stated in [16,Equation (35)] is formulated in terms of causal normal neighborhoods of smooth spacelike Cauchy surfaces (see also below) and the global Hadamard parametrix for distributions which are bisolutions of the vectorial Klein-Gordon equation. Our final goal is to prove an equivalence theorem of our definition of Hadamard state Definition 4.5 and the one adopted in [16]. As a first step, we translate the original Fewster-Pfenning's definition of a Hadamard state into an equivalent form which will turn out to be more useful for our comparison. The equivalence of the version stated below of Fewster-Pfenning's definition and the original one was established in [16, Section III C] (see also the comments under Definition 6.1). Definition 6.1. [Fewster-Pfenning's definition of Proca Hadamard state] Consider the globally hyperbolic spacetime pM, gq and a state ω : A g Ñ C for the Proca algebra of observables on pM, gq. ω is called Hadamard if it is quasifree and its two-point function has the form ωpâpfqâphqq " W g pf, Qhq (6.1) @f, h P Γ c pV g q, where Q : ΓpV g q Ñ ΓpV g q in the differential operator Q " Id`m´2pdδ g q. Above W g P Γ 1 c pV g b V g q is a Klein-Gordon distributional bisolution such that W g pf, gq´W g pg, fq " iG N pf, gq mod C 8 , Finally, [16] also contains a proof of the existence of Hadamard states for the Proca (and the Maxwell) field in globally hyperbolic spacetimes with compact Cauchy surfaces (whose first homology group is trivial when treating the Maxwell field). This proof establishes first the existence in ultrastatic spacetimes and next it exploits a standard deformation argument [56]. An (almost) equivalence theorem We are in a position to state and prove our equivalence result. Theorem 6.6. Consider the globally hyperbolic spacetime pM, gq and a quasifree state ω : A g Ñ C for the˚-algebra of observables on pM, gq of the real Proca field. Let ω 2 P Γ 1 c pV g b V g q be the twopoint function of ω. The following facts are true. (a) If ω is Hadamard according to Definition 6.1, then it is also Hadamard according to Definition 4.5. (b) If pM, gq admits a Proca quasifree Hadamard state according to Definition 6.1 and ω is Hadamard according to Definition 4.5, then ω is Hadamard in the sense of Definition 6.1. Proof. Tha following argument is identical to the one used in 4.7 to prove W F pG P q " W F pG N q, but we repeat it here to keep this section self-contained. First of all notice that, since ω 2 pf, gq " W g pf, Qgq, then viewing ω 2 and W g as bidistributions, we have ωpx, yq " pId x b Q y q W px, yq (where we have used the fact that Q is formally selfadjoint) taking Remark 4.4 into account). Now suppose that ω is Hadamard according to Definition 6.1. Since W g satisfies the microlocal spectrum condition and the differential operator I b Q is smooth, we have W F pω 2 q Ă W F pW g q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u . Notice that, in particular, k x and k y cannot vanish (simultaneously or separately) if they take part of W F pW g q. Let us prove the converse inclusion to complete the proof of (a). Again from known results, from ω 2 px, yq " pId x b Q y qW g px, yq, we have W F pW g q Ă CharpI b Qq Y W F pω 2 q . However, by direct inspection, one sees that CharpI b Qq " tpx, k x ; y, 0q | px, k x q P T˚M , y P Mu , so that W F pω 2 q Ă W F pW g q Ă W F pω 2 q Y tpx, k x ; y, 0q | px, k x q P T˚M , y P Mu . (6.4) However W F pW g q X tpx, k x ; y, 0q | px, k x q P T˚M , y P Mu " H and thus we can re-write the chain of inclusions (6.4) as W F pω 2 q Ă W F pW g q Ă W F pω 2 q so that W F pω 2 q " W F pW g q . This is the thesis of (a) because we have established that Definition 4.5 is satisfied by ω. To prove (b), let us assume that ω satisfies Definition 4.5. By hypotheses the antisymmetric part of ω 2 is´iG P . Let Ω be another quasifree state of the Proca field which satisfies Definition 6.1. Also the antisymmetric part of Ω 2 is´iG P . Due to Proposition 4.6, F px, yq :" ω 2 px, yq´Ω 2 px, yq . is a smooth function. Furthermore it is a symmetric bisolution of the Proca equation. In particular it therefore satisfies 13 F pf, dh p0q q " 0, where h p0q P Ω 0 c pMq, so that F pf, Qgq " F pf, gq`1 m 2 F pf, dpδ g gqq " F pf, gq . Collecting everything together, we can assert that, for some distributional bisolution of the Klein-Gordon equation W g which satisfies (6.2), (6.3), and is associated to the Hadamard state Ω, it holds ω 2 pf, gq " W g pf, Qgq`F pf, gq " W g pf, Qgq`F pf, Qgq . If we re-absorb F in the definition of W g , W 1 g pf, Qgq " W g pf, Qgq`F pf, Qgq . the new W 1 g is again a distributional bisolution of the Klein-Gordon equation which satisfies (6.2), (6.3) and ω 2 pf, gq " W 1 g pf, Qgq . In other words, the Hadamard state ω according to Definition 4.5 is also Hadamard in the sense of Definition 6.1 concluding the proof of (b). Remark 6.7. Regarding (b), the existence of Hadamard states in the sense of Definition 6.1 has been established in [16] for globally hyperbolic spacetimes whose Cauchy surfaces are compact: in those types of spacetimes at least, the two definitions are completely equivalent. We expect that actually the equivalence is complete, even dropping the compactness hypothesis (see the conclusion section). This issue will be investigated elsewhere. Conclusion and future outlook We conclude this paper by discussing some open issues which are raised in this paper and we leave for future works. On an ultrastatic spacetime M " RˆΣ, the one-parameter group of isometries given by timetranslations has an associated action on A g in terms of˚-algebras isomorphisms α u completely induced by α u pâpfqq :"âpf u q for every f P Γ c pMq, where f u pt, pq :" fpt´u, pq for every t, u P R and p P Σ. It is shall not be difficult to prove that the Hadamard state constructed in Theorem 3 is invariant under the action of α u ω µ pα u paqq " ω µ paq @u P R @a P A g It should be also true that the map R Q u Þ Ñ ω µ pbα u paqq P C is continuous for every a, b P A g which would assure (see, e.g. [43]) that α :" tα h u hPR is unitarily implementable by a strongly continuous unitary representation of R in the GNS representation of ω µ and that the vacuum vector of the Fock-GNS representation is left invariant under the said unitary representation. We expect that the selfadjoint generator of that unitary group has a positive spectrum where, necessarily, the vacuum state is an eigenvector with eigenvalue 0. In other words ω µ should be a ground state of α. We finally expect that ω µ is pure (on the Weyl algebra associated to the symplectic space ppKerPq X Γ sc pMq, σ pPq q and it is the unique quasifree algebraic state which is invariant under α. We can summarize the previous discussion in the following question. Question 7.1. Is the Hadamard state defined in Theorem 3 a ground state for the timetranslation? More precisely, is it the unique, pure, quasifree algebraic state which is invariant under action of α? Last, but not least, we have seen in Section 6 that if a globally hyperbolic manifold admits a Proca quasifree Hadamard state according to the definition of Fewster-Pfenning, then Definition 4.5 and 6.1 are equivalent. This is the case for example for globally hyperbolic spacetimes whose Cauchy surfaces are compact. We do expect to extend this result for the whole class of globally hyperbolic spacetime. Conjecture 7.2. Deifnition 4.5 and 6.1 are equivalent on any globally hyperbolic spacetime. As is evident from our quasi equivalence theorem, a complete equivalence would take place if a Hadamard state according to [16] is proved to exist for every globally hyperbolic spacetime. As a matter of fact, we expect that every globally hyperbolic spacetime pM, gq admits a quasifree Proca Hadamard state ω according to Fewster and Pfenning. This state should exist in every paracausally related ultrastatic spacetime pRˆΣ,´dt 2`h q with complete Cauchy surfaces of bounded geometry. With the same argument used for our existence proof of Hadamard states or the deformation argument exploited in [16], it should be possible to export this state to the original space pM, gq. We expect that the Hadamard Klein-Gordon bisolution for the real Proca field on pRˆΣ,´dt 2`h q used to define ω according to (6.1) in Definition 6.1 should have this form. W g pf, f 1 q :" µpG N f, G N f 1 q`i 2 σ pN q pG N f, G N f 1 q , f, f 1 P Γ c pRˆΣq , where N is the Klein-Gordon operator (3.2) associated to P and G N its causal propagator. The bilinear symmetric form µ :`pΩ 0 c pΣqq 2ˆp Ω 1 c pΣqq 2˘ˆ`p Ω 0 c pΣqq 2ˆp Ω 1 c pΣqq 2˘Ñ R is defined as in (5.21), but with the crucial difference that here its arguments are not restricted to C ΣˆCΣ . Klein-Gordon vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Issues with the quantization of the Proca field . . . . . . . . . . . . . . . . . . . . . 4 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and notation of geometric tools in spacetimes . . . . . . . . . . . . . . 8 2.2 Smooth forms, Hodge operators, and the Proca equation . . . . . . . . . . . . . . . 9 3 Møller Maps and Møller Operators 10 3.1 Linear fiber-preserving isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Klein-Gordon operator associated to a Proca operator and Green operators . . . . 11 Funding V.M. and D.V. acknowledge the support of the INFN-TIFPA national project "Bell". S.M. acknowledges the support of the INFN and the GNFM-INDAM. is a spacelike compact solution of the Proca equation(2.3), then it satisfies (5.2)-(5.4) and its Cauchy data (5.9) satisfy the constraints (5.10). On the other hand, if we have smooth compactly supported Cauchy data (5.9), then the two Klein-Gordon equations (5.2) and (5.3) admit unique spacelike compact smooth solutions which also satisfies (5.8) as a consequence. If the said Cauchy data satisfy the constraint (5.10), then also (5.4) is satisfied, because it is equivalent to the unique solution of (5.8) with zero Cauchy data. In that case, the two solutions A p0q and A p1q define a unique solution of the Proca equation with the said Cauchy data.We have established the following result completely extracted from the theory of normally hyperbolic equations.Proposition 5.1. Let pM, gq " pΣ,´dt b dt`hq be a smooth globally hyperbolic ultrastatic spacetime with dt past directed, where h is a smooth complete Riemannian metric on Σ. Consider the Cauchy problem on pM, gq for the smooth 1-form A satisfying the Proca equation (2.3) for m 2 ą 0, with smooth compactly supported Cauchy data (5.9) on Σ viewed as the t " 0 time slice. The Proca Cauchy problem for A with constraints (5.10) is equivalent, regarding existence and uniqueness of spacelike compact smooth solutions , to the double normally hyperbolic Klein-Gordon constrained Cauchy problem (5.2)-(5. adjoint in the Hilbert space H h . (b) The unique selfadjoint extension ∆ of ∆ satisfies ∆ " d δ`δd " Proposition 5.10. If the metric h on the time slice Σ is of bounded geometry, then the quasifree state ω µ : A g Ñ C defined in Proposition 5.8 is Hadamard according to Definition 4.5.Proof. Consider a pair of complex Klein-Gordon fields A p0q and A p1q in the ultrastatic spacetime pM, gq " pRˆΣ,´dt b dt`hq, with pΣ, hq a smooth complete Riemannian manifold of bounded geometry obeying the normally hyperbolic equations (5.2) and (5.3) in the respective vector bundles on M, according to Section 5.1. We stress that we now assume that the two fields are complex. Referring to[23, Chapter 4], we define the covariances, for j " 0, 1 pjq pπ pjq , H´1 pjq π pjq 1 q`h 7 pjq pa pjq , H pjq a pjq 1 q vol h`i 2 σ pjq pA pjq , A pjq 1 q (5.31) pjq pπ pjq 1 , H´1 pjq π pjq q`h 7 pjq pa pjq 1 , H pjq a pjq q vol h`i 2 σ pjq pA pjq 1 , A pjq q (5.32)λp jq pA pjq , A pjq 1 q :" 1 2 ż Σ h 7 λṕ jq pA pjq , A pjq 1 q :" 1 2 ż Σ h 7 We use throughout the convention of summation over the repeated indices. In this work to be a Hermitian or real scalar product does not include the positivity condition, though it is always assumed to be non-degenerate. The notion of wavefront set refers to distributions acting on complex valued test sections in view of the pervasive use of the Fourier transform. For this reason, when dealing with these notions we consider the natural complex extension of the involved distributions, by imposing that they are also C-linear. If ω2 indicates the distribution associated to the two-point function: ω2 " ωN 2˝T b T. Notice that, as σ pPq pA, A 1 q is non degenerate, we have that σ pPq pâ, Aq " 0 only if A " 0. It holds pB`CqA " BC`BA, but AB`AC Ă ApB`Cq. Below, α ą 0 otherwise p∆`cIq α is bounded in view of its spectral properties and (5.27) is enough to conclude the proof. It easily arises from spectral calculus using the fact that the complex conjugation is bijective from H h to H h , continuous, and commutes with ∆ pjq`m2 I. Where these open sets are named normal neighborhoods of smooth spacelike Cauchy surfaces, omitting "causal".12 Essentially because convex normal neighborhoods of points form a topological basis of any spacetime and in view of[44, Proposition 9] We are grateful to C. Fewster for this observation. Proof. Taking(3.8)into account, suppose that Γ sc pV g q X KerP Q A 1 " G P f whose Cauchy data are pa p0q 1 , π p0q 1 , a p1q 1 , π p1q 1 q P C Σ is such that σ pPq pA, A 1 q " 0 for all A " G P f P Γ sc pV g q X KerP " C Σ , we want to prove that A 1 " 0 namely, its initial conditions are p0, 0, 0, 0q. From (5.12), using the fact that g 7 is non-degenerate, we have that A 1 " G P f 1 " 0 so that its Cauchy data are the zero data in view of the well-posedness of the Cauchy problem Proposition 5.1.To conclude this section we prove that, when using Cauchy data in C Σ , the expression of σ pPq can be re-arranged in order to make contact with the analogous symplectic forms of the two Klein-Gordon fields A p0q and A p1q the solution A is made of, as discussed in Section 5.1. Indeed, remembering the constraint π p0q "´δ p1q h a p1q , and using the duality of δ and d, part of the integral in the right-hand side of (5.11) can be rearranged to żAs a consequence, if η i " 0 for i " 1 and η i "´1 for i " 0 and h 7 piq is h 7 for i " 1 and the pointwise product for i " 0,In other words, referring to the (Klein-Gordon) symplectic forms introduced in[48]for normally hyperbolic equations (5.2) and (5.3) σ pPq pA, A 1 q " σ p1q pA p1q , A p1q 1 q´σ p0q pA p0q , A p0q 1 q where σ pkq is the symplectic form for a normally hyperbolic field operator on a real vector bundle defined, e.g.,[48,Proposition 3.12]. A similar result is valid for the causal propagators. Decomposing f " f p0q dt`f p1q P Γ c pV g q where f p0q P Γ c pV p0q g q and f p1q P Γ c pV p1q g q, (5.12), the analogs for scalar and vector Klein Gordon fields[48]and (5.14) imply ż M g 7 pf, G P f 1 qvol g "where G piq , i " 0, 1 are the causal propagators for the normally hyperbolic operatorsaccording to the theory of[48]. Here ∆ p0q h coincides with the standard Laplace-Beltrami operator for scalar fields on Σ.Remark 5.5. With the same argument, the found results immediately generalize to the case of complex k-forms. More precisely, if the Cauchy data belong to C Σ`i C Σ ,where the left-hand side is again (5.11) evaluated for complex Proca fields, i.e., complex Cauchy data. Above, the bar denotes the complex conjugation and the Cauchy data of the considered complex Proca fields satisfy the constraints (5.10). Furthermore ż M g 7 pf, G P f 1 qvol g " ż M h 7 pf p1q , G p1q f p1q 1 qvol g´ż M f p0q G p0q f p0q 1 vol g G N being the causal propagator of the Klein-Gordon operator (3.2) and which satisfies the microlocal spectrum condition W F pW g q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py. k y q, k x Ź 0u . (6.3G N being the causal propagator of the Klein-Gordon operator (3.2) and which satisfies the mi- crolocal spectrum condition W F pW g q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u . (6.3) Sahmann -Verch's [53] generalization to vector (and spinor) fields of some classic Radzikowski results originally formulated for scalar fields. In practice, (a) if a distribution which is a bisolution of the vectorial Klein-Gordon equation and it is of Hadamard form in a normal causal neighborhoods of a smooth spacelike Cauchy surface, then it necessarily has the wavefront set of the form (6.3) ((a) [53, Theorem 5.8]) and its antisymmetric part satisfies (6.2) directly from the definition of parametrix; (b) if a distribution which is a bisolution of the vectorial Klein-Gordon equation satisfies (6.3) and (6.2), then it is of Hadamard form in some normal causal neighborhoods of a smooth spacelike Cauchy surface. Remark 6.2. The equivalence of Definition 6.1 and the original one stated in [16] relies on. see [53, Remark 5.9. (i)Remark 6.2. The equivalence of Definition 6.1 and the original one stated in [16] relies on Sah- mann -Verch's [53] generalization to vector (and spinor) fields of some classic Radzikowski results originally formulated for scalar fields. In practice, (a) if a distribution which is a bisolution of the vectorial Klein-Gordon equation and it is of Hadamard form in a normal causal neighborhoods of a smooth spacelike Cauchy surface, then it necessarily has the wavefront set of the form (6.3) ((a) [53, Theorem 5.8]) and its antisymmetric part satisfies (6.2) directly from the definition of parametrix; (b) if a distribution which is a bisolution of the vectorial Klein-Gordon equation satisfies (6.3) and (6.2), then it is of Hadamard form in some normal causal neighborhoods of a smooth spacelike Cauchy surface (see [53, Remark 5.9. (i)]). That result was already established for the Hadamard states of scalar and vector (including spinor) fields in [22, 41, 53] (see [40, 44] for a general recap for the KG scalar field). The pivotal tool is the already mentioned notion of causal normal neighborhood N of a smooth spacelike Cauchy surface Σ in a globally hyperbolic spacetime pM; gq. The notion introduced in [41] has been recently improved (closing a gap in the geometric definition of Hadamard states) in [44] 11 . The propagation results established in. For the Proca fields. it has been established in [16] the property of propagation of the Hadamard condition stated in the next proposition. 41,53] and [16] are valid with the improved notion of causal normal neighborhoods and Hadamard states of [44For the Proca fields, it has been established in [16] the property of propagation of the Hadamard condition stated in the next proposition. That result was already established for the Hadamard states of scalar and vector (including spinor) fields in [22, 41, 53] (see [40, 44] for a general recap for the KG scalar field). The pivotal tool is the already mentioned notion of causal normal neighborhood N of a smooth spacelike Cauchy surface Σ in a globally hyperbolic spacetime pM; gq. The notion introduced in [41] has been recently improved (closing a gap in the geometric definition of Hadamard states) in [44] 11 . The propagation results established in [41,53] and [16] are valid with the improved notion of causal normal neighborhoods and Hadamard states of [44]. Let N be a causal normal neighborhood of a Cauchy surface Σ of pM, gq. Suppose that the restriction of ω to pN , g| N q is Hadamard according to Definition 6.1. Then ω is Hadamard in pM, gq according to the same definition. make contact with our Proposition 4.7. One may only assume that pN , g| N q is globally hyperbolic also therein. That is a consequence of the following facts. (a) Every causal normal neighborhood N Ă M of a Cauchy surface Σ of pM, gq is. Proposition 6.3. Let ω : A g Ñ C be a quasifree state for the Proca field in the globally hyperbolic spacetime pM, gq. by definition [41, 44], a globally hyperbolic spacetime with respect to the restriction of the metric and Σ is also a Cauchy surface in pN , g| N qProposition 6.3. Let ω : A g Ñ C be a quasifree state for the Proca field in the globally hyperbolic spacetime pM, gq. Let N be a causal normal neighborhood of a Cauchy surface Σ of pM, gq. Suppose that the restriction of ω to pN , g| N q is Hadamard according to Definition 6.1. Then ω is Hadamard in pM, gq according to the same definition. make contact with our Proposition 4.7. One may only assume that pN , g| N q is globally hyperbolic also therein. That is a consequence of the following facts. (a) Every causal normal neighborhood N Ă M of a Cauchy surface Σ of pM, gq is, by definition [41, 44], a globally hyperbolic spacetime with respect to the restriction of the metric and Σ is also a Cauchy surface in pN , g| N q. Every smooth spacelike Cauchy surface admits a causal normal neighborhood. 41, 44Every smooth spacelike Cauchy surface admits a causal normal neighborhood [41, 44]. According to the proof of [41, Lemma 2.2 ] whose validity extends to [44], every neighborhood of a smooth spacelike Cauchy surface includes a causal normal neighborhood of that Cauchy surface 12. According to the proof of [41, Lemma 2.2 ] whose validity extends to [44], every neighbor- hood of a smooth spacelike Cauchy surface includes a causal normal neighborhood of that Cauchy surface 12 . In [16], it is an immediate consequence of (6.1) and the analogous feature of Klein-Gordon bisolutions (see the discussion on p. 4488 in [16]). b V g q be a pair of bisolutions of the Proca equation satisfying the Hadamard condition (6.1) for corresponding Klein-Gordon bisolutions which. 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[ "The U(5)-O(6) transition in the Interacting Boson Model and the E(5) critical point symmetry", "The U(5)-O(6) transition in the Interacting Boson Model and the E(5) critical point symmetry" ]
[ "J M Arias \nDepartamento de Física Atómica\nMolecular y Nuclear\nFacultad de Física\nUniversidad de Sevilla\nApartado 106541080SevillaSpain\n", "C E Alonso \nDepartamento de Física Atómica\nMolecular y Nuclear\nFacultad de Física\nUniversidad de Sevilla\nApartado 106541080SevillaSpain\n", "A Vitturi \nDipartimento di Fisica Galileo Galilei\nVia Marzolo\n", "J E García-Ramos \nDepartamento de Física Aplicada\nUniversidad de Huelva\n21071HuelvaSpain\n", "J Dukelsky \nInstituto de Estructura de la Materia\nCSIC\nSerrano 12328006MadridSpain\n", "A Frank \nInstituto de Ciencias Nucleares\nUNAM\nCircuito Exterior C.U\n04510México, MéxicoD.F\n", "\n35131PadovaItaly\n" ]
[ "Departamento de Física Atómica\nMolecular y Nuclear\nFacultad de Física\nUniversidad de Sevilla\nApartado 106541080SevillaSpain", "Departamento de Física Atómica\nMolecular y Nuclear\nFacultad de Física\nUniversidad de Sevilla\nApartado 106541080SevillaSpain", "Dipartimento di Fisica Galileo Galilei\nVia Marzolo", "Departamento de Física Aplicada\nUniversidad de Huelva\n21071HuelvaSpain", "Instituto de Estructura de la Materia\nCSIC\nSerrano 12328006MadridSpain", "Instituto de Ciencias Nucleares\nUNAM\nCircuito Exterior C.U\n04510México, MéxicoD.F", "35131PadovaItaly" ]
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The relation of the recently proposed E(5) critical point symmetry with the interacting boson model is investigated. The large-N limit of the interacting boson model at the critical point in the transition from U(5) to O(6) is obtained by solving the Richardson equations. It is shown explicitly that this algebraic calculation leads to the same results as the solution of the Bohr differential equation with a β 4 potential.
10.1103/physrevc.68.041302
[ "https://arxiv.org/pdf/nucl-th/0309005v1.pdf" ]
9,742,690
nucl-th/0309005
3b7c4336d4372a456c188a7375c867dd6609039f
The U(5)-O(6) transition in the Interacting Boson Model and the E(5) critical point symmetry 2 Sep 2003 J M Arias Departamento de Física Atómica Molecular y Nuclear Facultad de Física Universidad de Sevilla Apartado 106541080SevillaSpain C E Alonso Departamento de Física Atómica Molecular y Nuclear Facultad de Física Universidad de Sevilla Apartado 106541080SevillaSpain A Vitturi Dipartimento di Fisica Galileo Galilei Via Marzolo J E García-Ramos Departamento de Física Aplicada Universidad de Huelva 21071HuelvaSpain J Dukelsky Instituto de Estructura de la Materia CSIC Serrano 12328006MadridSpain A Frank Instituto de Ciencias Nucleares UNAM Circuito Exterior C.U 04510México, MéxicoD.F 35131PadovaItaly The U(5)-O(6) transition in the Interacting Boson Model and the E(5) critical point symmetry 2 Sep 2003 The relation of the recently proposed E(5) critical point symmetry with the interacting boson model is investigated. The large-N limit of the interacting boson model at the critical point in the transition from U(5) to O(6) is obtained by solving the Richardson equations. It is shown explicitly that this algebraic calculation leads to the same results as the solution of the Bohr differential equation with a β 4 potential. The study of phase transitions is one of the most exciting topics in Physics. Recently the concept of critical point symmetry has been proposed by Iachello [1]. These kind of symmetries apply when a quantal system undergoes transitions between traditional dynamical symmetries. In Ref. [1] the particular case of the Bohr Hamiltonian [2] in Nuclear Physics was worked out. In this case, in the situation in which the potential energy surface in the β-γ plane is γ-independent and the dependence in the β degree of freedom can be modeled by an infinite square well, the so called E(5) symmetry appears. This situation is expected to be realized in actual nuclei when they undergo a transition from spherical to γ-unstable deformed shapes. The E(5) symmetry is obtained within the formalism based on the Bohr hamiltonian, but it has also been used in connection with the Interacting Boson Model (IBM) [3]. Although this is not the form it was originally proposed [1], it has been in fact argued that moving from the spherical to the γ-unstable deformed case within the IBM one should reobtain, at the critical point in the transition, the predictions of the E(5) symmetry. This correspondence is supposed to be valid in the limit of large number N of bosons, but the calculations with the IBM should provide predictions for finite N as stated in Ref. [4]. In this letter, on one hand we calculate exactly the large N limit of the IBM at the critical point in the transition from U(5) (spherical case) to O(6) (deformed γ-unstable case). On the other hand, we solve the Bohr differential equation for a β 4 potential. Both calculations lead to the same results and are not close to those obtained by solving the Bohr equation for an infinite square well (E(5) symmetry). We also show with two schematic examples that the corrections arising from the finite number of bosons are important. With this in mind, the IBM calculations still provide a tool for including corrections due to the finite number of bosons. In Ref. [1] the Bohr Hamiltonian is considered for the case of a γ independent potential, described by an infinite square well in the β variable. In that case, the hamiltonian is separable in both variables and if we set Ψ(β, γ, θ i ) = f (β)Φ(γ, θ i )(1) where θ i stands for the three Euler angles, the Schrödinger equation can be split in two equations. The solutions of the (γ, θ i ) part were studied in Ref. [5] and tabulated in Ref. [6]. Iachello solved the β part and found that the f (β) functions are related to Bessel functions. The main results are illustrated in Table I and Fig. 1 of Ref. [1]. These results are obtained from a geometrical picture and we would like to investigate its relation with the interacting boson model. The geometrical interpretation of the abstract IBM hamiltonian can be obtained by introducing a coherent state [7][8][9] which allows to associate to it a geometrical shape in terms of the deformation variables (β, γ). The basic idea of this formalism is to consider that the pure quadrupole states are globally described by a boson condensate of the form |g; N, β, γ = 1 √ N! (Γ † g ) N |0 ,(2) where the basic boson is given by Γ † g = 1 √ 1 + β 2 s † + β cos γd † 0 + 1 √ 2 β sin γ(d † 2 + d † −2 ) ,(3) which depends on the β and γ shape variables. The energy surface is defined as E N (β, γ) = g; N, β, γ|Ĥ|g; N, β, γ ,(4) whereĤ is the IBM hamiltonian. Here we are interested in the case in which the hamiltonian undergoes a transition from U(5) to O(6) and, consequently, the corresponding potential energy surfaces are γ-independent. In order to investigate the geometrical limit of the IBM in the transitional class going from U(5) (spherical) to O(6) (deformed γ-unstable) the most general (up to two-body terms) IBM hamiltonian is, H = ε dnd + κ 0P †P + κ 1L ·L + κ 2Q χ=0 ·Q χ=0 + κ 3T3 ·T 3 + κ 4T4 ·T 4(5) wheren d is the d boson number operator, and P † = 1 2 (d † · d † − s † · s † ),(6)L = √ 10(d † ×d) (1) ,(7)Q χ=0 = (s † ×d + d † ×s) (2) ,(8)T 3 = (d † ×d) (3) ,(9)T 4 = (d † ×d) (4) .(10) The scalar product is defined asT L ·T L = M (−1) MT LMTL−M ,E(N, β) = N 1 + β 2 5κ 2 + β 2 (ε d + 6κ 1 + κ 2 + 7 5 κ 3 + 9 5 κ 4 ) + N(N − 1) (1 + β 2 ) 2 (1 − β 2 ) 2 4 κ 0 + 4β 2 κ 2 + 18 35 β 4 κ 4 .(11) The condition to find the critical point is d 2 E(N, β)/dβ 2 β=0 = 0(12) and gives the following relation among the hamiltonian parameters ε d = −6κ 1 + 4κ 2 − 7 5 κ 3 − 9 5 κ 4 + (N − 1)(κ 0 − 4κ 2 ).(13) Thus the most general energy surface at the critical point in the U(5)-O(6) phase transition is E crit (N, β) = 5Nκ 2 + N(N − 1) κ 0 4 + κ 0 − 4κ 2 + 18 35 κ 4 β 4 (1 + β 2 ) 2 .(14) These expressions are consistent with those obtained in Ref. [10] for a slightly different hamiltonian. Note that (14) completely defines the form of the potential up to a scale and an energy translation. The expansion of this critical energy surface around β = 0 is E crit (N, β) ≈ 5κ 2 N + κ 0 4 N(N − 1) + N(N − 1) κ 0 − 4κ 2 + 18 35 κ 4 β 4 − 2β 6 + . . . . (15) whose leading term is β 4 . Alternatively, one can carry out the transformation β 2 /(1 + β 2 ) → β 2 and findsβ 4 as the critical potential. In order to make some calculations to illustrate the large N limit in the IBM at the critical point in the U(5)-O(6) phase transition and the corresponding finite N corrections, we propose two schematic transitional hamiltonians. The first one iŝ H I = xn d + 1 − x N − 1P †P .(16) The corresponding energy surface is obtained from Eq. (11) with ε d = x, κ 0 = 1−x N −1 and all the rest of the parameters equal to 0, E I (N, β) = N   x β 2 1 + β 2 + 1 − x 4 1 − β 2 1 + β 2 2   .(17) The condition to localize the critical point, Eq. (13), gives in this case x I c = 0.5. In Fig. 1 we represent as an example the energy surfaces for the hamiltonian (16) (left panel) with three selections for the order parameter x: one at the critical point, one above that value and one below it. For x > x c an equilibrium spherical shape is obtained, while for x < x c the equilibrium shape is deformed. The value x c gives a flat β 4 surface close to β = 0. The second schematic hamiltonian we propose iŝ H II = xn d − 1 − x NQ χ=0 ·Q χ=0 ,(18) The corresponding energy surface is obtained from Eq. (11) with ε d = x, κ 2 = − 1−x N and all the rest of the parameters equal to 0, E II (N, β) = −(5 + β 2 ) 1 − x 1 + β 2 + N x β 2 1 + β 2 − 4(N − 1)(1 − x) β 2 (1 + β 2 ) 2 .(19) Condition (13) gives in this case the critical point x II c = 4N −8 5N −8 that in the large N limit gives 4/5. In Fig. 1 the corresponding energy surfaces are plotted in the right panel. Same comments as in the preceding case are in order. Thus, we conclude that, in the transition from spherical systems to γ-unstable deformed ones, the critical point in IBM should be associated to a β 4 potential rather that to an infinite square well. The question is then how different are the E(5) predictions from those obtained with a β 4 potential? In order to investigate this point we have solved numerically the Bohr hamiltonian for a potential β 4 . The results for energies are presented in Table I and in Fig. 2. Here we keep the label ξ used in the E(5) case. It is related to the label n β = n d −τ 2 , sometimes used in the U(5) classification, by n β = ξ − 1, where n d is the U(5) label and τ is the O(5) label. Particularly interesting are the energy ratios given in Table II which have been used in recent works to identify possible nuclei as critical. In this table the E(5) and β 4 values are shown for comparison. The labeling of the states is L ξ,τ . Besides the excitation energies, B(E2) transition probabilities can be calculated using the quadrupole operator T (E2) µ = t β D (2) µ0 (θ i ) cos γ + 1 √ 2 D (2) µ2 (θ i ) + D (2) µ−2 (θ i ) sin γ ,(20) where t is a scale factor. In Table II two important B(E2) ratios are given for E(5) and β 4 cases. In Fig. 2 the B(E2) values for a β 4 potential are shown besides the arrows. They are given normalized to the B(E2; 2 1,1 → 0 1,0 ) value which is taken as 100. Comparing Figs. 1 and Table I in Ref. [1] with the present Fig. 2 and Table I The last two panels labeled with R 1 and R 2 refer to the B(E2) ratios presented in Table II. From Fig. 3 it is clear that the finite N effects are important and depend on the precise form of the hamiltonian used. However, it is difficult to conclude whether E(5) or β 4 is the large N limit of the corresponding IBM hamiltonian. It is necessary to perform calculations with larger values of N. Fortunately, Dukelsky et al. [11] have recovered an exactly solvable model for pairing proposed by Richardson in the 60's [12]. Following Ref. [11] we have solved the Richardson's equations and obtained the exact eigenvalues for the hamiltonians (16) and (18) up to N = 1000, so approaching the large N limit of the corresponding IBM hamiltonians. Details of this method will be given in a longer publication. In Fig. 4 we present the results of these calculations for energy ratios up to N = 1000 and B(E2) ratios up to N = 40 together with the corresponding values for the E(5) symmetry and the β 4 potential. From this figure it clearly emerges that the large N limit for the studied IBM hamiltonians corresponds to the β 4 potential. Both hamiltonians Eq. (16) and Eq. (18) converge to the same results in the large N limit, although the corresponding corrections for finite N are quite different (see Fig. 3). We conclude that the large N limit of the IBM hamiltonian at the critical point in potential. These are relative to the transition 2 1,1 → 0 1,0 whose B(E2) value is taken as 100. E 4 1,2 /E 2 1,1 E 0 2,0 /E 2 1,1 E 0 1,3 /E 2 1,1 E 0 2,0 /E 0 1,3 R 1 = B(E2;4 1,2 →2 1,1 ) B(E2;2 1,1 →0 1,0 ) R 2 = B(E2;0 2,0 →2 1,1 ) B(E2;2 1,1 →0 1,0 ) E( whereT LM corresponds to the M component of the operatorT L . The operatorsd m = (−1) m d −m ands = s are introduced to ensure the correct tensorial character under spatial rotations. The corresponding energy surface is obtained from Eq. (4) we can observe important differences between E(5) and β 4 potentials. In order to see which is the actual large N limit of IBM we have performed calculations with the IBM codes for hamiltonians H I (Eq. 16) and H II (Eq. 18) at the critical point for different number of bosons. These codes allow to manage a small number of bosons, typically 20. In Fig. 3 the results of these calculations are shown with a full line for Eq. (16) and with a dashed line for Eq. (18). The values for E(5) and β 4 potentials are shown as dotted lines as references. the transition from U(5) (spherical) to O(6) (deformed γ-unstable) is represented in the geometrical model by a β 4 potential. The results are similar but not close to those of an infinite square well as in the E(5) critical point symmetry. The analysis of the IBM energy surface followed by an IBM calculation, as presented in Ref.[13], can provide the appropriate finite N corrections and thus lead to the identification of nuclei at the critical points. In that work a systematic study of the properties of the Ru isotopes allowed to select the appropriate form of the hamiltonian. Once it is fixed the construction of the energy surfaces identify the critical nucleus ( 104 Ru in that case). The corresponding IBM calculation for the critical nucleus then provides the correct finite N corrections. We believe that this is a fundamental step if we wish to robustly identify the spectroscopic properties that signal the presence of criticality in the atomic nucleus.TABLES FIG. 1 . 1Representation of the energy surfaces for N = 20 as functions of the shape parameter β obtained for two schematic hamiltonians, Eq. (16) (left panel) and Eq. (18) (right panel). In each case three values of the order parameter are presented, one at the critical value, one above and one below that value. The curves have been arbitrarily displaced in energy so as to show clearly the behavior. FIG. 2 . 2Schematic spectrum for a β 4 potential. Numbers close to the arrows are B(E2) values. FIG. 3 .FIG. 4 . 34Variation with the number of bosons (up to N = 20) of selected energy and B(E2) ratios for IBM calculations performed at the critical points of hamiltonian (16) (full line) and (18) (dashed line). The corresponding E(5) and β 4 values are marked with dotted lines. Same as Fig 3 but here the number of bosons runs up to 1000 in the energy ratios and up to 40 in the B(E2) ratios. TABLE I . IExcitation energies for a β 4 potential relative to the energy of the first excited state.ξ = 1 ξ = 2 ξ = 3 ξ = 4 τ = 0 0.00 2.39 5.15 8.20 τ = 1 1.00 3.63 6.56 9.75 τ = 2 2.09 4.92 8.01 11.34 τ = 3 3.27 6.26 9.50 12.95 TABLE II. Energy and B(E2) transition rate ratios in the E(5) symmetry and for the β 4 . F Iachello, Phys. Rev. Lett. 853580F. Iachello, Phys. Rev. Lett. 85, 3580 (2000). . A Bohr, B Mottelson, Nuclear Structure. IIA. Bohr and B. Mottelson, Nuclear Structure, vol II, Benjamin, Reading, Mass. 1975. F Iachello, A Arima, The Interacting Boson Model. CambridgeCambridge University PressF. Iachello and A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge, 1987. . R F Casten, N V Zamfir, Phys. Rev. Lett. 853584R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000). . L Wilets, M Jean, Phys. Rev. 102788L. Wilets and M. Jean, Phys. Rev. 102, 788 (1956). . D Bès, Nucl. Phys. 10373D. Bès, Nucl. Phys. 10, 373 (1959). . J N Ginocchio, M W Kirson, Nucl. Phys. 35031J. N. Ginocchio and M. W. Kirson, Nucl. Phys. A350, 31 (1980). . A E L Dieperink, O Scholten, F Iachello, Phys. Rev. Lett. 441747A. E. L. Dieperink, O. Scholten and F. Iachello, Phys. Rev. Lett. 44, 1747 (1980). . A Bohr, B Mottelson, Phys. Scripta. 22468A. Bohr and B. Mottelson, Phys. Scripta 22, 468 (1980). . A Frank, Phys. Rev. 39652A. Frank, Phys. Rev. C39, 652 (1989). . J Dukelsky, C Esebbag, P Schuck, Phys. Rev. Lett. 8766403J. Dukelsky, C. Esebbag and P. Schuck, Phys. Rev. Lett. 87, 066403 (2001). . R W W Richardson ; R, N Richardson, W Sherman ; R, Richardson, J. Math. Phys. (N.Y.). 31327Phys. Lett.R.W. Richardson, Phys. Lett. 3, 277 (1963). R.W. Richardson and N. Sherman, Nucl. Phys. 52, 221 (1964). R.W. Richardson, J. Math. Phys. (N.Y.) 9, 1327 (1968). . A Frank, C E Alonso, J M Arias, Phys. Rev. 6514301A. Frank, C.E. Alonso, and J.M. Arias, Phys. Rev. C65, 014301 (2001).
[]
[ "Comment on 'Invalidation of the Kelvin force in ferrofluids'", "Comment on 'Invalidation of the Kelvin force in ferrofluids'" ]
[ "A Engel \nInstitut für Theoretische Physik\nLehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag\nOtto-von-Guericke Universität\nPostfach 4120D-39016Magdeburg, Berlin, 1980)Germany\n", "D Landau \nInstitut für Theoretische Physik\nLehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag\nOtto-von-Guericke Universität\nPostfach 4120D-39016Magdeburg, Berlin, 1980)Germany\n", "E M Lifshitz \nInstitut für Theoretische Physik\nLehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag\nOtto-von-Guericke Universität\nPostfach 4120D-39016Magdeburg, Berlin, 1980)Germany\n" ]
[ "Institut für Theoretische Physik\nLehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag\nOtto-von-Guericke Universität\nPostfach 4120D-39016Magdeburg, Berlin, 1980)Germany", "Institut für Theoretische Physik\nLehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag\nOtto-von-Guericke Universität\nPostfach 4120D-39016Magdeburg, Berlin, 1980)Germany", "Institut für Theoretische Physik\nLehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag\nOtto-von-Guericke Universität\nPostfach 4120D-39016Magdeburg, Berlin, 1980)Germany" ]
[]
PACS: 75.50.Mm,In a recent letter [1] Odenbach and Liu claim that their experimental results for the force on a container filled with ferrofluid in an inhomogeneous external magnetic field invalidate the standard Kelvin expression f = µ 0 (M · ∇)H for the magnetic force density in a magnetizable medium. It is the purpose of this comment to point out that the described experiment measuring the total force on a magnetizable body cannot verify or falsify different expressions for the magnetic force density without taking into account the corresponding surface contributions.The magnetic force on a magnetizable body in thermodynamic equilibrium in an external magnetic field can be determined in a simple and unambiguous way. The change of the free energy of the body due to variations of the external field is well-known to be [2]where the integral is over the volume of the body, M(r) is its local magnetization, and H 0 (r) denotes the external field in the absence of the body. If the change in the field is due to a displacement of the body by an infinitesimal vector δr we have δH 0 = (δr · ∇) H 0 . At the same time the corresponding change in free energy is given by δF = −F · δr, where F is by definition the total force on the body. Using ∇ × H 0 = 0 we findThis is a generally valid expression, subject only to the constraint of thermodynamic equilibrium. In particular it does correctly describe the experimental findings reported in [1].By formal manipulations expression (0.2) can be decomposed into a surface and a volume part in various ways. Besides the decomposition advocated in [1] there is the standard possibility to use the Kelvin force density f = µ 0 (M · ∇) H in the volume and a surface integral over µ 0 M 2 n /2 with M n denoting the normal component of the magnetization [3]. If consistently used all these decompositions, including the latter one using the Kelvin force density, are equivalent to (0.2) and therefore describe the experimental findings equally well.The expression for the force suggested in [1] (their eq.(6)) differs from (0.2) by a factor (1 + χ)/(1 + Dχ) with D denoting the demagnetization factor. This is probably due to the fact that at the same time where demagnetization effects are taken into account also contributions from the surface integral ( in their case involving M 2 t ) matter. Since in the experiment D ∼ = 0.9694 the difference between their result (6) and the correct expression (0.2) is too small to cause noticable differences with the experiment.In conclusion the main aim of the letter namely to invalidate the Kelvin force on the basis of experimental facts was not accomplished. Moreover the variant expression (6) offered as alternative to describe the experiment is incomplete due to the neglect of surface contributions.Acknowledgement: Discussions with Hanns-Walter Müller and René Friedrichs are gratefully acknowledged.[1] S. Odenbach, M. Liu, Phys. Rev. Lett. 86, 328 (2001) [2] L.
10.1103/physrevlett.86.4978
[ "https://arxiv.org/pdf/cond-mat/0101374v1.pdf" ]
27,852,087
cond-mat/0101374
f14475d8686779a9a251d31364ccec191768cd18
Comment on 'Invalidation of the Kelvin force in ferrofluids' 24 Jan 2001 A Engel Institut für Theoretische Physik Lehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag Otto-von-Guericke Universität Postfach 4120D-39016Magdeburg, Berlin, 1980)Germany D Landau Institut für Theoretische Physik Lehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag Otto-von-Guericke Universität Postfach 4120D-39016Magdeburg, Berlin, 1980)Germany E M Lifshitz Institut für Theoretische Physik Lehrbuch der theoretischen Physik, Band VIII, Elektrodynamik der Kontinua (Akademie-Verlag Otto-von-Guericke Universität Postfach 4120D-39016Magdeburg, Berlin, 1980)Germany Comment on 'Invalidation of the Kelvin force in ferrofluids' 24 Jan 20011 PACS: 75.50.Mm,In a recent letter [1] Odenbach and Liu claim that their experimental results for the force on a container filled with ferrofluid in an inhomogeneous external magnetic field invalidate the standard Kelvin expression f = µ 0 (M · ∇)H for the magnetic force density in a magnetizable medium. It is the purpose of this comment to point out that the described experiment measuring the total force on a magnetizable body cannot verify or falsify different expressions for the magnetic force density without taking into account the corresponding surface contributions.The magnetic force on a magnetizable body in thermodynamic equilibrium in an external magnetic field can be determined in a simple and unambiguous way. The change of the free energy of the body due to variations of the external field is well-known to be [2]where the integral is over the volume of the body, M(r) is its local magnetization, and H 0 (r) denotes the external field in the absence of the body. If the change in the field is due to a displacement of the body by an infinitesimal vector δr we have δH 0 = (δr · ∇) H 0 . At the same time the corresponding change in free energy is given by δF = −F · δr, where F is by definition the total force on the body. Using ∇ × H 0 = 0 we findThis is a generally valid expression, subject only to the constraint of thermodynamic equilibrium. In particular it does correctly describe the experimental findings reported in [1].By formal manipulations expression (0.2) can be decomposed into a surface and a volume part in various ways. Besides the decomposition advocated in [1] there is the standard possibility to use the Kelvin force density f = µ 0 (M · ∇) H in the volume and a surface integral over µ 0 M 2 n /2 with M n denoting the normal component of the magnetization [3]. If consistently used all these decompositions, including the latter one using the Kelvin force density, are equivalent to (0.2) and therefore describe the experimental findings equally well.The expression for the force suggested in [1] (their eq.(6)) differs from (0.2) by a factor (1 + χ)/(1 + Dχ) with D denoting the demagnetization factor. This is probably due to the fact that at the same time where demagnetization effects are taken into account also contributions from the surface integral ( in their case involving M 2 t ) matter. Since in the experiment D ∼ = 0.9694 the difference between their result (6) and the correct expression (0.2) is too small to cause noticable differences with the experiment.In conclusion the main aim of the letter namely to invalidate the Kelvin force on the basis of experimental facts was not accomplished. Moreover the variant expression (6) offered as alternative to describe the experiment is incomplete due to the neglect of surface contributions.Acknowledgement: Discussions with Hanns-Walter Müller and René Friedrichs are gratefully acknowledged.[1] S. Odenbach, M. Liu, Phys. Rev. Lett. 86, 328 (2001) [2] L. In a recent letter [1] Odenbach and Liu claim that their experimental results for the force on a container filled with ferrofluid in an inhomogeneous external magnetic field invalidate the standard Kelvin expression f = µ 0 (M · ∇)H for the magnetic force density in a magnetizable medium. It is the purpose of this comment to point out that the described experiment measuring the total force on a magnetizable body cannot verify or falsify different expressions for the magnetic force density without taking into account the corresponding surface contributions. The magnetic force on a magnetizable body in thermodynamic equilibrium in an external magnetic field can be determined in a simple and unambiguous way. The change of the free energy of the body due to variations of the external field is well-known to be [2] δF = −µ 0 V d 3 r M(r) · δH 0 (r), (0.1) where the integral is over the volume of the body, M(r) is its local magnetization, and H 0 (r) denotes the external field in the absence of the body. If the change in the field is due to a displacement of the body by an infinitesimal vector δr we have δH 0 = (δr · ∇) H 0 . At the same time the corresponding change in free energy is given by δF = −F · δr, where F is by definition the total force on the body. Using ∇ × H 0 = 0 we find F = µ 0 V d 3 r (M · ∇) H 0 . (0.2) This is a generally valid expression, subject only to the constraint of thermodynamic equilibrium. In particular it does correctly describe the experimental findings reported in [1]. By formal manipulations expression (0.2) can be decomposed into a surface and a volume part in various ways. Besides the decomposition advocated in [1] there is the standard possibility to use the Kelvin force density f = µ 0 (M · ∇) H in the volume and a surface integral over µ 0 M 2 n /2 with M n denoting the normal component of the magnetization [3]. If consistently used all these decompositions, including the latter one using the Kelvin force density, are equivalent to (0.2) and therefore describe the experimental findings equally well. The expression for the force suggested in [1] (their eq.(6)) differs from (0.2) by a factor (1 + χ)/(1 + Dχ) with D denoting the demagnetization factor. This is probably due to the fact that at the same time where demagnetization effects are taken into account also contributions from the surface integral ( in their case involving M 2 t ) matter. Since in the experiment D ∼ = 0.9694 the difference between their result (6) and the correct expression (0.2) is too small to cause noticable differences with the experiment. In conclusion the main aim of the letter namely to invalidate the Kelvin force on the basis of experimental facts was not accomplished. Moreover the variant expression (6) offered as alternative to describe the experiment is incomplete due to the neglect of surface contributions.
[]
[ "Rethinking the Encoding of Satellite Image Time Series", "Rethinking the Encoding of Satellite Image Time Series" ]
[ "Xin Cai [email protected] \nSchool of Computing\nUlster University\n\n", "Yaxin Bi [email protected] \nSchool of Computing\nUlster University\n\n", "Peter Nicholl [email protected] \nSchool of Computing\nUlster University\n\n", "Roy Sterritt [email protected] \nSchool of Computing\nUlster University\n\n" ]
[ "School of Computing\nUlster University\n", "School of Computing\nUlster University\n", "School of Computing\nUlster University\n", "School of Computing\nUlster University\n" ]
[]
Representation learning of Satellite Image Time Series (SITS) presents its unique challenges, such as prohibitive computation burden caused by high spatio-temporal resolutions, irregular acquisition times, and complex spatio-temporal interactions, leading to highly-specialized neural network architectures for SITS analysis. Despite the promising results achieved by some pioneering work, we argue that satisfactory representation learning paradigms have not yet been established for SITS analysis, causing an isolated island where transferring successful paradigms or latest advances from Computer Vision (CV) to SITS is arduous. In this paper, we develop a unique perspective of SITS processing as a direct set prediction problem, inspired by the recent trend in adopting query-based transformer decoders to streamline the object detection or image segmentation pipeline, and further propose to decompose the representation learning process of SITS into three explicit steps: collect-update-distribute, which is computationally efficient and suits for irregularly-sampled and asynchronous temporal observations. Facilitated by the unique reformulation and effective feature extraction framework proposed, our models pre-trained on pixel-set format input and then fine-tuned on downstream dense prediction tasks by simply appending a commonly-used segmentation network have attained new state-of-the-art (SoTA) results on PASTIS dataset compared to bespoke neural architectures such as U-TAE[7]. Furthermore, the clear separation, conceptually and practically, between temporal and spatial components in the panoptic segmentation pipeline of SITS makes us to leverage the recent advances in CV, such as Mask2Former [2], a universal segmentation architecture, resulting in a noticeable 8.8 points increase in PQ compared to the best score reported so far. To facilitate further research, the code and models will be made publicly available.
10.48550/arxiv.2305.02086
[ "https://export.arxiv.org/pdf/2305.02086v1.pdf" ]
258,461,391
2305.02086
c0f5a27f26c93a5109d8a7103ac3536344cd164a
Rethinking the Encoding of Satellite Image Time Series Xin Cai [email protected] School of Computing Ulster University Yaxin Bi [email protected] School of Computing Ulster University Peter Nicholl [email protected] School of Computing Ulster University Roy Sterritt [email protected] School of Computing Ulster University Rethinking the Encoding of Satellite Image Time Series Representation learning of Satellite Image Time Series (SITS) presents its unique challenges, such as prohibitive computation burden caused by high spatio-temporal resolutions, irregular acquisition times, and complex spatio-temporal interactions, leading to highly-specialized neural network architectures for SITS analysis. Despite the promising results achieved by some pioneering work, we argue that satisfactory representation learning paradigms have not yet been established for SITS analysis, causing an isolated island where transferring successful paradigms or latest advances from Computer Vision (CV) to SITS is arduous. In this paper, we develop a unique perspective of SITS processing as a direct set prediction problem, inspired by the recent trend in adopting query-based transformer decoders to streamline the object detection or image segmentation pipeline, and further propose to decompose the representation learning process of SITS into three explicit steps: collect-update-distribute, which is computationally efficient and suits for irregularly-sampled and asynchronous temporal observations. Facilitated by the unique reformulation and effective feature extraction framework proposed, our models pre-trained on pixel-set format input and then fine-tuned on downstream dense prediction tasks by simply appending a commonly-used segmentation network have attained new state-of-the-art (SoTA) results on PASTIS dataset compared to bespoke neural architectures such as U-TAE[7]. Furthermore, the clear separation, conceptually and practically, between temporal and spatial components in the panoptic segmentation pipeline of SITS makes us to leverage the recent advances in CV, such as Mask2Former [2], a universal segmentation architecture, resulting in a noticeable 8.8 points increase in PQ compared to the best score reported so far. To facilitate further research, the code and models will be made publicly available. Introduction Recent years have witnessed a surge of interest in automating the monitoring of the Earth surface based on satellites with high revisit frequency, such as European Space Agency (ESA) Sentinel satellites. In particular, automated large-scale crop type mapping benefits most from leveraging complex temporal dynamics contained in satellite image time series (SITS), which can promote the fair allocation of agricultural subsidies and the regulation of the best crop practices being observed by farmers. However, applying deep learning models to extract representative features from SITS is non-trivial, e.g., some of which with a naïve concatenation of spatial and temporal encoders even struggle to surpass the performance of a random forest classifier [14], forcing researchers to devote great efforts to develop bespoke neural architectures. The pioneering work PSE+TAE [8]/PSE+L-TAE [6] has demonstrated a promising learning paradigm for SITS, where statistics of spectral values are first summarized across the spatial extent of parcels by Multi-Layer Perceptrons (MLPs) operated independently on unordered sets of pixels, which are then fed into a temporal encoder with self-attention to explore the underlying temporal patterns, i.e., respecting a spatio-then-temporal factorization order. With the empirical evidence provided by the recent work TSViT [31], however, it argues that the temporal-then-spatial factorization order is a more intuitive design choice for SITS analysis as spatial contexts in satellite imagery provide non-informative information in contrast to natural images. This line of research has demonstrated one important aspect when designing deep learning models for SITS: decoupling the learning framework into spatially and temporally separated components. However, the lack of flexibility to account for different input formats simultaneously, i.e., the pixel-set or image sequence format, imposes restrictions on PSE+TAE or TSViT on dealing with classification and downstream dense prediction tasks. Furthermore, the classical pretrain-finetune paradigm in CV, i.e., pre-trainining a classification model on large-scale datasets (e.g., Im-ageNet [3]) with fully-/self-supervised learning [5,9] and fine-tuning on downstream tasks such as object detection [24] or semantic segmentation [17], has not been successfully adopted in SITS analysis yet. Meanwhile, as pointed out by previous work [6,8], another great challenge for learning effective representations for SITS is to capture the complex temporal dynamics of crop phenological development, i.e., the particular timings of plant phenological events are key to identify different crop types [22]. However, recent work for SITS analysis [6][7][8]22] advocates adopting self-attention [33] as a core compute unit without questioning its validity for temporal modelling, especially considering its permutation-invariant nature. Based on the latest findings in time series forecasting [38,43], the capability of self-attention operations for modelling complex temporal relations is exaggerated due to a lack of rich semantics in numerical time series data. Modules with strong built-in priors or inductive biases on temporal ordering such as the classical exponential smoothing [38] or frequency analysis methods [44] have proven to be superior over the vanilla self-attention mechanism for temporal pattern extraction. But irregularity in the temporal axis which is prevalent in satellite image sequences, e.g., optical acquisitions obstructed by clouds, complicates the problem even further, which usually calls for imputation or interpolation as a preprocessing step [14] or developing an end-to-end learning framework which should reconcile potentially conflict optimization objectives [29] between interpolation and classification. Except for the validity of self-attention for temporal modelling that has been questioned recently, the quadratic space and time complexity w.r.t. the processed sequence length introduces extra computational concerns for model designs and limits its applicability to dense prediction tasks in SITS [7,31]. These two observations combined motivated us to rethink the existing encoding schemes for SITS: Do we really need to develop highly-specialized neural architectures for SITS? Can we transfer the successful paradigms from CV to SITS by designing a general representation learning framework in a conceptually clear way? More specifically, we propose to frame SITS as sets of observations, inspired by the formulation proposed by [10] for classifying irregularly-sampled and asynchronous time series, where each element is represented by its spectral signatures augmented with static or dynamic covariates such as date-specific temporal encodings. Facilitated by this unique perspective, we propose a simple yet effective representation learning framework for SITS processing by decomposing the encoding process into three steps: collect-update-distribute, which excludes the use of self-attention to circumvent its limitations. Furthermore, we showcase for the first time that pre-training a classification model on pixel-set format datasets and fine-tuning it on downstream dense prediction tasks, such as semantic or panoptic segmentation, with image sequences as input by simply appending a commonly-used semantic segmentation model from CV can lead to the new SoTA performance on PASTIS [7] compared to highly-specialized network architectures such as U-TAE [7]. Furthermore, we can directly introduce the latest universal image segmentation architecture Mask2Former [2] into panoptic segmentation of SITS without any modifications by simply letting it consume output feature maps encoded by our proposed representation learning models, leading to new SoTA on PASTIS. To sum up, the contributions of this work include: • We uniquely reformulate the representation of SITS from conventional spatiotemporal signals to sets of instances, which removes restrictions imposed on the existing model design for SITS analysis and opens up the possibility of leveraging pre-trained models on the resource economical pixel-set format and then being finetuned on downstream dense prediction tasks, which we argue is a more desirable way to replicate the successful pretrain-finetune paradigm in SITS. • We propose to explicitly decompose the representation learning process of SITS into three steps: collect-update-distribute, leading to a conceptually clear and computationally efficient learning framework, dubbed as Exchanger, for general feature extraction of SITS. • In contrast to the current work where temporal and spatial components are intricately interwoven with each other in the dense prediction pipeline, we argue that a clear separation of temporal and spatial encoders can greatly reduce the complexity in model design and facilitate leveraging the latest advances in CV, mitigating the gap between CV and SITS. • We have conducted extensive experiments to verify the effectiveness of our proposed model, which outperforms the previous SoTA models by a significant margin across semantic and panoptic segmentation tasks on PASTIS benchmark dataset. And the code and trained model weights will be released to facilitate future research. Related Work Encoding of SITS The high frequency revisit time of satellites enables the exploitation of rich temporal dynamics captured for more complex land use recognition, e.g., crop type mapping. Traditional machine learning methods [34] rely on handcrafted features where the encoding has not been properly tackled despite the heavy domain expertise required. Until recently, differential neural architectures have dominated the field. Specifically, Convolutional Neural Networks (CNNs) [23] and Recurrent Neural Networks (RNNs) [27] have been adopted as a de facto choice to encode spatial and temporal features, respectively. Furthermore, the convolutional-recurrent hybrid models [26] have been proposed to process SITS by viewing it as spatio-temporal signals. Despite the promising results attained, these early attempts have overlooked the significant differences between natural images/videos and SITS. The pioneering work PSE+TAE [8] has proposed to use MLPs to summarize spatial statistics given the lack of rich spatial semantics and self-attention to encode temporal patterns, followed by PSE+L-TAE [6] where a light-weight transformer decoder has been used to extract temporal features. Pixel-Set Encoder (PSE) is particularly effective for dealing with the irregularity in parcel geometry by simplifying parcel represen- tation from T × C × H × W to T × C × N , where T is the length of temporal sequence, C is the number channels, H/W denotes the height/width, and N denotes the number of pixels, and consequently requires significantly less storage memory [8] compared to the patch format. But, when it comes to downstream dense prediction tasks, TAE needs to be integrated into spatial encoders in a complicated manner as shown in the state-of-the-art model U-TAE [7], which prevents the replication of the successful pretrain-finetune paradigm. TSViT [31] is the first attempt to bridge the gap between SITS analysis and CV by incorporating a unique inductive bias into ViT [4], which is the temporal-then-spatial factorization based on the observation that spatial contexts provide little information for crop type recognition. However, the patch tokenization scheme in ViT is naturally built for images, therefore making TSViT incapable to directly consume unordered pixel-set format, which is a more desirable format for SITS classification. Furthermore, the intense computation required by self-attention, which is the workhorse in ViT, is exacerbated because the spatial dimension is maintained through the whole temporal learning process, which causes TSViT problematic for dense prediction tasks. Proposed Method In this section, we first reformulate the representation of SITS as sets of observations in contrast to the conventional spatio-temporal signals. Facilitated by this unique perspective, we can simplify the current encoding process of SITS by removing the need to specially account for the spatial dimension and further decompose temporal feature learning procedure into three explicit steps: collectupdate-distribute. Last, we give a specific network instantiation of our proposed framework. Definition 1. We describe satellite image sequences captured at a particular geo-referenced location with a certain spatial extent as a set S i of instances/sets S i = S 1 , . . . , S n , where each instance/set S j is comprised of a set of temporal acquisitions S j = s j t1 , . . . , s j tm . And we assume each observation s j t k is repre- sented by v j t k , p j t k , , where v j t k is feature embedding of sensor measurements, p j t k is temporal positional embedding for a particular acquisition time, and serves as a placeholder for other static or dynamic covariates such as geometric boundaries or modality information, opening up the possibility of arriving at a universal representation for SITS. [·] denotes an arbitrary operator to mix the features included in it such as summation or concatenation. Note that the superscript and subscript of s j t k denote a spatial and temporal identifier, respectively, and we omit the index i for differentiating parcels to avoid notational clutter. In contrast to the commonly-adopted representation of satellite observations as spatio-temporal signals X i ∈ R T ×C×H×W , we relax the constraints on spatial dimensions imposed by regular grids, for the spatial structure prior is not necessary for SITS processing and further restricts the flexibility when it comes to model design, thereby indicating more emphasis should be placed on the temporal dimension and the aggregation of spatial information can be flexibly dealt with according to output requirements of various tasks. Specifically, with such a more universal reformulation, the classification problem of SITS is intimately linked to Multiple Instance Learning (MIL) [11] where a single class label is assigned to a bag of instances with no ordering or strong dependencies among each other, treating each temporal sequence of observations sampled from different sub-locations within a parcel field as independent instances with uneven contributing weights to the final bag-level classification results. Concerning the dense prediction problem, the regular grid arrangement is only retained for matching the required output format rather than being used for mining highlevel spatial semantics. And we have observed in experiments that simply appending well-established semantic segmentation models such as U-Net [25] after first summarizing temporal information of SITS leads to superior performance to highly-specialized segmentation networks for SITS such as U-TAE [7], which reveals that rich semantics emerge after temporal processing of SITS and resonates with the temporal-then-spatial factorization order advocated in TSViT [31]. Temporal Context Clusters Rewarded by our reformulated representation of SITS, spatial modelling is not considered in SITS representation learning pipeline because of the weak dependencies in the spatial dimension. As for dense prediction tasks, mining high-level semantics can be accomplished by appending a semantic segmentation model after temporal feature extraction and summarization of SITS, which greatly simplifies the existing dense prediction model design for SITS where temporal encoding components are intricately interwoven with spatial encoding components. Motivated by the success of substituting self-attention with other temporal modelling blocks in time series analysis [38,44], we propose to use a set of learnable queries as an external memory module to exchange temporal information with the input, given that the extra complexity caused by the irregularity in SITS acquisition times, and therefore dub our model "Exchanger". Formally, we distil the representation learning process of SITS into three steps: collect-update-distribute, with the aid of a set of learnable queries or temporal context clusters, which is further split into two components: feature and position queries: C v ∈ R N ×d , C p ∈ R N ×d to avoid blemishing each other, where N is the number of clusters. COLLECT Given the input feature embeddings V ∈ R T ×d and temporal positional embeddings P ∈ R T ×d , temporal clusters C v first collect information from feature embeddings [v 1 , . . . , v T ] by calculating pair-wise similarities followed by a selective function S to filter out the least significant ones, which is formulated as follows: A 1 = cal simlarity ([C v , V ] , [C p , P ]) W = S (A 1 ) C v = C v + W V(1) where A 1 ∈ R N ×T is the affinity matrix and is further refined by the selective function S to obtain W to be multiplied by V , achieving the collection process. UPDATE Then temporal clusters are updated by solely relying on C v , C p to allow for global information exchange among different temporal segments and redundancy removal, which is formulated as follows: C v = Update (C v , C p )(2) DISTRIBUTE After updating the clusters, the more robust and representative features of temporal context clusters are distributed back by assigning each temporal element v i to C v j in a hard or soft manner, which is formulated as follows: A 2 = cal simlarity ([V , C v ] , [P , C p ]) I = assign (A 2 ) V = V + IC v(3) where A 2 ∈ R T ×N is the affinity matrix and each row of I ∈ R T ×N contains a hard index or soft probability vector to indicate the temporal context cluster to which each temporal element v i is assigned. The proposed temporal representation learning paradigm collect-updatedistribute is particularly effective for dealing with the irregularity and asynchronization in time series data as it imposes no prior assumption such as processing temporal observations in a sequential manner. The features of each temporal element can be updated by interacting with temporal context clusters and essential information flow among different temporal segments is realized through communication between context clusters, which is a more computationally efficient way for information exchange. Compared to the computation complexity of self-attention O T 2 d , it only requires O (N T d) where N T and therefore scales much better w.r.t. the number of temporal tokens. More importantly, the proposed representation learning framework for SITS can be seen as a generalization of current self-attention based models such as L-TAE [6] or TSViT [31]. To be concrete, L-TAE [6] is a lightweight transformer decoder where a set of learnable queries is used for extracting key features from outputs of the spatial encoder, which corresponds to the collect step we proposed. The lack of update and distribute steps renders L-TAE ineffective for encoding as there is no mechanism implemented for feature updating. The temporal encoder of TSViT [31] prepends a set of class tokens to input temporal elements and relies on selfattention for feature learning, which can be seen as a special case of our proposed framework where collect-update-distribute steps are implicitly realized through self-attention. The added external tokens and input temporal elements communicate with each other synchronously, which is more computationally intensive and conceptually vague compared to our proposed decomposition scheme. Network Instantiation Given the flexibility of the reformulated representation of SITS and the generalizability of our proposed feature learning framework, we decided to directly leverage recent advances in CV where object queries in the transformer decoder has been reinterpreted as cluster centres and cross-attention has been recast as a clustering operation [2,30,40,42], reviving the classical idea of framing image segmentation as a pixel grouping procedure rather than per-pixel classification. As clustering is essentially a quantization process where redundant information is gradually filtered out and therefore abstract concepts or high-level semantics may emerge, it has the potential for generic representation learning, not only limited to image segmentation tasks, as demonstrated by the recent pioneering work [20,41]. As the main focus of this paper is to establish an effective representation learning framework for SITS, we decided to borrow the core building unit Group Propagation Block (GP Block) from GPViT [41] to instantiate the idea, leaving the architectural invention for future work. We simply incorporate the construction of GP Block for completeness as follows and refer readers to the original work [41] for specific details: C v = Concat h Softmax 1 √ 2d C v W Q h V W K h T + 1 √ 2d C p U Q h P U K h T V W V h(4) where W Q,K,V h and U Q,K h are projection matrices for feature and position embeddings, respectively. Eq.(4) implements the collection process by using crossattention where the affinity matrix is calculated through scaled dot-product and the softmax function is used for selecting most relevant temporal elements. C v = C v + MLP 1 LayerNorm (C v ) T T C v = C v + MLP 2 (LayerNorm (C v ))(5) Eq.(5) implements the context cluster updating by using a MLPMixer [32] with one MLPs operated along the token dimension and another MLPs operated along the channel dimension. Z = Concat h Softmax 1 √ 2d VW Q h C vW K h T + 1 √ 2d PŨ Q h C pŨ K h T C vW V h Z = Concat (Z, V )W proj V = Z + FFN (Z ) (6) whereW Q,K,V h andŨ Q,K h are a different set of projection matrices for feature and position embeddings, respectively,W proj is for linear projection of the concatenated features to the same dimension as the input, and FFN is a feed-forward neural network. Eq.(6) implements the distribution process by using input temporal elements as queries to gather information from updated context clusters, performing cross-attention in the reversed direction. Experiments In this section, we perform extensive ablation studies to verify the effectiveness of our proposed representation learning framework for SITS and make comparisons with other SoTA models on semantic and panoptic segmentation tasks. In addition to the provided implementation details below, the code and trained model weights will be released. Datasets We choose PASTIS (Panoptic Agricultural Satellite TIme Series) 1 benchmark dataset [7] to evaluate the performance of our proposed model and make comparisons with other state-of-the-art models, which consists of 2433 sequences of multi-spectral images of shape 10 ×128×128 and each sequence contains temporal acquisitions taken between September 2018 and November 2019 with varying sequence lengths between 38 and 61, for a total of over 2 billion pixels. Furthermore, PASTIS covers four different regions of France with diverse climates and crop distributions, spanning over 4000 km 2 and including 18 crop types plus a background class. In addition to the spatio-temporal format T ×C ×H ×W with high-quality semantic and panoptic annotations, over 120,000 bounding boxes and pixel-precise masks, it is accompanied with a pixel-set format T × C × N dataset [8] for parcel-based crop type classification. As the pixel-set format saves up to 70% disk storage space [8], a significant proportion of GPU memory usage and training time, we argue that it is a desired format for (pre-)training classification networks whose representation learning performance is expected to be transferable to downstream tasks. We mainly use the 5-Fold splits officially provided by PASTIS for extensive ablation studies and model performance evaluation and additionally report semantic segmentation results on another dataset MTLCC [27]. The MTLCC dataset covers a large area of interest (AOI) of 102 km × 42 km north of Munich, Germany, with 17 distinct crop classes and temporal observations of two different lengths of 46 and 52 gathered in two growing seasons in 2016 and 2017. Despite the high temporal density, the individual samples in MTLCC have limited spatial resolutions of 24 × 24, which maybe is not sufficient to verify the effectiveness of dense prediction performance. Implementation Details Classification We train and validate the classification model on PASTIS pixelset format dataset. Based on the observation from [28,36] that an additional MLP projector is beneficial for reducing the transferability gap between unsupervised and supervised pre-training, we append the projector proposed in t-ReX [28] after the feature extractor Exchanger and use cosine softmax crossentropy loss. We use AdamW [18] optimizer, a batch size of 128, a weight decay of 0.005, an initial learning rate of 0.0002, and a step learning rate scheduler which decays the learning rate at 0.7 and 0.9 fractions of the total number of training steps by a factor of 10 to train models for 50 epochs on 4 V100 GPUs. We randomly drop temporal observations by uniformly sampling from the interval between 0.2 and 0.4 as a data augmentation strategy to counter the adverse effect of cloud obstruction, which has also been adopted in training semantic & panoptic segmentation models. Semantic Segmentation We then use the pre-trained model to initialize Exchanger which serves as the temporal encoder in the semantic segmentation pipeline, unless otherwise specified. For the Unet [25] used as the spatial encoder, we use the AdamW [18] optimizer, a batch size of 4, a weight decay of 0.005, an initial learning rate of 0.0002, and a poly decay learning rate scheduler to train models for 100 epochs on 4 V100 GPUs with Focal cross-entropy loss [16]. As it cannot fit a single input SITS sample with a spatial resolution of 128 × 128 and the temporal length of more than 30 into V100 GPU with 16G memory, we perform random crop with a crop size of 32 × 32 in training and test the model performance on full resolution on a A100 GPU. For concatenating the Exchanger with Mask2Former [2] framework, we mainly follow the settings in [2] only with the learning rate changed to 2 × 10 −5 . And we train models for 100 epochs with a random crop size enlarged to 64 × 64, a batch size of 1 on 8 V100 GPUs. Panoptic Segmentation We also use the pre-trained model to initialize Exchanger for panoptic segmentation, which is crucial for effective training with the Mask2Former framework. For using the Parcels-as-Points (PaPs) module as panoptic prediction head, we mainly follow the hyper-parameter settings used in [7] and use the AdamW [18] optimizer, a batch size of 4, a weight decay of 0.005, an initial learning rate of 0.0002, and a poly decay learning rate scheduler to train models for 100 epochs on 4 V100 GPUs. We also perform random crop with a crop size of 32 × 32 during the training and test on the full resolution on a single A100 GPU. For embedding the Exchanger into the Mask2Former framework [2], we follow the exact settings used in semantic segmentation except that we use a spatial resolution of 64 × 64 in testing by splitting the input and stitch the prediction together as we found empirically that the performance is particularly sensitive to spatial resolution variation. We first study the impact of several key design choices in Exchanger on PASTIS validation dataset compared to a strong baseline model where selfattention is employed to process temporal and spatial features as done in TSViT [31]. As seen in Tab.1, not incorporating position queries results in the worst performance with around an absolute 4% drop compared to all other models, indicating date-specific temporal embeddings are key to capture crop phenological profiles. Instead of mixing the content and position information in attention calculation, adopting untied content & position attention as proposed in TUPE [12] slightly improves F1-Score by 0.2%, which is set as the default choice for all the subsequent experiments, unless stated otherwise. Then we evaluate the performance of Exchanger w.r.t. the number of content & position tokens by increasing it from 4 to 8 to 16. As shown in Tab. 1, Exchanger has achieved the best scores across precision, recall and F1 metrics with 8 tokens. In contrast to the only 1 class token prepended to the input sequence in NLP, we hypothesize that requiring slightly more tokens for crop type recognition is due to the significant intra-class variation which we will show the latent embeddings in supplementary materials. Contradicting with fixing the number of tokens to that of classes needed to be identified in TSViT [31], we found that continually increasing the number of content/position queries did not bring the expected performance boost but with a noticeable increase in computational cost. When comparing Default(8) with its self-attention counterpart (Temp. Self-Attn (8)), it is easily seen that Exchanger can achieve nearly identical results with a similar number of parameters but with a drastic drop in computational cost (almost 50% saving in GFLOPs). Last, with stacking of two identical Exchanger blocks (2-Stages (8)), it reached a F1-Score of 83.1, which is on par with that obtained by Temp. & Spatio. Self-Attn (8) which is a modified TSViT [31] whilst being computationally-light (around 15% saving in GFLOPs). Additionally, the latter (Temp. & Spatio. Self-Attn (8)) can be seen as adding an attentive MIL pooling component [11] after the temporal self-attention block to identify key spatial instances given that the permutation-invariant nature of self-attention renders it effective as a set transformation function. However, we have demonstrated solely increasing the depth of Exchanger can bring a similar performance boost, enjoying the advantage that it can be reused in downstream tasks rather than being discarded in TSViT [31] for dense prediction. Based on the extensive ablation studies, it is safe to conclude that the proposed collect-update-distribute paradigm is a promising alternative for encoding SITS. Ablation Studies We then showcase the pretrain-finetune paradigm can be successfully transferred from CV to SITS analysis enabled by the reformulated SITS representation from spatio-temporal signals to sets of instances, allowing the backbone network to be pre-trained on resource economical pixel-set format and then finetuned on regular spatio-temporal grids for downstream dense prediction tasks. Specifically, as shown in Fig. 1, pre-trained Exchanger as backbone network appended with a commonly-used segmentation model Unet with randomly initialized weights has led to faster convergence, more stable training and higher validation accuracy than totally training from scratch. Comparison with SoTA Semantic Segmentation As shown in Tab. 2, coupling the Exchanger which serves as a pure temporal encoder with a plain Unet [25] which exclusively focuses on spatial semantic mining has easily led to 66.8% and 90.7% mIoU on The dissociation between temporal and spatial components further allows us to explore the potential of adopting the recently proposed powerful universal image segmentation framework Mask2Former [2] with PVT2 [35] as backbone and FPN [15] as the pixel decoder, resulting in a significant improvement of around an absolute 2.5% compared to the best results reported in the literature and a boost of about 1.1% compared to Exchanger+Unet but only with less than 10% increase in the computational cost. It is notable that all previous semantic segmentation models for SITS except for TSViT [31] feature a complicated composition of spatial and temporal components, hindering them from leverag-ing the latest advances in CV. Although TSViT [31] is the first fully-attentional neural architecture for SITS processing, it faces extra obstacles when deployed in the pretrain-finetune paradigm because of the patch tokenization layer which prevents it from being directly operated on the pixel-set format, and the selfattention operation can incur prohibitive computational cost for dense prediction tasks. Another marked fact is that the temporal-then-spatial processing order, which has been demonstrated is a more desirable inductive bias [31] for SITS analysis, would cause the temporal encoder to consume a drastic proportion of the requested computation, e.g., the Exchanger accounts for nearly 96% of the total computational cost in Exchanger+Unet, indicating the importance of taking the computational efficiency into consideration when developing models for SITS processing. And it should be pointed out that our proposed model only has a linear computational complexity O (N T d) w.r.t. the input sequence length. Panoptic Segmentation To further demonstrate the effectiveness of our proposed representation learning framework, we tested its performance on the panoptic segmentation task [13] on PASTIS, which unifies semantic and instance segmentation into a joint task and therefore delivers a holistic scene understanding vision system. Panoptic segmentation is already highly-challenging for natural images or videos, e.g., various specialized architectures [39] or sophisticated proxy tasks [1] have been developed to reconcile the inconsistencies between segmenting stuff and instance classes. The recognition of parcel instances has complicated the task to a greater extent because of the significant variation of parcel geometries and exploiting the complex and very subtle differences in plant phenological profiles to make accurate delineation across different crop types. Despite the pioneering effort made in [7] where a single-stage instance segmentation network CenterMask [37] has been adapted to a panoptic segmentation module named Parcels-as-Points (PaPs), the task still remains extremely difficult as the majority of existing panoptic segmentation networks proposed for natural images or videos is not particularly effective for directly processing SITS due to the reasons mentioned previously. We argue that a strong temporal encoder is key to extract high-level semantics from SITS, converting the low signal-to-noise ratio 4-D satellite data T × C × H × W to rich semantic 3-D feature maps C × H × W , which can be fed into off-the-shelf panoptic segmentation models. We report the class-averaged Segmentation Quality (SQ), Recognition Quality (RQ), and Panoptic Quality 2 (PQ) in Tab.3. It can be seen that Exchanger equipped with Unet [25] as the spatial encoder and the PaPs module [7] for panoptic prediction has increased RQ and PQ by a significant margin of 5.7% and 4.0%, respectively, compared to the previous state-of-the-art U-TAE+PaPs. Furthermore, it is prominent to see that Exchanger combined with Mask2Former [2] consistently outperforms Exchanger+Unet+PaPs by 4.3, 2.7 and 4.8 points in SQ, RQ, and PQ, respectively, setting a new state-of-the-art. Besides, it is noticeable that the required inference time on A100 GPU for Exchanger+Mask2Former is much lower because of the high parallelizability. Qualitative Results In this section, we present a qualitative comparison between previous SoTA model U-TAE+PaPs, Exchanger+Unet+PaPs and the first universal SITS segmentation architecture Exchanger+Mask2Former as a result of concatenating Exchanger as the temporal encoder with the recently proposed universal natural image segmentation framework Mask2Former [2]. As shown in Fig.2, U-TAE+PaPs can retrieve crop parcels almost as the same number as that of Ex-changer+PaPs but is more prone to error predictions, which indicates that the weaker representation learning capability of U-TAE. Coupling Exchanger with a more powerful segmentation architecture Mask2Former [2], the panoptic prediction quality is significantly improved in terms of crop type recognition accuracy and crop shape prediction consistent with the SQ and RQ metrics reported in Tab.3. Conclusion To conclude, in this paper, we first present a unique reformulation of SITS representation as sets of instances, which relaxes the constraints caused by traditional spatio-temporal grids and further enables designing models that can flexibly process both pixel-set and image sequence format. Then, we propose to explicitly decompose the representation learning procedure of SITS into three steps: collect-update-distribute, resulting in a conceptually clear and computationally efficient feature learning framework called Exchanger. Facilitated by the unique perspective of treating SITS and the effective encoding scheme proposed, we have demonstrated for the first time that transferring the successful pretrain-finetune paradigm from CV to SITS, leading to a streamlined semantic & panoptic segmentation pipeline and marked performance gains over the previous SoTA models. We show latent features from the output of stage-1 and stage-2 of Exchanger before the projector head in Fig.B.2.1. It can be seen first that the intra-class B.2 Visualisation of the Latent Features in Exchanger D.4 Domain Generalization for SITS In this section, we further present results of the Exchanger[2-stages w/ 8 tokens] evaluated on TimeMatch dataset [22] which is comprised of SITS from four different tiles: 33UVP (Austria), 32VNH (Denmark), 30TXT (mid-west France), and 31TCJ (southern France). We follow the naming convention adopted in [22] to refer to these four Sentinel-2 tiles as AT1, DK1, FR1, and FR2, respectively, and the leave-one-region-out evaluation protocol where one Sentinel-2 tile is held out for testing and the remaining three tiles are used for training. In addition to the specifically-curated dataset for evaluating spatial generalization capability of crop classifiers, authors in [22] proposed to use thermal positional encoding (TPE) to combat temporal shifts across different geographical locations where Growing Degree Days (GDD) have been used to replace calendar time, which has been proven to be effective in improving spatial generalizability. We directly use the TPE method proposed in [22] to modify the positional encoding component in Exchanger. Based on our empirical observations, it is favourable to set the dimension of positional embeddings to a relatively small number for better generalization performance, indicating the sensitivity to resolutions of frequencies in sine/cosine functions. As seen in Tab.D.4.1, our proposed model trained only for 20 epochs can achieve results comparable to those of PSE+LTAE [6] trained for 100 epochs in the original setup. But the highly-specialized architecture PSE+LTAE [6] still has demonstrated superiority to our model, which we leave as a future direction for improvement. 1 : 1Ablation studies of core design choices in Exchanger on PASTIS validation dataset with 5-Fold cross-validation. Default setting means use untied content & position attention as shown in Eq.(4)(6). The figure in parenthesis denotes the number of content/position queries used. Fig. 1 : 1Convergence analysis for Exchanger+Unet with pre-trained backbones or training from scratch on PASTIS validation dataset (Fold-1). The left figure shows the training and validation losses. The right figure shows the evaluation metric mIoU on the validation dataset. Fig. 2 : 2Qualitative comparison. We randomly sample 4 SITS sample from PASTIS Fold-1 validation dataset and present the panoptic prediction results from U-TAE+PaPs, Exchanger+Unet+PaPs, and Exchanger+Mask2Former. Please note the artefacts in the last column result from stitching 64 × 64 predictions to 128 × 128.A.1 Color Palette for PASTISFig. A.1.1: Color Palette used for visualising latent features, semantic & panoptic predictions on PASTIS. Fig. B. 2 . 1 : 21TSNE visualisations of latent features from stage-1 and stage-2 of Exchanger. Table Table 2 : 2Comparison with state-of-the-art models on PASTIS and MTLCC test dataset. The figure in parenthesis denotes the standard deviation across the official 5-Fold splits in PASTIS. FLOPs are calculated based on a single SITS sample with T × C × H × W = 30 × 10 × 128 × 128.mIoU (%) #Params(M) FLOPs PASTIS MTLCC FPN + ConvLSTM [21] 57.1 73.7 1.45 714 G Unet + ConvLSTM [19] 57.8 76.2 2.33 55 G Unet-3D [19] 58.4 75.2 1.55 92G U-TAE [7] 63.1 77.1 1.09 47 G TSViT [31] 65.4 84.8 2.16 558 G Exchanger+Unet 66.8(+1.2) 90.7 8.08 300 G Exchanger+Mask2Former 67.9(+1.2) 90.5 24.59 329 G PASTIS and MTLCC, surpassing the previous state-of-the-art results attained by TSViT [31] by 1.4 and 5.9 points respectively while only using 53% FLOPs. Table 3 : 3Comparison with state-of-the-art models on PASTIS test dataset. The figure in parenthesis denotes the standard deviation across the official 5-Fold splits in PASTIS. FLOPs are calculated based on a single SITS sample with T × C × H × W = 30 × 10 × 128 × 128. Inference Time (IT) is calculated on Fold-1 with around 490 sequences on a single A100 GPU.SQ RQ PQ #Params(M) FLOPs IT(s) Unet+ConvLSTM+PaPs [7] 80.2 43.9 35.6 2.50 55 G 660 U-TAE+PaPs [7] 81.5 53.2 43.8 1.26 47 G 207 Exchanger+Unet+PaPs 80.3(+0.1) 58.9(+0.6) 47.8(+0.4) 9.99 301 G 252 Exchanger+Mask2Former 84.6(+0.9) 61.6(+1.6) 52.6(+1.8) 24.63 332 G 154 Table D . D4.1: Leave-one-region-out spatial generalization results (macro F1 score). DK1 FR1 FR2 Avg. TPE-Recurrent 86.5 80.3 86.0 80.5 83.3 TPE-Recurrent 82.9 80.1 81.2 76.4 80.2AT1 PSE+LTAE [6] TPE-Fourier 84.7 79.0 77.3 80.0 80.3 Exchanger TPE-Fourier 84.1 77.8 84.2 77.6 80.9 https://github.com/VSainteuf/pastis-benchmark Note: we follow the evaluation protocol proposed in[7] where the calculation of PQ metric only involves thing classes. AcknowledgementsThe work is supported by Department for the Economy (DfE) international studentship at Ulster University (UU). 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[ "https://github.com/VSainteuf/pastis-benchmark" ]
[ "Capacity Bounds for the Gaussian Interference Channel", "Capacity Bounds for the Gaussian Interference Channel" ]
[ "Student Member, IEEEAbolfazl S Motahari [email protected] \nCoding & Signal Transmission Laboratory (\n\n", "Member, IEEEAmir K Khandani [email protected] \nCoding & Signal Transmission Laboratory (\n\n" ]
[ "Coding & Signal Transmission Laboratory (\n", "Coding & Signal Transmission Laboratory (\n" ]
[]
The capacity region of the two-user Gaussian Interference Channel (IC) is studied. Three classes of channels are considered: weak, one-sided, and mixed Gaussian IC. For the weak Gaussian IC, a new outer bound on the capacity region is obtained that outperforms previously known outer bounds. The sum capacity for a certain range of channel parameters is derived. For this range, it is proved that using Gaussian codebooks and treating interference as noise is optimal. It is shown that when Gaussian codebooks are used, the full Han-Kobayashi achievable rate region can be obtained by using the naive Han-Kobayashi achievable scheme over three frequency bands (equivalently, three subspaces). For the one-sided Gaussian IC, an alternative proof for the Sato's outer bound is presented. We derive the full Han-Kobayashi achievable rate region when Gaussian codebooks are utilized. For the mixed Gaussian IC, a new outer bound is obtained that outperforms previously known outer bounds. For this case, the sum capacity for the entire range of channel parameters is derived. It is proved that the full Han-Kobayashi achievable rate region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel parameters.
10.1109/isit.2008.4594986
[ "https://arxiv.org/pdf/0801.1306v1.pdf" ]
60,435
0801.1306
471d92dc2d4002b844145dea5d97b4defdc83ae3
Capacity Bounds for the Gaussian Interference Channel 8 Jan 2008 Student Member, IEEEAbolfazl S Motahari [email protected] Coding & Signal Transmission Laboratory ( Member, IEEEAmir K Khandani [email protected] Coding & Signal Transmission Laboratory ( Capacity Bounds for the Gaussian Interference Channel 8 Jan 20081Index Terms Gaussian interference channelscapacity regionsum capacityconvex regions The capacity region of the two-user Gaussian Interference Channel (IC) is studied. Three classes of channels are considered: weak, one-sided, and mixed Gaussian IC. For the weak Gaussian IC, a new outer bound on the capacity region is obtained that outperforms previously known outer bounds. The sum capacity for a certain range of channel parameters is derived. For this range, it is proved that using Gaussian codebooks and treating interference as noise is optimal. It is shown that when Gaussian codebooks are used, the full Han-Kobayashi achievable rate region can be obtained by using the naive Han-Kobayashi achievable scheme over three frequency bands (equivalently, three subspaces). For the one-sided Gaussian IC, an alternative proof for the Sato's outer bound is presented. We derive the full Han-Kobayashi achievable rate region when Gaussian codebooks are utilized. For the mixed Gaussian IC, a new outer bound is obtained that outperforms previously known outer bounds. For this case, the sum capacity for the entire range of channel parameters is derived. It is proved that the full Han-Kobayashi achievable rate region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel parameters. I. INTRODUCTION O NE of the fundamental problems in Information Theory, originating from [1], is the full characterization of the capacity region of the interference channel (IC). The simplest form of IC is the two-user case in which two transmitters aim to convey independent messages to their corresponding receivers through a common channel. Despite some special cases, such as very strong and strong interference, where the exact capacity region has been derived [2], [3], the characterization of the capacity region for the general case is still an open problem. A limiting expression for the capacity region is obtained in [4] (see also [5]). Unfortunately, due to excessive computational complexity, this type of expression does not result in a tractable approach to fully characterize the capacity region. To show the weakness of the limiting expression, Cheng and Verdú have shown that for the Gaussian Multiple Access Channel (MAC), which can be considered as a special case of the Gaussian IC, the limiting expression fails to fully characterize the capacity region by relying only on Gaussian distributions [6]. However, there is a point on the boundary of the capacity region of the MAC that can be obtained directly from the limiting expression. This point is achievable by using simple scheme of Frequency/Time Division (FD/TD). The computational complexity inherent to the limiting expression is due to the fact that the corresponding encoding and decoding strategies are of the simplest possible form. The encoding strategy is based on mapping data to a codebook constructed from a unique probability density and the decoding strategy is to treat the interference as noise. In contrast, using more sophisticated encoders and decoders may result in collapsing the limiting expression into a single letter formula for the capacity region. As an evidence, it is known that the joint typical decoder for the MAC achieves the capacity region [7]. Moreover, there are some special cases, such as strong IC, where the exact characterization of the capacity region has been derived [2], [3] where decoding the interference is the key idea behind this result. In their pioneering work, Han and Kobayashi (HK) proposed a coding strategy in which the receivers are allowed to decode part of the interference as well as their own data [8]. The HK achievable region is still the best inner bound for the capacity region. Specifically, in their scheme, the message of each user is split into two independent parts: the common part and the private part. The common part is encoded such that both users can decode it. The private part, on the other hand, can be decoded only by the intended receiver and the other receiver treats it as noise. In summary, the HK achievable region is the intersection of the capacity regions of two three-user MACs, projected on a two-dimensional subspace. The HK scheme can be directly applied to the Gaussian IC. Nonetheless, there are two sources of difficulties in characterizing the full HK achievable rate region. First, the optimal distributions are unknown. Second, even if we confine the distributions to be Gaussian, computation of the full HK region under Gaussian distribution is still difficult due to numerous degrees of freedom involved in the problem. The main reason behind this complexity is the computation of the cardinality of the time-sharing parameter. Recently, reference [9], Chong et al. has presented a simpler expression with less inequalities for the HK achievable region. Since the cardinality of the time-sharing parameter is directly related to the number of inequalities appearing in the achievable rate region, the computational complexity is decreased. However, finding the full HK achievable region is still prohibitively complex. Regarding outer bounds on the capacity region, there are three main results known. The first one obtained by Sato [10] is originally derived for the degraded Gaussian IC. Sato has shown that the capacity region of the degraded Gaussian IC is outer bounded by a certain degraded broadcast channel whose capacity region is fully characterized. In [11], Costa has proved that the capacity region of the degraded Gaussian broadcast channel is equivalent to that of the one-sided weak Gaussian IC. Hence, Sato outer bound can be used for the one-sided Gaussian IC as well. The second outer bound obtained for the weak Gaussian IC is due to Kramer [12]. Kramer outer bound is based on the fact that removing one of the interfering links enlarges the capacity region. Therefore, the capacity region of the two-user Gaussian IC is inside the intersection of the capacity regions of the underlying one-sided Gaussian ICs. For the case of weak Gaussian IC, the underlying one-sided IC is weak, for which the capacity region is unknown. However, Kramer has used the outer bound obtained by Sato to derive an outer bound for the weak Gaussian IC. The third outer bound due to Etkin, Tse, and Wang (ETW) is based on the Genie aided technique [13]. A genie that provides some extra information to the receivers can only enlarge the capacity region. At first glance, it seems a clever genie must provide some information about the interference to the receiver to help in decoding the signal by removing the interference. In contrast, the genie in the ETW scheme provides information about the intended signal to the receiver. Remarkably, reference [13] shows that their proposed outer bound outperforms Kramer bound for certain range of parameters. Moreover, using a similar method, [13] presents an outer bound for the mixed Gaussian IC. In this paper, by introducing the notion of admissible ICs, we propose a new outer bounding technique for the two-user Gaussian IC. The proposed technique relies on an extremal inequality recently proved by Liu and Viswanath [14]. We show that by using this scheme, one can obtain tighter outer bounds for both weak and mixed Gaussian ICs. More importantly, the sum capacity of the Gaussian weak IC for a certain range of the channel parameters is derived. The rest of this paper is organized as follows. In Section II, we present some basic definitions and review the HK achievable region when Gaussian codebooks are used. We study the time-sharing and the convexification methods as means to enlarge the basic HK achievable region. We investigate conditions for which the two regions obtained from time-sharing and concavification coincide. Finally, we consider an optimization problem based on extremal inequality and compute its optimal solution. In Section III, the notion of an admissible IC is introduced. Some classes of admissible ICs for the two-user Gaussian case is studied and outer bounds on the capacity regions of these classes are computed. We also obtain the sum capacity of a specific class of admissible IC where it is shown that using Gaussian codebooks and treating interference as noise is optimal. In Section IV, we study the capacity region of the weak Gaussian IC. We first derive the sum capacity of this channel for a certain range of parameters where it is proved that users should treat the interference as noise and transmit at their highest possible rates. We then derive an outer bound on the capacity region which outperforms the known results. We finally prove that the basic HK achievable region results in the same enlarged region by using either time-sharing or concavification. This reduces the complexity of the characterization of the full HK achievable region when Gaussian codebooks are used. In Section V, we study the capacity region of the one-sided Gaussian IC. We present a new proof for the Sato outer bound using the extremal inequality. Then, we present methods to simplify the HK achievable region such that the full region can be characterized. In Section VI, we study the capacity region of the mixed Gaussian IC. We first obtain the sum capacity of this channel and then derive an outer bound which outperforms other known results. Finally, by investigating the HK achievable region for different cases, we prove that for a certain range of channel parameters, the full HK achievable rate region using Gaussian codebooks is equivalent to that of the one-sided IC. Finally, in Section VII, we conclude the paper. A. Notations Throughout this paper, we use the following notations. Vectors are represented by bold faced letters. Random variables, matrices, and sets are denoted by capital letters where the difference is clear from the context. |A|, tr{A}, and A t represent the determinant, trace, and transpose of the square matrix A, respectively. I denotes the identity matrix. N and ℜ are the sets of nonnegative integers and real numbers, respectively. The union, intersection, and Minkowski sum of two sets U and V are represented by U ∪ V , U ∩ V , and U + V , respectively. We use γ(x) as an abbreviation for the function 0.5 log 2 (1 + x). II. PRELIMINARIES A. The Two-user Interference Channel Definition 1 (two-user IC): A two-user discrete memoryless IC consists of two finite sets X 1 and X 2 as input alphabets and two finite sets Y 1 and Y 2 as the corresponding output alphabets. The channel is governed by conditional probability distributions ω(y 1 , Definition 2 (capacity region of the two-user IC): A code (2 nR1 , 2 nR2 , n, λ n 1 , λ n 2 ) for the two-user IC consists of the following components for User i ∈ {1, 2}: y 2 |x 1 , x 2 ) where (x 1 , x 2 ) ∈ X 1 × X 2 and (y 1 , y 2 ) ∈ Y 1 × Y 2 .1) A uniform distributed message set M i ∈ [1, 2, ..., 2 nRi ]. 2) A codebook X i = {x i (1), x i (2), ..., x i (2 nRi )} where x i (·) ∈ X n i . 3) An encoding function F i : [1, 2, ..., 2 nRi ] → X i . 4) A decoding function G i : y i → [1, 2, ..., 2 nRi ]. 5) The average probability of error λ n i = P(G i (y i ) = M i ). A rate pair (R 1 , R 2 ) is achievable if there is a sequence of codes (2 nR1 , 2 nR2 , n, λ n 1 , λ n 2 ) with vanishing average error probabilities. The capacity region of the IC is defined to be the supremum of the set of achievable rates. Let C IC denote the capacity region of the two-user IC. The limiting expression for C IC can be stated as [5] C IC = lim n→∞ closure   P(X n 1 )P(X n 2 ) (R 1 , R 2 ) | R 1 ≤ 1 n I (X n 1 , Y n 1 ) R 2 ≤ 1 n I (X n 2 , Y n 2 )   .(1) In this paper, we focus on the two-user Gaussian IC which can be represented in standard form as [15], [16] y 1 = x 1 + √ ax 2 + z 1 , y 2 = √ bx 1 + x 2 + z 2 ,(2) where x i and y i denote the input and output alphabets of User i ∈ {1, 2}, respectively, and z 1 ∼ N (0, 1), z 2 ∼ N (0, 1) are standard Gaussian random variables. Constants a ≥ 0 and b ≥ 0 represent the gains of the interference links. Furthermore, Transmitter i, i ∈ {1, 2}, is subject to the power constraint P i . Achievable rates and the capacity region of the Gaussian IC can be defined in a similar fashion as that of the general IC with the condition that the codewords must satisfy their corresponding power constraints. The capacity region of the two-user Gaussian IC is denoted by C . Clearly, C is a function of the parameters P 1 , P 2 , a, and b. To emphasize this relationship, we may write C as C (P 1 , P 2 , a, b) as needed. Remark 1: Since the capacity region of the general IC depends only on the marginal distributions [16], the ICs can be classified into equivalent classes in which channels within a class have the same capacity region. In particular, for the Gaussian IC given in (2), any choice of joint distributions for the pair (z 1 , z 2 ) does not affect the capacity region as long as the marginal distributions remain Gaussian with zero mean and unit variance. Depending on the values of a and b, the two-user Gaussian IC is classified into weak, strong, mixed, one-sided, and degraded Gaussian IC. In Figure 1, regions in ab-plane together with their associated names are shown. Briefly, if 0 < a < 1 and 0 < b < 1, then the channel is called weak Gaussian IC. If 1 ≤ a and 1 ≤ b, then the channel is called strong Gaussian IC. If either a = 0 or b = 0, the channel is called one-sided Gaussian IC. If ab = 1, then the channel is called degraded Gaussian IC. If either 0 < a < 1 and 1 ≤ b, or 0 < b < 1 and 1 ≤ a, then the channel is called mixed Gaussian IC. Finally, the symmetric Gaussian IC (used throughout the paper for illustration purposes) corresponds to a = b and P 1 = P 2 . Among all classes shown in Figure 1, the capacity region of the strong Gaussian IC is fully characterized [3], [2]. In this case, the capacity region can be stated as the collection of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ(P 1 ), R 2 ≤ γ(P 2 ), R 1 + R 2 ≤ min {γ(P 1 + aP 2 ), γ(bP 1 + P 2 )} . B. Support Functions Throughout this paper, we use the following facts from convex analysis. There is a one to one correspondence between any closed convex set and its support function [17]. The support function of any set D ∈ ℜ m is a function σ D : ℜ m → ℜ defined as σ D (c) = sup{c t R|R ∈ D}.(3) Clearly, if the set D is compact, then the sup is attained and can be replaced by max. In this case, the solutions of (3) correspond to the boundary points of D [17]. The following relation is the dual of (3) and holds when D is closed and convex D = {R|c t R ≤ σ D (c), ∀ c}.(4) For any two closed convex sets D and D ′ , D ⊆ D ′ , if and only if σ D ≤ σ D ′ . C. Han-Kobayashi Achievable Region The best inner bound for the two-user Gaussian IC is the full HK achievable region denoted by C HK [8]. Despite having a single letter formula, C HK is not fully characterized yet. In fact, finding the optimum distributions achieving boundary points of C HK is still an open problem. We define G as a subset of C HK where Gaussian distributions are used for codebook generation. Using a shorter description of C HK obtained in [9], G can be described as follows. Let us first define G 0 as the collection of all rate pairs (R 1 , R 2 ) ∈ ℜ 2 + satisfying R 1 ≤ ψ 1 = γ P 1 1 + aβP 2 ,(5)R 2 ≤ ψ 2 = γ P 2 1 + bαP 1 ,(6)R 1 + R 2 ≤ ψ 3 = min {ψ 31 , ψ 32 , ψ 33 } ,(7)2R 1 + R 2 ≤ ψ 4 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ αP 1 1 + aβP 2 + γ βP 2 + b(1 − α)P 1 1 + bαP 1 ,(8)R 1 + 2R 2 ≤ ψ 5 = γ βP 2 1 + bαP 1 + γ P 2 + b(1 − α)P 1 1 + bαP 1 + γ αP 1 + a(1 − β)P 2 1 + aβP 2 ,(9) for fixed α ∈ [0, 1] and β ∈ [0, 1]. 1 ψ 3 is the minimum of ψ 31 , ψ 32 , and ψ 33 defined as ψ 31 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ βP 2 1 + bαP 1 ,(10)ψ 32 = γ αP 1 1 + aβP 2 + γ P 2 + b(1 − α)P 1 1 + bαP 1 ,(11)ψ 33 = γ αP 1 + a(1 − β)P 2 1 + aβP 2 + γ βP 2 + b(1 − α)P 1 1 + bαP 1 .(12) G 0 is a polytope and a function of four variables P 1 , P 2 , α, and β. To emphasize this relation, we may write G 0 (P 1 , P 2 , α, β) as needed. It is convenient to represent G 0 in a matrix form as G 0 = {R|AR ≤ Ψ(P 1 , P 2 , α, β)} where R = (R 1 , R 2 ) t , Ψ = (ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 ) t , and A = 1 0 1 2 1 0 1 1 1 2 t . Equivalently, G 0 can be represented as the convex hull of its extreme points, i.e., G 0 (P 1 , P 2 , α, β) = conv {r 1 , r 2 , . . . , r K }, where it is assumed that G 0 has K extreme points. It is easy to show that K ≤ 7. Now, G can be defined as a region obtained from enlarging G 0 by making use of the time-sharing parameter, i.e., G is the collection of all rate pairs R = (R 1 , R 2 ) t satisfying AR≤ q i=1 λ i Ψ(P 1i , P 2i , α i , β i ),(13) where q ∈ N and q i=1 λ i P 1i ≤ P 1 ,(14)q i=1 λ i P 2i ≤ P 2 ,(15)q i=1 λ i = 1,(16)λ i ≥ 0, (α i , β i )∈ [0, 1] 2 ; ∀i ∈ {1, . . . , q}.(17) It is easy to show that G is a closed, bounded and convex region. In fact, the capacity region C which contains G is inside the rectangle defined by inequalities R 1 ≤ γ(P 1 ) and R 2 ≤ γ(P 2 ). Moreover, (0, 0), (γ(P 1 ), 0), and (0, γ(P 2 )) are extreme points of both C and G . Hence, to characterize G , we need to obtain all extreme points of G that are in the interior of the first quadrant (the same argument holds for C ). In other words, we need to obtain σ G (c 1 , c 2 ), the support function of G , either when 1 ≤ c 1 and c 2 = 1 or when c 1 = 1 and 1 ≤ c 2 . We also define G 1 and G 2 obtained by enlarging G 0 in two different manners. G 1 is defined as G 1 (P 1 , P 2 ) = (α,β)∈[0,1] 2 G 0 (P 1 , P 2 , α, β).(18) G 1 is not necessarily a convex region. Hence, it can be further enlarged by the convex hull operation. G 2 is defined as the collection of all rate pairs R = (R 1 , R 2 ) t satisfying R = q ′ i=1 λ i R i(19) where q ′ ∈ N and AR i ≤ Ψ(P 1i , P 2i , α i , β i ),(20)q ′ i=1 λ i P 1i ≤ P 1 ,(21)q ′ i=1 λ i P 2i ≤ P 2 ,(22)q ′ i=1 λ i = 1,(23)λ i ≥ 0, (α i , β i )∈ [0, 1] 2 ; ∀i ∈ {1, . . . , q ′ }.(24) It is easy to show that G 2 is a closed, bounded and convex region. In fact, G 2 is obtained by using the simple method of TD/FD. To see this, let us divide the available frequency band into q ′ sub-bands where λ i represents the length of the i'th band and q ′ i=1 λ i = 1. User 1 and 2 allocate P 1i and P 2i in the i'th sub-band, respectively. Therefore, all rate pairs in G 0 (P 1i , P 2i , α i , β i ) are achievable in the i'th sub-band for fixed (α i , β i ) ∈ [0, 1] 2 . Hence, all rate pairs in q ′ i=1 λ i G 0 (P 1i , P 2i , α i , β i ) are achievable provided that q ′ i=1 λ i P 1i ≤ P 1 and q ′ i=1 λ i P 2i ≤ P 2 . Clearly, the chain of inclusions G 0 ⊆ G 1 ⊆ G 2 ⊆ G ⊆ C HK ⊆ C always holds. D. Concavification Versus Time-Sharing In this subsection, we follow two objectives. First, we aim at providing some necessary conditions such that G 2 = G . Second, we bound q and q ′ which are parameters involved in the descriptions of G and G 2 , respectively. However, we derive the required conditions for the more general case where there are M users in the system. To this end, assume an achievable scheme for an M -user channel with the power constraint P = [P 1 , P 2 , . . . , P M ] is given. The corresponding achievable region can be represented as D 0 (P, Θ) = {R|AR ≤ Ψ(P, Θ)} ,(25) where A is a K × M matrix and Θ ∈ [0, 1] M . D 0 is a polyhedron in general, but for the purpose of this paper, it suffices to assume that it is a polytope. Since D 0 is a convex region, the convex hull operation does not lead to a new enlarged region. However, if the extreme points of the region are not a concave function of P, it is possible to enlarge D 0 by using two different methods which are explained next. The first method is based on using the time sharing parameter. Let us denote the corresponding region as D which can be written as D = R|AR ≤ q i=1 λ i Ψ(P i , Θ i ), q i=1 λ i P i ≤ P, q i=1 λ i = 1, λ i ≥ 0, Θ i ∈ [0, 1] M ∀i ,(26) where q ∈ N. In the second method, we use TD/FD to enlarge the achievable rate region. This results in an achievable region D 2 represented as D 2 =    R = q ′ i=1 λ i R i |AR i ≤ Ψ(P i , Θ i ), q ′ i=1 λ i P i ≤ P, q ′ i=1 λ i = 1, λ i ≥ 0, Θ i ∈ [0, 1] M ∀i    ,(27) where q ′ ∈ N. We refer to this method as concavification. It can be readily shown that D and D 2 are closed and convex, and D 2 ⊆ D. We are interested in situations where the inverse inclusion holds. The support function of D 0 is a function of P, Θ, and c. Hence, we have σ D0 (c, P, Θ) = max{c t R|AR ≤ Ψ(P, Θ)}.(28) For fixed P and Θ, (28) is a linear program. Using strong duality of linear programming, we obtain σ D0 (c, P, Θ) = min{y t Ψ(P, Θ)|A t y = c, y ≥ 0}. In general,ŷ, the minimizer of (29), is a function of P, Θ, and c. We say D 0 possesses the unique minimizer property if y merely depends on c, for all c. In this case, we have σ D0 (c, P, Θ) =ŷ t (c)Ψ(P, Θ),(30) where A tŷ = c. This condition means that for any c the extreme point of D 0 maximizing the objective c t R is an extreme point obtained by intersecting a set of specific hyperplanes. A necessary condition for D 0 to possess the unique minimizer property is that each inequality in describing D 0 is either redundant or active for all P and Θ. (31) By fixing P, P i 's, Θ i 's, and λ i 's, the above maximization becomes a linear program. Hence, relying on weak duality of linear programming, we obtain σ D (c, P) ≤ min A t y=c,y≥0 y t q i=1 λ i Ψ(P i , Θ i ).(32) Clearly,ŷ(c), the solution of (29), is a feasible point for (32) and we have σ D (c, P) ≤ŷ t (c) q i=1 λ i Ψ(P i , Θ i ).(33) Using (30), we obtain σ D (c, P) ≤ q i=1 λ i σ D0 (c, P i , Θ i ).(34) Let us assumeR i is the maximizer of (28). In this case, we have σ D (c, P) ≤ q i=1 λ i c tR i .(35) Hence, we have σ D (c, P) ≤ c t q i=1 λ iRi .(36) By definition, q i=1 λ iRi is a point in D 2 . Therefore, we conclude σ D (c, P) ≤ σ D2 (c, P). This completes the proof. Corollary 1 (Han [18]): If D 0 is a polymatroid, then D=D 2 . Proof: It is easy to show that D 0 possesses the unique minimizer property. In fact, for given c,ŷ can be obtained in a greedy fashion independent of P and Θ. In what follows, we upper bound q and q ′ . Theorem 2: The cardinality of the time sharing parameter q in (26) is less than M + K + 1, where M and K are the dimensions of P and Ψ(P), respectively. Moreover, if Ψ(P) is a continuous function of P, then q ≤ M + K. Proof: Let us define E as E = q i=1 λ i Ψ(P i , Θ i )| q i=1 λ i P i ≤ P, q i=1 λ i = 1, λ i ≥ 0, Θ i ∈ [0, 1] M ∀i .(38) In fact, E is the collection of all possible bounds for D. To prove q ≤ M + K + 1, we define another region E 1 as E 1 = {(P ′ , S ′ )|0 ≤ P ′ , S ′ = Ψ(P ′ , Θ ′ ), Θ ′ ∈ [0, 1] M }.(39) From the direct consequence of the Caratheodory's theorem [19], the convex hull of E 1 denoted by conv E 1 can be obtained by convex combinations of no more than M + K + 1 points in E 1 . Moreover, if Ψ(P ′ , Θ ′ ) is continuous, then M + K points are sufficient due to the extension of the Caratheodory's theorem [19]. Now, we define the regionÊ aŝ E = {S ′ |(P ′ , S ′ ) ∈ conv E 1 , P ′ ≤ P}.(40) Clearly,Ê ⊆ E. To show the other inclusion, let us consider a point in E, say S = q i=1 λ i Ψ(P i , Θ i ). Since (P i , Ψ(P i , Θ i )) is a point in E 1 , q i=1 λ i (P i , Ψ(P i , Θ i )) belongs to conv E 1 . Having q i=1 λ i P i ≤ P, we conclude q i=1 λ i Ψ(P i , Θ) ∈Ê. Hence, E ⊆Ê. This completes the proof. Corollary 2 (Etkin, Parakh, and Tse [20]): For the M -user Gaussian IC where users use Gaussian codebooks for data transmission and treat the interference as noise, the cardinality of the time sharing parameter is less than 2M . Proof: In this case, D 0 = {R|R ≤ Ψ(P)} where both P and Ψ(P) have dimension M and Ψ(P) is a continuous function of P. Applying Theorem 2 yields the desired result. In the following theorem, we obtain an upper bound on q ′ . Theorem 3: To characterize boundary points of D 2 , it suffices to set q ′ ≤ M + 1. Proof: Let us assumeR is a boundary point of D 2 . Hence, there exists c such that σ D2 (c, P) = max R∈D2 c t R = c tR ,(41) whereR = q ′ i=1λ iRi and the optimum is achieved for the set of parametersΘ i ,λ i , andP i . The optimization problem in (41) can be written as σ D2 (c, P) =max q ′ i=1 λ i g(c, P i )(42) subject to: q ′ i=1 λ i = 1, q ′ i=1 λ i P i ≤ P, 0 ≤ λ i , 0 ≤ P i , ∀i ∈ {1, 2, . . . , q ′ }, where g(c, P) is defined as g(c, P) =max c t R (43) subject to: AR ≤ Ψ(P, Θ), 0 ≤ Θ ≤ 1, In fact, σ D2 (c, P) in (42) can be viewed as the result of the concavification of g(c, P) [19]. Hence, using Theorem 2.16 in [19], we conclude that q ′ ≤ M + 1. Remarkable point about Theorem 3 is that the upper bound on q ′ is independent of the number of inequalities involved in the description of the achievable rate region. Corollary 3: For the M -user Gaussian IC where users use Gaussian codebooks and treat the interference as noise, we have D 2 = D and q = q ′ = M + 1. E. Extremal Inequality In [14], the following optimization problem is studied: W = max QX≤S h(X + Z 1 ) − µh(X + Z 2 ),(44) where Z 1 and Z 2 are n-dimensional Gaussian random vectors with the strictly positive definite covariance matrices Q Z1 and Q Z2 , respectively. The optimization is over all random vectors X independent of Z 1 and Z 2 . X is also subject to the covariance matrix constraint Q X ≤ S, where S is a positive definite matrix. In [14], it is shown that for all µ ≥ 1, this optimization problem has a Gaussian optimal solution for all positive definite matrices Q Z1 and Q Z2 . However, for 0 ≤ µ < 1 this optimization problem has a Gaussian optimal solution provided Q Z1 ≤ Q Z2 , i.e., Q Z2 − Q Z1 is a positive semi-definite matrix. It is worth noting that for µ = 1 this problem when Q Z1 ≤ Q Z2 is studied under the name of the worse additive noise [21], [22]. In this paper, we consider a special case of (44) where Z 1 and Z 2 have the covariance matrices N 1 I and N 2 I, respectively, and the trace constraint is considered, i.e., W = max tr{QX}≤nP h(X + Z 1 ) − µh(X + Z 2 ).(45) In the following lemma, we provide the optimal solution for the above optimization problem when N 1 ≤ N 2 . Lemma 1: If N 1 ≤ N 2 , the optimal solution of (45) is iid Gaussian for all 0 ≤ µ and we have 1) For 0 ≤ µ ≤ N2+P N1+P , the optimum covariance matrix is P I and the optimum solution is W = n 2 log [(2πe)(P + N 1 )] − µn 2 log [(2πe)(P + N 2 )] .(46) 2) For N2+P N1+P < µ ≤ N2 N1 , the optimum covariance matrix is N2−µN1 µ−1 I and the optimum solution is W = n 2 log (2πe) N 2 − N 1 µ − 1 − µn 2 log µ(2πe)(N 2 − N 1 ) µ − 1 .(47) 3) For N2 N1 < µ, the optimum covariance matrix is 0 and the optimum solution is W = n 2 log(2πeN 1 ) − µn 2 log(2πeN 2 ). (48) Proof: From the general result for (44), we know that the optimum input distribution is Gaussian. Hence, we need to solve the following maximization problem: W =max 1 2 log ((2πe) n |Q X + N 1 I|) − µ 2 log ((2πe) n |Q X + N 2 I|)(49) subject to: 0 ≤ Q X , tr{Q X } ≤ nP. Since Q X is a positive semi-definite matrix, it can be decomposed as Q X = U ΛU t , where Λ is a diagonal matrix with nonnegative entries and U is a unitary matrix, i.e., U U t = I. Substituting Q X = U ΛU t in (49) and using the identities tr{AB} = tr{BA} and |AB + I| = |BA + I|, we obtain W =max 1 2 log ((2πe) n |Λ + N 1 I|) − µ 2 log ((2πe) n |Λ + N 2 I|) (50) subject to: 0 ≤ Λ, tr{Λ} ≤ nP. This optimization problem can be simplified as W =max n 2 n i=1 [log(2πe)(λ i + N 1 ) − µ log(2πe)(λ i + N 2 )](51)subject to: 0 ≤ λ i ∀i, n i=1 λ i ≤ nP. By introducing Lagrange multipliers ψ and Φ = {φ 1 , φ 2 , . . . , φ n }, we obtain L(Λ, ψ, Φ) = max n 2 n i=1 [log(2πe)(λ i + N 1 ) − µ log(2πe)(λ i + N 2 )] + ψ nP − n i=1 λ i + n i=1 φ i λ i . (52) N2−µN1 µ−1 P +N2 P +N1 N2 N1 1 P Variance µ Fig. 2. Optimum variance versus µ. The first order KKT necessary conditions for the optimum solution of (52) can be written as 1 λ i + N 1 − µ λ i + N 2 − ψ + φ i =0, ∀i ∈ {1, 2, . . . , n},(53)ψ nP − n i=1 λ i =0,(54)φ i λ i =0, ∀i ∈ {1, 2, . . . , n}.(55) It is easy to show that when N 1 ≤ N 2 , λ = λ 1 = . . . = λ n and the only solution for λ is λ =      P, if 0 ≤ µ ≤ N2+P N1+P N2−µN1 µ−1 , if N2+P N1+P < µ ≤ N2 N1 0, if N2 N1 < µ(56) Substituting λ into the objective function gives the desired result. In Figure 2, the optimum variance as a function of µ is plotted. This figure shows that for any value of µ ≤ P +N2 P +N1 , we need to use the maximum power to optimize the objective function, whereas for µ > P +N2 P +N1 , we use less power than what is permissible. Lemma 2: If N 1 > N 2 , the optimal solution of (45) is iid Gaussian for all 1 ≤ µ. In this case, the optimum variance is 0 and the optimum W is W = n 2 log(2πeN 1 ) − µn 2 log(2πeN 2 ). (57) Proof: The proof is similar to that of Lemma 1 and we omit it here. Corollary 4: For µ = 1, the optimal solution of (45) is iid Gaussian and the optimum W is W =    n 2 log P +N1 P +N2 , if N 1 ≤ N 2 n 2 log N1 N2 , if N 1 > N 2 .(58) We frequently apply the following optimization problem in the rest of the paper: f h (P, N 1 , N 2 , a, µ) = max tr{QX}≤nP h(X + Z 1 ) − µh( √ aX + Z 2 ),(59) where N 1 ≤ N 2 /a. Using the identity h(AX) = log(|A|) + h(X), (59) can be written as f h (P, N 1 , N 2 , a, µ) = n 2 log a + max tr{QX}≤nP h(X + Z 1 ) − µh(X + Z 2 √ a ).(60) Now, Lemma 1 can be applied to obtain f h (P, N 1 , N 2 , a, µ) =      1 2 log [(2πe)(P + N 1 )] − µ 2 log [(2πe)(aP + N 2 )] if 0 ≤ µ ≤ P +N2/a P +N1 1 2 log (2πe) N2/a−N1 µ−1 − µ 2 log aµ(2πe)(N2/a−N1) µ−1 if P +N2/a P +N1 < µ ≤ N2 aN1 1 2 log(2πeN 1 ) − µ 2 log(2πeN 2 ) if N2 aN1 < µ (61) y 1 y 2 y 1 y 2 f 1 f 2 ω(ỹ 1 ,ỹ 2 |x 1 , x 2 ) x 1 x 2 Fig. 3. An admissible channel. f 1 and f 2 are deterministic functions. III. ADMISSIBLE CHANNELS In this section, we aim at building ICs whose capacity regions contain the capacity region of the two-user Gaussian IC, i.e., C . Since we ultimately use these to outer bound C , these ICs need to have a tractable expression (or a tractable outer bound) for their capacity regions. Let us consider an IC with the same input letters as that of C and the output lettersỹ 1 andỹ 2 for Users 1 and 2, respectively. The capacity region of this channel, say C ′ , contains C if I(x n 1 ; y n 1 ) ≤I(x n 1 ;ỹ n 1 ), (62) I(x n 2 ; y n 2 ) ≤I(x n 2 ;ỹ n 2 ),(63) for all p(x n 1 )p(x n 2 ) and for all n ∈ N. One way to satisfy (62) and (63) is to provide some extra information to either one or to both receivers. This technique is known as Genie aided outer bounding. In [12], Kramer has used such a genie to provide some extra information to both receivers such that they can decode both users' messages. Since the capacity region of this new interference channel is equivalent to that of the Compound Multiple Access Channel whose capacity region is known, reference [12] obtains an outer bound on the capacity region. To obtain a tighter outer bound, reference [12] further uses the fact that if a genie provides the exact information about the interfering signal to one of the receivers, then the new channel becomes the one-sided Gaussian IC. Although the capacity region of the one-sided Gaussian IC is unknown for all ranges of parameters, there exists an outer bound for it due to Sato and Costa [23], [11] that can be applied to the original channel. In [13], Etkin et al. use a different genie that provides some extra information about the intended signal. Even though at first glance their proposed method appears to be far from achieving a tight bound, remarkably they show that the corresponding bound is tighter than the one due to Kramer for certain ranges of parameters. Next, we introduce the notion of admissible channels to satisfy (62) and (63). Definition 3 (Admissible Channel): An IC C ′ with input letter x i and output letterỹ i for User i ∈ {1, 2} is an admissible channel if there exist two deterministic functionsŷ n 1 = f 1 (ỹ n 1 ) andŷ n 2 = f 2 (ỹ n 2 ) such that I(x n 1 ; y n 1 ) ≤I(x n 1 ;ŷ n 1 ),(64)I(x n 2 ; y n 2 ) ≤I(x n 2 ;ŷ n 2 )(65) hold for all p(x n 1 )p(x n 2 ) and for all n ∈ N. E denotes the collection of all admissible channels (see Figure 3). Remark 2: Genie aided channels are among admissible channels. To see this, let us assume a genie provides s 1 and s 2 as side information for User 1 and 2, respectively. In this case,ỹ i = (y i , s i ) for i ∈ {1, 2}. By choosing f i (y i , s i ) = y i , we observe thatŷ i = y i , and hence, (64) and (65) trivially hold. To obtain the tightest outer bound, we need to find the intersection of the capacity regions of all admissible channels. Nonetheless, it may happen that finding the capacity region of an admissible channel is as hard as that of the original one (in fact, based on the definition, the channel itself is one of its admissible channels). Hence, we need to find classes of admissible channels, say F , which possess two important properties. First, their capacity regions are close to C . Second, either their exact capacity regions are computable or there exist good outer bounds for them. Since F ⊆ E , we have C ⊆ F C ′ .(66) Recall that there is a one to one correspondence between a closed convex set and its support function. Since C is closed and convex, there is a one to one correspondence between C and σ C . In fact, boundary points of C correspond to the solutions of the following optimization problem σ C (c 1 , c 2 ) = max (R1,R2)∈C c 1 R 1 + c 2 R 2 .(67) Since we are interested in the boundary points excluding the R 1 and R 2 axes, it suffices to consider 0 ≤ c 1 and 0 ≤ c 2 where c 1 + c 2 = 1. Admissible Channel f 2 (ỹ 22 ,ỹ 21 ) = (1 − √ g 2 )ỹ 22 + √ g 2ỹ21 y 1 y 2 f 1 (ỹ 1 ) =ỹ 1 y 1 x 2 z 21 z 22 √ a x 1 z 1 √ g 2 y 21 y 22 1 − √ g 2 √ b ′ Fig. 4. Class A1 admissible channels. Since C ⊆ C ′ , we have σ C (c 1 , c 2 ) ≤ σ C ′ (c 1 , c 2 ).(68) Taking the minimum of the right hand side, we obtain σ C (c 1 , c 2 ) ≤ min C ′ ∈F σ C ′ (c 1 , c 2 ),(69) which can be written as σ C (c 1 , c 2 ) ≤ min C ′ ∈F max (R1,R2)∈C ′ c 1 R 1 + c 2 R 2 .(70) For convenience, we use the following two optimization problems σ C (µ, 1) = max (R1,R2)∈C µR 1 + R 2 ,(71)σ C (1, µ) = max (R1,R2)∈C R 1 + µR 2 ,(72) where 1 ≤ µ. It is easy to show that the solutions of (71) and (72) correspond to the boundary points of the capacity region. In the rest of this section, we introduce classes of admissible channels and obtain upper bounds on σ C ′ (µ, 1) and σ C ′ (1, µ). A. Classes of Admissible Channels 1) Class A1: This class is designed to obtain an upper bound on σ C (µ, 1). Therefore, we need to find a tight upper bound on σ C ′ (µ, 1). A member of this class is a channel in which User 1 has one transmit and one receive antenna whereas User 2 has one transmit antenna and two receive antennas (see Figure 4). The channel model can be written as y 1 = x 1 + √ ax 2 + z 1 , y 21 = x 2 + √ b ′ x 1 + z 21 , y 22 = x 2 + z 22 ,(73) whereỹ 1 is the signal at the first receiver,ỹ 21 andỹ 22 are the signals at the second receiver, z 1 is additive Gaussian noise with unit variance, z 21 and z 22 are additive Gaussian noise with variances N 21 and N 22 , respectively. Transmitters 1 and 2 are subject to the power constraints of P 1 and P 2 , respectively. To investigate admissibility conditions in (64) and (65), we introduce two deterministic functions f 1 and f 2 as follows (see Figure 4) f 1 (ỹ n 1 )=ỹ n 1 , (74) f 2 (ỹ n 22 ,ỹ n 21 )= (1 − √ g 2 )ỹ n 22 + √ g 2ỹ n 21 ,(75) where 0 ≤ g 2 . For g 2 = 0, the channel can be converted to the one-sided Gaussian IC by letting N 21 → ∞ and N 22 = 1. Hence, Class A1 contains the one-sided Gaussian IC obtained by removing the link between Transmitter 1 and Receiver 2. Using f 1 and f 2 , we obtainŷ n 1 =x n 1 + √ ax n 2 + z n 1 ,(76)y n 2 = b ′ g 2 x n 1 + x n 2 + (1 − √ g 2 )z n 22 + √ g 2 z n 21 .(77) Hence, this channel is admissible if the corresponding parameters satisfy b ′ g 2 = b, (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1.(78) We further add the following constraints to the conditions of the channels in Class A1: b ′ ≤ N 21 , aN 22 ≤ 1.(79) Although these additional conditions reduce the number of admissible channels within the class, they are needed to get a closed form formula for an upper bound on σ C ′ (µ, 1). In the following lemma, we obtain the required upper bound. Lemma 3: For the channels modeled by (73) and satisfying (79), we have σ C ′ (µ, 1) ≤min µ 1 2 log [2πe(P 1 + aP 2 + 1)] − µ 2 2 log(2πe) + 1 2 log N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22 (80) + µ 2 f h P 1 , 1, N 21 , b ′ , 1 µ 2 + f h (P 2 , N 22 , 1, a, µ 1 ) subject to: µ 1 + µ 2 = µ, µ 1 , µ 2 ≥ 0. Proof: Let us assume R 1 and R 2 are achievable rates for User 1 and 2, respectively. Furthermore, we split µ into µ 1 ≥ 0 and µ 2 ≥ 0 such that µ = µ 1 + µ 2 . Using Fano's inequality, we obtain n(µR 1 + R 2 ) ≤µI(x n 1 ;ỹ n 1 ) + I(x n 2 ;ỹ n 22 ,ỹ n 21 ) + nǫ n =µ 1 I(x n 1 ;ỹ n 1 ) + µ 2 I(x n 1 ;ỹ n 1 ) + I(x n 2 ;ỹ n 22 ,ỹ n 21 ) + nǫ n (a) ≤ µ 1 I(x n 1 ;ỹ n 1 ) + µ 2 I(x n 1 ;ỹ n 1 |x n 2 ) + I(x n 2 ;ỹ n 22 ,ỹ n 21 ) + nǫ n =µ 1 I(x n 1 ;ỹ n 1 ) + µ 2 I(x n 1 ;ỹ n 1 |x n 2 ) + I(x n 2 ;ỹ n 21 |ỹ n 22 ) + I(x n 2 ;ỹ n 22 ) + nǫ n =µ 1 h(ỹ n 1 ) − µ 1 h(ỹ n 1 |x n 1 ) + µ 2 h(ỹ n 1 |x n 2 ) − µ 2 h(ỹ n 1 |x n 1 , x n 2 ) +h(ỹ n 21 |ỹ n 22 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 ) + h(ỹ n 22 ) − h(ỹ n 22 |x n 2 ) + nǫ n = µ 1 h(ỹ n 1 ) − µ 2 h(ỹ n 1 |x n 1 , x n 2 ) + µ 2 h(ỹ n 1 |x n 2 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 ) + h(ỹ n 21 |ỹ n 22 ) − h(ỹ n 22 |x n 2 ) + h(ỹ n 22 ) − µ 1 h(ỹ n 1 |x n 1 ) + nǫ n ,(81) where (a) follows from the fact that x n 1 and x n 2 are independent. Now, we separately upper bound the terms within each bracket in (81). To maximize the terms within the first bracket, we use the fact that Gaussian distribution maximizes the differential entropy subject to a constraint on the covariance matrix. Hence, we have µ 1 h(ỹ n 1 ) − µ 2 h(ỹ n 1 |x n 1 , x n 2 )= µ 1 h(x n 1 + √ ax n 2 + z n 1 ) − µ 2 h(z n 1 ) ≤ µ 1 n 2 log [2πe(P 1 + aP 2 + 1)] − µ 2 n 2 log(2πe).(82) Since b ′ ≤ N 21 , we can make use of Lemma 1 to upper bound the second bracket. In this case, we have µ 2 h(ỹ n 1 |x n 2 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 )= µ 2 h(x n 1 + z n 1 ) − 1 µ 2 h( √ b ′ x n 1 + z n 21 ) ≤ µ 2 nf h P 1 , 1, N 21 , b ′ , 1 µ 2 ,(83) where f h is defined in (61). We upper bound the terms within the third bracket as follows [13]: where (a) follows from the chain rule and the fact that removing independent conditions does not decrease differential entropy, (b) follows from the fact that Gaussian distribution maximizes the conditional entropy for a given covariance matrix, and (c) follows form Jenson's inequality. For the last bracket, we again make use of the definition of f h . In fact, since aN 22 ≤ 1, we have h(ỹ n 21 |ỹ n 22 ) − h(ỹ n 22 |x n 2 ) (a) ≤ n i=1 h(ỹ 21 [i]|ỹ 22 [i]) − h(z n 22 ) (b) ≤ n i=1 1 2 log 2πe N 21 + b ′ P 1 [i] + P 2 [i]N 22 P 2 [i] + N 22 − n 2 log (2πeN 22 ) (c) ≤ n 2 log 2πe N 21 + 1 n n i=1 b ′ P 1 [i] + 1 n n i=1 P 2 [i]N 22 1 n n i=1 P 2 [i] + N 22 − n 2 log (2πeN 22 ) ≤ n 2 log 2πe N 21 + b ′ P 1 + P 2 N 22 P 2 + N 22 − n 2 log (2πeN 22 ) ≤ n 2 log N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22 ,(84)f 1 (ỹ 11 ,ỹ 12 ) = (1 − √ g 1 )ỹ 11 + √ g 1ỹ12 √ a ′ x 2 x 1ỹ 2 y 12 y 11 f 2 (ỹ 2 ) =ỹ 2ŷ 1 y 2 √ g 1 1 − √ g 1 z 11 z 12 z 2 √ bh(ỹ n 22 ) − µ 1 h(ỹ n 1 |x n 1 )= h(x n 2 + z n 22 ) − µ 1 h( √ ax n 2 + z n 1 ) ≤ nf h (P 2 , N 22 , 1, a, µ 1 ).(85) Adding all inequalities, we obtain µR 1 + R 2 ≤ µ 1 2 log [2πe(P 1 + aP 2 + 1)] − µ 2 2 log(2πe) + 1 2 log N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22 +µ 2 f h P 1 , 1, N 21 , b ′ , 1 µ 2 + f h (P 2 , N 22 , 1, a, µ 1 ),(86) where the fact that ǫ n → 0 as n → ∞ is used to eliminate ǫ n form the right hand side of the inequality. Now, by taking the minimum of the right hand side of (86) over all µ 1 and µ 2 , we obtain the desired result. This completes the proof. 2) Class A2: This class is the complement of Class A1 in the sense that we use it to upper bound σ C (1, µ). A member of this class is a channel in which User 1 is equipped with one transmit and two receive antennas, whereas User 2 is equipped with one antenna at both transmitter and receiver sides (see Figure 5). The channel model can be written as y 11 = x 1 + z 11 , y 12 = x 1 + √ a ′ x 2 + z 12 , y 2 = x 2 + √ bx 1 + z 2 ,(87) whereỹ 11 andỹ 12 are the signals at the first receiver,ỹ 2 is the signal at the second receiver, z 2 is additive Gaussian noise with unit variance, z 11 and z 12 are additive Gaussian noise with variances N 11 and N 12 , respectively. Transmitter 1 and 2 are subject to the power constraints P 1 and P 2 , respectively. For this class, we consider two linear functions f 1 and f 2 as follows (see Figure 5): f 1 (ỹ n 11 ,ỹ n 12 )= (1 − √ g 1 )ỹ n 11 + √ g 1ỹ n 12 ,(88)f 2 (ỹ n 2 )=ỹ n 2 .(89) Similar to Class A1, when g 1 = 0, the admissible channels in Class A2 become the one-sided Gaussian IC by letting N 12 → ∞ and N 11 = 1. Therefore, we haveŷ n 1 =x n 1 + a ′ g 1 x n 2 + (1 − √ g 1 )z n 11 + √ g 1 z n 12 ,(90)y n 2 = √ bx n 1 + x n 2 + z n 2 .(91) We conclude that the channel modeled by (87) is admissible if the corresponding parameters satisfy a ′ g 1 = a, (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1.(92) x 2 Similar to Class A1, we further add the following constraints to the conditions of Class A2 channels: x 1 z 11 z 12 z 21 z 22ỹ 22 y 21 y 11 √ g 1 1 − √ g 1ŷ 1 y 2 1 − √ g 2 √ g 2 √ b ′ √ a ′ f 1 (ỹ 11 ,ỹ 12 ) = (1 − √ g 1 )ỹ 11 + √ g 1ỹ12 f 2 (ỹ 22 ,ỹ 21 ) = (1 − √ g 2 )ỹ 22 + √ g 2ỹ21a ′ ≤ N 12 , bN 11 ≤ 1.(93) In the following lemma, we obtain the required upper bound. Lemma 4: For the channels modeled by (87) and satisfying (93), we have σ C ′ (1, µ) ≤min µ 1 2 log [2πe(bP 1 + P 2 + 1)] − µ 2 2 log(2πe) + 1 2 log N 12 N 11 + a ′ P 2 N 11 + P 1 P 1 + N 11 (94) + µ 2 f h P 2 , 1, N 12 , a ′ , 1 µ 2 + f h (P 1 , N 11 , 1, b, µ 1 ) subject to: µ 1 + µ 2 = µ, µ 1 , µ 2 ≥ 0. Proof: The proof is similar to that of Lemma 3 and we omit it here. 3) Class B: A member of this class is a channel with one transmit antenna and two receive antennas for each user modeled by (see Figure 6)ỹ 11 = x 1 + z 11 , y 12 = x 1 + √ a ′ x 2 + z 12 , y 21 = x 2 + √ b ′ x 1 + z 21 , y 22 = x 2 + z 22 ,(95) whereỹ 11 andỹ 12 are the signals at the first receiver,ỹ 21 andỹ 22 are the signals at the second receiver, and z ij is additive Gaussian noise with variance N ij for i, j ∈ {1, 2}. Transmitter 1 and 2 are subject to the power constraints P 1 and P 2 , respectively. In fact, this channel is designed to upper bound both σ C (µ, 1) and σ C (1, µ). Next, we investigate admissibility of this channel and the conditions that must be imposed on the underlying parameters. Let us consider two linear deterministic functions f 1 and f 2 with parameters 0 ≤ g 1 and 0 ≤ g 2 , respectively, as follows (see Figure 6) f 1 (ỹ n 11 ,ỹ n 12 )= (1 − √ g 1 )ỹ n 11 + √ g 1ỹ n 12 ,(96)f 2 (ỹ n 22 ,ỹ n 21 )= (1 − √ g 2 )ỹ n 22 + √ g 2ỹ n 21 .(97) Therefore, we haveŷ n 1 =x n 1 + a ′ g 1 x n 2 + (1 − √ g 1 )z n 11 + √ g 1 z n 12 ,(98)y n 2 = b ′ g 2 x n 1 + x n 2 + (1 − √ g 2 )z n 22 + √ g 2 z n 21 .(99) To satisfy (64) and (65), it suffices to have a ′ g 1 = a, b ′ g 2 = b, (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1, (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1.(100) Hence, a channel modeled by (95) is admissible if there exist two nonnegative numbers g 1 and g 2 such that the equalities in (100) are satisfied. We further add the following two constraints to the equality conditions in (100): b ′ N 11 ≤ N 21 , a ′ N 22 ≤ N 12 .(101) Although adding more constraints reduces the number of the admissible channels, it enables us to compute an outer bound on σ C ′ (µ, 1) and σ C ′ (1, µ). Lemma 5: For the channels modeled by (95) and satisfying (101), we have σ C ′ (µ, 1) ≤µγ P 1 N 11 + P 1 a ′ P 2 + N 12 + γ P 2 N 22 + P 2 b ′ P 1 + N 21 +f h (P 2 , N 22 , N 12 , a ′ , µ) + µ 2 log((2πe)(a ′ P 2 + N 12 )) − 1 2 log((2πe)(P 2 + N 22 )),(102)σ C ′ (1, µ) ≤γ P 1 N 11 + P 1 a ′ P 2 + N 12 + µγ P 2 N 22 + P 2 b ′ P 1 + N 21 +f h (P 1 , N 11 , N 21 , b ′ , µ) + µ 2 log((2πe)(b ′ P 1 + N 21 )) − 1 2 log((2πe)(P 1 + N 11 )). (103) Proof: We only upper bound σ C ′ (µ, 1) and an upper bound on σ C ′ (1, µ) can be similarly obtained. Let us assume R 1 and R 2 are achievable rates for User 1 and User 2, respectively. Using Fano's inequality, we obtain n(µR 1 + R 2 ) ≤µI(x n 1 ;ỹ n 11 ,ỹ n 12 ) + I(x n 2 ;ỹ n 22 ,ỹ n 21 ) + nǫ n =µI(x n 1 ;ỹ n 12 |ỹ n 11 ) + µI(x n 1 ;ỹ n 11 ) +I(x n 2 ;ỹ n 21 |ỹ n 22 , ) + I(x n 2 ;ỹ n 22 ) + nǫ n =µh(ỹ n 12 |ỹ n 11 ) − µh(ỹ n 12 |x n 1 ,ỹ n 11 ) + µh(ỹ n 11 ) − µh(ỹ n 11 |x n 1 ) +h(ỹ n 21 |ỹ n 22 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 ) + h(ỹ n 22 ) − h(ỹ n 22 |x n 2 ) + nǫ n = µh(ỹ n 12 |ỹ n 11 ) − µh(ỹ n 11 |x n 1 ) + h(ỹ n 21 |ỹ n 22 ) − h(ỹ n 22 |x n 2 ) + µh(ỹ n 11 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 ) + h(ỹ n 22 ) − µh(ỹ n 12 |x n 1 ,ỹ n 11 ) + nǫ n .(104) Next, we upper bound the terms within each bracket in (104) separately. For the first bracket, we have µh(ỹ n 12 |ỹ n 11 ) − µh(ỹ n 11 |x n 1 ) (a) ≤ µ n i=1 h(ỹ 12 [i]|ỹ 11 [i]) − µn 2 log (2πeN 11 ) (b) ≤ µ n i=1 1 2 log 2πe N 12 + a ′ P 2 [i] + P 1 [i]N 11 P 1 [i] + N 11 − µn 2 log (2πeN 11 ) (c) ≤ µn 2 log 2πe N 12 + 1 n n i=1 a ′ P 2 [i] + 1 n n i=1 P 1 [i]N 11 1 n n i=1 P 1 [i] + N 11 − µn 2 log (2πeN 11 ) ≤ µn 2 log 2πe N 12 + a ′ P 2 + P 1 N 11 P 1 + N 11 − µn 2 log (2πeN 11 ) = µn 2 log N 12 N 11 + a ′ P 2 N 11 + P 1 P 1 + N 11 ,(105) where (a) follows from the chain rule and the fact that removing independent conditions increases differential entropy, (b) follows from the fact that Gaussian distribution optimizes conditional entropy for a given covariance matrix, and (c) follows form Jenson's inequality. Similarly, the terms within the second bracket can be upper bounded as h(ỹ n 21 |ỹ n 22 ) − h(ỹ n 22 |x n 2 ) ≤ n 2 log N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22 .(106) Using Lemma 1 and the fact that N 11 ≤ N 21 /b ′ , the terms within the third bracket can be upper bounded as µh(ỹ n 11 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 )= µ h(x n 1 + z n 11 ) − 1 µ h( √ b ′ x n 1 + z n 21 ) ≤ µnf h P 1 , N 11 , N 21 , b ′ , 1 µ .(107) Since 1 ≤ µ, from (61) we obtain µh(ỹ n 11 ) − h(ỹ n 21 |x n 2 ,ỹ n 22 ) ≤ µn 2 log((2πe)(P 1 + N 11 )) − n 2 log((2πe)(b ′ P 1 + N 21 )).(108) For the last bracket, again we use Lemma 1 to obtain h(ỹ n 22 ) − µh(ỹ n 12 |x n 1 ,ỹ n 11 )= h(x n 2 + z n 22 ) − µh( √ a ′ x n 2 + z n 12 ) ≤ nf h (P 2 , N 22 , N 12 , a ′ , µ). Adding all inequalities, we have µR 1 + R 2 ≤ µ 2 log N 12 N 11 + a ′ P 2 N 11 + P 1 P 1 + N 11 + 1 2 log N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22 + µ 2 log((2πe)(P 1 + N 11 )) − 1 2 log((2πe)(b ′ P 1 + N 21 )) + f h (P 2 , N 22 , N 12 , a ′ , µ),(110) where the fact that ǫ n → 0 as n → ∞ is used to eliminate ǫ n from the right hand side of the inequality. By rearranging the terms, we obtain µR 1 + R 2 ≤µγ P 1 N 11 + P 1 a ′ P 2 + N 12 + γ P 2 N 22 + P 2 b ′ P 1 + N 21 +f h (P 2 , N 22 , N 12 , a ′ , µ) + µ 2 log((2πe)(a ′ P 2 + N 12 )) − 1 2 log((2πe)(P 2 + N 22 )). This completes the proof. A unique feature of the channels within Class B is that for 1 ≤ µ ≤ P2+N12/a ′ P2+N22 and 1 ≤ µ ≤ P1+N21/b ′ P1+N11 , the upper bounds in (102) and (103) become, respectively, µR 1 + R 2 ≤µγ P 1 N 11 + P 1 a ′ P 2 + N 12 + γ P 2 N 22 + P 2 b ′ P 1 + N 21(111) and R 1 + µR 2 ≤γ P 1 N 11 + P 1 a ′ P 2 + N 12 + µγ P 2 N 22 + P 2 b ′ P 1 + N 21 .(112) On the other hand, if the receivers treat the interference as noise, it can be shown that R 1 = γ P 1 N 11 + P 1 a ′ P 2 + N 12(113) and R 2 = γ P 2 N 22 + P 2 b ′ P 1 + N 21(114) are achievable. Comparing upper bounds and achievable rates, we conclude that the upper bounds are indeed tight. In fact, this property is first observed by Etkin et al. in [13]. We summarize this result in the following theorem: Theorem 4: The sum capacity in Class B is attained when transmitters use Gaussian codebooks and receivers treat the interference as noise. In this case, the sum capacity is C ′ sum =γ P 1 N 11 + P 1 a ′ P 2 + N 12 + γ P 2 N 22 + P 2 b ′ P 1 + N 21 .(115) Proof: By substituting µ = 1 in (111), we obtain the desired result. 4) Class C: Class C is designed to upper bound σ C (µ, 1) for the mixed Gaussian IC where 1 ≤ b. Class C is similar to Class A1 (see Figure 4), however we impose different constraints on the parameters of the channels within Class C. These constraints assist us in providing upper bounds by using the fact that at one of the receivers both signals are decodable. For channels in Class C, we use the same model that is given in (73). Therefore, similar to channels in Class A1, this channel is admissible if the corresponding parameters satisfy b ′ g 2 = b, (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1.(116) Next, we change the constraints in (79) as b ′ ≥ N 21 , aN 22 ≤ 1. Through this change of constraints, the second receiver after decoding its own signal will have a less noisy version of the first user's signal, and consequently, it is able to decode the signal of the first user as well as its own signal. Relying on this observation, we have the following lemma. Lemma 6: For a channel in Class C, we have σ C ′ (µ, 1) ≤ µ − 1 2 log (2πe(P 1 + aP 2 + 1)) + 1 2 log 2πe P 2 N 22 P 2 + N 22 + b ′ P 1 + N 21 − 1 2 log(2πeN 21 ) − 1 2 log(2πeN 22 ) + f h (P 2 , N 22 , 1, a, µ − 1). (118) Proof: Since the second user is able to decode both users' messages, we have R 1 ≤ 1 n I(x n 1 ;ỹ n 1 ),(119)R 1 ≤ 1 n I(x n 1 ;ỹ n 21 ,ỹ n 22 |x n 2 ),(120)R 2 ≤ 1 n I(x n 2 ;ỹ n 21 ,ỹ n 22 |x n 1 ),(121)R 1 + R 2 ≤ 1 n I(x n 1 , x n 2 ;ỹ n 21 ,ỹ n 22 ).(122) From aN 22 ≤ 1, we have I(x n 1 ;ỹ n 1 ) ≤ I(x n 1 ;ỹ n 21 |x n 2 ) = I(x n 1 ;ỹ n 21 ,ỹ n 22 |x n 2 ). Hence, (120) is redundant. It can be shown that µR 1 + R 2 ≤ µ − 1 n I(x n 1 ;ỹ n 1 ) + 1 n I(x n 1 , x n 2 ;ỹ n 21 ,ỹ n 22 ).(123) Hence, we have µR 1 + R 2 ≤ µ − 1 n h(ỹ n 1 ) − µ − 1 n h(ỹ n 1 |x n 1 ) + 1 n h(ỹ n 21 ,ỹ n 22 ) − 1 n h(ỹ n 21 ,ỹ n 22 |x n 1 , x n 2 ) = µ − 1 n h(ỹ n 1 ) + 1 n h(ỹ n 21 |ỹ n 22 ) − 1 n h(ỹ n 21 ,ỹ n 22 |x n 1 , x n 2 ) + 1 n h(ỹ n 22 ) − µ − 1 n h(ỹ n 1 |x n 1 )(124) Next, we bound the different terms in (124). For the first term, we have µ − 1 n h(ỹ n 1 ) ≤ µ − 1 2 log (2πe(P 1 + aP 2 + 1)) .(125) The second term can be bounded as 1 n h(ỹ n 21 |ỹ n 22 ) ≤ 1 2 log 2πe P 2 N 22 P 2 + N 22 + b ′ P 1 + N 21 .(126) The third term can be bounded as 1 n h(ỹ n 21 ,ỹ n 22 |x n 1 , x n 2 ) = 1 2 log(2πeN 21 ) + 1 2 log(2πeN 22 ). The last terms can be bounded as 1 n h(ỹ n 22 ) − µ − 1 n h(ỹ n 1 |x n 1 )= 1 n h(x n 2 + z n 22 ) − µ − 1 n h( √ ax n 2 + z 1 ) (128) ≤ f h (P 2 , N 22 , 1, a, µ − 1).(129) Adding all inequalities, we obtain the desired result. IV. WEAK GAUSSIAN INTERFERENCE CHANNEL In this section, we focus on the weak Gaussian IC. We first obtain the sum capacity of this channel for a certain range of parameters. Then, we obtain an outer bound on the capacity region which is tighter than the previously known outer bounds. Finally, we show that time-sharing and concavification result in the same achievable region for Gaussian codebooks. A. Sum Capacity In this subsection, we use the Class B channels to obtain the sum capacity of the weak IC for a certain range of parameters. To this end, let us consider the following minimization problem: W =min γ P 1 N 11 + P 1 a ′ P 2 + N 12 + γ P 2 N 22 + P 2 b ′ P 1 + N 21 (130) subject to: a ′ g 1 = a b ′ g 2 = b b ′ N 11 ≤ N 21 a ′ N 22 ≤ N 12 (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [a ′ , b ′ , g 1 , g 2 , N 11 , N 12 , N 22 , N 21 ]. The objective function in (130) is the sum capacity of Class B channels obtained in Theorem 4. The constraints are the combination of (100) and (101) where applied to confirm the admissibility of the channel and to validate the sum capacity result. Since every channel in the class is admissible, we have C sum ≤ W . Substituting S 1 = g 1 N 12 and S 2 = g 2 N 21 , we have W =min γ (1 − √ g 1 ) 2 P 1 1 − S 1 + g 1 P 1 aP 2 + S 1 + γ (1 − √ g 2 ) 2 P 2 1 − S 2 + g 2 P 2 bP 1 + S 2 (131) subject to: b(1 − S 1 ) (1 − √ g 1 ) 2 ≤ S 2 < 1 a(1 − S 2 ) (1 − √ g 2 ) 2 ≤ S 1 < 1 0 < [g 1 , g 2 ]. By first minimizing with respect to g 1 and g 2 , the optimization problem (131) can be decomposed as W =min W 1 + W 2(132) subject to: 0 < S 1 < 1, 0 < S 2 < 1. where W 1 is defined as W 1 =min g1 γ (1 − √ g 1 ) 2 P 1 1 − S 1 + g 1 P 1 aP 2 + S 1 (133) subject to: b(1 − S 1 ) S 2 ≤ (1 − √ g 1 ) 2 , 0 < g 1 . Similarly, W 2 is defined as W 2 =min g2 γ (1 − √ g 2 ) 2 P 2 1 − S 2 + g 2 P 2 bP 1 + S 2 (134) subject to: a(1 − S 2 ) S 1 ≤ (1 − √ g 2 ) 2 , 0 < g 2 . The optimization problems (133) and (134) are easy to solve. In fact, we have W 1 =      γ P1 1+aP2 if √ b(1 + aP 2 ) ≤ S 2 (1 − S 1 ) γ bP1 S2 + (1− √ b(1−S1)/S2) 2 P1 aP2+S1 Otherwise (135) W 2 =      γ P2 1+bP1 if √ a(1 + bP 1 ) ≤ S 1 (1 − S 2 ) γ aP2 S1 + (1− √ a(1−S2)/S1) 2 P2 bP1+S2 Otherwise(136) From (135) and (136), we observe that for S 1 and S 2 satisfying √ b(1 + aP 2 ) ≤ S 2 (1 − S 1 ) and √ a(1 + bP 1 ) ≤ S 1 (1 − S 2 ), the objective function becomes independent of S 1 and S 2 . In this case, we have W = γ P 1 1 + aP 2 + γ P 2 1 + bP 1 ,(137) which is achievable by treating interference as noise. In the following theorem, we prove that it is possible to find a certain range of parameters such that there exist S 1 and S 2 yielding (137). Theorem 5: The sum capacity of the two-user Gaussian IC is C sum = γ P 1 1 + aP 2 + γ P 2 1 + bP 1 ,(138) for the range of parameters satisfying √ bP 1 + √ aP 2 ≤ 1 − √ a − √ b √ ab .(139) Proof: Let us fix a and b, and define D as D = (P 1 , P 2 )|P 1 ≤ S 1 (1 − S 2 ) b √ a − 1 b , P 2 ≤ S 2 (1 − S 1 ) a √ b − 1 a , 0 < S 1 < 1, 0 < S 2 < 1 .(140) In fact, if D is feasible then there exist 0 < S 1 < 1 and 0 < S 2 < 1 satisfying √ b(1 + aP 2 ) ≤ S 2 (1 − S 1 ) and √ a(1 + bP 1 ) ≤ S 1 (1 − S 2 ) . Therefore, the sum capacity of the channel for all feasible points is attained due to (137). We claim that D = D ′ , where D ′ is defined as D ′ = (P 1 , P 2 )| √ bP 1 + √ aP 2 ≤ 1 − √ a − √ b √ ab .(141) To show D ′ ⊆ D, we set S 1 = 1 − S 2 in (140) to get (P 1 , P 2 )|P 1 ≤ S 1 b √ a − 1 b , P 2 ≤ 1 − S 1 a √ b − 1 a , 0 < S 1 < 1 ⊆ D.(142) It is easy to show that the left hand side of the above equation is another representation of the region D ′ . Hence, we have D ′ ⊆ D. To show D ⊆ D ′ , it suffices to prove that for any (P 1 , P 2 ) ∈ D, √ bP 1 + √ aP 2 ≤ 1− √ a− √ b √ ab holds. To this end, we introduce the following maximization problem: J = max (P1,P2)∈D √ bP 1 + √ aP 2 ,(143) which can be written as J = max (S1,S2)∈(0,1) 2 S 1 (1 − S 2 ) + S 2 (1 − S 1 ) √ ab − 1 √ a − 1 √ b .(144) It is easy to show that the solution to the above optimization problem is J = 1 √ ab − 1 √ a − 1 √ b .(145) Hence, we deduce that D ⊆ D ′ . This completes the proof. Remark 3: The above sum capacity result for the weak Gaussian IC (see also [24]) has been established independently in [25] and [26]. As an example, let us consider the symmetric Gaussian IC. In this case, the constraint in (139) becomes P ≤ 1 − 2 √ a 2a √ a .(146) In Figure 7, the admissible region for P , where treating interference as noise is optimal, versus √ a is plotted. For a fixed P and all 0 ≤ a ≤ 1, the upper bound in (130) and the lower bound when receivers treat the interference as noise are plotted in Figure 8. We observe that up to a certain value of a, the upper bound coincides with the lower bound. a Fig. 7. The shaded area is the region where treating interference as noise is optimal for obtaining the sum capacity of the symmetric Gaussian IC. B. New Outer Bound For the weak Gaussian IC, there are two outer bounds that are tighter than the other known bounds. The first one, due to Kramer [12], is obtained by relying on the fact that the capacity region of the Gaussian IC is inside the capacity regions of the two underlying one-sided Gaussian ICs. Even though the capacity region of the one-sided Gaussian IC is unknown, there exists an outer bound for this channel that can be used instead. Kramers' outer bound is the intersection of two regions E 1 and E 2 . E 1 is the collection of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ (1 − β)P ′ βP ′ + 1/a ,(147)R 2 ≤ γ(βP ′ ),(148) for all β ∈ [0, β max ], where P ′ = P 1 /a + P 2 and β max = P2 P ′ (1+P1) . Similarly, E 2 is the collection of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ(αP ′′ ),(149)R 2 ≤ γ (1 − α)P ′′ αP ′′ + 1/b ,(150) for all α ∈ [0, α max ], where P ′′ = P 1 + P 2 /b and α max = P1 P ′′ (1+P2) . The second outer bound, due to Etkin et al. [13], is obtained by using Genie aided technique to upper bound different linear combinations of rates that appear in the HK achievable region. Their outer bound is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ(P 1 ), (151) R 2 ≤ γ(P 2 ),(152)R 1 + R 2 ≤ γ(P 1 ) + γ P 2 1 + bP 1 ,(153)R 1 + R 2 ≤ γ(P 2 ) + γ P 1 1 + aP 2 ,(154)R 1 + R 2 ≤ γ aP 2 + P 1 1 + bP 1 + γ bP 1 + P 2 1 + aP 2 ,(155) 2R 1 + R 2 ≤ γ(P 1 + aP 2 ) + γ bP 1 + P 2 1 + aP 2 + 0.5 log 1 + P 1 1 + bP 1 ,(156)R 1 + 2R 2 ≤ γ(bP 1 + P 2 ) + γ aP 2 + P 1 1 + bP 1 + 0.5 log 1 + P 2 1 + aP 2 .(157) In the outer bound proposed here, we derive an upper bound on all linear combinations of the rates. Recall that to obtain the boundary points of the capacity region C , it suffices to calculate σ C (µ, 1) and σ C (1, µ) for all 1 ≤ µ. To this end, we make use of channels in A1 and B classes and channels in A2 and B classes to obtain upper bounds on σ C (µ, 1) and σ C (1, µ), respectively. In order to obtain an upper bound on σ C (µ, 1), we introduce two optimization problems as follows. The first optimization problem is written as W 1 (µ) =min µ 1 2 log [2πe(P 1 + aP 2 + 1)] − µ 2 2 log(2πe) + 1 2 log N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22 (158) + µ 2 f h P 1 , 1, N 21 , b ′ , 1 µ 2 + f h (P 2 , N 22 , 1, a, µ 1 ) subject to: µ 1 + µ 2 = µ b ′ g 2 = b b ′ ≤ N 21 aN 22 ≤ 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [µ 1 , µ 2 , b ′ , g 2 , N 22 , N 21 ]. In fact, the objective of the above minimization problem is an upper bound on the support function of a channel within Class A1 which is obtained in Lemma 3. The constraints are the combination of (78) and (79) which are applied to guarantee the admissibility of the channel and to validate the upper bound obtained in Lemma 3. Hence, σ C (µ, 1) ≤ W 1 (µ). By using a new variable S = (1 − √ g 2 ) 2 N 22 , we obtain W 1 (µ) =min µ 1 2 log [2πe(P 1 + aP 2 + 1)] + 1 2 log (1 − √ g 2 ) 2 ( 1 − S + bP 1 g 2 S + P 2 (1 − √ g 2 ) 2 P 2 + S ) (159) + µ 2 f h P 1 , 1, 1 − S g 2 , b g 2 , 1 µ 2 + f h (P 2 , S (1 − √ g 2 ) 2 , 1, a, µ 1 ) − µ 2 2 log(2πe) subject to: µ 1 + µ 2 = µ S ≤ 1 − b S ≤ (1 − √ g 2 ) 2 a 0 ≤ [µ 1 , µ 2 , S, g 2 ]. The second optimization problem is written as W 2 (µ) =min µγ P 1 N 11 + P 1 a ′ P 2 + N 12 + γ P 2 N 22 + P 2 b ′ P 1 + N 21 + f h (P 2 , N 22 , N 12 , a ′ , µ)(160)+ µ 2 log((2πe)(a ′ P 2 + N 12 )) − 1 2 log((2πe)(P 2 + N 22 )) subject to: a ′ g 1 = a b ′ g 2 = b b ′ N 11 ≤ N 21 a ′ N 22 ≤ N 12 (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [a ′ , b ′ , g 1 , g 2 , N 11 , N 12 , N 22 , N 21 ]. For this problem, Class B channels are used. In fact, the objective is the upper bound on the support function of channels within the class obtained in Lemma 5 and the constraints are defined to obtain the closed form formula for the upper bound and to confirm that the channels are admissible. Hence, we deduce σ C (µ, 1) ≤ W 2 (µ). By using new variables S 1 = g 1 N 12 and S 2 = g 2 N 21 , we obtain W 2 (µ) =min µγ (1 − √ g 1 ) 2 P 1 1 − S 1 + g 1 P 1 aP 2 + S 1 + γ (1 − √ g 2 ) 2 P 2 1 − S 2 + g 2 P 2 bP 1 + S 2 (161) + f h P 2 , 1 − S 2 (1 − √ g 2 ) 2 , S 1 g 1 , a g 1 , µ + µ 2 log (2πe)( aP 2 + S 1 g 1 ) − 1 2 log (2πe)(P 2 + 1 − S 2 (1 − √ g 2 ) 2 ) subject to: b(1 − S 1 ) (1 − √ g 1 ) 2 ≤ S 2 < 1 a(1 − S 2 ) (1 − √ g 2 ) 2 ≤ S 1 < 1 0 < [g 1 , g 2 ]. In a similar fashion, one can introduce two other optimization problems, sayW 1 (µ) andW 2 (µ), to obtain upper bounds on σ C (1, µ) by using the upper bounds on the support functions of channels in Class A2 and Class B. Theorem 6 (New Outer Bound): For any rate pair (R 1 , R 2 ) achievable for the two-user weak Gaussian IC, the inequalities µ 1 R 1 + R 2 ≤ W (µ 1 ) = min{W 1 (µ 1 ), W 2 (µ 1 )},(162)R 1 + µ 2 R 2 ≤W (µ 2 ) = min{W 1 (µ 2 ),W 2 (µ 2 )},(163) hold for all 1 ≤ µ 1 , µ 2 . To obtain an upper bound on the sum rate, we can apply the following inequality: Fig. 9. G 0 for the weak Gaussian IC. r 1 , r 2 , r 3 , and r 4 are extreme points of G 0 in the interior of the first quadrant. C sum ≤ min 1≤µ1,µ2 (µ 2 − 1)W (µ 1 ) + (µ 1 − 1)W (µ 2 ) µ 1 µ 2 − 1 . (164) 23 r 3 r 2 r 1 r 4 R 2 R 1 r ′ 1 R 1 + 2R 2 = ψ 5 R 2 = ψ 2 R 1 + R 2 = ψ 3 2R 1 + R 2 = ψ 4 R 1 = ψ 1 r ′ 4 r ′ 5 r ′ 2 r ′ 3 r ′ 6 C. Han-Kobayashi Achievable region In this sub-section, we aim at characterizing G for the weak Gaussian IC. To this end, we first investigate some properties of G 0 (P 1 , P 2 , α, β). First of all, we show that none of the inequalities in describing G 0 is redundant. In Figure 9, all possible extreme points are shown. It is easy to prove that r ′ i / ∈ G 0 for i ∈ {1, 2, . . . , 6}. For instance, we consider r ′ 6 = 2ψ4−ψ5 3 , 2ψ5−ψ4 3 . Since ψ 31 + ψ 32 + ψ 33 = ψ 4 + ψ 5 (see Section II.C), we have ψ 3 = min{ψ 31 , ψ 32 , ψ 33 } ≤ 1 3 (ψ 31 + ψ 32 + ψ 33 ) = 1 3 (ψ 4 + ψ 5 ). However, 1 3 (ψ 4 + ψ 5 ) is the sum of the components of r ′ 6 . Therefore, r ′ 6 violates (7) in the definition of the HK achievable region. Hence, r ′ 6 / ∈ G 0 . As another example, let us consider r ′ 1 = (ψ 1 , ψ 3 − ψ 1 ). We claim that r ′ 1 violates (8). To this end, we need to show that ψ 4 ≤ ψ 3 + ψ 1 . However, it is easy to see that ψ 4 ≤ ψ 31 + ψ 1 , ψ 4 ≤ ψ 32 + ψ 1 , and ψ 4 ≤ ψ 33 + ψ 1 reduce to 0 ≤ (1 − α)(1 − b + β(1 − ab)P 2 ), 0 ≤ (1 − β)(1 − a + (1 − ab)P 1 ), and 0 ≤ (1 − α)(1 − β)aP 2 , respectively. Therefore, r ′ 1 / ∈ G 0 . We conclude that G has four extreme points in the interior of the first quadrant, namely r 1 = (ψ 1 , ψ 4 − 2ψ 1 ),(165)r 2 = (ψ 4 − ψ 3 , 2ψ 3 − ψ 4 ),(166)r 3 = (2ψ 3 − ψ 5 , ψ 5 − ψ 3 ),(167)r 4 = (ψ 5 − 2ψ 2 , ψ 2 ).(168) Most importantly, G 0 possesses the unique minimizer property. To prove this, we need to show thatŷ, the minimizer of the optimization problem σ D0 (c 1 , c 2 , P 1 , P 2 , α, β)= max{c 1 R 1 + c 2 R 2 |AR ≤ Ψ(P 1 , P 2 , α, β)} = min{y t Ψ(P 1 , P 2 , α, β)|A t y = (c 1 , c 2 ) t , y ≥ 0},(169) is independent of the parameters P 1 , P 2 , α, and β and only depends on c 1 and c 2 . We first consider the case (c 1 , c 2 ) = (µ, 1) for all 1 ≤ µ. It can be shown that for 2 < µ, the maximum of (169) is attained at r 1 regardless of P 1 , P 2 , α, and β. Therefore, the dual program has the minimizerŷ = (µ − 2, 0, 0, 1, 0) t which is clearly independent of P 1 , P 2 , α, and β. In this case, we have σ D0 (µ, 1, P 1 , P 2 , α, β) = (µ − 2)ψ 1 + ψ 4 , 2 < µ. For 1 ≤ µ ≤ 2, one can show that r 2 andŷ = (0, 0, 2−µ, µ−1, 0) t are the maximizer and the minimizer of (169), respectively. In this case, we have σ D0 (µ, 1, P 1 , P 2 , α, β) = (2 − µ)ψ 3 + (µ − 1)ψ 4 , 1 ≤ µ ≤ 2. Next, we consider the case (c 1 , c 2 ) = (1, µ) for all 1 ≤ µ. Again, it can be shown that for 2 < µ and 1 ≤ µ ≤ 2, y = (0, µ − 2, 0, 0, 1) t andŷ = (0, 0, 2 − µ, 0, µ − 1) t minimizes (169), respectively. Hence, we have σ D0 (1, µ, P 1 , P 2 , α, β)= (µ − 2)ψ 2 + ψ 5 , if 2 < µ,(172)σ D0 (1, µ, P 1 , P 2 , α, β)= (2 − µ)ψ 3 + (µ − 1)ψ 5 , if 1 ≤ µ ≤ 2. (173) We conclude that the solutions of the dual program are always independent of P 1 , P 2 , α, and β. Hence, G 0 possesses the unique minimizer property. Theorem 7: For the two-user weak Gaussian IC, time-sharing and concavification result in the same region. In other words, G can be fully characterized by using TD/FD and allocating power over three different dimensions. Proof: Since G 0 possesses the unique minimizer property, from Theorem 1, we deduce that G = G 2 . Moreover, using Theorem 3, the number of frequency bands is at most three. To obtain the support function of G 2 , we need to obtain g(c 1 , c 2 , P 1 , P 2 , α, β) defined in (43). Since G 0 possesses the unique minimizer property, (43) can be simplified. Let us consider the case where (c 1 , c 2 ) = (µ, 1) for µ > 2. It can be shown that for this case g = max (α,β)∈[0,1] 2 (µ − 2)ψ 1 (P 1 , P 2 , α, β) + ψ 4 (P 1 , P 2 , α, β). Substituting into (42), we obtain σ G2 (µ, 1, P 1 , P 2 ) =max 3 i=1 λ i [(µ − 2)ψ 1 (P 1i , P 2i , α i , β i ) + ψ 4 (P 1i , P 2i , α i , β i )](175) subject to: 3 i=1 λ i = 1 3 i=1 λ i P 1i ≤ P 1 3 i=1 λ i P 2i ≤ P 2 0 ≤ λ i , 0 ≤ P 1i , 0 ≤ P 2i , ∀i ∈ {1, 2, 3} 0 ≤ α i ≤ 1, 0 ≤ β i ≤ 1, ∀i ∈ {1, 2, 3}. For other ranges of (c 1 , c 2 ), a similar optimization problem can be formed. It is worth noting that even though the number of parameters in characterizing G is reduced, it is still prohibitively difficult to characterize boundary points of G . In Figures (10) and (11), different bounds for the symmetric weak Gaussian IC are plotted. As shown in these figures, the new outer bound is tighter than the previously known bounds. V. ONE-SIDED GAUSSIAN INTERFERENCE CHANNELS Throughout this section, we consider the one-sided Gaussian IC obtained by setting b = 0, i.e, the second receiver incurs no interference from the first transmitter. One can further split the class of one-sided ICs into two subclasses: the strong one-sided IC and the weak one-sided IC. For the former, a ≥ 1 and the capacity region is fully characterized [16]. In this case, the capacity region is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ(P 1 ), R 2 ≤ γ(P 2 ), R 1 + R 2 ≤ γ(P 1 + aP 2 ). For the latter, a < 1 and the full characterization of the capacity region is still an open problem. Therefore, we always assume a < 1. Three important results are proved for this channel. The first one, proved by Costa in [11], states that the capacity region of the weak one-sided IC is equivalent to that of the degraded IC with an appropriate change of parameters. The second one, proved by Sato in [10], states that the capacity region of the degraded Gaussian IC is outer bounded by the capacity region of a certain degraded broadcast channel. The third one, proved by Sason in [16], characterizes the sum capacity by combining Costa's and Sato's results. In this section, we provide an alternative proof for the outer bound obtained by Sato. We then characterize the full HK achievable region where Gaussian codebooks are used, i.e., G . A. Sum Capacity For the sake of completeness, we first state the sum capacity result obtained by Sason. Theorem 8 (Sason): The rate pair γ P1 1+aP2 , γ(P 2 ) is an extreme point of the capacity region of the one-sided Gaussian IC. Moreover, the sum capacity of the channel is attained at this point. B. Outer Bound In [10], Sato derived an outer bound on the capacity of the degraded IC. This outer bound can be used for the weak one-sided IC as well. This is due to Costa's result which states that the capacity region of the degraded Gaussian IC is equivalent to that of the weak one-sided IC with an appropriate change of parameters. Theorem 9 (Sato): If the rate pair (R 1 , R 2 ) belongs to the capacity region of the weak one-sided IC, then it satisfies R 1 ≤ γ (1−β)P 1/a+βP , R 2 ≤ γ(βP ),(176) for all β ∈ [0, 1] where P = P 1 /a + P 2 . Proof: Since the sum capacity is attained at the point where User 2 transmits at its maximum rate R 2 = γ(P 2 ), other boundary points of the capacity region can be obtained by characterizing the solutions of σ C (µ, 1) = max {µR 1 + R 2 |(R 1 , R 2 ) ∈ C } for all 1 ≤ µ. Using Fano's inequality, we have n(µR 1 + R 2 ) ≤µI(x n 1 ; y n 1 ) + I(x n 2 ; y n 2 ) + nǫ n =µh(y n 1 ) − µh(y n 1 |x n 1 ) + h(y n 2 ) − h(y n 2 |x n 2 ) + nǫ n =[µh(x n 1 + √ ax n 2 + z n 1 ) − h(z n 2 )] + [h(x n 2 + z n 2 ) − µh( √ ax n 2 + z n 1 )] + nǫ n (a) ≤ µn 2 log [2πe(P 1 + aP 2 + 1)] − n 2 log(2πe) + [h(x n 2 + z n 2 ) − µh( √ ax n 2 + z n 1 )] + nǫ n (b) ≤ µn 2 log [2πe(P 1 + aP 2 + 1)] − n 2 log(2πe) + nf h (P 2 , 1, 1, a, µ) + nǫ n , where (a) follows from the fact that Gaussian distribution maximizes the differential entropy for a given constraint on the covariance matrix and (b) follows from the definition of f h in (59). Depending on the value of µ, we consider the following two cases: 1-For 1 ≤ µ ≤ P2+1/a P2+1 , we have µR 1 + R 2 ≤ µγ P 1 1 + aP 2 + γ(P 2 ).(177) In fact, the point γ P1 1+aP2 , γ(P 2 ) which is achievable by treating interference as noise at Receiver 1, satisfies (177) with equality. Therefore, it belongs to the capacity region. Moreover, by setting µ = 1, we deduce that this point corresponds to the sum capacity of the one-sided Gaussian IC. This is in fact an alternative proof for Sason's result. 2-For P2+1/a P2+1 < µ ≤ 1 a , we have µR 1 + R 2 ≤ µ 2 log (P 1 + aP 2 + 1) + 1 2 log 1/a − 1 µ − 1 − µ 2 log aµ(1/a − 1) µ − 1 .(178) Equivalently, we have µR 1 + R 2 ≤ µ 2 log (aP + 1)(µ − 1) µ(1 − a) + 1 2 log 1/a − 1 µ − 1 ,(179) where P = P 1 /a + P 2 . Let us define E 1 as the set of all rate pairs (R 1 , R 2 ) satisfying (179), i.e. E 1 = (R 1 , R 2 )|µR 1 + R 2 ≤ µ 2 log (aP + 1)(µ − 1) µ(1 − a) + 1 2 log 1/a − 1 µ − 1 , ∀ P 2 + 1/a P 2 + 1 < µ ≤ 1 a .(180) We claim that E 1 is the dual representation of the region defined in the statement of the theorem, see (4). To this end, we define E 2 as E 2 = (R 1 , R 2 )|R 1 ≤ γ (1 − β)P 1/a + βP , R 2 ≤ γ(βP ), ∀β ∈ [0, 1](181) We evaluate the support function of E 2 as σ E2 (µ, 1) = max {µR 1 + R 2 |(R 1 , R 2 ) ∈ E 2 } .(182) It is easy to show that β = 1/a−1 P (µ−1) maximizes the above optimization problem. Therefore, we have σ E2 (µ, 1) = µ 2 log (aP + 1)(µ − 1) µ(1 − a) + 1 2 log 1/a − 1 µ − 1 .(183) Since E 2 is a closed convex set, we can use (4) to obtain its dual representation which is indeed equivalent to (180). This completes the proof. C. Han-Kobayashi Achievable Region In this subsection, we characterize G 0 , G 1 , G 2 , and G for the weak one-sided Gaussian IC. G 0 can be characterized as follows. Since there is no link between Transmitter 1 and Receiver 2, User 1's message in the HK achievable region is only the private message, i.e., α = 1. In this case, we have ψ 1 = γ P 1 1 + aβP 2 ,(184)ψ 2 = γ(P 2 ),(185)ψ 31 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ),(186)ψ 32 = γ P 1 1 + aβP 2 + γ(P 2 ),(187)ψ 33 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ),(188)ψ 4 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ P 1 1 + aβP 2 + γ(βP 2 ),(189)ψ 5 = γ(βP 2 ) + γ(P 2 ) + γ P 1 + a(1 − β)P 2 1 + aβP 2 ,(190) It is easy to show that ψ 3 = min{ψ 31 , ψ 32 , ψ 33 } = ψ 31 , ψ 31 + ψ 1 = ψ 4 , ψ 31 + ψ 2 = ψ 5 . Hence, G 0 can be represented as all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aβP 2 ,(191)R 2 ≤ γ(P 2 ),(192)R 1 + R 2 ≤ γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ).(193) We claim that G 2 = G . To prove this, we need to show that G 0 possesses the unique minimizer property. G 0 is a pentagon with two extreme points in the interior of the first quadrant, namely r 1 and r 2 where r 1 = γ P 1 1 + aβP 2 , γ (1 − β)aP 2 1 + P 1 + βaP 2 + γ(βP 2 ) ,(194)r 2 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ) − γ(P 2 ), γ(P 2 ) .(195) Using above, it can be verified that G 0 possesses the unique minimizer property. Next, we can use the optimization problem in (42) to obtain the support function of G . However, we only need to consider (c 1 , c 2 ) = (µ, 1) for µ > 1. Therefore, we have g(µ, 1, P 1 , P 2 , β) = max 0≤β≤1 µγ P 1 1 + βaP 2 + γ(βP 2 ) + γ (1 − β)aP 2 1 + P 1 + βaP 2 .(196) Substituting into (42), we conclude that boundary points of G can be characterized by solving the following optimization problem: W =max 3 i=1 λ i µγ P 1i 1 + β i aP 2i + γ(β i P 2i ) + γ (1 − β i )aP 2i 1 + P 1i + β i aP 2i(197) subject to: 3 i=1 λ i = 1 3 i=1 λ i P 1i ≤ P 1 3 i=1 λ i P 2i ≤ P 2 0 ≤ β i ≤ 1, ∀i ∈ {1, 2, 3} 0 ≤ [P 1i , P 2i , λ i ], ∀i ∈ {1, 2, 3}. For the sake of completeness, we provide a simple description for G 1 in the next lemma. Lemma 7: The region G 1 can be represented as the collection of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aβ ′ P 2 ,(198)R 2 ≤ γ(β ′ P 2 ) + γ a(1 − β ′ )P 2 1 + P 1 + aβ ′ P 2 ,(199) for all β ′ ∈ [0, 1]. Moreover, G 1 is convex and any point that lies on its boundary can be achieved by using superposition coding and successive decoding. Proof: Let E denote the set defined in the above lemma. It is easy to show that E is convex and E ⊆ G 1 . To prove the inverse inclusion, it suffices to show that the extreme points of G 0 , r 1 and r 2 (see (194) and (195)) are inside E for all β ∈ [0, 1]. By setting β ′ = β, we see that r 1 ∈ E. To prove r 2 ∈ E, we set β ′ = 1. We conclude that r 2 ∈ E if the following inequality holds γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ) − γ(P 2 ) ≤ γ P 1 1 + aP 2 ,(200) for all β ∈ [0, 1]. However, (200) reduces to 0 ≤ (1 − a)(1 − β)P 2 which holds for all β ∈ [0, 1]. Hence, G 1 ⊆ E. Using these facts, it is straightforward to show that the boundary points G 1 are achievable by using superposition coding and successive decoding. Figure 12 compares different bounds for the one-sided Gaussian IC. VI. MIXED GAUSSIAN INTERFERENCE CHANNELS In this section, we focus on the mixed Gaussian Interference channel. We first characterize the sum capacity of this channel. Then, we provide an outer bound on the capacity region. Finally, we investigate the HK achievable region. Without loss of generality, we assume a < 1 and b ≥ 1. 29 A. Sum Capacity Theorem 10: The sum capacity of the mixed Gaussian IC with a < 1 and b ≥ 1 can be stated as C sum = γ (P 2 ) + min γ P 1 1 + aP 2 , γ bP 1 1 + P 2 . (201) Proof: We need to prove the achievability and converse for the theorem. Achievability part: Transmitter 1 sends a common message to both receivers, while the first user's signal is considered as noise at both receivers. In this case, the rate R 1 = min γ P 1 1 + aP 2 , γ bP 1 1 + P 2(202) is achievable. At Receiver 2, the signal from Transmitter 1 can be decoded and removed. Therefore, User 2 is left with a channel without interference and it can communicate at its maximum rate which is R 2 = γ(P 2 ).(203) By adding (202) and (203), we obtain the desired result. Converse part: The sum capacity of the Gaussian IC is upper bounded by that of the two underlying one-sided Gaussian ICs. Hence, we can obtain two upper bounds on the sum rate. We first remove the interfering link between Transmitter 1 and Receiver 2. In this case, we have a one-sided Gaussian IC with weak interference. The sum capacity of this channel is known [16]. Hence, we have C sum ≤ γ(P 2 ) + γ P 1 1 + aP 2 .(204) By removing the interfering link between Transmitter 2 and Receiver 1, we obtain a one-sided Gaussian IC with strong interference. The sum capacity of this channel is known. Hence, we have C sum ≤ γ (bP 1 + P 2 ) ,(205) which equivalently can be written as C sum ≤ γ(P 2 ) + γ bP 1 1 + P 2 .(206) By taking the minimum of the right hand sides of (204) and (206), we obtain C sum ≤ γ (P 2 ) + min γ P 1 1 + aP 2 , γ bP 1 1 + P 2 . This completes the proof. Remark 4: In an independent work [25], the sum capacity of the mixed Gaussian IC is obtained for a certain range of parameters, whereas in the above theorem, we characterize the sum capacity of this channel for the entire range of its parameters (see also [24]). By comparing γ P1 1+aP2 with γ bP1 1+P2 , we observe that if 1 + P 2 ≤ b + abP 2 , then the sum capacity corresponds to the sum capacity of the one-sided weak Gaussian IC, whereas if 1 + P 2 > b + abP 2 , then the sum capacity corresponds to the sum capacity of the one-sided strong IC. Similar to the one-sided Gaussian IC, since the sum capacity is attained at the point where User 2 transmits at its maximum rate R 2 = γ(P 2 ), other boundary points of the capacity region can be obtained by characterizing the solutions of σ C (µ, 1) = max {µR 1 + R 2 |(R 1 , R 2 ) ∈ C } for all 1 ≤ µ. B. New Outer Bound The best outer bound to date, due to Etkin et al. [13], is obtained by using the Genie aided technique. This bound is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ(P 1 ),(208)R 2 ≤ γ(P 2 ),(209)R 1 + R 2 ≤ γ(P 2 ) + γ P 1 1 + aP 2 ,(210)R 1 + R 2 ≤ γ(P 2 + bP 1 ),(211) 2R 1 + R 2 ≤ γ(P 1 + aP 2 ) + γ bP 1 + P 2 1 + aP 2 + γ P 1 1 + bP 1 . 30 The capacity region of the mixed Gaussian IC is inside the intersection of the capacity regions of the two underlying onesided Gaussian ICs. Removing the link between Transmitter 1 and Receiver 2 results in a weak one-sided Gaussian IC whose outer bound E 1 is the collection of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ (1 − β)P ′ βP ′ + 1/a ,(213)R 2 ≤ γ(βP ′ ),(214) for all β ∈ [0, β max ], where P ′ = P 1 /a + P 2 and β max = P2 P ′ (1+P1) . On the other hand, removing the link between Transmitter 2 and Receiver 1 results in a strong one-sided Gaussian IC whose capacity region E 2 is fully characterized as the collection of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ(bP 1 ),(215)R 2 ≤ γ (P 2 ) ,(216)R 1 + R 2 ≤ γ(bP 1 + P 2 ).(217) Using the channels in Class C, we upper bound σ C (µ, 1) based on the following optimization problem: W (µ) =min µ − 1 2 log (2πe(P 1 + aP 2 + 1)) + 1 2 log 2πe P 2 N 22 P 2 + N 22 + b ′ P 1 + N 21 (218) − 1 2 log(2πeN 21 ) − 1 2 log(2πeN 22 ) + f h (P 2 , N 22 , 1, a, µ − 1) subject to: b ′ g 2 = b b ′ ≥ N 21 aN 22 ≤ 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [b ′ , g 2 , N 22 , N 21 ]. By substituting S = g 2 N 21 , we obtain W (µ) =min µ − 1 2 log (2πe(P 1 + aP 2 + 1)) + 1 2 log 2πe P 2 (1 − S) (1 − √ g 2 ) 2 P 2 + 1 − S + bP 1 + S g 2 (219) − 1 2 log 2πeS g 2 − 1 2 log 2πe(1 − S) (1 − √ g 2 ) 2 + f h P 2 , 1 − S (1 − √ g 2 ) 2 , 1, a, µ − 1 subject to: S < 1 a(1 − S) ≤ (1 − √ g 2 ) 2 0 ≤ [S, g 2 ]. Hence, we have the following theorem that provides an outer bound on the capacity region of the mixed Gaussian IC. Theorem 11: For any rate pair (R 1 , R 2 ) achievable for the two-user mixed Gaussian IC, (R 1 , R 2 ) ∈ E 1 E 2 . Moreover, the inequality µR 1 + R 2 ≤ W (µ)(220) holds for all 1 ≤ µ. C. Han-Kobayashi Achievable Region In this subsection, we study the HK achievable region for the mixed Gaussian IC. Since Receiver 2 can always decode the message of the first user, User 1 associates all its power to the common message. User 2, on the other hand, allocates βP 2 and (1 − β)P 2 of its total power to its private and common messages, respectively, where β ∈ [0, 1]. Therefore, we have ψ 1 = γ P 1 1 + aβP 2 ,(221)ψ 2 = γ(P 2 ),(222) ψ 31 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ), ψ 32 = γ(P 2 + bP 1 ), ψ 33 = γ a(1 − β)P 2 1 + aβP 2 + γ(βP 2 + bP 1 ), ψ 4 = γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 + bP 1 ),(225) ψ 5 = γ(βP 2 ) + γ(P 2 + bP 1 ) + γ a(1 − β)P 2 1 + aβP 2 . Due to the fact that the sum capacity is attained at the point where the second user transmits at its maximum rate, the last inequality in the description of the HK achievable region can be removed. Although the point r ′ 5 = (ψ 3 − γ(P 2 ), γ(P 1 )) in Figure 9 may not be in G 0 , this point is always achievable due to the sum capacity result. Hence, we can enlarge G 0 by removing r 3 and r 4 . Let us denote the resulting region as G ′ 0 . Moreover, one can show that r ′ 2 , r ′ 3 , r ′ 4 , and r ′ 6 are still outside G ′ 0 . However, for the mixed Gaussian IC, it is possible that r ′ 1 belongs to G ′ 0 . In Figure 13, two alternative cases for the region G ′ 0 along with the new labeling of its extreme points are plotted. The new extreme points can be written as r 1 = (ψ 1 , ψ 4 − 2ψ 1 ), r 2 = (ψ 1 , ψ 3 − ψ 1 ), r 3 = (ψ 4 − ψ 3 , 2ψ 3 − ψ 4 ), r 4 = (ψ 3 − ψ 2 , ψ 2 ). In fact, we have either G ′ 0 = conv{r 1 , r 3 , r 4 } or G ′ 0 = conv{r 2 , r 4 }. To simplify the characterization of G 1 , we consider three cases: Case I: 1 + P 2 ≤ b + abP 2 . Case II: 1 + P 2 > b + abP 2 and 1 − a ≤ abP 1 . Case III: 1 + P 2 > b + abP 2 and 1 − a > abP 1 . Case I (1 + P 2 ≤ b + abP 2 ): In this case, ψ 3 = ψ 31 . Moreover, it is easy to verify that ψ 31 + ψ 1 ≤ ψ 4 which means (8) is redundant for the entire range of parameters. Hence, G ′ 0 = conv{r 2 , r 4 } consists of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aβP 2 ,(228)R 2 ≤ γ (P 2 ) ,(229) R 1 + R 2 ≤ γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ), where β ∈ [0, 1]. Using a reasoning similar to the one used to express boundary points of G 1 for the one-sided Gaussian IC, we can express boundary points of G 1 as R 1 ≤ γ P 1 1 + aβP 2 ,(231) R 2 ≤ γ(βP 2 ) + γ a(1 − β)P 2 1 + P 1 + aβP 2 , for all β ∈ [0, 1]. Theorem 12: For the mixed Gaussian IC satisfying 1 ≤ ab, region G is equivalent to that of the one sided Gaussian IC obtained from removing the interfering link between Transmitter 1 and Receiver 2. Proof: If 1 ≤ ab, then 1 + P 2 ≤ b + abP 2 holds for all P 1 and P 2 . Hence, G ′ 0 (P 1 , P 2 , β) is a pentagon defined by (228), (229), and (229). Comparing with the corresponding region for the one-sided Gaussian IC, we see that G ′ 0 is equivalent to G 0 obtained for the one-sided Gaussian IC. This directly implies that G is the same for both channels. Case II (1 + P 2 > b + abP 2 and 1 − a ≤ abP 1 ): In this case, ψ 3 = min{ψ 31 , ψ 32 }. It can be shown that G 1 is the union of three regions E 1 , E 2 , and E 3 , i.e, G 0 = E 1 E 2 E 3 . Region E 1 is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aβP 2 ,(233)R 2 ≤ γ(βP 2 ) + γ a(1 − β)P 2 1 + P 1 + aβP 2 .(234) for all β ∈ [0, b−1 (1−ab)P2 ]. Region E 2 is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ bP 1 1 + βP 2 ,(235)R 2 ≤ γ P 1 + a(1 − β)P 2 1 + aβP 2 + γ(βP 2 ) − γ bP 1 1 + βP 2 .(236)R 2 ≤ γ (P 2 ) ,(237) R 1 + R 2 ≤ γ(bP 1 + P 2 ). (239) Case III (1 + P 2 > b + abP 2 and 1 − a > abP 1 ): In this case, ψ 3 = min{ψ 31 , ψ 32 }. Similar to Case II, we have G 1 = E 1 E 2 E 3 , where regions E 1 , E 2 , and E 3 are defined as follows. Region E 1 is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aβP 2 ,(240) R 2 ≤ γ(βP 2 ) + γ a(1 − β)P 2 1 + P 1 + aβP 2 . for all β ∈ [0, b−1 (1−ab)P2 ]. Region E 2 is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aβP 2 ,(242) R 2 ≤ γ a(1 − β)P 2 1 + P 1 + aβP 2 + γ(βP 2 + bP 1 ) − γ P 1 1 + aβP 2 . for all β ∈ [ b−1 (1−ab)P2 , 1]. Region E 3 is the union of all rate pairs (R 1 , R 2 ) satisfying R 1 ≤ γ P 1 1 + aP 2 ,(244)R 2 ≤ γ (P 2 ) ,(245)R 1 + R 2 ≤ γ(bP 1 + P 2 ).(246) Remark 5: Region E 3 in Case II and Case III represents a facet that belongs to the capacity region of the mixed Gaussian IC. It is important to note that, surprisingly, this facet is obtainable when the second transmitter uses both the common message and the private message. Different bounds are compared for the mixed Gaussian IC for Cases I, II, and III in Figures 14, 15, and 16, respectively. VII. CONCLUSION We have studied the capacity region of the two-user Gaussian IC. The sum capacities, inner bounds, and outer bounds have been considered for three classes of channels: weak, one-sided, and mixed Gaussian IC. We have used admissible channels as the main tool for deriving outer bounds on the capacity regions. For the weak Gaussian IC, we have derived the sum capacity for a certain range of channel parameters. In this range, the sum capacity is attained when Gaussian codebooks are used and interference is treated as noise. Moreover, we have derived a new outer bound on the capacity region. This outer bound is tighter than the Kramer's bound and the ETW's bound. Regarding inner bounds, we have reduced the computational complexity of the HK achievable region. In fact, we have shown that when Gaussian codebooks are used, the full HK achievable region can be obtained by using the naive HK achievable scheme over three frequency bands. For the one-sided Gaussian IC, we have presented an alternative proof for the Sato's outer bound. We have also derived the full HK achievable region when Gaussian codebooks are used. For the mixed Gaussian IC, we have derived the sum capacity for the entire range of its parameters. Moreover, we have presented a new outer bound on the capacity region that outperforms ETW's bound. We have proved that the full HK achievable region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel gains. Fig. 16. Comparison between different bounds for the mixed Gaussian IC when 1 + P 2 > b + abP 2 and 1 − a > abP 1 (Case III) for P 1 = 7, P 2 = 700, a = 0.01, and b = 1.5. We have also derived a facet that belongs to the capacity region for a certain range of parameters. Surprisingly, this facet is obtainable when one of the transmitters uses both the common message and the private message. Fig. 1 . 1Classes of the two-user ICs. Theorem 1 : 1If D 0 possesses the unique minimizer property, then D = D 2 . Proof: Since D 2 ⊆ D always holds, we need to show D ⊆ D 2 which can be equivalently verified by showing σ D ≤ σ D2 . The support function of D can be written as σ D (c, P) = max c t R|R ∈ D . Fig. 5 . 5Class A2 admissible channels. Fig. 6 . 6Class B admissible channels. Fig. 8 . 8The upper bound obtained by solving (130). The lower bound is obtained by treating the interference as noise. Fig. 10 . 10Comparison between different bounds for the symmetric weak Gaussian IC when P = 7 and a = 0.2. Fig. 11 . 11Comparison between different bounds for the symmetric weak Gaussian IC when P = 100 and a = 0.1. 1 Fig. 12 . 112Comparison between different bounds for the one-sided Gaussian IC when P 1 = 1, P 2 = 7, and a = 0.4. Fig. 13 . 13The new region G ′ 0 which is obtained by enlarging G 0 . for all β ∈ [ b−1 (1−ab)P2 , (b−1)P1+(1−a)P2 (1−ab)P1P2+(1−a)P2 ].Region E 3 is the union of all rate pairs (R 1 , R 2 ) satisfying Fig. 14 . 14Comparison between different bounds for the mixed Gaussian IC when 1 + P 2 ≤ b + abP 2 (Case I) for P 1 = 7, P 2 = 7, a = 0.6, and b = 2. Fig. 15 . 15Comparison between different bounds for the mixed Gaussian IC when 1 + P 2 > b + abP 2 and 1 − a ≤ abP 1 (Case II) for P 1 = 7, P 2 = 7, a = 0.4, and b = 1.5. In the HK scheme, two independent messages are encoded at each transmitter, namely the common message and the private message. α and β are the parameters that determine the amount of power allocated to the common and private messages for the two users, i.e., αP 1 , βP 2 and (1 − α)P 1 , (1 − β)P 2 of the total power is used for the transmission of the private/common messages to the first/second users, respectively. Two-way communication channels. 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[ "Selforganisation and sympathetic cooling of multispecies ensembles in a cavity", "Selforganisation and sympathetic cooling of multispecies ensembles in a cavity" ]
[ "Tobias Grießer \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria\n", "Wolfgang Niedenzu \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria\n", "Helmut Ritsch \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria\n" ]
[ "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria" ]
[]
We predict concurrent selforganisation and cooling of multispecies ensembles of laser-illuminated polarisable particles within a high-Q cavity mode. Resonant collective scattering of laser light into the cavity creates optical potentials which above a threshold pump power transforms a homogeneous particle distribution to a crystalline order for all constituents. Adding extra particles of any mass and temperature always lowers the pump power required for selfordering and allows to concurrently trap atoms, for which high phase-space densities are readily available, in combination with many other kind of atoms, molecules or even polarisable nanoparticles. Collective scattering leads to energy exchange between the different species without direct collisional interactions. We analytically calculate the threshold condition, energy fluxes and the resulting equilibrium phase-space distributions and show that cavity-mediated energy transfer enhances cooling of heavy particles by adding light particles forming a cold reservoir. Extensive numerical many-body simulations support the results of our kinetic analytic model.
10.1088/1367-2630/14/5/053031
[ "https://arxiv.org/pdf/1106.2340v1.pdf" ]
119,011,302
1106.2340
8a8edf92f65e31639dd67fe7c4ba5460e1d7a80b
Selforganisation and sympathetic cooling of multispecies ensembles in a cavity 12 Jun 2011 Tobias Grießer Institut für Theoretische Physik Universität Innsbruck Technikerstraße 256020InnsbruckAustria Wolfgang Niedenzu Institut für Theoretische Physik Universität Innsbruck Technikerstraße 256020InnsbruckAustria Helmut Ritsch Institut für Theoretische Physik Universität Innsbruck Technikerstraße 256020InnsbruckAustria Selforganisation and sympathetic cooling of multispecies ensembles in a cavity 12 Jun 2011(Dated: June 14, 2011) We predict concurrent selforganisation and cooling of multispecies ensembles of laser-illuminated polarisable particles within a high-Q cavity mode. Resonant collective scattering of laser light into the cavity creates optical potentials which above a threshold pump power transforms a homogeneous particle distribution to a crystalline order for all constituents. Adding extra particles of any mass and temperature always lowers the pump power required for selfordering and allows to concurrently trap atoms, for which high phase-space densities are readily available, in combination with many other kind of atoms, molecules or even polarisable nanoparticles. Collective scattering leads to energy exchange between the different species without direct collisional interactions. We analytically calculate the threshold condition, energy fluxes and the resulting equilibrium phase-space distributions and show that cavity-mediated energy transfer enhances cooling of heavy particles by adding light particles forming a cold reservoir. Extensive numerical many-body simulations support the results of our kinetic analytic model. I. INTRODUCTION Laser-induced light forces are routinely used to trap and manipulate a large class of polarisable particles from atoms, molecules up to larger objects as microspheres, nanoparticles or even protozoae [1]. Laser cooling, however, has so far been limited to a finite class of atomic species [2], very few kinds of molecules [3] or individual vibration modes of nanomechanical objects [4]. Successful cooling requires specific setups with well-chosen laser frequencies and field configurations so that the number of laser-cooled species has only slowly grown in the past years [5]. In principle selforganisation and cooling by coherent light scattering in cavities provides a general alternative to trap and cool any kind of polarisable particles, which can be injected into an optical resonator [6][7][8]. In practise, however, the required particle phase-space densities and laser intensities to reach a useful regime so far have been achieved only for atomic ensembles [9][10][11], where the theoretical predictions were fully confirmed and showed very fast cooling to sub-Doppler temperatures [12]. However, as the required phase-space densities are hard to achieve for molecules [13], we propose here to generalise the scheme by introducing ensembles of different species and temperatures simultaneously into a single optical resonator. We show that under quite general conditions all species are simultaneously trapped and cooled using only a single laser frequency without the need of direct interparticle interaction. As our central claim we predict that the simultaneous presence of any other species will always increase the total scattering rate for each particle and thus improve the total performance of cooling and trapping for each individual species. * Electronic address: [email protected] As the most interesting case we study a mixture of a precooled and dense enough atomic ensemble with a hotter and much smaller ensemble of molecules or nanospheres. While it would be impossible to reach the selforganisation threshold for the latter alone, combined trapping and sympathetic cooling can be readily achieved without the need of collisions. In fact the different particles could be trapped at different locations within the cavity. Overall this provides for a general route to cool new particle species and also allows a simple general setup for simultaneous multispecies cooling and trapping without the need of a tailored laser for each species. As for a single species, multispecies cooling can be significantly improved using several cavity modes [7,14]. Similarly, a combination of traditional laser cooling methods for atoms with sympathetic cavity cooling could be envisaged to enhance the performance of the combined system. The paper is organised as follows: After introducing the basic setup and model equations, we calculate the required threshold intensity and joint phase-space density to reach selforganisation and enter the superradiant scattering regime. Then we derive approximate expressions for the cooling dynamics and the energy flow between the different ensembles as well as the asymptotic equilibrium phase-space distributions. These results are finally checked by numerical simulations of the selforganisation and cooling dynamics using a particle model. II. MODEL Let us consider a mixed dilute gas of S species of N s (s = 1, . . . , S) polarisable particles each inside a high-Q optical resonator. They are illuminated by a standing wave of counterpropagating off-resonant laser beams that transversely cross the cavity and are close to resonance with a single mode as sketched in fig. 1. Particles within the overlap region of pump laser and cavity mode reso- nantly scatter light from the one into the other. Pump laser and cavity light give rise to a dynamical optical potential for each species, modifying the particles' distributions. For simplicity we approximate the pump field in the interaction region by a plane standing wave and consider particle motion along the cavity axis only. In single species calculations this approximation proved sufficient to explain the essential physics of selforganisation and cooling [9,15,16]. An almost ideal implementation of this model could be realised by confining the multiple ensembles into one-dimensional tubes created by two crossed retro-reflected pump laser beams [17]. Extensions of the theory presented in this work to 3d-motion and spatially dependent mode functions are straight-forward and lead to only minor quantitative changes in the appropriate limits. In terms of the effective laser pump amplitude η s , the light shift per photon U 0,s and the cavity field amplitude α, the combined optical potential for the particles along the cavity axis is given by [15] Φ s = η s (α + α * ) sin(kx) + U 0,s |α| 2 sin 2 (kx). (1) Combining the one-particle position and momentum variables z = (x, p) and introducing the one-particle Hamiltonian function H s (z, α, α * ) = p 2 2m s + Φ s (x, α, α * ),(2) the semi-classical model equations [18] in Klimontovich's formulation [19] then read (s = 1, . . . , S) ∂f K,s ∂t + ∂H s ∂p ∂f K,s ∂x − ∂H s ∂x ∂f K,s ∂p = 0. (3a)α = (−κ + i∆ c )α − i S s=1 N s ∂H s ∂α * f K,s d 2 z − √ κ ξ. (3b) Here, f K,s (z, t) is the so-called Klimontovich distribution satisfying f K,s (z, 0) = N −1 s Ns js=1 δ(z −z js ) with {z js } a set of initial phase points, ξ denotes white noise modelling the fields' quantum fluctuations with ξ(t)ξ * (t ′ ) = δ(t − t ′ ), ξ(t)ξ(t ′ ) = 0. κ > 0 designates the cavity decay rate and ∆ c = ω p − ω c is the mismatch between the pump frequency ω p and the cavity resonance frequency ω c . The equations (3) are equivalent to a set of stochastic differential equations (SDEs) for the particles' positions and momenta z js (t) and the mode amplitude α(t). III. SELFORGANISATION THRESHOLD Let us decompose the distributions according to f K,s (z, t) = f s (z, t) + δf s (z, t) with f s (z, t) = f K,s (z, t) denoting the average over a statistical ensemble of similar initial conditions {z js }, α(0) and the realisations of the noise process ξ. The ensemble-averaged Klimontovich distributions, called one-particle distribution functions, fulfil ∂f s ∂t + p m s ∂f s ∂x − ∂ Φ s ∂x ∂f s ∂p = ∂δΦ s ∂x ∂δf s ∂p .(4) These equations are exact but not particularly useful as such because they do not form a closed set. However, for N s → ∞, statistical correlations become negligible during the initial stage of the time evolution and hence the one-particle distributions satisfy Vlasov's equation [20] ∂f s ∂t + p m s ∂f s ∂x − ∂Φ s ∂x ∂f s ∂p = 0,(5a) where Φ s = Φ s (x, α , α * ). In the rest of this work we shall for convenience omit the ensemble-average brackets. These equations together with the average form of eq. (3b) for the ensemble-averaged mode amplitudė α = (−κ + i∆ c )α− −i s N s U 0,s α sin 2 (kx) + η s sin(kx) f s (x, p) dxdp. (5b) represent the essence of the Vlasov kinetic theory of polarisable particles in a resonator describing the initial evolution purely due to the mean field interaction [21]. Note that spatially homogeneous initial distributions f s (x, p, 0) = f 0,s (p) with zero cavity field α(0) = 0 are equilibrium states of (5). In any finite ensemble, however, density fluctuations cause light scattering and the particles experience friction and diffusion. This is mathematically described by the correlation term in (4), which induces a slow "collisional" evolution of the Vlasov and after selforganisation (solid) as determined from numerical simulations of the particle system [22]. The circles show the predictions of eq. (9). (b) shows the evolution of the individual order parameters θs = | fs sin(kx)dxdp| and the corresponding final values given by the adiabatic theory. Parameters: N1 = 10 4 , N2 = 500, m2 = 10m1, kBT1 = 10 4 κ, kBT2 = 2.5 × 10 5 κ, η1 = 2.4κ, η2 = 27.4κ and ωR,1 = 10 −2 κ. equilibria (5) towards a new equilibrium. As a central question we now determine the stability of uniform equilibria, i.e. whether small perturbations are damped or amplified in the course of time. Linearising equations (5) around a spatially homogeneous steady state and solving the resulting equations according to Landau [23], one can find the conditions for a dynamical instability under quite general conditions. Obviously, for trapping to occur the effective detuning must be negative, i.e. δ : = ∆ c − 1 2 s N s U 0,s < 0. For convenience we rescale the steady-state distributions as f 0,s (p) = (Lm s v s ) −1 G s p mvs , where v s > 0 is a typical velocity of the sth species, L is the cavity length and we assume that these distributions decay monotonously with |p|. Then, such an equilibrium is unstable if and only if S s=1 N s η 2 s k B T s P ∞ −∞ G ′ s (u) −2u du > κ 2 + δ 2 |δ| ,(6) where P denotes the Cauchy principal value and k B T s = m s v 2 s /2. In that case, initial density perturbations amplify and the cavity mode amplitude grows exponentially in time at a rate γ > 0 that solves (γ + κ) 2 + δ 2 = S s=1 N s η 2 s δ 2k B T s ∞ −∞ u G ′ s (u)du (γ/kv s ) 2 + u 2 .(7) This growth finally ceases and particles and field reach a quasistationary selforganised state. As a central result of this work let us emphasise here, that the right hand side of eq. (6) only depends on cavity parameters and all terms in the sum on the left hand side are manifestly positive. Hence adding any extra species will always lower the power needed to start the selforganisation process, regardless of temperature and polarisability of the additional particles. For thermal momentum distributions the integrals in (6) are unity and the condition gets particularly simple. At higher temperatures, where (k min v s ) 2 ≫ κ 2 +δ 2 , the first term in the denominators of eq. (7) can be neglected and the field amplitude's growth rate is given by γ = −κ + s |δ| k B T s N s η 2 s − δ 2 1/2 .(8) A glance at this expression shows, that the instability also grows at a larger rate the more terms contribute to the sum. Hence, both the required power and time needed to achieve selforganisation is lowered by combining several species. Obviously, if one can reach the threshold with one species, the system certainly still selforganises if one adds a second species. Let us remark that there exists a dynamical instability for positive effective frequency mismatch δ as well, but it is connected to heating and does not lead to an ordered distribution. Numerical simulations indicate that the quasi-equilibrium state into which the system evolves in case of instability is close to a BGK solution [24] of (5a) and (5b). A BGK solution is a stationary solution of (5) where all one-particle distributions depend on position and momentum solely via the Hamiltonian functions f s (z, t) = F s (H s ) and α(t) = α 0 . The real-valued functions F s are essentially arbitrary and the steady-state mode amplitude α 0 needs to be selfconsistently determined from (5b). Let us note that in the weak coupling regime, i.e. | s N s U 0,s | ≪ |δ|, the single-particle actions (10) are nearly invariant during selforganisation for a wide range of parameters and in this case it is therefore possible to relate the functions F s to the unperturbed uniform and unstable states f 0,s (p) to obtain the selforganised state f so s (z) as f so s (x, p) = f 0,s (J s )(9) where J s = kI s for untrapped and J s = kI s /2 for trapped orbits [25]. For an illustration the reader may consult fig. 2. IV. KINETIC THEORY FOR THE COOLING AND ENERGY FLUX Let us now turn to describe the system evolution beyond eqs. (5) in the weak-coupling limit where δ ≈ ∆ c . To this end we introduce the single-particle action belonging to the instantaneous average potential seen by the sth species, I s = ± 1 2π 2m s [H s − Φ s (x ′ )] dx ′ ,(10) in which Φ s (x, α) ≃ 2 η s Re (α) sin(kx) as a valid approximation in this limit. The corresponding angle variable θ s can be obtained from the generating function S s = ± x 2m s [H s − Φ s (x ′ ) ] dx ′ as θ s = ∂Ss ∂Is . Starting from any initial condition, in the long time limit the oneparticle distribution functions to a good approximation become functions of the single-particle Hamiltonians H s and thus actions I s alone. Statistical fluctuations slowly modify these distributions in such a way, that the system evolves towards equilibrium in a sequence of BGK states, f s (x, p, t) ≃ f s (I s , t) [26]. Defining as well as U s (I s , α) = 1 2π 2π 0 ∂I s ∂t dθ s(12) as the average variation of the action along a "frozen" orbit, we obtain the following nonlinear Fokker-Planck equations for the one-particle distributions f s (I s , t) and the average mode amplitude α(t) ∂f s ∂t + U s ∂f s ∂I s = ∂ ∂I s A s f s + B s ∂f s ∂I s + s ′ C BL [f s , f s ′ ] (13) α = −2πi κ − iδ s N s η s g 0,s f s dI s .(14) The collision operator, i.e. the r.h.s. of (13), consists of two contributions due to the fluctuation and decay of the mode amplitude A s [f s ] = −4 δη 2 s κω s n n 2 |g n,s | 2 |D(inω s )| 2 (15a) B s [f s ] = 2 η 2 s κ n n 2 |g n,s | 2 |D(inω s )| 2 κ 2 + δ 2 + n 2 ω 2 s (15b) and a generalised Balescu-Lenard term [27,28] C BL [f s , f l ] = 8π 2 2 δ 2 N l η 2 l η 2 s n,m n|g n,s | 2 |D(inω s )| 2 × × |g ′ m,l | 2 δ (nω s − mω ′ l ) n ∂f s ∂I s f ′ l − m ∂f l ∂I ′ l f s dI ′ l ,(16) where the prime means taking the function at I l = I ′ l and ω s is the nonlinear frequency defined by ω s = ∂H s /∂I s . For spatially uniform ensembles the actions reduce to I s → p/k and the expressions for the coefficients (15) given in [22] are recovered. In the above expressions, D(s) denotes the so-called dispersion relation given by D(s) = (s + κ) 2 + δ 2 − i4π δ n,s N s η 2 s n|g n,s | 2 ∂fs ∂Is s + inω s dI s (17) if Re(s) > 0. Apart from the integral terms in the dispersion relation D and the dependence of the nonlinear frequencies ω s and functions g n,s and U s on the common mean field amplitude α, the cavity-mediated interspecies interaction is described more explicitly by the Balescu-Lenard collision operator (16). It has been derived before by several authors [29,30] and accounts for the interspecies heat flow, which is mediated by the cavity field. As noted before by these authors C BL involves resonant actions I l , I s , i.e. orbits such that the condition nω l (I l ) = mω s (I s ) is fulfilled. This term is a possible source of sympathetic cooling is such a setup. A. Joint equilibrium states The set of possible equilibria of (13) and (14) can be divided into two classes: spatially homogeneous with vanishing average field and inhomogeneous with nonzero photon number. The first exist only for effective red detuning 2δ < −ω R,s and are stable only below the threshold determined by (6). The corresponding phasespace distributions can be explicitly calculated to give q-Gaussians f s,eq (x, p) ∼ exp qs − p 2 2m s k B T * ,(18) where the q-exponential function is given by exp q (u) = [1 + (1 − q)u] 1 1−q and we have set q s = 1 + ω R,s |δ| , k B T * = κ 2 + δ 2 4|δ| .(19) Their minimal "thermal" energy thus is given by k B T * = κ/2 and is reached if δ = −κ. Interestingly, we have a very small but finite minimum kinetic energy in very good resonators but of course if κ ∼ ω R,s quantum effects have to be taken into account. As q s → 1 the q-Gaussian becomes an ordinary Gaussian. For an example the reader is referred to fig. 3. It follows that as long as the joint equilibrium {f s,eq (x, p), s = 1 . . . S} is Vlasov-stable according to (6), the individual equilibrium distributions are independent of each other because all energy exchange currents cease. The cooling rates, however, are modified by energy exchange between different sorts of particles, described by the generalised Balescu-Lenard operator (16) and can considerably shorten the time for a given species to reach its steady state. Above threshold, stable and strongly trapped equilibria require −δ ≫ ω R,s and exist already if the uniform state is only weakly unstable. These are Maxwell-Boltzmann distributions f s,eq (x, p) ∼ exp − H s (z, α, α * ) k B T kin,s ,(20) with kinetic temperatures k B T kin,s := p 2 m s = k B T * + ω 2 0,s |δ| ,(21) where the trap frequencies are given by ω 2 0,s = 4η s ω R,s | Re α|. Again α is determined from (14) using g 0,s ≃ k 2 Is 2msω0,s in this limit. Here the equilibria are indeed modified by the presence of additional species through the selfconsistent cavity field but the mutual interaction is not enough to equalise the kinetic temperatures and, as in the uniform case, all equilibrium inter-species heat fluxes vanish. Figure 4 shows an example of a jointly selforganised steady state. Let us note that for any given deeply trapped species the equilibrium uncertainty product ∆x∆p = k B T kin,s /ω 0,s is bounded from below by , and thus by twice the minimal value for a particle in a classical potential. The additional uncertainty may therefore be attributed to the quantum fluctuations of the mode amplitude. The energy per particle E s can be shown to be ∆x∆p ≥ ,(22)E s = ∆x∆pω 0,s ≥ ω 0,s(23) which is again twice the usual value. The minimal uncertainty state, which coincides with the minimal energy state, is attained if 2ω 0,s ≫ κ for a detuning δ = −2ω 0,s . These findings remain correct in an entirely quantum-mechanical treatment because for deeply trapped particles and thus approximately harmonic potentials, the semiclassical equations are exactly equivalent to the quantum equations. B. Sympathetic cooling Let us finally examine the energy flow per particlė Q 2→1 from species one to species two for two spatially homogeneous ensembles a little closer. If we assume that species one is already cold, i.e. 2k B T kin,1 / κ ≪ κ/ω R,1 and far from instability, the inter-species heat flow is estimated to bė Q 2→1 = − N 1 m 1 N 2 m 2Q 1→2 ≃ N 1 η 2 2 η 2 1 4 √ π δ 2 (κ 2 + δ 2 ) 2 × × ω R,1 k B T 1 1 − T 2 T 1 1 + m 1 T 2 m 2 T 1 −3/2 ,(24) where we wrote T s instead of T kin,s for simplicity. Not surprisingly it is maximal if δ = −κ. The proportionality of the energy flow to the number of cold particles N 1 immediately hints towards a sympathetic cooling scheme, in which a cold ensemble is coupled to a smaller number of hotter and heavier particles whose cooling rate is enhanced due to this exchange currentQ 2→1 . From eq. (16) one sees that if at least one of the systems is spatially nonuniform (ordered), the inter-species heat flows are effectively suppressed due to the loss of resonances and sympathetic cooling gets inefficient. Hence in order to get a useful inter-species energy transfer one needs to have a large number of cold particles without crossing the instability threshold. This behaviour is exhibited in fig. 5, where we see that the kinetic energy of the heavy particles decays slower when selforganisation starts and a field is being built up. However, as the field and thus the potential grows the spatial confinement of the particles is continuously increased. This behaviour can be expected to improve using a larger number of non-degenerate modes and several pump frequencies. In this manner, each mode contributes to the energy transfer and dissipation without receiving enough scattered photons to actually trap particles. V. CONCLUSIONS We have shown analytically and in simulations that the selforganisation threshold condition for an ensemble of several different species of particles inside a transversally pumped standing wave resonator is strictly reduced below the value for each of the species separately. Hence the joint selforganisation threshold power is the lower the more species present. If the threshold can be reached with one species, many species can be added and simultaneously trapped and cooled. In the long-time limit the achievable temperatures are only limited by the resonator linewidth and get close to the quantum ground state for deep traps. In the multi-species case the cooling time of a given (heavy) species can be reduced due to energy exchange with a second already colder (light) species. In general one only needs a single intense and narrow laser, frequency-stabilised relative to a cavity mode, to simultaneously trap and cool a large number of different particles within the same volume. The method thus can be applied to atomic gas mixtures, atom-molecule mixtures or even microbeads in a dilute atomic gas. Using more complex setups involving several laser frequencies and modes can be expected to significantly enhance the The sympathetic cooling effect can be observed initially but as the system crosses the selforganisation threshold the inter-species heat flow ceases. This can be inferred from the kinetic temperature curve of the heavier particles (solid blue) becoming parallel to the curve depicting the heavy particles alone (dashed blue). Parameters: N1 = 320, N2 = 500, √ N1η1 = 207ωR,1, √ N2η2 = 258ωR,1, κ = 200ωR,1 and δ = −κ. cooling and lead to more complex distributions of the particles. The method can be easily generalised to moving ensembles in arbitrary cavity geometries, e.g. ring resonators. Here the cavity field mediated interaction of the ensembles transfers a stopping force applied to one ensemble to any other particles. As the general effect has been successfully experimentally demonstrated for single-species setups [12,31,32], we are confident that the multispecies generalisation proposed here are well within reach of current technology. This might include even atomic hydrogen, which could be stopped sympathetically with a Helium beam. FIG. 1 : 1(Colour online) Setup. A multispecies ensemble of particles within a cavity transversely illuminated by a laser close to resonance with a single cavity mode. Above threshold the particles order in a regular pattern which optimise scattering into the resonator. online) Joint selforganisation of two species starting from a perturbed uniform state above the instability threshold(6), such that species one is six times critical, whereas species two is far below its proper threshold.Figure (a) shows the position distributions in the final state, (c) and (d) the momentum distributions initially (dashed) kx)e −inθs dθ s FIG. 3 : 3(Colour online) Simulated (solid) and analytical (circles) steady-state momentum distributions for two different species m2 = 40m1 below threshold averaged over 250 realisations. The distribution of the lighter particles (a) is given by a q-Gaussian with q1 = 1.4. The dashed curves represent Gaussians corresponding to p 2 /ms = kBT * . As q2 = 1.01 the distribution of the heavier particles (b) is indistinguishable from a Gaussian. Parameters: N1 = 300, N2 = 200, √ N1η1 = √ N2η2 = 800ωR,1, κ = 100ωR,1, ∆c = −2.6ωR,1 and N1U0,1 = N2U0,2 = −0.1ωR,1. FIG. 4 : 4(Colour online) Selforganised steady-state momentum distributions of (a) species one and (b) species two with m2 = 10m1. In (c) we show the time evolution of the kinetic temperatures and the photon number. The dashed lines represent the predictions of eq. (20) and eq. (21). The dashdotted line shows the maximally possible photon number. The initial rise in the kinetic temperatures originates from the fast initial growth of the cavity intensity above threshold. Parameters: N1 = 300, N2 = 200, √ N1η1 = √ N2η2 = 600ωR,1, κ = 100ωR,1, δ = −κ and ER,1 = ωR,1. online) Time evolution of the kinetic temperatures for two species, one heavy and the other lightweight. Upper plot: Optimal cooling curve of the heavy species alone (blue dashed) vs. cooling curve in the presence of a lighter species (blue solid). 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[ "Interfacial adsorption in two-dimensional Potts models", "Interfacial adsorption in two-dimensional Potts models" ]
[ "E Carlon ", "F Iglói ", "W Selke ", "F Szalma ", "\nResearch Institute for Solid State Physics and Optics\nInstitute for Theoretical Physics\nLaboratoire de Physique des Matériaux\nUniversité Henri Poincaré\nB. P. 239P.O.Box 49F-54506, H-1525Nancy I, Vandoeuvre-lès-Nancy Cedex, BudapestFrance, Hungary\n", "\nInstitut für Theoretische Physik\nSzeged University\nTechnische HochschuleH-6720, D-52056Szeged, AachenHungary, Germany\n" ]
[ "Research Institute for Solid State Physics and Optics\nInstitute for Theoretical Physics\nLaboratoire de Physique des Matériaux\nUniversité Henri Poincaré\nB. P. 239P.O.Box 49F-54506, H-1525Nancy I, Vandoeuvre-lès-Nancy Cedex, BudapestFrance, Hungary", "Institut für Theoretische Physik\nSzeged University\nTechnische HochschuleH-6720, D-52056Szeged, AachenHungary, Germany" ]
[]
The interfacial adsorption W at the first-order transition in two-dimensional q-state Potts models is studied. In particular, findings of Monte Carlo simulations and of density-matrix renormalization group calculations, at q = 16, are consistent with the analytic result, obtained in the Hamiltonian limit at large values of q, that W ∝ t −1/3 on approach to the bulk critical temperature Tc, t = |Tc − T |/Tc. In addition, the numerical findings allow to estimate corrections to scaling. Our study supports and specifies a previous conclusion by Bricmont and Lebowitz based on lowtemperature expansions.
10.1023/a:1004542105635
[ "https://export.arxiv.org/pdf/cond-mat/9902182v1.pdf" ]
119,377,121
cond-mat/9902182
4d57a354b71c98986106bd745a6015e21e77f2a8
Interfacial adsorption in two-dimensional Potts models arXiv:cond-mat/9902182v1 12 Feb 1999 E Carlon F Iglói W Selke F Szalma Research Institute for Solid State Physics and Optics Institute for Theoretical Physics Laboratoire de Physique des Matériaux Université Henri Poincaré B. P. 239P.O.Box 49F-54506, H-1525Nancy I, Vandoeuvre-lès-Nancy Cedex, BudapestFrance, Hungary Institut für Theoretische Physik Szeged University Technische HochschuleH-6720, D-52056Szeged, AachenHungary, Germany Interfacial adsorption in two-dimensional Potts models arXiv:cond-mat/9902182v1 12 Feb 1999Potts modelinterfacial adsorptionMonte Carlo simulationsdensity-matrix renormalization groupHamiltonian limit The interfacial adsorption W at the first-order transition in two-dimensional q-state Potts models is studied. In particular, findings of Monte Carlo simulations and of density-matrix renormalization group calculations, at q = 16, are consistent with the analytic result, obtained in the Hamiltonian limit at large values of q, that W ∝ t −1/3 on approach to the bulk critical temperature Tc, t = |Tc − T |/Tc. In addition, the numerical findings allow to estimate corrections to scaling. Our study supports and specifies a previous conclusion by Bricmont and Lebowitz based on lowtemperature expansions. Introduction The interface between two phases may become unstable against the appearance of a third phase. This wetting phenomenon has been studied in various circumstances, considering different materials and geometries, both experimentally and theoretically. 1 Good candidates for modelling wetting are q-state Potts models 2,3 , where the two phases correspond to distinct boundary states, say, 1 and q, at opposite sides of the system. In that case one observes an excess adsorption of the non-boundary states at the interface between '1' rich and 'q' rich domains. In particular, in two dimensions, the interfacial adsorption, W , is found to diverge on approach to the critical temperature T c like W ∝ t −ω , where t = |T c − T |/T c . For q= 3 and 4, the bulk transition is continuous and ω can be expressed in terms of canonical bulk critical exponents, ω = ν −β. 2 For larger values of q, the transition is of first order. On general, phenomenological grounds 4 , one may expect then ω = 1/3, as observed in Monte Carlo (MC) simulations on the two-dimensional Blume-Capel model. 5 However, previous simulations on the two-dimensional 20-state Potts model suggested that ω may be significantly larger than 1/3, albeit a pronounced curvature in the corresponding log-log plots for W (t) was noticed indicating that the asymptotic behavior may not have been reached. 3 A reasonable explanation of the numerical findings was then offered by Bricmont and Lebowitz. 6 Based on low-temperature expansions, the critical region is argued to be extremely narrow in the Potts case, in contrast to the Blume-Capel case. 6 However, a numerical confirmation remained to be done. Motivated by a recent intrigiung analysis of the interfacial tension of the q-state Potts model in two dimensions employing a field-theoretic representation for configurations of the interface 7 (extending prior considerations 2 ), we decided to reconsider the somewhat irritating problem on the value of ω for q > 4. Indeed, advances in methods and computer facilities allow now to explore numerically the critical region more deeply than some years ago. In addition, we dealt with the issue analytically by studying the Hamiltonian limit of the Potts model for large values of q exactly. The outline of the article is as follows. In the next Section, the model is defined, and the numerical methods of our choice, MC simulations and density-matrix renormalization group (DMRG) calculations, are introduced. Then, we discuss the results obtained from those methods. In Section 4, the analytical findings in the Hamiltonian limit are presented. A summary concludes the paper. Model and numerical methods The two-dimensional q-state Potts model is described by the Hamiltonian H = −J (ij),(i ′ j ′ ) δ nij ,n i ′ j ′(1) summing over neighboring sites (ij), (i ′ j ′ ) on a L × M rectangular lattice, with the Potts variable n ij = 1, 2, ...q. In the thermodynamic limit, L, M −→ ∞, the model displays a bulk phase transition at the critical temperature k B T c /J = 1/ ln( √ q + 1) 8 , where k B is the Boltzmann constant. The transition is of continuous type at q ≤ 4, while it is of first order at larger values of q. To introduce an interface, we add a column of M sites on the left side boundary, fixing the Potts variable to be in the state n L = 1, and another column on the right side boundary with n R = q. The Potts variables on the top and at the bottom of the lattice may be connected by periodic boundary conditions ('periodic case'). Alternatively, the interface may be pinned at the lower and upper boundaries by inserting there additional boundary rows with fixed states being '1' on the left half and 'q' on the right half (pinned case). By examining typical Monte Carlo equilibrium configurations below T c , it is seen that an excess of non-boundary states is adsorbed at the interface, as illustrated in Fig. 1. This wetting phenomenon may be described quantitatively by the net interfacial adsorption W per unit length W = 1/M (ij),nb (< δ nb,nij > 1:q − < δ nb,nij > 1:1 )(2) summing over all L × M sites in the inner part of the system; the < > brackets refer to thermal averages; the index nb denotes the non-boundary states, nb = 2, ..., q − 1; the subscripts 1:q and 1:1 refer to systems with corresponding fixed states at the boundaries, i.e. with and without interface. The net adsorption W is closely related to the density profiles We computed profiles and net adsorption numerically, using Monte Carlo techniques 9 and the density-matrix renormalization group method. 10 In the MC simulations, the standard single-variable flip algorithm was applied, for system sizes L ≤ 256 and M ≤ 2000, studying mostly the periodic case, augmented by a few runs for the pinned case. To obtain accurate equilibrium data, we typically averaged over several realizations (using different random numbers), each consisting of 10 6 Monte Carlo steps per site, including at least 10 5 steps for equilibration. Error bars resulted from averaging over the ensemble of realizations. Usually, we set q = 16, where the bulk phase transition is strongly first order with a bulk correlation length at T c of a few lattice spacings. 11 To search for possible q-dependences, we also simulated models with q= 20 and 40. Furthermore, because clusters, formed by neighboring sites in the same state, seem to play an interesting role in the wetting phenomenon, see Fig. 1, we also computed cluster properties in the MC study. In the DMRG method, one considers strip-like lattices, i.e. M −→ ∞, while L is finite. The algorithm, introduced by White in 1992 12 for the study of the low-lying spectrum of quantum spin chains, has been extended in several directions. 10 In the present study we follow Nishino's 13 formulation of the method adapted to treat classical twodimensional systems, where the approach is used for constructing iteratively approximate transfer matrices, starting from strips of small width (say, L = 8) which can be also handled numerically exactly. At each DMRG step the strip width is increased, and the configurational space is truncated efficiently through the projection into smaller subspaces with the help of appropriate density matrices. We do not need to describe details of the widely used DMRG procedure here: a good introduction, together with recent developments, can be found in Ref. 10. The DMRG technique has already been applied to the two-dimensional Potts model, both at continuous, q ≤ 4, 14,15 and first-order transitions, q > 4 16 . In the former case, bulk and surface critical exponents have been calculated with a high degree of accuracy 15 . In the present study we are interested in rather large values of q, where the standard DMRG method is less suitable. An appropriate, powerful variant, the "pseudo -spin" version, has been introduced in Ref. 16. It enables one to treat systems with special values of q, namely q = p 2 , p = 2, 3, . . .. We used that algorithm in the current work. Most calculations were performed at q = 16, augmented by some calculations at the q = 9, for strips of widths up to L = 80. We kept up to m = 60 states per block with a typical truncation error of ǫ ≈ 10 −6 . From the dominant eigenvector of the transfer matrix the profiles 15 and the interfacial adsorption W were calculated. Numerical results The crucial quantity, which we computed numerically, is the net adsorption W as a function of the system size, L, M , and temperature t, W (L, M, t). The main aim is to analyse its critical behavior as t −→ 0 or T −→ T c , in the thermodynamic limit L, M −→ ∞. For that, one may extrapolate the numerical data for W to that limit, or one may study systems for which finite-size effects can be neglected. We also tried to perform a finite-size scaling analysis on W (L, M, t) 5 , see below. In any event, accurate data are needed. Their quality can be conveniently tested by comparing results obtained from the MC simulations and the DMRG calculations, as illustrated in Figs. 2 and 3. In Fig. 2, profiles, Eq. (3), are shown for the 16-state Potts model with and without interface, demonstrating again the excess adsorption of non-boundary states at the interface. In Fig. 3, the increase of the net adsorption W with increasing width L of the Potts model, at various temperatures, is displayed. In both figures, the finite-size effect arising from the length M of the system can be disregarded. In fact, M is infinite in the DMRG approach. In the simulations (periodic case), M was checked to be sufficiently large so that W approached closely W (L, M = ∞, t), with the characteristic crossover value depending, of course, on the width L and the distance from criticality t = |T c − T |/T c . As exemplified in Figs. 2 and 3, data from both methods do, indeed, agree nicely, being obviously accurate and reliable. Slight systematic deviations seem to become significant only for quite wide systems, say, L > 60. A reasonable extrapolation of our data for W to the thermodynamic limit is feasible for temperatures T ≤ 0.998T c , see Fig. 3. Closer to T c , both numerical methods would require much larger system sizes demanding extremely large storage and/or computing time. In the thermodynamic limit, one expects W ∝ t −ω as t −→ 0. To monitor the approach to the asymptotic behavior, one may consider the effective exponent ω ef f (t) = −d ln W/d ln t(5) with ω = ω ef f (t = 0). In Fig. 4, numerical estimates of the effective exponent are shown. At t ≥ 0.02, we determined ω ef f from MC data for systems being large enough to disregard finite-size effects. Actually, finite-size effects are much stronger in the pinned case than in the periodic case, and only MC data for the latter case are included, with L = 64 and M ≥ 400. At t ≤ 0.01, estimates are based on extrapolating MC and DMRG results for W to the thermodynamic limit, see Fig. 3. Note that the net adsorption is calculated at discrete temperatures, t i , and the effective exponent may be approximated by ω ef f (t) = − ln(W (t i )/W (t i+1 ))/ ln(t i /t i+1 ), with t= (t i t i+1 ) , and t i+1 < t i . Error bars in Fig. 4 stem from a proliferation of the error in the net adsorption W . Evidently, ω ef f depends strongly on the distance from criticality t. For instance, in the range 0.1 > t > 0.002, it changes from about 1.2 to about 0.5 when moving towards T c . Accordingly, an average critical exponent in that interval would be supposedly significantly larger than the true asymptotic critical exponent ω, as observed before. 3 Presuming ω = 1/3, see Refs. 4 and 6 as well as Section 4, it follows from Fig. 4 that the asymptotic regime is very narrow, and corrections to scaling are quite pronounced. To quantify these corrections, one may postulate the standard ansatz W (t) = W 0 t −1/3 (1 + at x + bt 2x + ...)(6) The coefficients can be calculated from chi-square fits to the numerical data near T c . Using, for instance, a fit to the points in the interval 0.02 ≤ t ≤ 0.1 (shown in Fig. 4), leads to a net adsorption W reproducing very well the numerical findings both closer to T c and further away from T c . Eventually, systematic deviations show up when further lowering the temperature, reflecting the need for additional correction terms in Eq. (6) in that region, see Fig. 4. The exponent x characterising the corrections to scaling is found to be rather small, x = 0.14 ± 0.06. Because of its smallness, we included the leading, with the exponent x, and subleading, 2x, terms in the ansatz (6). The error bars arise from using a variety of plausible fitting intervals and points. From MC simulations of Potts models with larger number of states, q = 20 and 40, one may conclude that the corrections to scaling, at t > 0.01, are rather insensitive to the concrete value of q. From general considerations 4,5 , one expects two diverging lengths at the interface, in the direction parallel to the interface, ξ , and perpendicular to it, ξ ⊥ , with ξ ⊥ ∝ t −ω and ξ ∝ t −2ω , as t −→ 0, see also Section 4. For strip-like systems, M −→ ∞, the following finite-size scaling expressions can be then motivated 5 W (L, M = ∞, t) = t −ω w 1 (Lt ω )(7) in the limit of Lt ω >> 1, and W (L, M = ∞, t = 0) ∝ L(8) for L >> 1. Indeed, the DMRG results indicate that Eq. (8) seems to be satisfied rather well already for strips of moderate width, say, L ≥ 16, see also Ref. 3. On the other hand, the numerical data do not suffice to establish the scaling form (7) with ω = 1/3. In fact, a 'reasonable' collapse of our data on an apparent scaling function w 1 might be achieved with a somewhat larger value, ω ≈ 1/2. Careful inspection, however, reveals that systematic deviations from a unique scaling curve w 1 set in for large arguments Lt ω . Thence, larger systems close to T c had to be studied (which are, at present, out of reach, because of limitations in storage and computing time). Indeed, ω ≈ 1/2 is merely an upper bound of the true value of ω. This observation corroborates the above mentioned finding on ω ef f : One has to include corrections to scaling to demonstrate consistency of the numerical data with the theoretically expected asymptotic behavior, due to the narrowness of the critical region. In closing the Section on the MC and DMRG results, we remark that the thermally averaged largest cluster of non-boundary states, as computed in the simulations, seems to diverge by approaching T c from below. A detailed analysis would be desirable, but it is beyond the scope of the present study. The Hamiltonian limit for large values of q In the following we consider the Hamiltonian limit of the Potts model with strong vertical and weak horizontal couplings 17 . The transfer matrix in the vertical direction has the form T = exp(−Ĥ), with the one-dimensional HamiltonianĤ 18Ĥ = − L−1 i=1 δ ni,ni+1 − h L i=1 q−1 k=1 M k i .(9) n i is the Potts variable on site i and M k i denotes the flip operation M k i |n i = |n i + k, mod q . The strength of the transverse field, h, at the transition point is h c = 1/q. Quantities of physical interest are derived from the ground state, |Ψ 0 , and from the energies of the ground state and the first excited state, E 0 and E 1 , of (9). The Hamiltonian limit of the Potts model has been treated recently 16 for free boundary conditions. In that case, the solution has a remarkably simple form in the vicinity of the transition point for large values of q. Repeating the same type of considerations for models with an interface, fixing the variables at the boundaries in the states '1' and 'q' (1:q) (as before), one finds that the ground state sector of the Hamiltonian (9) is spanned by the vectors |ψ i,j = |11 . . . 1n b n b . . . n b qq . . . q ,(10) where a non-boundary state is given by |n b = 1/ √ q(|2 + |3 + . . . + |q − 1 ), and the positions of the domain walls separating the boundary and non-boundary states are denoted by i (= 1, 2, . . . , L) and j = (i, i + 1, . . . , L). The diagonal matrix-elements of these states, ψ i,j |Ĥ|ψ i,j = −L − (j − i)t, with t(= hq − 1, |t| ≪ 1) being the distance from the critical point, are smaller by an amount of O(1) compared to any other states, like those containing boundary states in the domain of non-boundary (nb) states. The Hamiltonian in the ground state sector, spanned by the vectors (10), can be written asĤ g = −(j − i)t − h √ q[(a + + a − ) + (b + + b − )] ,(11) up to a constant; the operators, a ± and b ± , which move the positions of the domain walls in (10), are defined as a ± |ψ i,j = |ψ i±1,j , 1 < i < j; a + |ψ i,i = a − |ψ 1, j = 0 b ± |ψ i,j = |ψ i,j±1 , i < j < L; b − |ψ i,i = b + |ψ i,L = 0 .(12) In the continuum limit, when L ≫ 1, i ≫ 1 and j ≫ 1, but x = i/L = O(1) and y = j/L = O(1), the Hamiltonian (11) can be written in the form of a differential operator H g ψ(x, y) = − h √ q L 2 ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + tL(y − x) ψ(x, y) = Eψ(x, y) ,(13) with the boundary condition 0 ≤ x ≤ y ≤ 1. At the critical point, t = 0, the solution of the eigenvalue problem (13) reads ψ(x, y) = 2[sin(πk 1 x) sin(πk 2 y) − sin(πk 2 x) sin(πk 1 y)] ,(14) with k 1 = 1, 2, . . . and k 2 = k 1 + 1, k 1 + 2, . . .. For the ground state, ψ 0 (x, y), one has k 1 = 1 and k 2 = 2, whereas for the first excited state k 1 = 1 and k 2 = 3. Thus the energy gap is ∆E = E 1 − E 0 = 5π 2 h √ qL −2 , and the correlation length parallel to the interface behaves as ξ ∼ (∆E) −1 ∼ L 2 . Since the correlation length perpendicular to the interface is, at the critical point, limited by the width of the system, ξ ⊥ ∼ L, one arrives at ξ ∼ ξ 2 ⊥ , in agreement with the form mentioned above. The density profiles satisfy the relations n 1:q (1, x) = n 1:q (q, 1 − x) and n 1:q (x) = nb n 1:q (nb, x) = 1 − n 1:q (1, x) − n 1:q (q, x). From the ground state, one obtains n 1:q (1, x) = 1 x dx ′ 1 x ′ dy [ψ 0 (x, y)] 2 = 4 π 2 π 2 (1 − x) + 1 4 sin 2πx π 2 (1 − x) + 1 8 sin 4πx − 4 9 sin 6 πx .(15) For small x, one finds n 1: q (1, x) = 1 − (10π 2 /3)x 3 + O(x 6 ) , whereas for x close to one, the profile behaves like 10 . The profile of non-boundary states is symmetric and its maximal value is given by n 1:q (1/2) = 1/2 + 32/(9π 2 ) = 0.86025. For the (1 : 1) boundary condition, the profile of non-boundary states tends to zero for large values of q. Therefore the interfacial adsorption W , Eq. (2), at the transition point may be approximated by n 1:q (1, x) ∼ (1 − x)W L = 1 0 n 1:q (x)dx = 1 3 + 35 72π 2 = 0.3826 .(16) Thus, at the transition point, W is, indeed, proportional to L, see Eq. (8). Note that the prefactor seems to depend on q, being, at q = 16, about 0.3, according to the DMRG calculations. Below the critical point, t = 0, we consider the eigenvalue equation (13) in terms of the new variables x + = (y+x)/ √ 2 and x − = (y − x)/ √ 2. Then − h √ q 2 L 2 ∂ 2 ∂x 2 + + ∂ 2 ∂x 2 − + tL √ 2x − ψ(x + , x − ) = Eψ(x + , x − ) ,(17) with the boundary condition 0 ≤ x − ≤ x + ≤ 1/ √ 2 and 0 ≤ x − ≤ √ 2 − x + ≤ 1/ √ 2. Now the eigenfunction ψ can be written as ψ(x + , x − ) = φ + (x + )φ − (x − ). φ + (x + ) satisfies the free-particle equation −d 2 φ + /dx 2 + = L 2 E + /(2h √ q)φ + ; φ − (x − ) is the solution of the Schrödinger equation of a particle in a linear potential − h √ q 2 L 2 ∂ 2 ∂x 2 − + +tL √ 2x − φ − (x − ) = E − φ(x − ) .(18) Equation (18) leads to bound states, and the energy scale (both for the ground state and the excited states) is set by ǫ ∼ t 2/3 . Hence the temperature dependence of the parallel correlation length is given by ξ ∼ (∆E) −1 ∼ t −2/3 , in accordance with the phenomenological considerations 4 . On the other hand, the extent of the bound states sets the length scale ξ ⊥ ∼ t −1/3 , which is then proportional to the interfacial adsorption, W. Thence, in the Hamiltonian limit for large values of q, one has ω = 1/3. Summary In this article, critical interfacial properties of two-dimensional q-state Potts models at the bulk first-order phase transition have been studied. We applied two numerical methods, Monte Carlo simulations and the density-matrix renormalization group approach, mainly at q = 16. Furthermore, we considered analytically the model in its Hamiltonian limit at large values of q. The different methods lead to a consistent description of the critical behavior. The interfacial adsorption W diverges on approach to the phase transition temperature as W ∝ t −ω , ω = 1/3, with a very narrow asymptotic region. The strong corrections to scaling are characterised by a small exponent, x = 0.14 ± 0.06, which seems to depend (if at all) only weakly on the number of Potts states, q. At the critical point, W diverges linearly with the width of the system (being indefinitely long in the direction parallel to the interface). The proportionality factor has been calculated in the Hamiltonian limit. The value of the critical exponent ω, ω = 1/3, is typical for wetting phenomena at bulk transitions of first order in two dimensions. It follows from general, phenomenological considerations, based on an effective interface Hamiltonian 4 , as well as from calculations on various microscopic multi-state models, such as Potts and the Blume-Capel models. The strong corrections to scaling and the narrowness of the critical region are, on the other hand, features which are specific for Potts models. They have been predicted before 6 by using low-temperature arguments, and they have been quantified in this study. variation of the density of state s by going from the left side, fixed in state b 1 , to the right side, fixed in state b 2 , of the lattice, summing over each column, j = 1, ...M , with i running from 1 to L. Obviously, W = i,nb (n 1:q (nb, i) − n 1:1 (nb, i)) FIG. 1 .FIG 1Typical Monte Carlo equilibrium configuration of the two-dimensional 16-state Potts model with an interface, at T = 0.99Tc. The '1' domain is on the left hand side, and the '16' domain on the right hand side. Shown are clusters of distinct states. A MC system of size L = 60 and M = 60, periodic case, was simulated, but only a part is depicted. , b = 1, at T = 0.99Tc and L = 24, as obtained from the DMRG method (open circles) and MC simulations (periodic case with M = 400; full diamonds). FIG. 3. Net adsorption, W , vs. inverse width, 1/L, of the lattice at (from bottom to top) t = 0.01, 0.005, 0.002, 0.001, and 0.0005, depicting DMRG (open circles) and MC (periodic case with M ≤ 1000; full diamonds) data. FIG. 4. Temperature, t, dependence of the effective exponent of the net adsorption W , ω ef f (t), for numerical data 'free of finite-size effects', see text. The dashed curve corresponds to the fit to Eq. (6), with W0 = 3.280, a = −1.977, b = 0.939, and x = 0.165, quantifying the corrections to scaling. AcknowledgementsF. Sz. would like to thank the DAAD for a scholarship enabling his visit at the RWTH Aachen. The work of F. I. and F. Sz . S Dietrich, Phase Transitions and Critical Phenomena. C. Domb and J. L. Lebowitz121Academic PressS. Dietrich in "Phase Transitions and Critical Phenomena", Vol.12, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1988) p.1; Liquids at Interfaces. M Schick, Proceedings of the Les Houches Summer School Lectures. J. Chavrolin, J. F. Joanny, and J. Zinn-Justinthe Les Houches Summer School LecturesAmsterdamElsevier415M. Schick in "Liquids at Interfaces", Proceedings of the Les Houches Summer School Lectures, J. Chavrolin, J. F. Joanny, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1990), p. 415. . W Selke, W Pesch, Z. Physik B. 47335W. Selke and W. Pesch, Z. Physik B 47: 335 (1982); . W Selke, D A Huse, Z. Physik B. 50113W. Selke and D. A. 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Proceedings of the First International DMRG Workshop. I. Peschel, K. Hallberg, and X. Wangthe First International DMRG WorkshopHeidelbergSpringerin printI. Peschel, K. Hallberg, and X. Wang, eds., "Proceedings of the First International DMRG Workshop", Lecture Notes in Physics (Springer, Heidelberg, 1999), in print. . A Klümper, A Schadschneider, J Zittartz, Z. Physik B. 76247A. Klümper, A. Schadschneider, and J. Zittartz, Z. Physik B 76: 247 (1989). . S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett. 69: 2863 (1992). . T Nishino, J. Phys. Soc. Jpn. 643598T. Nishino, J. Phys. Soc. Jpn 64: 3598 (1995). . T Nishino, K Okunishi, J. Phys. Soc. Jpn. 65891T. Nishino and K. Okunishi, J. Phys. Soc. Jpn 65: 891 (1996). . E Carlon, F Iglói, Phys. Rev. B. 577877E. Carlon and F. Iglói, Phys. Rev. B 57: 7877 (1998). . F Iglói, E Carlon, Phys. Rev. B. 593783F. Iglói and E. Carlon, Phys. Rev. B 59: 3783 (1999). . J B Kogut, Rev. Mod. Phys. 51659J. B. Kogut, Rev. Mod. Phys. 51: 659 (1979). . J Sólyom, P Pfeuty, Phys. Rev. B. 24218J. Sólyom and P. Pfeuty, Phys. Rev. B 24: 218 (1981).
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[ "Network Visualization of ChatGPT Research: a study based on term and keyword co-occurrence network analysis", "Network Visualization of ChatGPT Research: a study based on term and keyword co-occurrence network analysis" ]
[ "Deep Kumar ", "Kirtania Librarian ", "Bankura Sammilani College " ]
[]
[]
The main objective of this paper is to identify the major research areas of ChatGPT through term and keyword co-occurrence network mapping techniques. For conducting the present study, total of 577 publications were retrieved from the Lens database for the network visualization. The findings of the study showed that "chatgpt" occurrence in maximum number of times followed by its related terms such as artificial intelligence, large language model, gpt, study etc. This study will be helpful to library and information science as well as computer or information technology professionals.
10.2139/ssrn.4406624
[ "https://export.arxiv.org/pdf/2304.01948v1.pdf" ]
257,921,317
2304.01948
e5823924627d207bb514da2c9e9254d5951f91b2
Network Visualization of ChatGPT Research: a study based on term and keyword co-occurrence network analysis Deep Kumar Kirtania Librarian Bankura Sammilani College Network Visualization of ChatGPT Research: a study based on term and keyword co-occurrence network analysis Network VisualizationTerm co-occurrenceKeyword co-occurrenceArtificial IntelligenceChatGPTBibliometrics The main objective of this paper is to identify the major research areas of ChatGPT through term and keyword co-occurrence network mapping techniques. For conducting the present study, total of 577 publications were retrieved from the Lens database for the network visualization. The findings of the study showed that "chatgpt" occurrence in maximum number of times followed by its related terms such as artificial intelligence, large language model, gpt, study etc. This study will be helpful to library and information science as well as computer or information technology professionals. together frequently in a text corpus, indicating a relationship or association between them (Scott & Tribble, 2006;Hunston, 2007;Santoni & Pourabbas, 2016;Lozano et al, 2019). Term and keyword co-occurrence analysis can be a powerful tool for identifying meaningful patterns in text data and gaining insights into the relationships between different concepts and keywords (Bondi, 2010;Cech, 2017;Radhakrishnan, Erbis, Isaacs & Kamarthi, 2017;Weerasekara et al, 2022). These network visualization tools can be helpful in analyzing the usage and impact of ChatGPT's keywords in academic publications. Recently two research papers have been carried out on ChatGPT and bibliometrics (Kirtania, 2023;Farhat, Sohail & Madsen, 2023) and here ChatGPT is discussed with advantages, limitations and trustworthiness. Visualizing this type of networks for analyzing the impact of ChatGPT's research and its keywords in the academic community, and can help researchers gain valuable insights into the broader research landscape. The present paper makes a small attempt to do this. Objective: The main objective of this study is to identify the major research areas of ChatGPT via term and keyword co-occurrence network mapping techniques. Limitation: There are not many publications on Chat GPT as it is a very recent topic. This is a limitation of the current study because, the more publications on this type of work, the more valid the research will be. Visualization of the Results Major Findings of the Study The major findings of this study are: I. 6908 terms were obtained from 577 publications by extracting the term co-occurrence of title and abstract filed. 50 most occurrencewords out of 6908 terms are given in Appendix 1 where terms, their occurrences time, relevance score and percentage are given. From that table it can be seen that the term chatgpt score the highest position with 1276 occurrence, followed by question (215), study (208), model (188) etc. The strong relationship between these terms is easily seen from the Network Visualization image (Fig 1, 2 & 3). II. Keyword Co-occurrence mapping (Fig 4) of all keywords shows that artificial intelligence is at the first place followed by chatgpt, humans, machine learning, ethics, publishing etc. III. Keyword Co-occurrence mapping (Fig 5) of author keywords shows that chatgpt, artificial intelligence and machine learning are the most use term followed by ethics, publishing, ai, chatbot, medical education etc. IV. Top research areas of ChatGPT shows that most papers are published under the field of computer science (209), followed by artificial intelligence (123) and psychology (115) respectively. In addition, many other topics are already being worked on with ChatGPT, as it can be easily seen from the word cloud (Fig 5) provided by the Lens database. From the subject trends, it is easy to say that the application of ChatGPT in various topics is currently being worked by the researcher globally and more research will be done on this topic in the future. Conclusion: Network visualization, co-occurrence network analysis has become a very important subject in the current context. With the help of this we can understand the relationship of various entities and we can easily analyze the research based relationship between them. In the present paper, a visualization network analysis of all the publications published on ChatGPT has been carried out. The present paper shows that keywords related to ChatGPT are the most used and these words have the highest occurrence and relevance score. ChatGPT is a completely recent topic and there is still a lot of research to be done on this topic. It is easy to expect that there will be much more work on Chat GPT in the future, and more network analysis of that work. That is, it can be easily said that more research work can be done on this topic in the future. Methodology: The Lens (https://www.lens.org/) database was searched with scholarly works under ChatGPT and 577 research outputs were considered for the present study. The Lens or Patent Lens is an online patent and scholarly literature search database and analytics tool which providing access to a global corpus of scholarly literature metadata with citation indexing. Those 577 publications were downloaded in excel format. Later those excel files were used in network analysis through VOSviewer (https://www.vosviewer.com/) software and keywords were extracted from there. Finally, the data visualization analysis has been done with the help of VOSviewer software and Word Cloud of major research area completed with the help of Lens database. Fig 1 : 1Network Visualization of term co-occurrence of title and abstract Fig 2: Overlay Visualization of term co-occurrence of title and abstract Fig 3: Network Visualization of term co-occurrence of abstract Fig 4: Network Visualization of Keyword co-occurrence of all keywords Fig 5: Network Visualization of Keyword co-occurrence of author keywords Fig 5: Word cloud of Top fields of study (https://www.lens.org/) Conceptual structure and perspectives on entrepreneurship education research: A bibliometric review. G Aparicio, T Iturralde, A Maseda, European research on management and business economics. 253Aparicio, G., Iturralde, T., & Maseda, A. (2019). Conceptual structure and perspectives on entrepreneurship education research: A bibliometric review. European research on management and business economics, 25(3), 105-113. Perspectives on keywords and keyness. M Bondi, Keyness in texts. Bondi, M. (2010). Perspectives on keywords and keyness. Keyness in texts, 1-20. Exploring emerging topics in social informatics: an online real-time tool for keyword co-occurrence analysis. F Cech, Social Informatics: 9th International Conference. Oxford, UKSpringer International PublishingProceedings, Part II 9Cech, F. (2017). Exploring emerging topics in social informatics: an online real-time tool for keyword co-occurrence analysis. In Social Informatics: 9th International Conference, SocInfo 2017, Oxford, UK, September 13-15, 2017, Proceedings, Part II 9 (pp. 527- 536). Springer International Publishing. The bibliometric analysis of scholarly production: How great is the impact?. O Ellegaard, J A Wallin, Scientometrics. 105Ellegaard, O., & Wallin, J. A. (2015). The bibliometric analysis of scholarly production: How great is the impact?. Scientometrics, 105, 1809-1831. How Trustworthy is ChatGPT? The Case of Bibliometric Analyses. F Farhat, S S Sohail, D Ø Madsen, 10.20944/preprints202303.0479.v1Farhat, F., Sohail, S. S., & Madsen, D. Ø. (2023) How Trustworthy is ChatGPT? The Case of Bibliometric Analyses. Preprints.org https://doi.org/10.20944/preprints202303.0479.v1. Semantic prosody revisited. S Hunston, International journal of corpus linguistics. 122Hunston, S. (2007). Semantic prosody revisited. International journal of corpus linguistics, 12(2), 249-268. Bibliometric visualisation: An application in tourism crisis and disaster management research. Y Jiang, B W Ritchie, P Benckendorff, Current Issues in Tourism. 2216Jiang, Y., Ritchie, B. W., & Benckendorff, P. (2019). Bibliometric visualisation: An application in tourism crisis and disaster management research. Current Issues in Tourism, 22(16), 1925-1957. ChatGPT as a Tool for Bibliometrics Analysis: Interview with ChatGPT. D K Kirtania, Kirtania, D. K. (2023). ChatGPT as a Tool for Bibliometrics Analysis: Interview with ChatGPT. Available at SSRN https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4391794 Revealing research themes and trends in knowledge management: From 1995 to 2010. Knowledge-Based Systems. M R Lee, T T Chen, 28Lee, M. R., & Chen, T. T. (2012). Revealing research themes and trends in knowledge management: From 1995 to 2010. Knowledge-Based Systems, 28, 47-58. A bibliometric analysis and visualization of medical big data research. H Liao, M Tang, L Luo, C Li, F Chiclana, X J Zeng, Sustainability. 101166Liao, H., Tang, M., Luo, L., Li, C., Chiclana, F., & Zeng, X. J. (2018). A bibliometric analysis and visualization of medical big data research. Sustainability, 10(1), 166. Complex network analysis of keywords co-occurrence in the recent efficiency analysis literature. S Lozano, L Calzada-Infante, B Adenso-Díaz, S García, Scientometrics. 120Lozano, S., Calzada-Infante, L., Adenso-Díaz, B., & García, S. (2019). Complex network analysis of keywords co-occurrence in the recent efficiency analysis literature. Scientometrics, 120, 609-629. Reviewing the transport domain: An evolutionary bibliometrics and network analysis. A Najmi, T H Rashidi, A Abbasi, S Travis Waller, Scientometrics. 110Najmi, A., Rashidi, T. H., Abbasi, A., & Travis Waller, S. (2017). Reviewing the transport domain: An evolutionary bibliometrics and network analysis. Scientometrics, 110, 843-865. Constructing bibliometric networks: A comparison between full and fractional counting. A Perianes-Rodriguez, L Waltman, N J Van Eck, Journal of informetrics. 104Perianes-Rodriguez, A., Waltman, L., & Van Eck, N. J. (2016). Constructing bibliometric networks: A comparison between full and fractional counting. Journal of informetrics, 10(4), 1178-1195. Science mapping and visualization tools used in bibliometric & scientometric studies: An overview. P Pradhan, Pradhan, P. (2017). Science mapping and visualization tools used in bibliometric & scientometric studies: An overview. https://ir.inflibnet.ac.in/bitstream/1944/2132/1/INFLIBNET%20NEWSLETTER%20 Novel keyword cooccurrence network-based methods to foster systematic reviews of scientific literature. S Radhakrishnan, S Erbis, J A Isaacs, S Kamarthi, PloS one. 123172778Radhakrishnan, S., Erbis, S., Isaacs, J. A., & Kamarthi, S. (2017). Novel keyword co- occurrence network-based methods to foster systematic reviews of scientific literature. PloS one, 12(3), e0172778. Evaluating impact from research: A methodological framework. M S Reed, M Ferre, J Martin-Ortega, R Blanche, R Lawford-Rolfe, M Dallimer, J Holden, Research Policy. 504104147Reed, M. S., Ferre, M., Martin-Ortega, J., Blanche, R., Lawford-Rolfe, R., Dallimer, M., & Holden, J. (2021). Evaluating impact from research: A methodological framework. Research Policy, 50(4), 104147. Automatic detection of words associations in texts based on joint distribution of words occurrences. D Santoni, E Pourabbas, Computational Intelligence. 324Santoni, D., & Pourabbas, E. (2016). Automatic detection of words associations in texts based on joint distribution of words occurrences. Computational Intelligence, 32(4), 535-560. M Scott, C Tribble, Textual patterns: Key words and corpus analysis in language education. John Benjamins Publishing22Scott, M., & Tribble, C. (2006). Textual patterns: Key words and corpus analysis in language education (Vol. 22). John Benjamins Publishing. Past, current, and future perspectives on eco-tourism: A bibliometric review between. Z T Shasha, Y Geng, H P Sun, W Musakwa, L Sun, Environmental Science and Pollution Research. 27Shasha, Z. T., Geng, Y., Sun, H. P., Musakwa, W., & Sun, L. (2020). Past, current, and future perspectives on eco-tourism: A bibliometric review between 2001 and 2018. Environmental Science and Pollution Research, 27, 23514-23528. Trends in Adopting Industry 4.0 for Asset Life Cycle Management for Sustainability: A Keyword Co-Occurrence Network Review and Analysis. S Weerasekara, Z Lu, B Ozek, J Isaacs, S Kamarthi, Sustainability. 141912233Weerasekara, S., Lu, Z., Ozek, B., Isaacs, J., & Kamarthi, S. (2022). Trends in Adopting Industry 4.0 for Asset Life Cycle Management for Sustainability: A Keyword Co- Occurrence Network Review and Analysis. Sustainability, 14(19), 12233. Mapping the scholarly literature found in Scopus on "research data management": A bibliometric and data visualization approach. L Zhang, N Eichmann-Kalwara, Journal of Librarianship and Scholarly Communication. 71Zhang, L., & Eichmann-Kalwara, N. (2019). Mapping the scholarly literature found in Scopus on "research data management": A bibliometric and data visualization approach. Journal of Librarianship and Scholarly Communication, 7(1). . 10.7710/2162-3309.2266https://doi.org/10.7710/2162-3309.2266
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[ "Higgs Portals for Thermal Dark Matter -EFT Perspectives and the NMSSM", "Higgs Portals for Thermal Dark Matter -EFT Perspectives and the NMSSM", "Higgs Portals for Thermal Dark Matter -EFT Perspectives and the NMSSM", "Higgs Portals for Thermal Dark Matter -EFT Perspectives and the NMSSM" ]
[ "Sebastian Baum [email protected] \nDepartment of Physics\nThe Oskar Klein Centre for Cosmoparticle Physics\nStockholm University\n10691Alba Nova, StockholmSweden\n\nKTH Royal Institute of Technology and Stockholm University\nRoslagstullsbacken 2310691Nordita, StockholmSweden\n", "Marcela Carena [email protected] \nFermi National Accelerator Laboratory\nP. O. Box 50060510BataviaILUSA\n\nEnrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA\n", "Nausheen R Shah [email protected] \nDepartment of Physics & Astronomy\nWayne State University\n48201DetroitMIUSA\n", "Carlos E M Wagner [email protected] \nEnrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA\n\nHEP Division\nArgonne National Laboratory\n9700 Cass Ave60439ArgonneILUSA\n", "Sebastian Baum [email protected] \nDepartment of Physics\nThe Oskar Klein Centre for Cosmoparticle Physics\nStockholm University\n10691Alba Nova, StockholmSweden\n\nKTH Royal Institute of Technology and Stockholm University\nRoslagstullsbacken 2310691Nordita, StockholmSweden\n", "Marcela Carena [email protected] \nFermi National Accelerator Laboratory\nP. O. Box 50060510BataviaILUSA\n\nEnrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA\n", "Nausheen R Shah [email protected] \nDepartment of Physics & Astronomy\nWayne State University\n48201DetroitMIUSA\n", "Carlos E M Wagner [email protected] \nEnrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA\n\nHEP Division\nArgonne National Laboratory\n9700 Cass Ave60439ArgonneILUSA\n" ]
[ "Department of Physics\nThe Oskar Klein Centre for Cosmoparticle Physics\nStockholm University\n10691Alba Nova, StockholmSweden", "KTH Royal Institute of Technology and Stockholm University\nRoslagstullsbacken 2310691Nordita, StockholmSweden", "Fermi National Accelerator Laboratory\nP. O. Box 50060510BataviaILUSA", "Enrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA", "Department of Physics & Astronomy\nWayne State University\n48201DetroitMIUSA", "Enrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA", "HEP Division\nArgonne National Laboratory\n9700 Cass Ave60439ArgonneILUSA", "Department of Physics\nThe Oskar Klein Centre for Cosmoparticle Physics\nStockholm University\n10691Alba Nova, StockholmSweden", "KTH Royal Institute of Technology and Stockholm University\nRoslagstullsbacken 2310691Nordita, StockholmSweden", "Fermi National Accelerator Laboratory\nP. O. Box 50060510BataviaILUSA", "Enrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA", "Department of Physics & Astronomy\nWayne State University\n48201DetroitMIUSA", "Enrico Fermi Institute and Kavli Institute for Cosmological Physics\nUniversity of Chicago\n60637ChicagoILUSA", "HEP Division\nArgonne National Laboratory\n9700 Cass Ave60439ArgonneILUSA" ]
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We analyze a low energy effective model of Dark Matter in which the thermal relic density is provided by a singlet Majorana fermion which interacts with the Higgs fields via higher dimensional operators. Direct detection signatures may be reduced if blind spot solutions exist, which naturally appear in models with extended Higgs sectors. Explicit mass terms for the Majorana fermion can be forbidden by a Z 3 symmetry, which in addition leads to a reduction of the number of higher dimensional operators. Moreover, a weak scale mass for the Majorana fermion is naturally obtained from the vacuum expectation value of a scalar singlet field. The proper relic density may be obtained by the s-channel interchange of Higgs and gauge bosons, with the longitudinal mode of the Z boson (the neutral Goldstone mode) playing a relevant role in the annihilation process. This model shares many properties with the Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) with light singlinos and heavy scalar and gauge superpartners. In order to test the validity of the low energy effective field theory, we compare its predictions with those of the ultraviolet complete NMSSM. Extending our framework to include Z 3 neutral Majorana fermions, analogous to the bino in the NMSSM, we find the appearance of a new bino-singlino well tempered Dark Matter region.
10.1007/jhep04(2018)069
[ "https://arxiv.org/pdf/1712.09873v2.pdf" ]
113,398,818
1712.09873
f0553edfa249f88ab7c7ea6d81d939a194a58c7e
Higgs Portals for Thermal Dark Matter -EFT Perspectives and the NMSSM 17 Apr 2018 Sebastian Baum [email protected] Department of Physics The Oskar Klein Centre for Cosmoparticle Physics Stockholm University 10691Alba Nova, StockholmSweden KTH Royal Institute of Technology and Stockholm University Roslagstullsbacken 2310691Nordita, StockholmSweden Marcela Carena [email protected] Fermi National Accelerator Laboratory P. O. Box 50060510BataviaILUSA Enrico Fermi Institute and Kavli Institute for Cosmological Physics University of Chicago 60637ChicagoILUSA Nausheen R Shah [email protected] Department of Physics & Astronomy Wayne State University 48201DetroitMIUSA Carlos E M Wagner [email protected] Enrico Fermi Institute and Kavli Institute for Cosmological Physics University of Chicago 60637ChicagoILUSA HEP Division Argonne National Laboratory 9700 Cass Ave60439ArgonneILUSA Higgs Portals for Thermal Dark Matter -EFT Perspectives and the NMSSM 17 Apr 2018 We analyze a low energy effective model of Dark Matter in which the thermal relic density is provided by a singlet Majorana fermion which interacts with the Higgs fields via higher dimensional operators. Direct detection signatures may be reduced if blind spot solutions exist, which naturally appear in models with extended Higgs sectors. Explicit mass terms for the Majorana fermion can be forbidden by a Z 3 symmetry, which in addition leads to a reduction of the number of higher dimensional operators. Moreover, a weak scale mass for the Majorana fermion is naturally obtained from the vacuum expectation value of a scalar singlet field. The proper relic density may be obtained by the s-channel interchange of Higgs and gauge bosons, with the longitudinal mode of the Z boson (the neutral Goldstone mode) playing a relevant role in the annihilation process. This model shares many properties with the Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) with light singlinos and heavy scalar and gauge superpartners. In order to test the validity of the low energy effective field theory, we compare its predictions with those of the ultraviolet complete NMSSM. Extending our framework to include Z 3 neutral Majorana fermions, analogous to the bino in the NMSSM, we find the appearance of a new bino-singlino well tempered Dark Matter region. Introduction While the Standard Model (SM) is extremely successful in describing the known particle interactions, it fails to explain the large scale structure of the Universe, since it does not provide a good Dark Matter (DM) candidate. The simplest addition to the SM particle content would be in the form of a SM gauge singlet and in this work we shall concentrate on the particular example of a fermion as our DM candidate. In recent years, DM-nucleon scattering experiments such as LUX, XENON1T and PandaX-II have set stringent bounds on the possible couplings of DM to SM particles [1][2][3][4][5][6]. In particular, the coupling of the 125 GeV Higgs boson to DM is significantly constrained. In addition, the vector coupling of DM to the Z gauge boson must be very small (see for instance Ref. [7]), therefore we shall concentrate on singlet Majorana fermions, which couple only axially to the Z boson. Such a fermion has the same gauge quantum numbers as a right-handed neutrino. One can define a matter parity, based on the (B −L) quantum numbers of particles, namely P = (−1) 3(B−L) , and demand interactions to be invariant under such a parity. Assuming that the DM carries no baryon B or lepton L numbers, this forbids all renormalizable interactions of the DM with SM particles, while allowing all SM Yukawa terms and Majorana masses for the right-handed neutrinos. Since the coupling of DM to SM mediators is strongly constrained, we shall consider extending the SM by additional particles which can mediate interactions between DM and SM particles. Experimental precision tests of the SM strongly constrain extensions of the SM gauge sector, while far less is known about the SM Higgs sector. A well studied extension of the SM Higgs sector are so-called two Higgs doublet models (2HDMs) [8], which consist of adding a second Higgs doublet, as commonly found in models that provide a dynamical origin of the electroweak symmetry breaking (EWSB) mechanism. The interactions of DM may quite generally be described by a set of non-renormalizable operators, including Majorana fermion bilinears and SM gauge invariant operators. The lower dimensional operators involve interactions with the Higgs fields and constitute a simple generalization of the socalled Higgs portal models [9][10][11][12]. As we shall see, the extended Higgs sector allows for the existence of blind spots where the interaction of the Higgs bosons with DM particles may be reduced, satisfying direct detection constraints, while still allowing for the possibility of obtaining the observed (thermal) relic density [13][14][15][16][17][18][19]. We shall require DM to be a weakly interacting massive particle (WIMP). In order to obtain a weak scale mass for the Majorana fermion in a natural way, we demand it to proceed from the vacuum expectation value (vev) of a singlet scalar field, which develops in the process of EWSB. The absence of explicit masses may be the result of the presence of an explicit Z 3 symmetry, which also reduces the allowed number of higher dimensional operators and leads to a redefinition of the blind spot condition. A possible realization of these class of models is provided by supersymmetric (SUSY) extensions of the SM [20] which also allow for a dynamical explanation for the weak scale [20][21][22]. A particular virtue of the SUSY framework is that the stability of the Higgs mass parameter under quantum corrections can be ensured. In minimal extensions, the SM-like Higgs boson is naturally light [23][24][25], and corrections to electroweak precision and flavor observables tend to be small, leading to good agreement with observations. Additionally, low scale SUSY leads to the unification of couplings at high energies and provides a natural DM candidate, namely the lightest neutralino. Among the simplest SUSY extensions, the Next-to-Minimal Supersymmetric extension of the SM (NMSSM) [26], fulfills all of the above properties while additionally containing a rich Higgs and neutralino spectrum. This may have an important impact on low energy observables. In particular, if the lighter neutralinos and neutral Higgs bosons are mainly singlets, they would be predominantly produced in association with heavier Higgs bosons or from the cascade decays of other SUSY particles, and therefore can easily avoid current direct experimental constraints [27][28][29][30][31][32][33]. The light neutralino in the NMSSM is naturally mostly singlino-like, but with a nonnegligible Higgsino component. Hence, its spin independent direct DM detection (SIDD) cross section is mediated predominantly via the SM-like Higgs boson. The current bounds on the SIDD cross section lead to relevant constraints on the couplings of DM to the SMlike Higgs boson, and demand the theory to be in the proximity of blind spots, where the contributions from the non-standard Higgs bosons become also relevant. The proper relic density may also be obtained; the thermal annihilation cross section is dominated by either resonant contributions of the Higgs bosons, or non-resonant Z boson exchange contributions, with subdominant contributions from the light CP-even or the CP-odd Higgs bosons, the latter also having a large singlet component. In addition, a bino-like neutralino region with non-negligible Higgsino component may be present. In such a case, a sufficiently large thermal annihilation cross section yielding the proper relic density can be obtained by co-annihilation with the next-to-lightest neutralino, generally the singlino. In Section 2 we use the language of Effective Field Theories (EFT) to outline the generic requirements of a model with singlet Majorana fermion DM and identify the required extended Higgs sector. In particular, we show the correlations of EFT parameters necessary to simultaneously obtain a thermal relic density, satisfy SIDD constraints, and accommodate a phenomenologically consistent Higgs sector. In Section 3 we discuss the NMSSM as a possible ultraviolet completion of our EFT model, and demonstrate the mapping of EFT parameters to NMSSM parameters utilizing a top-down EFT approach. In Section 4 we use the mature numerical tools available for the NMSSM to study the DM phenomenology, taking into account current collider and astrophysical constraints, as well as projections for the future. We identify two viable regions of parameter space with different DM phenomenology: 1) a new well tempered DM region, where the DM candidate is mostly bino-like and thermal production proceeds via resonant annihilation or co-annihilation with the singlino-like state, and 2) the region where the DM candidate is mostly singlino-like and the thermal relic density is mainly achieved via interactions mediated by the longitudinal mode of the Z boson, the neutral Goldstone mode. Much of the phenomenology in both regions can be understood from the properties of the EFT worked out in Section 2, although some details are only found in complete models such as the NMSSM. We reserve Section 5 for our conclusions. An EFT for Singlet Dark Matter As motivated in the introduction, we will consider a model of SM singlet Majorana fermion DM, which has no renormalizable interactions with SM particles. In order to couple DM to the SM, we consider a 2HDM Higgs sector, more specifically, we shall take two Higgs doublets with opposite hypercharges Y , H u with Y = +1/2 and H d with Y = −1/2, which are naturally responsible for generating masses for the up and down-type quarks, respectively, as in type II 2HDMs. Since non-renormalizable interactions are suppressed by the scale associated with the masses of a heavy sector that was integrated out, one expects the dominant interactions to be associated with lower dimensional operators. Including operators of dimension d ≤ 5, the generic Lagrangian density describing interactions of a Majorana fermion χ with the two Higgs doublets H u , H d is L = − χχ µ δH u ·H d + γ(H † d H d + H † u H u ) − m χ 2 χχ + h.c. ,(2.1) where we have imposed a symmetry H d ↔ H u and used a dot notation for SU (2) products, H u ·H d = H + u H − d − H 0 u H 0 d . (2.2) Assuming, as usual, that both Higgs doublets acquire vevs, H d = v d , H u = v u , with (v 2 d + v 2 u ) = (174 GeV) 2 and tan β = v u /v d ,H SM = √ 2Re sin βH 0 u + cos βH 0 d , (2.3) G 0 = √ 2Im sin βH 0 u − cos βH 0 d , (2.4) H NSM = √ 2Re cos βH 0 u − sin βH 0 d ,(2. 5) A NSM = √ 2Im cos βH 0 u + sin βH 0 d ,(2.6) where H 0 d and H 0 u denote the neutral components of the respective Higgs doublets. These may be related to the usual type II 2HDM Higgs bosons by the relations H i d = ij H j * 1 , H i u = H i 2 . (2.7) The H SM interaction eigenstate has the same couplings to SM particles as a SM Higgs boson, G 0 is the (neutral) Goldstone mode making up the longitudinal polarization of the Z boson after EWSB, and H NSM and A NSM are the non SM-like CP-even and CP-odd states, respectively. In particular, note that the Higgs basis fields are defined such that all the SM vev is acquired by the field corresponding to the neutral component of H SM , hence H SM = √ 2v and H NSM = 0. Since the observed 125 GeV Higgs state h appears to be close to SM-like in nature [41,42], the interactions of χ with h may be obtained from the above, approximating h ∼ H SM to first order. Ignoring the charged Higgs fluctuations, we obtain, at linear order in the fields, H u ·H d → − v 2 2 s 2β − v √ 2 s 2β H SM + c 2β H NSM + iA NSM . (2.8) Hence, the SM-like Higgs coupling to DM becomes g χχh g χχH SM = √ 2v µ (δ sin 2β − 2γ) . (2.9) The interaction of a Majorana fermion with the SM-like Higgs boson listed above may be suppressed in three scenarios: 1) suppression of the couplings δ and γ; 2) large values of µ v; and 3) a particular correlation of the two couplings δ and γ resulting in g χχh ∼ 0. The last scenario, the so-called blind spot solution, is given by sin 2β = 2γ/δ . (2.10) It is interesting to consider a model in which there are no explicit mass terms or scales and hence the Lagrangian is scale invariant. In such a situation, a natural way to generate the mass m χ and the scale µ is via the vev of a singlet S = S + 1 √ 2 H S + iA S . Hence, without loss of generality we can define m χ = 2κ S and µ = λ S , where κ and λ are dimensionless parameters. The absence of explicit scale dependence could be understood as originating from a Z 3 symmetry, under which all scalar and fermion fields transform like Ψ → exp[2πi/3] Ψ (therefore also µ → exp[2πi/3] µ). Besides forbidding explicit fermion mass terms, imposing such a Z 3 symmetry also forbids certain interactions. The remaining d ≤ 5 terms are L = − χχ µ (δH u ·H d ) − κSχχ + h.c. ,(2.11) resulting in the following DM-Higgs sector interactions: g χχH SM = √ 2v µ δ sin 2β , g χχH NSM = √ 2v µ δ cos 2β , g χχA NSM = i √ 2v µ δ , g χχH S = ig χχA S = − √ 2κ . (2.12) Imposing the Z 3 symmetry removes the possibility of a blind spot as defined in Eq. (2.10). The contributions from the χχS coupling to either the thermal annihilation cross section relevant for the relic density or the SIDD cross section is further suppressed by singlet-doublet mixing since the singlet S does not couple to SM particles beyond the Higgs sector. Hence, the dominant contributions to the SIDD and the thermal annihilation cross section will be proportional to δ 2 . Barring accidental cancellations between contributions from different Higgs bosons, the coupling δ must be suppressed in order to satisfy the stringent bounds from direct detection experiments. Hence, since current data implies that the dimension d = 5 operators must be suppressed, we will include d = 6 operators in the following. As we shall demonstrate, this will again allow for blind spot solutions to appear, enabling the suppression of the SIDD cross section. In addition, we find relevant contributions to the annihilation cross section from d = 6 operators which will allow us to obtain sufficiently large annihilation cross sections to avoid over-closure of the Universe. The most relevant d = 6 operators are suppressed by powers of m χ /µ with respect to the d = 5 ones, and thus become most relevant if the ratio m χ /µ is not very small. One could inquire about the impact of higher dimensional d > 6 operators in such a case. We shall address this question later by considering an ultraviolet completion of the EFT. Although the qualitative features found in the EFT remain valid in the complete theory, the precise quantitative predictions will indeed be affected to some degree by higher dimensional terms. Assuming that the d > 4 terms in Eq. (2.11) originate from a theory where a heavier SU (2)-doublet Dirac fermion with mass µ has been integrated out, we can write all the allowed d = 6 operators which would arise from integrating out such a field. Ignoring the charged gauge boson interactions, we get L = − δ χχ µ (H u ·H d ) 1 − λŜ µ − κSχχ 1 + ξ H † d H d + H † u H u |µ| 2 + h.c. + α |µ| 2 χ † H † uσ µ i∂ µ − g 1 s W (T 3 − Qs 2 W )Z µ (χH u ) + χ † H † dσ µ i∂ µ − g 1 s W (T 3 − Qs 2 W )Z µ (χH d ) ,(2.13) where S = µ/λ +Ŝ, Q and T 3 are the charge and weak isospin operators, s W ≡ sin θ W with the weak mixing angle θ W , and g 1 = e/ cos θ W is the hypercharge coupling. Note, that the term proportional toŜ (the fluctuations of S) in the δ-term arises because this originally d = 5 term was actually suppressed by 1/λS, which we have expanded around the vev of S, yielding 1/λS = 1/µ − λŜ/µ 2 + O(S 2 /µ 3 ). On the other hand, all the d = 6 terms arising from integrating out a Dirac fermion are suppressed by 1/λ 2 S † S = 1/|µ| 2 + O(Ŝ|/µ 3 ) instead of 1/λ 2 S 2 = 1/µ 2 + O(Ŝ|/µ 3 ) . Moreover, we have not included terms involving higher powers of the singlet field, since they are not expected to arise from integrating out a Dirac doublet fermion. The DM interactions with singlets are dominated by the tree level coupling κ, and the only modification from such terms would be a redefinition of the Sχχ coupling κ → κ(1 + O(m χ /µ). Observe, that if dealing with on-shell χ fields, there is a redundancy in the above terms, since the application of the equation of motion on the terms proportional to the derivative of χ will lead to terms proportional to the χ mass, which also appear from the κ-term when inserting the vev of the field S. g χχh g χχH SM = √ 2v µ δ sin 2β − (ξ − α)m χ µ * , (2.14) where the dependence on αm χ results from the application of the equations of motion. In general, we calculate the on-shell relationships by using the fact that, ignoring total derivatives, i(∂ µ Φ)χ † iσ µ χ = −iΦχ † iσ µ ( ← ∂ µ + → ∂ µ )χ = im χ Φχχ + h.c. ,(2.15) where Φ is a real scalar field. Note, that the direct expansion of the derivative terms proportional to α in Eq. (2.13) leads to interactions with the CP-even Higgs bosons when the derivative is acting on the Majorana fermion fields, and to derivative interactions with the CP-odd Higgs states when the derivative is acting on the Higgs doublets, as required by hermiticity. We see that the blind spot for the cancellation of the coupling of H SM to pairs of DM now occurs for sin 2β = (ξ − α)m χ µ * δ ,(2.γ = (ξ − α)m χ 2µ * . (2.17) The χ interactions with H NSM are g χχH NSM = √ 2v µ δ cos 2β . (2.18) Note, that there are no terms proportional to γ (or m χ ) and therefore there is no blindspot such as the one for H SM in Eq. (2.16); instead g χχH NSM → 0 for tan β → 1. On the other hand, the interactions with the CP-even singlet state are given by g χχH S = − √ 2 v 2 2µ 2 δλ sin 2β + κ 1 + (ξ − α)v 2 |µ| 2 . (2.19) Here, the dependence on α comes from a field renormalization of χ necessary to retain a canonical kinetic term for χ when including dimension d ≤ 6 operators. In principle, this field renormalization introduces corrections to all couplings of χ. However, we are only considering operators of d ≤ 6. The modification from the field renormalization is suppressed by |µ| −2 , hence, this correction is only relevant for the renormalizable χχŜ interactions. The interactions of χ with the CP-odd scalars are easy to read from the above as well. For instance, although the Goldstone interactions involve derivatives of the Goldstone fields, for on-shell χ's one can use Eq. (2.15) to obtain the interaction with the (neutral) Goldstone mode g χχG 0 = −i √ 2m χ v |µ| 2 α cos 2β . (2.20) The orthogonal state, A NSM , also has relevant interactions with DM, namely g χχA NSM = i √ 2v µ δ + m χ µ * α sin 2β . (2.21) Finally, the interactions of the CP-odd singlet state A S are analogous to its CP-even counterpart, g χχA S = i √ 2 v 2 2µ 2 δλ sin 2β + κ 1 + (ξ − α)v 2 |µ| 2 . (2.22) Higgs Sector In the previous section we have motivated a structure for the scalar sector consisting of two Higgs doublets and one singlet, all three of which acquire a vev. We can define the extended Higgs Basis, {H SM , H NSM , H S } for the CP-even states and {A NSM , A S } for the CP-odd states, where the doublet components are as defined in Eqs. (2.3)-(2.6), and the singlet S = S +Ŝ = µ/λ+ 1 √ 2 H S + iA S does not get rotated [29]. These interaction eigenstates mix into mass eigenstates. We denote the CP-even mass eigenstates as h i = {h, H, h S }, h i = S SM h i H SM + S NSM h i H NSM + S S h i H S ,(2.23) and the CP-odd states as a i = {A, a S }, a i = P NSM a i A NSM + P S a i A S . (2.24) The mixing angles S j i and P j i are obtained from the diagonalization of the corresponding mass matrices. We can write the (symmetric) squared mass matrix of the CP-even Higgs for the doublet-like eigenstate, and S SM h S S S h S = − M 2 S,12 M 2 S,23 − ηM 2 S,13 m 2 h S − M 2 S,22 2 M 2 S,12 2 − m 2 h S − M 2 S,11 m 2 h S − M 2 S,22 , (2.32) S NSM h S S S h S = − ηM 2 S,12 M 2 S,13 − M 2 S,23 m 2 h S − M 2 S,11 2 M 2 S,12 2 − m 2 h S − M 2 S,11 m 2 h S − M 2 S,22 ,(2.33) for the singlet-like mass eigenstate. If we use the approximate eigenmasses, we find for the SM-like mass eigenstate S SM h ≈ 1 , S NSM h S SM h , S S h S SM h = O( , η) ,(2.34) and for the other mass eigenstates S SM H ≈ S SM h S ≈ 0 , (2.35) − S S H S NSM H ≈ S NSM h S S S h S ≈ 2M 2 S,23 M 2 S,22 − M 2 S,33 + M 2 S,22 − M 2 S,33 2 + 4 M 2 S,23 2 , (2.36) S NSM H ≈ S S h S ≈   1 + S NSM h S S S h S 2   −1/2 . (2.37) EFT: Relic Density In the absence of co-annihilation, the thermally averaged annihilation cross section for a pair of DM particles at temperature T can be expanded as σ χχ v ≡ σ (χχ → SM) v = a + b v 2 + O( v 4 ) = a + 6b x + O( 1 x 2 ) ,(2.38) where x = m χ /T . After integrating over the thermal history of the Universe until the freeze-out temperature T F = m χ /x F , the thermal relic density is obtained Ωh 2 = 0.12 80 g * 1/2 x F 25 2.3 × 10 −26 cm 3 /s σv x F , σv x F ≡ a + 3b x F . (2.39) The interactions of the singlet fermion χ with SM particles depicted in Fig. 1 arise via the couplings to the extended Higgs basis states given in Eqs.(2.14)-(2.22) and the mixing of extended Higgs basis states into mass eigenstates, g χχh i = S SM h i g χχH SM + S NSM h i g χχH NSM + S S h i g χχH S , g χχa i = P NSM a i g χχA NSM + P S a i g χχA S . (2.40) The singlet states H S and A S do not couple to SM particles, thus, assuming a type II 2HDM Yukawa structure, the couplings of the mass eigenstates to up-type quarks are given by g uh i = S SM h i + S NSM h i tan β m u √ 2v , g ua i = i P NSM a i tan β m u √ 2v ,(2.41) and to down-type quarks by g dh i = S SM h i − S NSM h i tan β m d √ 2v , g da i = iP NSM a i tan β m d √ 2v . (2.42) For completeness, we record the couplings to pairs of vector bosons g W + W − h i = 2m 2 W v S SM h i , g ZZh i = m 2 Z v S SM h i , g W + W − a i = g ZZa i = 0 . (2.43) The contribution to σv x F from annihilation into pairs of quarks (χχ → qq) from the s-channel exchange of the CP-even Higgs bosons is given by σv qq,p x F = 3 2π 3 4 T m χ 1 − m 2 q m 2 χ 3/2 i A qq h i 2 , A qq h i = −g χχh i g qh i m χ m 2 h i − 4m 2 χ ,(2.44) and from the exchange of CP-odd Higgs boson by σv qq,s x F = 3 2π 1 − m 2 q m 2 χ 1/2 i A qq a i 2 , A qq a i = −g χχa i g qa i m χ m 2 a i − 4m 2 χ , (2.45) for |m Φ i − 2m χ | Γ Φ i with Γ Φ i the width of the Higgs mass eigenstate Φ i . Note, that there is no interference between the contributions listed in Eqs. (2.44) and (2.45) since the scalar Higgs bosons exchange contribution is p-wave suppressed while the annihilation cross section via pseudoscalar Higgs bosons is s-wave. For typical freeze-out temperatures m χ /T F 25, the contribution from CP-even Higgs bosons to σv x F is suppressed by 3T F /4m χ ∼ 1/30 compared to the contribution from CP-odd Higgs bosons, as long as m q /m χ 1 such that the kinematic correction from the quark mass is irrelevant. Besides via Higgs bosons, (χχ → qq) annihilations can also be mediated by the schannel exchange of Z bosons. This is accounted for by extending the sum in Eq. (2.45) to include the s-wave amplitudes mediated by both the longitudinal polarization of the Z boson, i.e. the (neutral) Goldstone mode G 0 , A qq G 0 = −g χχG 0 g qG 0 m χ m 2 Z − 4m 2 χ , (2.46) as well as the transversal polarizations of the Z boson A qq Z = − m q m χ g χχZ g qZ m χ m 2 Z − 4m 2 χ . (2.47) The couplings of the Goldstone mode to up-type and down-type quarks, respectively, are 48) and the relevant axial-vector coupling of the transversal polarizations of the Z boson to quarks are g uG 0 = i m u √ 2v , g dG 0 = −i m d √ 2v ,(2.g uZ = −g dZ = g 1 4s W . (2.49) The coupling of the Z boson to the Majorana fermion can be read off from Eqs. (2.13) or (A.1) g χχZ = − v 2 |µ| 2 α g 1 2s W cos 2β . (2.50) Note, that the s-wave contribution to the annihilation cross section from the transversal polarization of the Z boson is suppressed with respect to that of its longitudinal polarization (the neutral Goldstone mode) by A Z /A G 0 = −(m 2 Z /4m 2 χ ) ∼ −0.023 × (m χ /300 GeV) −2 . All the amplitudes appearing in Eqs. (2.44)-(2.47) are proportional to the Yukawa couplings. Due to the hierarchy of the Yukawa couplings, the contribution to the thermal cross section from (χχ → qq) annihilations will be dominated by the heaviest accessible quarks, i.e. top-quarks for m χ > m t ∼ 173 GeV and bottom quarks for lighter m χ . In the latter case the p-wave contribution from the transversal polarization of the Z may become relevant, since in contrast to its s-wave contribution listed above it is not chirality suppressed (hence, not proportional to the Yukawa couplings). It is interesting to consider the size of the thermally averaged cross section obtainable via (χχ → qq) annihilation. For example, if we assume m χ > m t , such that (χχ → tt) is kinematically allowed, and assume the dominant annihilation channel to be via the (neutral) Goldstone mode, for m 2 t m 2 χ ,σv qq x F ∼ 2 × 10 −26 cm 3 s g χχG 0 0.1 2 m χ 300 GeV −2 . (2.51) Hence, the correct relic density Ωh 2 ∼ 0.12 [43] is obtained from (χχ → qq) annihilations for couplings |g χχG 0 | ∼ 0.1. In addition to the (χχ → qq) annihilation discussed above, there may be relevant contributions to σv x F from (χχ → Φ i Φ j ) annihilations, where Φ i denotes a scalar or pseudoscalar Higgs mass eigenstate. Such processes can be mediated either via diagrams with a Higgs or a Z boson in the s-channel, or via t/u-channel exchange of the Majorana fermion χ or the (heavy) SU (2)-doublet Dirac fermion we integrated out. In our EFT, the last possibility proceeds via the χχΦ i Φ j contact interaction terms in Eq. (2.13). Regardless of the type of diagram, the annihilation into a pair of CP-even (χχ → h i h j ) or CP-odd Higgs bosons (χχ → a i a j ) is p-wave suppressed, while (χχ → h i a j ) annihilations are swave. The corresponding s-wave contribution to the thermally averaged annihilation cross section is given by σv ha x F = 1 64πm 2 χ 1 − m h i + m a j 2 4m 2 χ 1 − m h i − m a j 2 4m 2 χ 1/2 k A h i a j k 2 , (2.52) where the sum includes the s-wave amplitudes mediated by CP-odd scalars Φ k = {a S , A, G 0 } in the s-channel A h i a j Φ k = −2m χ g χχΦ k g h i a j Φ k m 2 Φ k − 4m 2 χ , (2.53) the amplitude mediated by transversally polarized Z bosons in the s-channel A h i a j Z = −g χχZ g 1 s W P NSM a j S NSM h i m 2 h i − m 2 a j m 2 Z − 4m 2 χ , (2.54) the amplitudes mediated by the Dirac fermion in the t/u-channel proceeding via contact interaction terms after integrating out the Dirac fermion Ψ 55) and the amplitude from the t/u-channel exchange of the Majorana fermion χ A h i a j Ψ = −2g χχh i a j m χ ,(2.A h i a j χ = −2g χχh i g χχa j   1 + 2m 2 a j 4m 2 χ − m 2 h i + m 2 a j   .Φ i Φ j Φ k (g Φ i Φ j G 0 ) are the dimensionful trilinear couplings between different Higgs mass eigenstates (between the neutral Goldstone mode and two Higgs mass eigenstates), which are not related to the parameters of our EFT, but arise from the Higgs potential (see Appendices of Ref. [29] for details). The g χχΦ i Φ j are the (χχΦ j Φ k ) couplings of dimension (mass) −1 which can be read off from the Lagrangian Eqs. (2.13) or (A.1) taking into account the mixing of the Higgs mass eigenstates, Eqs. (2.23), (2.24). After accounting for suppression of couplings arising from the requirement of an m h = 125 GeV SM-like Higgs mass eigenstate and from approximately satisfying the blindspot condition, the most relevant final states for the (χχ → Φ i Φ j ) processes are (χχ → a S h) and (χχ → a S h S ). If kinematically accessible, they can compete with (χχ → tt) annihilation. For both these channels, the amplitudes mediated by the singlet-like CP-odd a S in the s-channel may play an important role. However, their relevance to the total cross section is dictated by the coupling strengths g ha S a S and g h S a S a S respectively, which as mentioned above are not related to our EFT parameters. Hence for simplicity, we will assume that these couplings are small, rendering these processes irrelevant for the relic density. Ignoring such Higgs exchange diagrams, the amplitude A h i a j Ψ mediated by a t/u-channel Dirac fermion, which we integrated out, is most relevant for the final state (χχ → a S h). Ignoring the kinematic correction in Eq. (2.52) which is relevant only very close to threshold (m h i + m a i = 2m χ ), canonical values of the thermally averaged annihilation cross section σv x F ∼ 2 × 10 −26 cm 3 s −1 can be achieved for |g χχha S | ≈ g χχH SM A S ≈ −i v µ λδ µ sin 2β + 2κξ µ * ∼ 4 × 10 −4 GeV −1 , (2.57) where we have assumed the mixing of the CP-odd Higgs bosons to be small. This corresponds to couplings δλ sin 2β + 2κξ ∼ µ 700 GeV 2 . (2.58) For the channel (χχ → a S h S ) the processes associated with a Majorana fermion χ in the t/u-channel can be relevant. Assuming again for simplicity that these processes dominate compared to the one associated to the interchange of a singlet pseudoscalar (i.e. g h S a S a S is small), neglecting the threshold corrections for (m h S + m a S ≈ 2m χ ), and corrections from singlet-doublet mixing (the latter potentially leading to O(1) suppression), the thermally averaged annihilation cross section σv x F ∼ 2 × 10 −26 cm 3 s −1 can be achieved for a DM coupling to the singlets |g χχa S | ≈ |g χχh S | ∼ 0.2 m χ 300 GeV 1/2 . (2.59) This implies |κ| ∼ 0.15 m χ 300 GeV 1/2 , (2.60) where we assumed v µ for the estimate on κ such that g χχh S ∼ √ 2κ. Annihilations into pairs of vector bosons [χχ → ZZ(W + W − )] do not play an important role for obtaining the thermal relic density as long as m χ > m t . Final states consisting of two vector bosons with longitudinal polarizations are p-wave suppressed since they correspond to annihilations into a pair of CP-odd scalars (i.e. the neutral and charged Goldstone modes for the Z and W bosons, respectively). Annihilations into a pair of transversally polarized vector bosons or one transversally polarized and one longitudinally polarized vector boson are s-wave. However, such annihilations proceeding via t/u-channel exchange of the neutral (charged) components of the SU (2)-doublet fermion we integrated out correspond to χχZZ (χχW + W − ) contact interaction terms in our EFT which would only appear at dimension d ≥ 7 and hence are strongly suppressed. For m χ > m t , annihilations mediated by an s-channel Higgs or Z boson are also suppressed: the coupling of the s-channel mediator to one transversally and one longitudinally (a pair of transversally) polarized vector bosons is proportional to the gauge couplings (squared). The couplings of the Higgs and Goldstone bosons to quarks, instead, are proportional to the top Yukawa coupling. The Higgs mediated channel is furthermore also p-wave suppressed. To summarize, the proper value of the thermally averaged annihilation cross section σv x F ∼ 2×10 −26 cm 3 s −1 leading to the observed relic density can be easily obtained when (χχ → tt) annihilations are kinematically allowed, i.e. when m χ > m t . The annihilation cross section will then typically be dominated by annihilations into top quarks mediated by the neutral Goldstone mode for which the proper value of σv x F is achieved for g χχG 0 ∼ 0.1, cf. Eq. (2.51). Annihilation into pairs of vector bosons is suppressed because it proceeds through smaller couplings. If kinematically allowed, the annihilation cross section into pairs of Higgs mass eigenstates may become large enough, cf. Eqs. (2.58), (2.59), although the annihilation into pairs of top quarks tends to be competitive or dominant unless the EFT parameters conspire to suppress the g χχG 0 coupling. For lighter DM candidates, m χ < m t , achieving a sufficiently large annihilation cross section σv x F ∼ 2 × 10 −26 cm 3 s −1 is more difficult. In this case, (χχ → bb) annihilation is dominated by the Z boson mediated p-wave contribution, and couplings are usually not sufficiently large to obtain the proper annihilation cross section. Annihilation into pairs of Higgs bosons requires large couplings between the different Higgs mass eigenstates in order to be sufficiently effective; in addition it is not easy to obtain a light enough Higgs mass spectrum (m h i + m a i < 2m χ ) while simultaneously satisfying collider constraints. Annihilation into pairs of vector bosons usually does not achieve sufficiently large cross sections either since they are controlled by gauge couplings. Hence, for m χ < m t avoiding over-closure of the Universe requires large couplings in the Higgs sector unless the annihilation cross sections via a Higgs boson Φ (a Z boson) in the s-channel is boosted via resonant annihilation, 2m χ ≈ m Φ (2m χ ≈ m Z ). EFT: Direct Detection Elastic (χq − χq) scattering relevant for direct detection proceeds via the same diagrams as annihilation in the early Universe depicted in Fig. 1, but interpreting them as t-channel exchanges of Higgs bosons. The exchange of CP-odd Higgs bosons leads to spin-dependent scattering, and the contribution of the Goldstone mode G 0 is furthermore suppressed by q 2 /m 2 Z , where q is the momentum transfer. In contrast, the exchange of CP-even Higgs bosons leads to spin-independent scattering. Since bounds on the spin-independent DMnucleon scattering (SIDD) cross section [1][2][3][4] are much stronger than those on the spindependent scattering (SDDD) cross section [5,6] we focus on SIDD in the remainder of this section. Summing coherently over the contribution from all CP-even Higgs mass eigenstates, the χ-proton SIDD cross section can be written as σ SI p = 2m 2 p π m p m χ m p + m χ 2    i=h,H,h S F u a u m u i + F d a d m d i    2 , (2.61) where m p is the mass of the proton, and the form factors (at zero momentum transfer) are 2 F u = f p T u + 4 27 1 − q=u,d,s f p T q ≈ 0.15 and F d = f p T d +f p T s + 2 27 1 − q=u,d,s f p T q ≈ 0. 13. The (a q /m q ) i parametrize the contribution to the scattering amplitude from one Higgs mass eigenstate, a q m q i = − 1 √ 2 1 m 2 h i g qh i m q g χχh i , (2.62) where the g χχh i and g qh i are given in Eqs. (2.40) and (2.41). As discussed in Section 2.1, the observation of the 125 GeV Higgs boson at the Large Hadron Collider (LHC) with couplings close to that of a SM Higgs implies that our model must contain a CP-even Higgs eigenstate h with m h ≈ 125 GeV and composition S SM h ≈ 1, {S NSM h , S S h } 1. In addition, to avoid bounds on additional Higgs bosons from the LHC and the Large Electron-Positron Collider (LEP), the remaining CP-even mass eigenstates H and h S must be either heavy m H m h or dominantly composed of H S . The coupling of the H NSM Higgs boson to down-type quarks is enhanced by tan β, and therefore at large values of tan β the suppression induced by its large mass may be compensated by an enhancement of the coupling. This case allows for effective destructive interference between the H SM and H NSM contributions to the SIDD cross section [15]. For values of tan β = O(1), as we shall use in our work, the contribution of the non-standard Higgs bosons to the SIDD cross section will either be suppressed by (a q /m q ) 2 ∝ 1/m 4 (a q /m q ) 2 ∝ {(S SM h i ) 2 , (S NSM h i ) 2 } ≤ (1 − S S h i ) 2 . Under such conditions, the SIDD cross section will be dominated by the contribution from the SM-like state h. Hence, the current stringent bounds on the SIDD cross section lead to relevant constraints on g χχh . The bounds on g χχh may be estimated by considering the SIDD cross section, taking into account only the diagrams with an h in the t-channel. We find from Eq. (2.61) σ SI p h = 2m 2 p π m p m χ m p + m χ 2 F u a u m u h + F d a d m d h 2 ∼ 5 × 10 −9 pb g χχh 0.1 2 m h 125 GeV −4 , (2.63) while the experimental limit is σ SI p (m χ = 300 GeV) 3.3 × 10 −10 pb [4]. Hence, values of g χχh 0.025 are necessary to fulfill these constraints. This range of values of g χχh may be compared with the values of g χχG 0 ∼ 0.1 necessary to obtain the observed relic density as discussed in the previous section, cf. Eq. (2.51). In general, the couplings of χ to the Higgs mass eigenstates are expected to be of similar magnitude, or larger, than g χχG 0 , since they arise at the same order, or lower, in our EFT expansion. In particular, the coupling g χχh arises from dimension d = 5 operators with the leading contribution suppressed by v/µ, while g χχG 0 arises via d = 6 operators and is suppressed by m χ v/|µ| 2 . We therefore conclude that under the requirement of obtaining an acceptable relic density, the values of −0.04 −0.02 0.00 0.02 0.04 g χχH SM −0.4 −0.2 0.0 0.2 0.4 g χ χ H N S M − i g χ χ A N S M − ig χ χ G0 m χ = 300 GeV; m H NSM = m A NSM = 500 GeV; tan β = 2; σ SI p ≤ 3.0 × 10 −10 pb; σv xF = 2.3 × 10 −26 cm 3 s −1 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 (ξ − α) m χ v/|µ| 2 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 δ v / µ α m χ v / | µ | 2 m χ = 300 GeV; m H NSM = m A NSM = 500 GeV; tan β = 2; σ SI p ≤ 3.0 × 10 − Bounds on Couplings and Parameters of the EFT In the previous section we argued that in order to suppress the SIDD cross section below experimental limits, the model must sit close to the blind spot, Eq. (2.16). It is also possible that the amplitude mediated by the h interferes destructively with those mediated by the other CP-even mass eigenstates H and h S , or both mechanisms may be at work simultaneously. As argued above, the amplitudes of the diagrams mediated by H and h S are in general suppressed with respect to the amplitude of the h-mediated diagram, such that destructive interference will only be effective when the contribution from the SMlike Higgs h is already suppressed by proximity to the blind spot condition. In order to demonstrate these properties, we shall briefly consider the simple case in which the singlet fields are heavy and play no role in the low energy effective theory. Fig. 2 shows a representative example, illustrating the bounds on the parameters of our EFT and the couplings of DM to the CP-even and CP-odd Higgs bosons necessary to satisfy the SIDD experimental constraints and obtain the proper relic density concurrently. We used characteristic values of the DM mass, m χ = 300 GeV, and the non-standard Higgs boson masses m H NSM = m A NSM = 500 GeV, and moderate values of tan β = 2. The singlets are decoupled {m H S , m A S } 500 GeV and perfect alignment is assumed in the doublet sector such that the Higgs basis states coincide with mass eigenstates. In the left panel we show the couplings of the DM to the non-standard Higgs bosons as a function of the DM coupling to the SM-like Higgs boson. As explained in the last section, ignoring the other CP-even Higgs bosons contributions, the coupling to the SMlike Higgs boson needs to be suppressed to satisfy the SIDD constraints, g χχh 0.025. As can be seen in Fig 2, this bound may be slightly relaxed in the presence of destructive interference with other Higgs bosons. The orange shaded region denotes the values of the couplings of DM to the CP-even Higgs bosons consistent with the SIDD bounds, with the boundaries denoting the values of these couplings saturating the current SIDD bound σ SI p = 3 × 10 −10 pb (dashed and solid lines). The couplings of the CP-odd Higgs bosons are constrained only by the demand of obtaining a proper relic density. 3 The solid and dashed ellipses show the values of g χχA NSM (black) and g χχG 0 (blue) needed to obtain Ωh 2 ∼ 0.12, consistent with the SM-like Higgs couplings and the boundary values of the heavy CPeven Higgs couplings denoted by the solid and dashed lines, respectively. Thick and thin lines represent two different solutions for the CP-odd/Goldstone couplings for each set of values of the CP-even Higgs bosons couplings, with points in each of the ellipses in one-to-one correspondence with similar points in the other ellipses. The clear correlation between the couplings to the Goldstone mode and to the heavy CP-odd Higgs boson may be understood from the fact that for the given Higgs and DM masses, the suppression associated with the propagator contributing to the annihilation cross section is about a factor 3 weaker for the heavy CP-odd Higgs boson than for the Goldstone mode of mass m Z : 3 × 1/ m 2 A NSM − 4m 2 χ ∼ 1/ m 2 Z − 4m 2 χ . For couplings of DM to the CP-even Higgs bosons in the shaded area, for which the SIDD cross section would be smaller than the experimental limit, the couplings of the CP-odd Higgs boson and the Goldstone mode would take intermediate values between the ones represented by the two ellipses. The right panel of Fig. 2 shows the three independent combinations of the parameters of the EFT that play a role in the determination of the relic density and the direct detection constraints. The shaded regions, solid lines, and dashed lines are in one-to-one correspondence with those in the left panel, and the shown correlations may be easily understood from the dependence of the couplings on these parameters, Eqs. (2.14), (2.18), (2.20) and (2.21). NMSSM Ultraviolet Completion The EFT model discussed in the previous section extends the SM particle content by an additional SU (2)-doublet Higgs field, as well as a complex scalar field and a Majorana fermion both of which transform as singlets under the SM gauge group. There is also an additional SU (2)-doublet Dirac fermion which is assumed to be heavy and integrated out, yielding interactions of the Majorana fermion with the Higgs doublets. This particle content is very similar to the Higgs and neutralino sector of the NMSSM. Hence, decoupling the gluinos as well as the SUSY partners of the SM fermions from the theory, the NMSSM can serve as an ultraviolet completion for the EFT, preserving all the essential features discussed above. In addition, a range of mature numerical tools are available for the NMSSM which allows us to compute particle spectra and couplings and subsequently study the DM and collider phenomenology of the model. As we shall see, the NMSSM also contains a second region of parameter space where the DM candidate does not transform under the Z 3 symmetry, however, this region can nonetheless quite simply be mapped onto our EFT. In the SUSY case such a Z 3 symmetry is defined so that all chiral superfields transform by e 2πi/3 , while gauge superfields transform trivially. Reviews of the NMSSM can be found in Refs. [26,48]. The LHC phenomenology of the NMSSM Higgs and neutralino sectors has recently been investigated in Refs. [16,[27][28][29][30][31][32][33][49][50][51][52][53], and studies of the DM phenomenology include Refs. [16,19,[52][53][54][55][56][57]. The superpotential of the Z 3 -invariant NMSSM contains no dimensionful parameters and is given by [26] W = ij λ S( H u ) i ( H d ) j − Y u ū Q i ( H u ) j − Y d d Q i ( H d ) j − Y e ē L i ( H d ) j + κ 3 S 3 ,(3.p i = {λ, κ, tan β, µ, A λ , A κ },(3.2) where λ and κ are the dimensionless couplings appearing in the superpotential Eq. (3.1), µ ≡ λ S with S the vev of S, and A λ and A κ are the dimensionful trilinear soft SUSY breaking couplings. Assuming CP conservation, one can without loss of generality choose all parameters to be real, and furthermore λ and tan β to be positive, while κ and the dimensionful parameters µ, A λ and A κ can have both signs [26]. The Higgs sector is analogous to that of our EFT model, and can be rotated to the extended Higgs basis using Eqs. (2.3)-(2.6). Including the usual soft SUSY breaking and F -and D-terms [26], and ignoring radiative corrections for presentational purposes, see e.g. Ref. [29], the symmetric squared mass matrix for the neutral CP-even Higgs bosons in the basis {H SM , H NSM , H S } is 4 M 2 S =    m 2 Z c 2 2β + λ 2 v 2 s 2 2β − m 2 Z − λ 2 v 2 s 2β c 2β 2λvµ 1 − M 2 A 4µ 2 s 2 2β − κ 2λ s 2β M 2 A + m 2 Z − λ 2 v 2 s 2 2β −λvµc 2β M 2 A 2µ 2 s 2β + κ λ 1 2 λ 2 v 2 s 2β M 2 A 2µ 2 s 2β − κ λ + κµ λ Aκ + 4κµ λ   ,(3. 3) where c β ≡ cos β, s β ≡ sin β, and we have introduced M 2 A ≡ 2µ sin 2β A λ + κµ λ . (3.4) In the basis {A NSM , A S }, the symmetric tree-level squared mass matrix for the CP-odd neutral Higgs bosons is M 2 P =   M 2 A 1 √ 2 λv M 2 A 2µ s 2β − 3κµ λ 1 √ 2 λv M 2 A 2µ s 2β − 3κµ λ 1 2 λ 2 v 2 s 2β M 2 A 4µ 2 s 2β + 3κ 2λ − 3κAκµ λ   . (3.5) Radiative corrections may be relevant and are incorporated at the two-loop level in the numerical results obtained with NMSSMTools [58]. As in the case of our EFT model, the NMSSM must contain a CP-even 125 GeV mass eigenstate mostly composed out of H SM to accommodate the SM-like Higgs boson observed at the LHC [41,42]. This can be achieved either by making the remaining CPeven states H and h S heavy, 5 M S,22 , M S, 33 M S,11 , the decoupling limit, or by setting M 2 S,12 ≈ M 2 S,13 ≈ 0, the alignment-without-decoupling limit [29,32]. Perfect alignment is achieved for [29] λ 2 = m 2 h − m 2 Z cos(2β) 2v 2 sin 2 β , M 2 A µ 2 = 4 sin 2 2β 1 − κ 2λ sin 2β ,(3.6) and in the alignment limit the mass of the SM-like Higgs mass eigenstate is given by m 2 h = M 2 S,11 = m 2 Z c 2 2β + λ 2 v 2 s 2 2β + ∆(mt h ) ,(3.7) where ∆(mt h ) are radiative corrections that are common to the MSSM. Note, that with respect to the MSSM, one obtains an additional contribution (λvs 2β ) 2 to m 2 h which allows for a 125 GeV SM-like Higgs boson mass without the need for large radiative corrections for moderate values of tan β 3 if λ ∼ 0.7. Intriguingly, the first alignment condition in Eq. (3.6), which suppresses the H SM − H NSM mixing, is satisfied for the same values of λ, namely 0.6 λ 0.7. Alignment with the singlet [the second condition in Eq. (3.6)] is also easily achieved by judicious choices of M A and µ. The neutralino sector of the NMSSM consists of the superpartners of the neutral electroweak gauge bosons, the bino B and neutral wino W 3 , the neutral Higgsinos H 0 d and 4 Note, that Ref. [29] uses the parameter M Z ≡ m 2 Z − λ 2 v 2 and the v = 246 GeV convention, while we use v = 174 GeV. 5 Note, that we use the same notation for the Higgs states as in Section 2. H 0 u belonging to the respective Higgs doublets, and the singlino S, the fermionic component of S. In the basis { B, W 3 , H 0 d , H 0 u , S}, the symmetric tree level neutralino mass matrix is M χ 0 =        M 1 0 −m Z s W c β m Z s W s β 0 M 2 m Z c W c β −m Z c W s β 0 0 −µ −λvs β 0 −λvc β 2κµ/λ        ,(3.8) where M 1 and M 2 are the bino and wino soft SUSY breaking masses. The neutralino mass eigenstates are given in terms of the interaction eigenstates by . χ i = N i1 B + N 12 W 3 + N i3 H 0 d + N i4 H 0 u + N i5 S ,(3.N 13 N 11 = g 1 √ 2 v µ s β + (m χ 1 /µ) c β 1 − (m χ 1 /µ) 2 , N 14 N 11 = − g 1 √ 2 v µ c β + (m χ 1 /µ) s β 1 − (m χ 1 /µ) 2 ,(3.N 13 N 15 = λ v µ (m χ 1 /µ) s β − c β 1 − (m χ 1 /µ) 2 , N 14 N 15 = λ v µ (m χ 1 /µ) c β − s β 1 − (m χ 1 /µ) 2 ,(3. In terms of the mixing angles, the couplings of the lightest neutralino to the Higgs basis states are g χ 1 χ 1 H SM = √ 2λN 15 (N 13 s β + N 14 c β ) + (g 1 N 11 − g 2 N 12 ) (N 13 c β − N 14 s β ) , (3.12) g χ 1 χ 1 H NSM = √ 2λN 15 (N 13 c β − N 14 s β ) − (g 1 N 11 − g 2 N 12 ) (N 13 s β + N 14 c β ) , (3.13) g χ 1 χ 1 H S = ig χ i χ j A S = √ 2 [λN 13 N 14 − κN 15 N 15 ] ,(3. 14) From these, the couplings to the Higgs mass eigenstates can be obtained using Eq. (2.40) and the mixing angles of the Higgs mass eigenstates. The singlino will play the role of the Majorana singlet in our EFT model. However, the NMSSM contains a second SU (2)-singlet neutralino, the bino. Unlike the singlino, the bino does not transform under the Z 3 symmetry since it stems from a gauge superfield. Besides allowing for an explicit mass term ( 1 2 M 1 B B + h.c.), this also results in different couplings to the Higgs doublets and the singlet than those found for the singlino. The region where the bino is the DM candidate can nonetheless be connected to our EFT in a straightforward way, as we will see in the following section. g χ 1 χ 1 A NSM = i √ 2λN 15 (N 13 c β + N 14 s β ) − (g 1 N 11 − g 2 N 12 ) (N 13 s β − N 14 c β ) , (3.15) g χ 1 χ 1 G 0 = i √2λN Top-down EFT In order to connect the NMSSM to our EFT from Section 2, we can construct a top-down EFT for the NMSSM Higgs and neutralino sector by integrating out the Higgsinos. This approach is valid as long as the Higgsino mass is large compared to the mass of the lightest neutralino. We will also assume the winos to be heavy, M 2 {M 1 , µ}. Neglecting the Yukawa terms and ignoring the charged current interactions, the terms in the Lagrangian involving the neutral components of the Higgsinos are L ⊃ ( H 0 u ) † iσ µ ∂ µ + i g 1 2 sin θ W Z µ H 0 u + ( H 0 d ) † iσ µ ∂ µ − i g 1 2 sin θ W Z µ H 0 d + λS H 0 u H 0 d + λH 0 d S − g 1 √ 2 (H 0 u ) † B H 0 u + λH 0 u S + g 1 √ 2 (H 0 d ) † B H 0 d + h.c. . (3.17) The corresponding equation of motion for H 0 u is 0 = iσ µ ∂ µ + i g 1 2 sin θ W Z µ H 0 u + λS † ( H 0 d ) † + λ(H 0 d ) † S † − g 1 √ 2 H 0 u B † ,(3.18) and for H 0 20) and d 0 = iσ µ ∂ µ − i g 1 2 sin θ W Z µ H 0 d + λS † ( H 0 u ) † + λ(H 0 u ) † S † + g 1 √ 2 H 0 d ( B) † .(H 0 d = − 1 λ 2 S † S iσ µ ∂ µ − i g 1 2 sin θ W Z µ λ(H 0 u ) † S † + g 1 √ 2 (H 0 d ) B † − 1 λS λH 0 d S − g 1 √ 2 (H 0 u ) † B ,(3.H 0 u = − 1 λ 2 S † S iσ µ ∂ µ + i g 1 2 sin θ W Z µ λ(H 0 d ) † S † − g 1 √ 2 H 0 u B † − 1 λS λH 0 u S + g 1 √ 2 (H 0 d ) † B . (3.21) Substituting these into Eq. (3.17) and keeping terms of dimension d ≤ 6 (the same order used for our generic EFT in section 2), we find L ⊃ 1 λS −λ 2 H 0 d H 0 u S S + λg 1 √ 2 (H 0 u ) † H 0 u − (H 0 d ) † H 0 d B S + g 2 1 2 (H 0 d ) † (H 0 u ) † B B + h.c. + 1 λ 2 S † S λ(H 0 d ) † S † − g 1 √ 2 H 0 u B † iσ µ ∂ µ − i g 1 2 sin θ W Z µ λH 0 d S − g 1 √ 2 (H 0 u ) † B + 1 λ 2 S † S λ(H 0 u ) † S † + g 1 √ 2 H 0 d B † iσ µ ∂ µ + i g 1 2 sin θ W Z µ λH 0 u S + g 1 √ 2 (H 0 d ) † B − κS S S + M 1 2 B B + h.c. , (3.22) where in the last line we have included the standard bino mass term and the singletsinglino-singlino interaction term. Note, that because the singlino transforms under the Z 3 symmetry, it only gets a mass when the singlet S acquires a vev. In contrast, the bino does not transform under the Z 3 and hence a soft SUSY breaking mass term M 1 is allowed. Expanding S around its vev S → µ/λ +Ŝ and correspondingly 1 λS → 1 µ − λŜ µ 2 + O Ŝ 2 µ 3 , 1 λ 2 S † S → 1 |µ| 2 + O Ŝ 2 µ 3 , (3.23) we can appreciate the similarities with the Lagrangian of our EFT model. In particular, the singlino has the same structure for the couplings as the Majorana fermion χ in Section 2, and we can map the couplings in Eq. (2.13) directly to those in the NMSSM via of (H 0 d )(H 0 u )χχ, which can be compensated for by changing the sign of the coupling of the binos to the CP-odd states coming from the Higgs doublets, i.e. A NSM and G 0 . Keeping this in mind, we can map the couplings of the bino in Eq. (3.22) to those in the EFT, Eq. (2.13), via δ = −α → −λ 2 , λ → λ , κ → κ , ξ → 0 .δ = α → g 2 1 2 , λ → λ , κ = ξ → 0 . (3.26) The blind spot condition for the bino region is then sin 2β = −m χ /µ . (3.27) Note, that the bino couples with characteristic strength g 2 1 /2 ≈ 0.06 to Higgs doublet states, whereas the singlino couples to Higgs doublet states with characteristic strength λ 2 ∼ 0.4, recalling that the presence of the 125 GeV SM-like Higgs implies λ ∼ 0.6. The couplings of pairs of (on-shell) singlinos to the extended Higgs basis states and the (neutral) Goldstone mode can be directly read off from Eq. g S SH SM = − √ 2v µ λ 2 sin 2β + √ 2m χ v |µ| 2 λ 2 , (3.28) g S SH NSM = − √ 2v µ λ 2 cos 2β , (3.29) g S SH S = ig S SA S = v 2 √ 2µ 2 λ 3 sin 2β − √ 2κ 1 − v 2 |µ| 2 λ 2 , (3.30) g S SA NSM = −i √ 2v µ λ 2 + i √ 2m χ v |µ| 2 λ 2 sin 2β , (3.31) g S SG 0 = −i √ 2m χ v |µ| 2 λ 2 cos 2β . (3.32) Similarly, the couplings of pairs of (on-shell) binos are given by 37) in agreement with the mapping of parameters in Eq. (3.26), keeping in mind the switch in the sign of the (χχA NSM ) and (χχG 0 ) couplings and the additional mass term m χ = M 1 . Note that Eq. (3.22) also induces bino-singlino-Higgs couplings which might be relevant for thermal production via co-annihilation when the bino and singlino are approximately mass degenerate, M 1 ≈ 2κµ/λ. For on-shell binos and singlinos, these couplings are g B BH SM = v √ 2µ g 2 1 sin 2β + m χ v √ 2|µ| 2 g 2 1 , (3.33) g B BH NSM = v √ 2µ g 2 1 cos 2β , (3.34) g B BH S = ig B BA S = − v 2 2 √ 2µ 2 λg 2 1 sin 2β , (3.35) g B BA NSM = −i v √ 2µ g 2 1 − i m χ v √ 2|µ| 2 g 2 1 sin 2β , (3.36) g B BG 0 = i m χ v √ 2|µ| 2 g 2 1 cos 2β ,(3.g B SH SM = − 2v µ λg 1 cos 2β , (3.38) g B SH NSM = 2v µ λg 1 sin 2β + m B − m S v |µ| 2 λg 1 , (3.39) g B SH S = ig B SA S = v 2 µ 2 λ 2 g 1 cos 2β , (3.40) g B SA NSM = i m B + m S v |µ| 2 λg 1 cos 2β , (3.41) g B SG 0 = i m B + m S v |µ| 2 λg 1 sin 2β . Dark Matter phenomenology Besides serving as an ultraviolet completion of our EFT, the NMSSM also provides a convenient computational basis since mature numerical tools are available for the analysis of both collider and DM phenomenology. In this section, we study the phenomenological properties of the NMSSM, going beyond the EFT validity conditions. Doing so, we can identify those properties that are shared with the EFT approach, while also determining the differences between the EFT and the full theory predictions. We use NMSSMTools 5.1.2 [58-62] to compute NMSSM spectra and couplings and to subsequently test parameter points against a subset of the constraints implemented in NMSSMTools (see Ref. [58] for details). In particular, points are excluded if they feature an unphysical global minimum, soft Higgs masses much larger than the SUSY scale, and if the lightest neutralino is not the lightest SUSY particle. Furthermore, we require the spectrum to contain an m h ≈ 125 GeV Higgs boson with couplings to SM particles compatible with those of the SM-like Higgs observed at the LHC. We also require compatibility with constraints from LEP, Tevatron, and the LHC on additional Higgs bosons and sparticles as implemented in NMSSMTools. For points passing these constraints we compute the relic density and the direct detection cross section with micrOMEGAs 4.3.5 [44-47, 63, 64]. We perform a random scan over 10 9 parameter points, drawing the parameters from linear-flat distributions over the ranges listed in Table 1. The choice of parameter ranges is motivated by the phenomenology of a SM-like Higgs: for tan β ≤ 5, a 125 GeV SMlike Higgs boson is obtained without the need for large radiative corrections to its mass, cf. Eq. (3.7). The range for A λ is chosen to be larger than those for µ and A κ because approximate alignment implies, from Eqs. (3.4) and (3.6), A λ = 2µ 1 sin 2β − κ λ . (4.1) The range of the bino mass M 1 is chosen such that we obtain both the case where the lightest neutralino χ 1 is mostly bino-like (i.e. when M 1 < {µ, 2κµ/λ}) and the case where the bino component of χ 1 is negligible (i.e. when min(µ, 2κµ/λ) M 1 ). In addition to the parameters listed in Table 1, we scan over the stop mass M 3 Q = M 3 U in the range [0.75; 2.5] TeV and, for definiteness, we set the stop and sbottom mixing parameters X t ≡ (A t −µ cot β) = 0 and X b ≡ (A b −µ tan β) = 0, 6 allowing for moderate radiative corrections to the Higgs masses. We decouple the remaining SUSY particles by setting all remaining sfermion mass parameters to 3 TeV, the gluino mass parameter to M 3 = 2 TeV, and, in order to minimize the wino component of the lightest neutralino, the wino mass parameter to M 2 = 10 TeV. Note, that due to this choice of parameters we also satisfy all LHC bounds on sfermions and gluinos. Due to our choice of parameters M 2 {M 1 , |µ|, |2κµ/λ|}, the heaviest neutralino χ 5 will be wino like, while the wino component of the lighter mass eigenstates will be negligible. Furthermore, since we chose |κ| ≤ 0.3, the singlino mass parameter |2κµ/λ| will practically always be smaller than the Higgsino mass parameter |µ|, while we chose the range of the bino mass parameter M 1 such that the bino can be both lighter and heavier than the singlino and the Higgsinos. This also ensures that the Higgsinos with mass parameter µ are always heavier than the lightest neutralino χ 1 , such that we omit the phenomenologically disfavored Higgsino DM region and we can map results onto our EFT, where m χ 1 /µ appears as an expansion parameter. Hence, our DM candidate, the lightest neutralino, will be either mostly singlino-like or mostly bino-like with small Higgsino admixture facilitating Higgs mediated processes. In all figures presented in this section, we compute the couplings of the neutralinos to Higgs mass eigenstates and of the SM particles to the Higgs mass eigenstates from the mixing angles N ij for the neutralinos and S ij (P ij ) for the CP-even (CP-odd) Higgs bosons as output from NMSSMTools. Computing the couplings from mixing angles, Eqs. (3.12)-(3.16), takes into account sub-leading effects from the (small) admixture of additional neutralino components. These would only appear in the EFT through higher dimensional operators, or for wino effects from including operators arising from integrating out an SU (2)-triplet fermion in addition to the operators from integrating out an SU (2)-doublet fermion. Furthermore, using the output from NMSSMTools for the mixing angles also captures effects induced by the running of the parameters from the SUSY scale to the electroweak scale. In the NMSSM, apart from the processes discussed in detail in section 2 , the annihilation cross section can be enhanced either by resonant annihilation or by co-annihilation [65]. For the non-resonant annihilation cross sections, we checked numerically that for points with an acceptable relic density Ωh 2 ∼ 0.12, annihilations into top quarks typically make up for O(80 %) of the thermally averaged cross section σv x F , while annihilations into pairs of Higgs bosons usually account for O(20 %) of σv x F . This can be quiet generically understood to be due to the difficulty in obtaining a Higgs spectrum light enough to allow for the second final state, while evading Higgs phenomenology and collider constraints. Concentrating on (χχ → tt), Fig. 3 shows the SIDD cross section vs. the contribution to the thermal annihilation cross section from the diagrams mediated by the Goldstone mode obtained from Eq. (2.45) when taking into account only the Goldstone amplitude in Eq. (2.46). The left panel shows all points from our scan passing the Higgs and LHC constraints described above but before requiring the correct relic density or compatibility with direct detection limits. The dominant composition of the DM candidate is color coded and denoted as bino B if N 2 11 ≥ 0.95 (red), singlino S if N 2 15 ≥ 0.95 (blue). Similarly, points are denoted as mixed if the sum of the square of corresponding mixing angles is N 2 1i ≥ 0.95 but none of the individual contributions is sufficiently large to put them in one of the previous categories. The behavior shown can be understood quite intuitively from our findings in sections 2 and 3.1: first of all, we note that both the SIDD and the annihilation cross section for mostly bino DM are smaller than those for mostly singlino DM because the couplings to Higgs bosons are proportional to g 2 1 /2 ≈ 0.06 for binos while for singlinos the couplings are proportional to λ 2 ∼ 0.4. Secondly, for all the different DM compositions, a striking feature of this plot is the relative independence of the SIDD cross section and σv G , which can be suppressed independently from each other. The SIDD cross section is suppressed close to the blind spot conditions m χ /µ → ± sin 2β. However, in this case the coupling to the Goldstone mode g χχG 0 ∝ m χ cos 2β/|µ| 2 remains sizable and thus the corresponding contribution to the annihilation cross section is not suppressed. Note, that the contributions to the thermal annihilation cross section from the Higgs mass eigenstates are usually smaller than those from the Goldstone mode: while the couplings are typically of the same order, the contributions from the CP-odd eigenstates are suppressed by A a i A G 0 ∼ P NSM a i tan β × m 2 a i − 4m 2 χ m 2 Z − 4m 2 χ 2 ,(4.2) and the contributions from CP-even Higgs bosons are p-wave suppressed. As argued in section 2.2, the contribution from the transverse polarizations of the Z boson is generally also much smaller than the contribution from the Goldstone mode. The right panel of Fig. 3 shows the points from our scan having approximately the correct relic density Ωh 2 = 0.12 ± 50 % and satisfying bounds from SIDD and SDDD experiments [3][4][5][6]. For mostly singlino DM, the contribution from the Goldstone mode mediated amplitudes to the thermally averaged annihilation cross section is of the order of the canonical value σv ∼ 2 × 10 −26 cm 2 /s yielding the observed relic density Ωh 2 ∼ 0.12 [43] 7 . In contrast, we find that for mostly bino DM the Goldstone mode does not mediate sufficiently large amplitudes to avoid over-closure of the Universe. Hence in the bino DM scenario, both resonant annihilation and co-annihilation play a very important role. In Fig. 4 we show points from our numerical scan which have an acceptable relic density, Ωh 2 = 0.12 ± 50 %, with the color coding indicating the possibility of resonant annihilation. In the right panel of Fig. 4 we show points with mostly singlino DM candidate in the (singlino mass)-(Higgsino mass) plane, demonstrating that neither co-annihilation nor resonant annihilation is relevant for the singlino region (M 1 is always large in this region). In the left panel of Fig. 4 we show bino-like points in the (bino mass)-(singlino mass) plane. We find that points either resonantly annihilate via the Z boson or one of the Higgs mass eigenstates, or, feature binos approximately mass degenerate with the singlino such that co-annihilation yields the correct relic density. In the latter case, it is in fact the annihilations of the mostly singlino like m χ 2 which set the relic density. We have thus found a new well tempered bino region: m χ 1 is mostly bino-like with very small couplings, evading direct detection constraints easily. However due to the presence of an almost mass degenerate singlino-like m χ 2 (which does not play a role in direct detection) which has significantly larger couplings, an observationally consistent relic density is easily obtained. The value of the µ parameter, and consequently the Higgsinos, tends to be about the same order as shown for the singlino-like DM in the right panel. In the left panel of Fig. 5 we show the SIDD cross section vs. m χ /(µ sin 2β) for points from our scan passing the Higgs and collider constraints described above but before requiring the correct relic density or compatibility with direct detection limits. For bino DM, the SIDD cross section is suppressed when the blind spot condition m χ /µ = − sin β is approximately satisfied, while for all other compositions of the DM candidate, usually singlino dominated, the SIDD cross section is suppressed for m χ /µ ≈ sin β. Note that for bino DM candidates we also find suppression of the SIDD cross section for m χ /µ ≈ 0, which corresponds to M 1 µ for which we do not find any parameter points with an acceptable relic density in our scan. This is because in this region neither co-annihilation with the singlino nor resonant annihilation with the Higgs bosons is possible, one of which would be required to boost the thermal annihilation cross section to avoid over-closure of the Universe. In the right panel of Fig. 5 we show the SIDD cross section vs. the DM coupling to the SM-like 125 GeV Higgs mass eigenstate. Besides the coupling to the H SM Higgs basis state, which vanishes at the respective blind spots, this takes into account the contributions to the coupling from the (small) admixtures of the H NSM and H S interaction eigenstates to the 125 GeV mass eigenstate. We find that for mostly bino points the SIDD cross section is very tightly correlated with the coupling to the 125 GeV SM-like Higgs mass eigenstate, and SIDD cross sections satisfying the current experimental bounds can be achieved by suppression of the g B Bh coupling. In the case of mostly singlino DM we find this correlation to be looser, indicating that the SIDD cross sections must be suppressed by additional mechanisms. In Fig. 6 we show the contributions to the SIDD cross section when taking into account only one Higgs mass eigenstate at a time as obtained from Eq. (2.61) ignoring the sum over the CP-even Higgs mass eigenstates. We show these contributions plotted against the full SIDD cross section in the left (right) panel for mostly bino (singlino) points from our dataset satisfying all Higgs/collider constraints described above and featuring an acceptable relic density. For mostly bino DM, we find that SIDD cross sections as small as σ SI p ∼ 10 −13 pb can be obtained by suppression of the coupling to the SM-like Higgs mass eigenstate alone. Destructive interference between different Higgs mass eigenstates is needed only for even For mostly singlino DM, as shown in the right panel of Fig. 6, destructive interference between different Higgs mass eigenstates is almost always required to satisfy the experimental bounds on the SIDD cross section. This can be understood from the typical strength of singlino couplings σ SI p ∝ g 2 S Sh ∝ λ 4 ∼ 0.1, while binos couple with characteristic strength σ SI p ∝ g 2 B Bh ∝ g 4 1 /4 ∼ 0.004. In addition, compared to bino DM, singlino DM has a much larger coupling to the scalar singlet state H S due to the presence of the tree-level coupling κ, cf. Eqs. (3.30) and (3.35). Hence we see the necessity of destructive interference between the contributions from the singlet like mass eigenstate h S and the SM-like mass eigenstate h to suppress the SIDD cross section below the experimental limits. Although blindspot cancellation or destructive interference arguably require some fine tuning, we stress that we readily find points in our dataset with SIDD cross sections below σ SI p 10 −13 pb, out of reach of direct detection experiments for the foreseeable future. Such small cross sections are challenging to probe with current direct detection strategies due to the presence of the neutrino floor. In Fig. 7 we show the constraints from direct detection experiments for points from our scan with an acceptable relic density and satisfying collider constraints. The left panel shows the SIDD cross section vs. the DM mass. Note, that almost all parameter points with DM masses below the top mass m χ < m t ≈ 173 GeV are ruled out by current SIDD Figure 7. Left: The SIDD cross section vs. the mass of the DM candidate m χ for points from our parameter scan passing the required experimental collider constraints with acceptable relic density Ωh 2 = 0.12±50 %. The solid black and dashed red lines indicate the most constraining experimental upper limits on the SIDD cross section from XENON1T [3] and PandaX-II [4], respectively. Right: SIDD vs. SDDD cross section in units of the respective observed limit for the same points. For SIDD cross section, at each respective DM mass we use the stronger of the two limits from XENON1T and PandaX-II. For SDDD scattering we use the more constraining of the current bounds for either SDDD scattering of neutrons from LUX [5], or SDDD scattering of protons from PICO-60 [6]. To guide the eye we indicate the current bounds with thin dashed lines; points lying in the lower left quadrant satisfy all current direct detection bounds. The color coding in both panels is the same as in Fig. 3. constraints. The right panel shows a comparison of the constraining power of current SIDD and SDDD experiments [3][4][5][6] for the same points. As noted above, current experimental SIDD limits are much more constraining for our model than SDDD limits: as we can see in Fig. 7, parameter points satisfying SIDD limits almost always satisfy current constraints on the SDDD scattering cross section from direct detection experiments, while the reverse is not true. Fig. 7 also shows that improvements in SDDD experiments will probe the remaining parameter space of our model more efficiently than a correspondingly large improvement in the bounds from SIDD. In particular, improving the sensitivity of SDDD scattering by approximately two orders of magnitude with respect to current bounds will probe most of the singlino region. A naïve estimate of the sensitivity can be obtained by rescaling current bounds from LUX on the spin-dependent WIMP-neutron scattering cross section which were obtained with an exposure of = 129.5 kg yr [5]. The XENON1T experiment will have a sensitivity roughly one order of magnitude better assuming an exposure of = 1 t yr, while XENONnT could probe most of the singlino region assuming an exposure of = 20 t yr. Finally, let us mention that out of the points passing the main phenomenological constraints described at the beginning of this section, which in particular lead to a SM-like Higgs boson compatible with LHC measurements and evade collider constraints as implemented in NMSSMTools, ∼ 15 % have an acceptable relic density Ωh 2 = 0.12 ± 50 %, and of the points with the correct relic density ∼ 35 % satisfy SDDD and SIDD constraints. Indirect Detection In Section 4 we have focused on direct detection constraints on our model. Any DM candidate is also subject to constraints from indirect detection experiments. Due to large astrophysical uncertainties on indirect detection constraints arising from charged particle production, we focus on constraints from photon emission here. The currently strongest indirect detection constraints on WIMP DM come from Fermi-LAT and MAGIC analyses of Milky Way satellite galaxies, ruling out canonical values of the thermally averaged annihilation cross sections σv ∼ 2 × 10 −26 cm 3 s −1 for DM masses m χ 100 GeV if the annihilation is dominantly into pairs of b quarks or τ leptons [69,70]. For DM masses m χ m t indirect detection bounds are much weaker, constraining thermal annihilation cross sections much larger than the typical values σv x F ∼ 2×10 −26 cm 3 s −1 consistent with an acceptable relic density. Relevant constraints might further arise from the annihilation of DM matter particles captured via elastic scattering in the Sun, potentially giving rise to a flux of neutrinos observable at Earth. Under certain assumptions [71], most relevant for our case if DM annihilates dominantly into pairs of W bosons, current bounds from DM capture and annihilation yield the most constraining upper limits on the spin dependent DM-proton scattering cross section [72][73][74][75][76]. The parameter region in our model yielding an acceptable relic density and compatible with current SIDD constraints typically features DM masses heavier than the top quark m χ m t ≈ 173 GeV. Furthermore, SIDD constraints require the coupling of DM to the SMlike Higgs boson to be suppressed which in turn implies that (χχ → W + W − ) annihilation is suppressed. This is because of all the Higgs basis states, only H SM couples to pairs of W bosons. Thus, current indirect detection limits do not pose relevant constraints on our model since the annihilation modes most constraining for indirect detection bounds (χχ →bb/τ + τ − /W + W − ) are suppressed for our preferred region of parameter space. Note however, that DM with mass m χ > m t decaying dominantly into pairs of top quarks may explain the excess of photons emitted from the Galactic Center observed by Fermi-LAT [77][78][79][80]. Collider Constraints Our EFT DM model extends the neutral particle content of the SM by four Higgs mass eigenstates, two of them CP-even and two CP-odd, and a SM singlet Majorana fermion. The Dirac fermion we integrated out may be accessible at the LHC if its mass µ is below the TeV scale [53]. Note, that in our NMSSM ultraviolet completion we set the masses of all additional particles such as the SUSY partners of the SM fermions and the charged superpartners of the SM gauge bosons to values ≥ 2 TeV such that they are not directly accessible at the LHC. 8 Furthermore, DM is coupled to the SM via the Dirac fermion (the Higgsinos in the case of the NMSSM) which in turn couples to the Higgs and the Z bosons. Thus, this model can be tested at the LHC either by DM pair production via Z or Higgs bosons, or, via production of the additional Higgs bosons. As discussed previously in Sections 2.1 and 3, the observation of a 125 GeV Higgs boson with couplings compatible with those of a SM Higgs [41,42] requires the presence of an m h ∼ 125 GeV Higgs mass eigenstate approximately aligned with the H SM Higgs basis interaction eigenstate. Direct detection limits furthermore strongly constrain the coupling of DM to both the SM-like Higgs and the Z boson, such that the additional decay width of the Z boson or the SM-like Higgs into DM particles must be small, even before taking into account the fact that m χ m h /2 > m Z /2 in our model. The non-observation of signals from additional Higgs bosons at the LHC furthermore implies that additional Higgs bosons must either have masses heavier than 2m t ∼ 350 GeV, or, be dominantly composed of the additional singlet interaction states H S or A S such that their production cross section is suppressed. A comparison of the constraining power of conventional searches for additional Higgs bosons decaying into pairs of SM particles can for example be found in the Appendix of Ref. [32]. Note, that most of the conventional Higgs searches such as (gg → H/A → τ + τ − /γγ/Zh) will only have small increases in sensitivity at future LHC runs because they are increasingly dominated by systematic errors. On the other hand, models with extended Higgs sectors and potentially large couplings between different Higgs mass eigenstates such as ours can be probed effectively by decays of heavy Higgs bosons into lighter Higgs bosons or a light Higgs and a Z boson. Due to the presence of the SM-like Higgs, decays into pairs of SM-like Higgs bosons (H → hh) or into a SM-like Higgs boson and a Z (A → Zh) are suppressed since the corresponding couplings are proportional to the mixing of the H SM Higgs basis state with the H NSM and H S states. If kinematically allowed, the branching ratios of (H → hh S /Za S ) and (A → Zh S /ha S ) can however be sizeable. The collider signatures arising through such decays can be effective probes of our model [29,32,33]. Conclusions Current bounds from direct detection experiments [1][2][3][4][5][6] have set stringent limits on WIMP DM scenarios where the relic density proceeds from thermal production mediated by Higgs and gauge bosons. In particular, direct detection experiments strongly constrain the coupling of the SM-like Higgs boson to DM particles and the vector-like coupling of DM to Z bosons [7]. The latter constraint is naturally satisfied for Majorana DM, since Majorana fermions couple to Z bosons only via axial couplings. In this article, we explore an EFT describing the interactions of Majorana fermion WIMPs with an extended Higgs sector, comprised of a type II 2HDM and a SM gauge singlet. This model can be interpreted as an extension of Higgs portal models. In particular, we study the case where the EFT mass scale is identified with the mass of a heavy SU (2)-doublet Dirac fermion integrated out from the theory. Furthermore, we assume that all explicit mass terms are forbidden, which from the LHC allow for much smaller masses of the SUSY particles, probeable at ongoing and future LHC runs. may be realized by imposing a Z 3 symmetry. In such a case the new fermions acquire mass through the vev of the singlet field. If this vev is generated by the same mechanism that induces electroweak symmetry breaking, we naturally obtain a weak scale mass for the DM candidate. In addition, the Z 3 symmetry reduces the number of allowed operators involving the singlet Majorana fermion and the doublet and singlet scalar fields, such that we can easily extend the EFT to dimension 6, including derivative operators. We derive the low energy Higgs spectrum and the couplings of DM to the neutral Higgs and gauge bosons in this model, and use these results to compute the relic density and spin independent direct detection cross section. We find blind spot conditions that allow for the evasion of spin independent direct DM detection constraints, while yielding the observed relic density. These blind spot conditions can be easily characterized in terms of the EFT parameters. In order to test the validity of the results derived in our EFT model, we compare them to those obtained in the supersymmetric NMSSM, which features an analogous Higgs sector. We demonstrate that in the case of heavy gauge and fermion superpartners the NMSSM can be reduced to our EFT model at low energies, such that we can use it as an explicit computational basis. The Majorana singlet in our EFT is identified with the singlino in the NMSSM, and the EFT scale with the mass of the heavier Higgsinos. We show that the qualitative features obtained in the EFT are preserved in the complete theory, while the precise values of the couplings, relic density, and cross sections are modified by the presence of small corrections which could be included by dimension d > 6 operators in the EFT description. We also discuss the case of a Z 3 invariant Majorana fermion DM candidate, e.g. the bino in the NMSSM, and show that our EFT model can be generalized in a straightforward manner to include this scenario. Both in the EFT and in the NMSSM we show that the coupling of (singlino-like) DM to the SM-like Higgs is constrained to values below g χχh 0.1 by direct detection experiments, while simultaneously consistency with the DM relic density can be obtained through thermal annihilation via couplings of DM to the remaining Higgs and gauge bosons. The neutral Goldstone mode, comprising the longitudinal mode of the Z boson, plays a prominent role in obtaining the thermal annihilation cross section. Moreover, in order to evade direct detection bounds, not only the coupling to the SM-like Higgs boson must be reduced, but in addition destructive interference of the SM-like Higgs boson with the singlet and/or non SM-like CP-even doublet Higgs boson is required to further suppress the cross section. In the NMSSM, when considering the case of light binos, we find a new well tempered DM region. For bino-like DM the couplings to the Higgs bosons tend to be smaller than for singlino-like one and direct detection bounds are hence evaded by a mild proximity to the blind spot conditions for the DM coupling to the SM-like Higgs state. The relic density in the well tempered region is obtained via co-annihilation of the bino with the singlino. Beyond the well tempered region, the correct relic density for bino-like DM may also be obtained via resonant annihilation with either the Z boson or a Higgs mass eigenstate. In both the cases of bino-like and singlino-like DM, we find that the current constraints coming from SDDD are subdominant compared to those coming from SIDD. However, we also find that an improvement of two orders of magnitude in the bounds on the SDDD cross sections could efficiently probe most models consistent with a blind spot in SIDD. Finally, collider searches for the Majorana DM particles are mostly restricted to the usual channels of single SM-particles plus missing energy, in particular those associated with Higgs bosons. Going beyond the effective theory, these searches may be complemented by missing energy searches in the production and decay of the Dirac doublet fermion and, in the NMSSM, by related searches for heavy superpartners. A EFT Lagrangian Rescaling the Majorana fermion field χ in order to retain a canonical kinetic term and keeping terms to order O(µ −2 ), the EFT Lagrangian (2.13) reads in terms of the Higgs basis states, Eqs. (2.3)-(2.6), L = δ 2µ χχ s 2β 2 2v 2 + 2 √ 2vH SM + (H SM ) 2 − (H NSM ) 2 − (A NSM ) 2 + (G 0 ) 2 +c 2β √ 2vH NSM + H SM H NSM + A NSM G 0 +i √ 2vA NSM + H SM A NSM − H NSM G 0 + h.c. − λδ 2 √ 2µ 2 χχ H S + iA S × s 2β 2 2v 2 + 2 √ 2vH SM + (H SM ) 2 − (H NSM ) 2 − (A NSM ) 2 + (G 0 ) 2 +c 2β √ 2vH NSM + H SM H NSM + A NSM G 0 +i √ 2vA NSM + H SM A NSM − H NSM G 0 + h.c. − κ − καv 2 |µ| 2 µ λ + 1 √ 2 H S + iA S χχ + h.c. − κξ 2|µ| 2 µ λ + 1 √ 2 H S + iA S χχ × 2v 2 + 2 √ 2vH SM + (H SM ) 2 + (G 0 ) 2 + (H NSM ) 2 + (A NSM ) 2 + h.c. + α |µ| 2 g 1 2s W Z µ χ †σµ χ × c 2β 2 −2v 2 − 2 √ 2vH SM − (H SM ) 2 + (H NSM ) 2 + (A NSM ) 2 − (G 0 ) 2 +s 2β √ 2vH NSM + H SM H NSM + A NSM G 0 + α |µ| 2 χ †σµ χ is 2β v √ 2 i∂ µ A NSM + H SM i∂ µ A NSM + H NSM i∂ µ G 0 −ic 2β v √ 2 i∂ µ G 0 + H SM i∂ µ G 0 − H NSM i∂ µ A NSM + α 2|µ| 2 χ † iσ µ ∂ µ χ 2 √ 2vH SM + (H SM ) 2 + (G 0 ) 2 + (H NSM ) 2 + (A NSM ) 2 . (A.1) The couplings of the Majorana fermions to the Higgs basis states and the Z boson can be read off from here. Note, that couplings g χχΦ i Φ j ... to Higgs basis states written explicitely in this work are normalized as g χχΦ i Φ j . Figure 1 . 1Exemplary diagrams illustrating the interactions of singlet DM χ with SM particles. The left diagram (a) depicts interactions arising via the tree-level interaction of a pair of DM singlets χ with the scalar singlet states H S and A S , which mix (indicated by the cross) with the Higgs basis states from the Higgs doublets H SM , H NSM and A NSM respectively. The right diagram (b) appears via d ≥ 5 interactions (indicated by the solid black disc) of pairs of χ's with the doublet-like Higgs states arising when integrating out a heavy Dirac fermion SU (2)-doublet. . (2.53)-(2.56), the g χχΦ i are the couplings of pairs of χ's to the Higgs mass eigenstates given in Eq. (2.40) [Eq. (2.20) and (2.50) for the coupling to the neutral Goldstone mode g χχG 0 and the transversal polarizations of the Z boson g χχZ , respectively], the g in the case of mostly singlet states, by their small doublet components − 1 Figure 2 . 1210 pb; σv xF = 2.3 × 10 −26 cm 3 s EFT parameters and couplings of DM to the CP-even and CP-odd Higgs bosons required to obtain the correct thermal relic density while concurrently satisfying SIDD constraints, for tan β = 2, m χ = 300 GeV, m H NSM = m A NSM = 500 GeV, and decoupled singlet states. Left: The orange shaded-region bounded by solid and dashed lines represents the CP-even Higgs bosons couplings consistent with the SIDD bounds, while the blue and black ellipses denote the couplings of DM to the (neutral) Goldstone mode and the heavy CP-odd Higgs boson yielding Ωh 2 ∼ 0.12 for CP-even Higgs couplings denoted by the corresponding solid or dashed lines, with thick and thin lines denoting two different solutions. Right: Values of the EFT parameters consistent with the couplings shown in the left panel. Dashed and solid lines, as well as the shaded areas shown in this panel are in one-to-one correspondence with similar lines and areas shown in the left panel. the coupling g χχh necessary to fulfill the constraints on the SIDD cross section may only be obtained if the blind spot condition, Eq. (2.16), is approximately fulfilled. have written the SU (2) indices explicitly. The Yukawa couplings Y u , Y d , and Y e should be understood to be matrices, and the left-handed quark (lepton) doublets Q ( L) as well as the up-type ū and down-type right-handed quarks (leptons) d ( ē) as vectors in family space. H u and H d are the usual Higgs doublet supermultiplets, and S is a chiral supermultiplet which transforms as a singlet under the SM gauge group. The Higgs sector of the NMSSM consists of three neutral CP-even Higgs bosons, the real components of H u , H d , and S, two CP-odd neutral Higgs bosons, the imaginary components of S and A NSM = √ 2 Im cos βH 0 u + sin β H 0 d , and one charged Higgs H ± , with tan β = v u /v d and v u (v d ) the vev of H u (H d ). The remaining components of the Higgs doublets make up the longitudinal components of the W and Z bosons after electroweak symmetry breaking. The Higgs sector is controlled by the parameters 15 (N 13 s β − N 14 c β ) + (g 1 N 11 − g 2 N 12 ) (N 13 c β + N 14 s β ) .(3.16) relation between the δ and α couplings δ = −α arises because scalar and fermion components of chiral supermultiplets (vector and fermion components in the case of gauge supermultiplets) share the same couplings in SUSY models. The mapping above leads to the blind spot condition [cf. Eq. (2.16)] sin 2β = m χ /µ . (3.25) In contrast to the singlino, the bino couples to different combinations of the Higgs doublets and the singlet. Such interactions would be obtained by writing down the EFT for the Higgs doublets and the singlet transforming under the Z 3 , while assuming the Majorana fermion χ transforms trivially and has a Majorana mass m χ = M 1 . In particular, comparing Eqs. (3.22) and (2.13) we see that the bino couples to (H 0 d ) † (H 0 u ) † B B instead (3.22), or from Eqs. (2.14),(2.18)-(2.22), using the mapping of the parameters in Eq. (3.24), singlino-like (bino-like) DM with small Higgsino admixture, the couplings in Eqs. (3.28)-(3.32) [Eqs. (3.33)-(3.37)] give to good approximation the same numerical results as those in Eqs. (3.10)-(3.16). The differences between the two may be understood as originating from corrections due to higher dimensional operators in the EFT associated with the expansion of the denominator 1 − (m χ /µ) 2 in Eqs. (3.10) and (3.11) in powers of m 2 χ /µ 2 . Keeping only the first term, 1/µ 2 , when replacing these expressions into Eqs. (3.12)-(3.16) is sufficient to reproduce the couplings obtained from the d = 5 and d = 6 operators, Eqs. (3.28)-(3.37). Figure 3 . 3SIDD cross section σ SI p vs. the contribution of the longitudinal mode of the Z boson (i.e. the Goldstone mode of the Higgs doublets) to the thermally averaged annihilation cross section as defined in Eq.(2.45) with the amplitude given in Eq. (2.46). In the left panel, we show points from our parameter scan passing collider constraints, while points shown in the right panel are also required to have the correct relic density Ωh 2 = 0.12±50 % and satisfy bounds from direct detection experiments. The color coding indicates the compositions of the DM candidate as indicated in the legend. We denote points as purely bino B if N 2 11 ≥ 0.95, Higgsino H if N 2 13 +N 2 14 ≥ 0.95, or singlino S if N 2 15 ≥ 0.95. Similarly, points are denoted as mixed if the sum of the square of corresponding mixing angles is N 2 1i ≥ 0.95 but none of the individual contributions is sufficiently large to put them in one of the previous categories. Figure 4 . 4Left: Points from our parameter scan with bino-like lightest neutralino and relic density Ωh 2 = 0.12 ± 50 % in the bino mass (M 1 ) -singlino mass (2κµ/λ) plane. Right: Points with singlino-like lightest neutralino in the singlino mass (2κµ/λ) -Higgsino mass (µ) plane. For both panels, the color code indicates points where the lightest neutralino can pair-annihilate resonantly with the s-channel mediator with mass ≈ 2m χ as indicated in the legend. Figure 5 . 5Left: The SIDD cross section σ SI p for the points passing the required experimental collider constraints vs. m χ /(µ sin 2β), where the blind-spot conditions are satisfied for m χ /(µ sin 2β) = +1(−1) for the singlino-Higgsino (bino-Higgsino) case. Right: The spin independent cross section for the same points vs. the coupling of the DM candidate to the SM-like Higgs mass eigenstate. The color coding is the same as ifFig. 3. Figure 6 . 6SIDD cross section σ SI p vs. the contribution σ SI p i assuming only one CP-even Higgs mass eigenstate h i = {h, H, h S } multiplied by the sign of its amplitude, cf. Eq. (2.61). The dashed diagonal lines indicate σ SI p i = σ SI p . Hence, if the contribution from one of the h i lies on the diagonal lines and the contributions from the remaining mass eigenstates lie within the triangle, the SIDD cross section is dominantly mediated by that mass eigenstate. On the contrary, if the contributions lie outside the dashed diagonal lines, they interfere destructively to yield the total SIDD cross section. The left (right) panel shows parameter points where the lightest neutralino is bino (singlino) like. For both cases, we show points from our parameter scan which satisfy Ωh 2 = 0.12 ± 50 %. smaller cross sections. we can define the Higgs basis[34][35][36][37][38][39][40] 1 Another important point to note from Eq. (2.13) is that the presence of derivative terms allows for interactions between the Goldstone G 0 and DM, absent in Eq. (2.11), which as we shall see turn out to be relevant for the thermal annihilation cross section. For the convenience of the reader, we write Eq. (2.13) in terms of the Higgs basis states in the Appendix A, Eq. (A.1).From Eq. (2.13), the coupling of the DM particles to the SM-like Higgs is given by Dirac fermion with mass µ. This allows us to use the equation of motion for H 0 u to integrate out H 0 d and vice versa. Keeping only terms leading to dimension d ≤ 6 operators when substituting the equations of motion into the Lagrangian, we obtain3.19) Due to the Higgsino mass term λS H 0 u H 0 d → µ H 0 u H 0 d when S acquires a vev, and because of their identical couplings, we can interpret H 0 u as the right-handed and H 0 d as the left- handed component of a Table 1. NMSSM parameter ranges used in NMSSMTools scan.tan β [1.5; 5] λ [0.5; 0.7] κ [−0.3; +0.3] µ [−0.75; +0.75] TeV A λ [−1.5; +1.5] TeV A κ [−0.75; +0.75] TeV M 1 [0; 1] TeV .. ≡ ∂L ∂χ 2 ∂Φ i ∂Φ j . . . , (A.2) while couplings involving Z bosons are normalized as g χχZΦ i ... ≡ 1 2 ∂L ∂χ 2 ∂Z ∂Φ i . . . . (A.3) Note, that there are different conventions in the literature for the Higgs basis differing by an overall sign of H NSM and A NSM . Note, that there are considerable uncertainties on the form factors. In this work, we use the default values used by micrOMEGAs 4.3.5 {f p T u , f p T d , f p T s } = {0.0153, 0.0191, 0.0447} [44-47]. While we are mainly interested in the DM-Higgs couplings which are not directly affected by the form factors, a different choice for the values of the form factors can be compensated by (small) redefinitions of other parameters. Note, that the masses of the Higgs bosons are chosen such that (χχ → hA) annihilation are kinematically forbidden, hence, the relic density is set by (χχ → tt) annihilation mediated dominantly by G 0 and A NSM . This choice minimizes radiative corrections to soft breaking parameters such as A λ and to the elements of the Higgs mass matrices, such that we can easily match analytical tree level results to numerical results from NMSSMTools. Including non-minimal third generation squark mixing would not affect our results, since we are only interested in the phenomenology of the Higgs and neutralino sectors for which the corresponding radiative corrections can be compensated for by shifts of the tree level parameters. 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[]
[ "The Galactic habitable zone around M and FGK stars with chemical evolution models with dust", "The Galactic habitable zone around M and FGK stars with chemical evolution models with dust" ]
[ "E Spitoni \nDipartimento di Fisica\nSezione di Astronomia\nUniversità di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly\n", "⋆ ", "L Gioannini \nDipartimento di Fisica\nSezione di Astronomia\nUniversità di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly\n", "F Matteucci \nDipartimento di Fisica\nSezione di Astronomia\nUniversità di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly\n\nI.N.A.F. Osservatorio Astronomico di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly\n" ]
[ "Dipartimento di Fisica\nSezione di Astronomia\nUniversità di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly", "Dipartimento di Fisica\nSezione di Astronomia\nUniversità di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly", "Dipartimento di Fisica\nSezione di Astronomia\nUniversità di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly", "I.N.A.F. Osservatorio Astronomico di Trieste\nvia G.B. Tiepolo 11I-34131TriesteItaly" ]
[]
Context. The Galactic habitable zone is defined as the region with highly enough metallicity to form planetary systems in which Earth-like planets could be born and might be capable of sustaining life surviving to the destructive effects of nearby supernova explosion events. Aims. Galactic chemical evolution models can be useful tools for studying the galactic habitable zones in different systems. Our aim here is to find the Galactic habitable zone using chemical evolution models for the Milky Way disc, adopting the most recent prescriptions for the evolution of dust and for the probability of finding planetary systems around M and FGK stars. Moreover, for the first time, we will express those probabilities in terms of the dust-to-gas ratio of the ISM in the solar neighborhood as computed by detailed chemical evolution models. Methods. At a fixed Galactic time and Galactocentric distance we determine the number of M and FGK stars having Earths (but no gas giant planets) which survived supernova explosions, using the formalism of our Paper I. Results. The probabilities of finding terrestrial planets but not gas giant planets around M stars deviate substantially from the ones around FGK stars for supersolar values of [Fe/H]. For both FGK and M stars the maximum number of stars hosting habitable planets is at 8 kpc from the Galactic Centre, if destructive effects by supernova explosions are taken into account. At the present time the total number of M stars with habitable planets are ≃ 10 times the number of FGK stars. Moreover, we provide a sixth order polynomial fit (and a linear one but more approximated) for the relation found with chemical evolution models in the solar neighborhood between the [Fe/H] abundances and the dust-to-gas ratio.Conclusions. The most likely Galactic zone to find terrestrial habitable planets around M and FGK stars is the annular region 2 kpc wide centred at 8 kpc from the Galactic center (the solar neighborhood). We also provide the probabilities of finding Earth-like planets as the function of the ISM dust-to-gas ratio using detailed chemical evolution models results.
10.1051/0004-6361/201730545
[ "https://arxiv.org/pdf/1705.01297v2.pdf" ]
119,088,218
1705.01297
0c6f0e0c843417f884449dd176d2b3d87f28b30e
The Galactic habitable zone around M and FGK stars with chemical evolution models with dust 9 May 2017 May 10, 2017 E Spitoni Dipartimento di Fisica Sezione di Astronomia Università di Trieste via G.B. Tiepolo 11I-34131TriesteItaly ⋆ L Gioannini Dipartimento di Fisica Sezione di Astronomia Università di Trieste via G.B. Tiepolo 11I-34131TriesteItaly F Matteucci Dipartimento di Fisica Sezione di Astronomia Università di Trieste via G.B. Tiepolo 11I-34131TriesteItaly I.N.A.F. Osservatorio Astronomico di Trieste via G.B. Tiepolo 11I-34131TriesteItaly The Galactic habitable zone around M and FGK stars with chemical evolution models with dust 9 May 2017 May 10, 2017Received xxxx / Accepted xxxxAstronomy & Astrophysics manuscript no. spitoni c ESO 2017Galaxy: abundances -Galaxy: evolution -planets and satellites: general -ISM: general Context. The Galactic habitable zone is defined as the region with highly enough metallicity to form planetary systems in which Earth-like planets could be born and might be capable of sustaining life surviving to the destructive effects of nearby supernova explosion events. Aims. Galactic chemical evolution models can be useful tools for studying the galactic habitable zones in different systems. Our aim here is to find the Galactic habitable zone using chemical evolution models for the Milky Way disc, adopting the most recent prescriptions for the evolution of dust and for the probability of finding planetary systems around M and FGK stars. Moreover, for the first time, we will express those probabilities in terms of the dust-to-gas ratio of the ISM in the solar neighborhood as computed by detailed chemical evolution models. Methods. At a fixed Galactic time and Galactocentric distance we determine the number of M and FGK stars having Earths (but no gas giant planets) which survived supernova explosions, using the formalism of our Paper I. Results. The probabilities of finding terrestrial planets but not gas giant planets around M stars deviate substantially from the ones around FGK stars for supersolar values of [Fe/H]. For both FGK and M stars the maximum number of stars hosting habitable planets is at 8 kpc from the Galactic Centre, if destructive effects by supernova explosions are taken into account. At the present time the total number of M stars with habitable planets are ≃ 10 times the number of FGK stars. Moreover, we provide a sixth order polynomial fit (and a linear one but more approximated) for the relation found with chemical evolution models in the solar neighborhood between the [Fe/H] abundances and the dust-to-gas ratio.Conclusions. The most likely Galactic zone to find terrestrial habitable planets around M and FGK stars is the annular region 2 kpc wide centred at 8 kpc from the Galactic center (the solar neighborhood). We also provide the probabilities of finding Earth-like planets as the function of the ISM dust-to-gas ratio using detailed chemical evolution models results. Introduction The Galactic habitable zone (GHZ) has been defined as the region with sufficiently high abundances of heavy elements to form planetary systems in which terrestrial planets could be found and might be capable of sustaining life. Therefore, the chemical evolution of the Galaxy plays a key role to properly model the GHZ evolution in space and in time. The minimum metallicity needed for planetary formation, which would include the formation of a planet with Earth-like characteristics (firstly discussed by Gonzalez et al. 2001) has been fixed at the value of 0.1 Z ⊙ by the theoretical work of Johnson & Li 2012). In the last years several purely chemical evolution models (Lineweaver 2001, Lineweaver et al. 2004, Prantzos 2008, Carigi et al 2013) have studied the habitable zones of our Galaxy as functions of the Galactic time and Galactocentric distances. In particular Spitoni et al. (2014, hereafter Paper I), in which radial gas flows were included, confirmed the previous results of Lineweaver at al. (2004) and found that the maximum number of stars which ⋆ email to: [email protected] can host habitable terrestrial planets are in solar neighborhood, i.e. the region centered at 8 kpc, and 2 kpc wide. Recently, the GHZ has been also studied in the cosmological context (ΛCDM) by Forgan et al. (2015), Gobat & Hong (2016), Zackrisson et al. (2016), and Vukotić et al. (2016) showing which kind of halos can give rise to galactic structures in which habitable planets could be formed. In most of the models mentioned above (both for purely chemical evolution and cosmological models), it was considered the probability of forming planetary systems in which terrestrial planets are found without any gas giant planets or hot Jupiters, because in principle the last two planet types could destroy Earths during their evolution. The terms giant planets, gas giants, or simply Jupiters, refer to large planets, typically >10 M ⊕ , that are not composed primarily of rock or other solid matter. When orbiting close to the host star they are referred to as hot Jupiters or very hot Jupiters. The "core accretion model" of giant planet formation is the most widely accepted in the literature, and the next stochastic migration due to turbulent fluctuations in the disc could destroy the terrestrial planets. Rice & Armitage (2003) showed that the formation of Jupiter can be accelerated by almost an order of magnitude if the growing core executes a random walk with an amplitude of ≈ 0.5 AU. Nowadays, about 3 thousands of planets have been discovered and the statistics is good enough to confirm that most of planetary systems host planets which are not present in our solar system, such as hot Jupiters or super-Earths. In this paper we retain the assumption that gas giant planets could destroy, during their evolution, terrestrial ones (we are aware that the real effects are still uncertain). On this basis we study the GHZ using the most updated probabilities related to the formation of gas giant planets as functions of [Fe/H] abundance values as well as the stellar mass for FGK and M stars. It is recent the discovery of a planetary system around the M star Trappist-1 (Gillon at al. 2016(Gillon at al. , 2017 composed by seven terrestrial planets characterized by equilibrium temperature low enough to make possible the presence of liquid water. This detection makes the habitability around M stars even more interesting. In this paper we also compute the probabilities related to the formation of gas giant planets as functions of [Fe/H] abundance values and the stellar mass for FGK and M stars with a detailed chemical evolution model for the Milky Way with dust evolution. Even though, we know very little about the formation of planetary systems: in particular, it is not well understood the transition from a protoplanetary disc to a planetary system. In this transition, dust and gas rapidly evolve in very different ways due to many processes (Armitage 2013) such as dust growth, gas photoevaporization (Alexander et al. 2014), gas accretion onto the star (Gammie 1996). Dust plays a fundamental role in the formation of the first planetesimals, as it represents the solid compounds of the matter which can form rocky planetesimals and therefore planets. The first fundamental step to understand and explain the origin of the observed diversity of exoplanetary systems, is to measure the stellar disc properties, especially the disc mass. Dust grains of µm are directly observed in protoplanetary discs and then, dust coagulation increase their size up to mm (Dullemond & Dominik 2005). On the other hand, observations of the gas are almost forbidden, because it is relatively cool and in molecular form. In most cases, the mass of the gas is set starting from the one of the dust and by assuming a value for the dust-to-gas ratio. Unfortunately, this practice has several uncertainties (Williams & Best 2014). Usually, models of planetary formation use an average value of 10 −2 for the initial condition of the dust-to-gas ratio in the protoplanetary disc (Bohlin et al. 1978). Furthermore, the formation rate of gas giant planets seems to be related to the metallicity of the hosting stars (Fisher & Valenti 2005, Johnson et al. 2010, Mortier et al. 2012, Gaidos & Mann 2014. Moreover, even if Buhhave et al. (2012) showed that planets with radii smaller than 4 R ⊕ do not present any metallicity correlation, the theoretical work of Johnson & Li (2012) predicted that first Earth-like planets likely formed from circumstellar discs with metallicities Z ≥ 0.1 Z ⊙ . With this work, providing the time evolution of the dust, we discuss the connection between the metallicity of the ISM and the dust-to-gas ( D G ) ratio (especially for the solar neighborhood). In this way, we can link the metallicity of stars, which is observationally related to the probability of the presence of hosted planets, with the initial dust-to-gas ratio of the protoplanetary discs (the dust-to-gas ratio of the ISM at the instant of the protoplanetary disc formation). The paper is organized as follows: in Section 2 we present the probabilities of terrestrial planets around M and FGK stars. In Section 3 we describe the Milky Way chemical evolution model with dust and we present the main results in Section 4. Finally, our conclusions are summarized in Section 5. 2. The probabilities of terrestrial planets around M and FGK stars Buchhave et al. (2012) who analyzed the mission Kepler, found that the frequencies of the planets with earth-like sizes are almost independent of the metallicity of the host star up to [Fe/H] abundance values smaller than 0.5 dex. In agreement with these observations, Prantzos (2008) fixed the probability of forming Earth-like planets (P F E , where F E stands for Forming Earths) at value of 0.4 for [Fe/H] ≥ -1 dex, otherwise P F E = 0 for smaller values of [Fe/H]. This assumption was also adopted by Carigi et al. (2013) and Paper I. The value of P F E = 0.4 was chosen to reproduce the metallicity integrated probability of Lineweaver et al. (2001). In Paper I it was considered the case in which gaseous giant planets with same host star can destroy terrestrial planets (i.e. during their migration path). Armitage (2003) pointed out the potentially hazardous effects of the gas giant planet migration on the formation of Earth-like ones, and suggested that these planets preferentially exist in systems where massive giants did not migrate significantly. Matsumura et al. (2013) studying the orbital evolution of terrestrial planets when gas giant planets become dynamically unstable, showed that Earth-like planets far away from giants can also be removed. On the other hand, various numerical simulations found that the formation of earths is not necessarily prevented by the gas giant planet migration, when eccentricity excitation timescales for (proto-)terrestrial planets are long compared to migration timescales of giant planets (e.g., Mandell & Sigurdsson 2003;Lufkin et al. 2006;Raymond et al. 2006). The adopted probability of formation of a gaseous giant planet as the function of the iron abundance in the host star is taken by Fischer & Valenti (2005) as: P GGP ([Fe/H]) = 0.03 × 10 2.0[Fe/H] . (1) A possible theoretical explanation is that the high metallicity observed in some stars hosting giant planets represent the original composition that protostellar and protoplanetary molecular clouds were formed. In this scenario, the higher the metallicity of the primordial cloud, the proportion of dust to gas in the protoplanetary disc. This facilitates the condensation and accelerates the protoplanetary accretion before the disc gas is lost (Pollack et al. 1996). Giant planets are subsequently formed by runaway accretion of gas onto such rocky cores with M ≈ 10M ⊕ , rather than by gravitational instabilities in a gaseous disk which predicts formation much less sensitive to metallicity (Boss, 2002). The novelty of this work is to consider the following new probabilities for gas giant planets formation found by Gaidos & Mann (2014) and used by Zackrisson et al. (2016) around FGK and M stars which are functions also of the masses of the hosting stars: P GGP/F GK ([Fe/H], M ⋆ ) = 0.07×10 1.8[Fe/H] M ⋆ M ⊙ ,(2) for the FGK stars, and P GGP/M ([Fe/H], M ⋆ ) = 0.07 × 10 1.06[Fe/H] M ⋆ M ⊙ ,(3) and for M stars, where M ⋆ is the mass of the host star in units of solar masses. We assume that the range of masses spanned by M type stars is the following one: 0.08 ≤ M⋆ M⊙ ≤ 0.45. For the F GK the range is: 0.45 ≤ M⋆ M⊙ ≤ 1.40. The probability of forming terrestrial planets around FGK/M stars but not gaseous giant planets is given by: P E/F GK,M = P F E × (1 − P GGP/F GK,M ).(4) Here, we make the conservative assumption of Prantzos (2008) and Paper I that the P F E probability is constant at the value of 0.4 for all stellar types, including M and FGK stars. On the other hand, Zackrisson et al. (2016) presented results where P F E = 0.4 around FGK stars, and P F E = 1 around M stars. The possibility of finding habitable planets around Mdwarf has long been debated, due to differences between the unique stellar and planetary environments around these low-mass stars, as compared to hotter, more luminous Sunlike stars (Shields et al. 2016). The presence of multiple rocky planets (Howard et al. 2012), with roughly a third of these rocky M-dwarf planets orbiting within the habitable zone, supports the hypothesis of the presence of liquid water on their surfaces. On the other hand, flare activity, synchronous rotation, and the likelihood of photosynthesis could have a severe inpact on the habitability of planets hosted by M dwarf stars (Tarter et al., 2007). We define P GHZ (F GK/M, R, t) as the fraction of all FGK/M stars having around Earths (but no gas giant planets) which survived supernova explosions as a function of the Galactic radius and time: P GHZ (F GK/M, R, t) = = t 0 SF R(R, t ′ )P E/F GK,M (R, t ′ )P SN (R, t ′ )dt ′ t 0 SF R(R, t ′ )dt ′ .(5) This quantity must be interpreted as the relative probability to have complex life around one star at a given position, as suggested by Prantzos (2008). In eq. (5) SF R(R, t ′ ) is the star formation rate (SFR) at the time t ′ and Galactocentric distance R, and P SN (R, t ′ ) is the probability of surviving supernova explosion. We know that hard radiation originated by close-by SN explosions could lead to the depletion of the ozone layer in terrestrial atmosphere. At this point, the ultraviolet radiation from the host star can penetrate the atmosphere, altering and damaging the DNA and eventually causing the total sterilization of the planet (Gehrels et al. 2003). For this quantity we refer to the "case 2" model of Paper I in which the SN destruction is effective if the SN rate at any time and at any radius has been higher than twice the average SN rate in the solar neighborhood during the last 4.5 Gyr of the Milky Way life (we call it < RSN SV >). Therefore, we impose that if SN rate is larger than 2× < RSN SV > then P SN (R, t) = 0 else P SN (R, t) = 1. We also show results when SN effects are not taken into account, in this case we simply impose P SN (R, t) = 1 at any time and galactic radius. The "case 2" condition is almost the same as that used by Carigi et al. (2013) to describe their best models, motivated by the fact that life on Earth has proven to be highly resistant, and the real effects of SN explosions on life are still extremely uncertain . For < RSN SV > we adopt the value of 0.01356 Gyr −1 pc −2 using the results of the S2IT model of Spitoni & Matteucci (2011) and Paper I. Finally, we define the total number of stars formed at a certain time t and Galactocentric distance R hosting Earth-like planet with life N ⋆ lif e (F GK/M, R, t), as: N ⋆ lif e = P GHZ × N ⋆tot ,(6) where N ⋆tot (F GK/M, R, t) is the total number of stars created up to time t at the Galactocentric distance R. The Milky Way chemical evolution model with dust To trace the chemical evolution of the Milky Way we adopt an updated version of the two-infall model of Paper I in which we consider the dust evolution using the new prescriptions of Gioannini et al. (2017). The two infall model of Paper I The chemical evolution model of Paper I is based on the classical two-infall model of Chiappini et al. (2001). We describe here the main characteristics of the model. We define G i (t) = G(t)X i (t) as the fractional mass of the element i at the time t in the ISM, where X i (t) represents the abundance of the element i in the ISM at the time t. The temporal evolution of G i (t) in the ISM is described by the following expression: G i (t) = −ψ(t)X i (t) + R i (t) +Ġ i,inf (t).(7) The first term in the right side of eq. (7) represents the rate at which the fraction of the element i is subtracted by the ISM due to the SFR process. R i (t) is the returned mass fraction of the element i injected into the ISM from stars thanks to stellar winds and SN explosions. This term takes into account nucleosynthesis prescriptions concerning stellar yields and supernova progenitor models. The third term of eq. (7) represents the rate of the infall of the element i. The infalling gas is not pre-enriched and has a pure primordial composition. The two-infall approach is a sequential model in which the halo-thick disc and the thin disc form by means of two independent infall episodes of primordial gas following this infall rate law: G i,inf (t) = a(r)e −t/τH + b(r)e −(t−tmax)/τD (r) ,(8) where τ H is the typical timescale for the formation of the halo and thick disc and it is fixed to the value of 0.8 Gyr, while t max = 1 Gyr is the time for the maximum infall onto the thin disc. The coefficients a(r) and b(r) are obtained by imposing a fit to the observed current total surface mass density in the thin disc as a function of Galactocentric distance given by: Σ(r) = Σ 0 e −R/RD ,(9) where Σ 0 = 531 M ⊙ pc −2 is the central total surface mass density and R D = 3.5 kpc is the scale length. Moreover, the formation timescale of the thin disc τ D (r) is assumed to be a function of the Galactocentric distance, leading to an inside-out scenario for the Galaxy disc build-up. In particular, we assume that: τ D (r) = 1.033 R (kpc) − 1.267 Gyr.(10) The Galactic thin disc is approximated by several independent rings, 2 kpc wide, without exchange of matter between them. A threshold gas density of 7 M ⊙ pc −2 in the SF process (Kennicutt 1989(Kennicutt , 1998Martin & Kennicutt 2001;Schaye 2004) is also adopted for the disc. The halo has a constant surface mass density as a function of the Galactocentric distance at the present time equal to 17 M ⊙ pc −2 and a threshold for the star formation in the halo phase of 4 M ⊙ pc −2 , as assumed for the model B of Chiappini et al. (2001). (2014) and Paper I. The color code indicates the Galactocentric distance. The assumed IMF is the one of Scalo (1986), which is assumed constant in time and space. The adopted law for the SFR is a Schmidt (1959) like one: Ψ ∝ νΣ k gas (r, t), here Σ gas (r, t) is the surface gas density with the exponent k equal to 1.5 (see Kennicutt 1998;and Chiappini et al. 1997). The quantity ν is the efficiency of the star formation process, and is constant and fixed to be equal to 1 Gyr −1 . The chemical abundances are normalized to the Asplund et al. (2009) solar values. Evolution of dust Our chemical evolution model also traces the dust evolution in the interstellar medium (ISM). Defining G i,dust (t), we can thus write the equation for dust evolution as follows: G i,dust (t) = −ψ(t)X i,dust (t) + δ i R i (t) + G i,dust (t) τ accr − G i,dust (t) τ destr .(12) The right hand of this equation contains all the processes which govern the so called "dust cycle" : the first term represents the amount of dust removed from the ISM due to star formation, the second takes into account dust pollution by stars while the third and fourth terms represent dust accretion and destruction in the ISM, respectively. In this work, we used the same prescriptions used in Gioannini et al. (2017). Dust production are provided by taking into account condensation efficiencies δ i 1 , as provided by Piovan et al. (2011). 1 The condensation efficiency (δi) represents the fraction of an element i expelled from a star which goes into the dust phase of the ISM. The dust yields δ i R i (t) are not only metallicity dependent but also depend on the mass of the progenitor star. In this work we consider as dust producers Type II SNe (M ⋆ > 8M ⊙ ) and low-intermediate mass stars (1.0M ⊙ < M ⋆ < 8.0M ⊙ ). Concerning dust accretion and destruction we calculated the metallicity dependent timescales for these processes (τ accr and τ destr ) as described in Asano et al. (2013). For a more detailed explanation on dust prescriptions or dust chemical evolution model we address the reader to Gioannini et al. (2017). Results First, in this Section we present the main results of our Milky Way chemical evolution model in presence of dust. Moreover, the P E/F GK,M probabilities of finding Earthlike planets but not gas giant ones around FGK and M stars of Gaidos & Mann (2014) computed with detailed chemical evolution models for the Galactic disc at different Galactocentric distances are shown. We express those probabilities in terms of the dust-to-gas ratio D G obtained by our ISM chemical evolution models. Finally, we present the maps of habitability of our Galaxy as functions of the galactic time and Galactocentric distances in terms of the total number of FGK and M stars which could host habitable Earth like planets and not gas giant planets. The Milky Way disc in presence of dust In Fig. 1 we show the time evolution of the star formation rate (SFR) (panel A), of [Fe/H] abundances (panel B), of SN rates (panel C), and finally of the total dust (panel D) as functions of the Galactocentric distance. In panel A) we see the effect of the inside-out formation on the SFR in the thin disc phase. During the halo-thick disc phase (up to 1 Gyr since the beginning of the star formation) all the Galactocentric distances show the same star formation history (for all radii we assume the same surface gas density and same formation time-scales in the halo-thick disk phase, for details see Section 3). In the inner regions the SFR in the thin disc phase is higher because of the larger gas density and shorter time-scales of gas accretion compared to the outer regions. Indeed, in the outer regions it is more evident the effect of the threshold in the gas density: the SFR goes to zero when the gas density is below the threshold. We notice also that the in the halo-thick disc phase we have the same SFR history at all Galactocentric distances. In panel B) it is shown the "age-metallicity" relation in terms of [Fe/H] ratio vs Galactic time. It is clear also in this case the effects of the inside-out formation: the inner regions exhibit a faster and more efficient chemical enrichment with higher values of [Fe/H]. Actually, at early times, in correspondence of the beginning of the second infall of gas (thin disc phase) there is a drop of the [Fe/H] abundance values. This drop is more evident in the inner regions of the Galactic disc. This is due to the fact that in the inner regions the second infall of primordial gas related to the thin disc phase is more massive and on shorter timescales, therefore the chemical abundances are more diluted at the beginning of the thin disc phase in the inner Galactic regions, compared to the external ones. In panel C) of Fig. 1 we present the total SN rates as functions of the Galactic time and Galactocentric distances. With the red line we label the limit adopted in this paper and in Paper I to take into account the destruction effects of SN explosions on the Galactic habitable zones modeling (2× < RSN SV >). Above this SN rate limit we assume that there is zero probability to have life on a terrestrial planet. Finally in panel D) of Fig. 1 we show the time evolution of the total surface mass density of dust at different Galactocentric radii. Dust production by stars is the main source of dust in the early phases of the Milky Way evolution. For this reason, in the inner regions, the dust amount is higher because of the large production by Type II SNe, during the initial burst of star formation, as visible in panel A). On the other hand, dust accretion becomes important at later epochs, and the dust mass tends to increase at all Galactocentric distances. The observed oscillation of the model occurs when the rates of dust accretion and dust destruction are comparable. In fact, in this case there is a gain of the total mass surface density of dust provoked by the dust growth, rapidly followed by a decreasing due to the dust destruction rate, which exceeds dust accretion. This turnover between those processes occurs especially in the quiescent phases of the Galactic evolution. The computed probabilities P GGP/F GK,M with the two infall chemical evolution model To compute the probabilities P GGP/F GK,M presented in eqs. (2) and (3) with our chemical evolution model we consider the weighted values on the IMF using the following expressions: < P GGP/F GK (R, t) > IMF = 0.07 × 10 1.8[Fe/H](R,t) model < M ⋆,F GK M ⊙ > IMF ,(13) for the FGK stars, and < P GGP/M (R, t) > IMF = 0.07 × 10 1.06[Fe/H](R,t) model < M ⋆,M M ⊙ > IMF ,(14) for M stars. The [Fe/H](R, t) model quantity is the computed iron abundance with our chemical evolution model adopting the Scalo (1986) IMF at the Galactic time t and Galactocentric distance R. Therefore, to compute < P GGP/M > IMF and <P GGP/F GK > IMF quantities we have only to know the weighted stellar mass on the IMF in the mass range of M stars and FGK stars, respectively. The weighted stellar masses on the Scalo (1986) The dust-to-gas ratio D G In this Subsection we provide a useful theoretical tool to set the proper initial conditions for the formation of protoplanetary discs. As underlined in Section 1, while dust grains of µm are directly observed in protoplanetary discs, on the other hand, the amount of gas mass is set starting from the one of the dust and by assuming a value for the dust-to-gas ratio. Unfortunately, this practice has several uncertainties (Williams & Best 2014). In this subsection, we connect the evolution of the dust to gas ratios at different Galactocentric distances with the chemical enrichment expressed in terms of [Fe/H]. Because of the well known "age-metallicity" relation reported in the Panel C) of Fig. 1, the dust-to-gas ratio ( D G ) vs [Fe/H] abundance ratio relation can be seen as a time evolution for the dust-to-gas ratio (D/G)(t). In the upper Panel of Fig. 4 is presented the evolution of the dust-to-gas ratio ( D G ) as a function of [Fe/H] at different Galactocentric distances. As expected, higher metallicities are reached in the inner radii, where the star formation is higher. The dust-to-gas ratio ( D G ) increases in time for two reasons: the first is that dust production in star forming regions is high, especially from Type II SNe, while the second is related to dust accretion. Dust accretion is a very important process occurring in the ISM and it becomes the most important one as the critical metallicity is reached 2 . As dust grains are formed by metals, dust accretion becomes more efficient as the metallicity in the ISM increases. For this reason at high values of [Fe/H], we found higher values of dust-to-gas ratio. The relation between the [Fe/H] and the dust-to-gas ratio is important because it can provide the probability of planet formation depending on the amount of dust in the ISM and, on the other hand, provides an estimate of the dust-to-gas ratio in the ISM during the formation of a protoplanetary disc. The solar dust-to-gas ratio D G ⊙ value predicted by our model (model value computed in the solar neighborhood at 9.5 Gyr) is 0.01066. In the lower panel of Fig. 4 we show the D G as a function of the [Fe/H] values only for the shell centered at 8 kpc and 2 kpc wide (the solar neighborhood). In the same plot the sixth degree polynomial fit which follows exactly the [Fe/H] vs D G in the range of [Fe/H] between -1 dex and 0.5 dex is presented. The expression of this fit is the following one: [Fe/H] = 6 n=0 α n D G n(17) All the coefficients α n and n are reported in the footnote 3 . We found that for [Fe/H] values higher than -0.6 dex a linear fit is able to reproduce pretty well the computed [Fe/H] vs D G relation in the solar vicinity. The equation of the linear fit reported in the lower panel of Fig. 4 is: [Fe/H] = 96.49 D G − 0.92.(18) This relation is important to connect the dust-to-gas ratio with the [Fe/H] abundance. If we combine it with eq. (4) we obtain the probability of having terrestrial planets but not gas giant ones depending on the amount of dust in the ISM. The P GHZ values around FGK and M stars In the upper panels of Fig. 5 (A and B) we show the evolution in time of the P GHZ values for FGK and M stars as functions of the Galactocentric distance in the case where SN destruction effects are not taken into account. We notice that the < P GHZ/M > IMF and < P GHZ/F GK > IMF probabilities are identical at large Galactocentric distances. This is due to the fact that, as shown in Fig. 3, the < P E/F GK > IMF and < P E/M > IMF probabilities are similar for sub-solar values of [Fe/H]. The chemical evolution in the outer parts of the Galaxy, because of the inside-out formation, is slow and with longer time scales. The maximum values of [Fe/H] are smaller than in the inner region and sub-solar (see the "age-metallicity" relation reported in panel B of Fig. 1). Moreover, in the inner regions < P GHZ,M > IMF and < P GHZ,F GK > IMF probabilities become to be different only for Galactic times larger than 8 Gyr. As expected the higher probabilities are related to the M stars. Even if the < P E/F GK > IMF and < P E/M > IMF probabilities are substantially different for [Fe/H]> 0.2 (see Fig. 3), the two associated P GHZ probabilities are similar. This is due to the definition of P GHZ (t): at each Galactic time the SFR × P E quantity is integrated from 0 to t. In other words, we are weighting the P E quantity on the SFR. From panel C) of Fig. 1 it is clear that the peak of the SFR in the inner regions (annular region between 3 and 7 kpc) is around 3 Gyr. From the "age-metallicity" relation reported in panel B) of Fig. 1 stated above, the P E.F GK and P E.M values are almost the same. This is the reason why the two P GHZ probabilities are similar even at late time in the inner regions. In the lower panels of Fig.5 (C and D) we show that, the P GHZ probabilities for FGK and M stars as functions of the Galactocentric distances and time, when the SN destruction effects have been taken into account, are almost identical. As found in Paper I the Galactocentric distance with the highest probability that a star (FGK or M type) hosts a terrestrial planet but not gas giant ones is 10 kpc. THE GHZ maps for FGK and M stars In order to recover the total number of stars at the Galactic time t and Galactocentric distance R hosting habitable planets, it is required the total number of stars created up to time t at the Galactocentric distance R (the N ⋆tot quantity in eq. 6). Our chemical evolution model very well reproduces the observed total local stellar surface mass density of 35 ± 5 M ⊙ pc −2 (Gilmore et al. 1989, Spitoni et al. 2015. In fact, our predicted value for the total surface mass density of stars is 35.039 M ⊙ pc −2 . Morever, we find that this value concerning only to M stars is 24.923 M ⊙ pc −2 and the one for FGK stars is 9.579 M ⊙ pc −2 . In Fig. 6 we show the number of FGK stars (upper panel) and M stars (lower panel) hosting habitable terrestrial planets but not gas giant planets as functions of the Galactic time and Galactocentric radius (the quantity N ⋆ lif e of eq. 6). We notice that for both FGK and M stars, the GHZ maps, in terms of the total number of stars hosting planets with life, peaks at 8 kpc. On the other hand, as we have seen above the maximum fraction of stars which can host habitable terrestrial planets peaks at 10 kpc (see panels C and D of Fig. 5). The reason why the GHZ peaks at galactocentric distances smaller than in the case when it is expressed in terms of fraction of stars is the following one: in the external regions the number of stars formed at any time is smaller than in the inner regions because of the the smaller SFR. This is in agreement with the results of Prantzos (2008) and Paper I. We see that, at the present time, in the solar neighborhood the number N ⋆,M,lif e /N ⋆,F GK,lif e =10.60. This ratio is consistent with the IMF we adopt in our model. In fact, the ratio between the fraction of M stars over FGK stars (by number) in a newborn population adopting a Scalo IMF is: Conclusions In this work we investigated the Galactic habitable zone of the Milky Way adopting the most updated prescriptions for the probabilities of finding terrestrial planets and gas giant planets around FGK and M stars. To do that we adopted a chemical evolution model for the Milky Way which follows the evolution of the chemical abundances both in the gas and dust. The main results can be summarized as follow: -Adopting the Scalo (1986) IMF the probabilities of finding gas giant planets around FGK and M stars computed with the two-infall chemical evolution model of Paper I begin to be different for supersolar values of [Fe/H]. In particular, substantial differences are present in the annular region centred at 4 kpc from the Galactic centre. -We provide for the first time a sixth order polynomial fit (and a linear one but more approximated) for the relation found in the chemical evolution model in the solar neighborhood between the [Fe/H] abundances and the dust-to-gas ratio D G . With this relation it is possible to express the Gaidos & Mann (2014) probabilities of finding gaseous giant planets around FGK or M stars in terms of the gas to dust ratio D G . -We provide a useful theoretical tool to set the proper initial condition for the formation of protoplanetary disc connecting the evolution of the dust-to-gas ratios at different Galactocentric distances with the chemical enrichment expressed in terms of [Fe/H]. -The probabilities that a FGK or M star could host habitable planets are roughly identical. Slightly differences arise only at Galactic times larger than 9 Gyr where the probability of finding gas giant planets around FGK becomes substantially different from the one associated to M stars. -As found by Paper I, adopting the same prescriptions for the destructive effect from close-by SN explosions, the larger number of FGK and M stars with habitable planets are in the solar neighborhood. -At the present time the total number of M stars with habitable terrestrial planets without gas giant ones are ≃ 10 times the number of FGK stars. This result is consistent with the Scalo (1986) IMF adopted here. The Gaia mission with its global astrometry, will be crucial for the study of exoplanets. We recall the relation used in this work between the frequency of gas giant planets and the metallicity of the host star was obtained by means of the Doppler effect with the method of radial velocity, and it will be possible with Gaia to test whether this result is an observational bias or is related to real physical processes. It is estimated that Gaia will be able to find out up to 10 4 giant planets in the solar vicinity with its global astrometry with distances spanning the range between 0.5 and 4.5 AU from the host star. Fig. 1 . 1Panel A): The SFR as the function of the Galactic time; Panel B): The time evolution of the [Fe/H] abundances with the two-infall chemical evolution model for the Milky Way disc (the "age-metallicity" relation); Panel C): The evolution in time of the Type II SN rates plus the Type Ia SN rates. With the red line we label the quantity 2× < RSN SV > which represents the minimum SN rate value (adopted in this work and in Paper I) to have destruction effects from SN explosions; Panel D): The time evolution of the total dust surface mass density. The color code in the four panels indicates the different Galactocentric distances. Fig. 2 . 2The probabilities <P GGP/F GK > IMF and <P GGP/M > IMF to find gas giant planets around FGK and M stars, respectively as functions of the abundance ratio [Fe/H] using our chemical evolution model for the Milky Way disc and adopting the prescriptions given byGaidos & Mann (2014). The color code indicates the Galactocentric distance. Fig. 4 . 4Upper panel: The evolution of the abundance ratio [Fe/H] as the function of the dust-to-gas ratio D G predicted by our chemical evolution model of the Milky Way disc. As in Fig. 1 the color code indicates different Galactocentric distances. Lower panel With the black solid line we report the [Fe/H] ratio vs D G computed at 8 kpc (solar neighborhood) using the chemical evolution model of the Milky Way. With the green solid line we show the fit obtained by mean of a sixth order polynomial fit. With the red dashed line we report the linear fit at 8 kpc. In both panels with the orange triangle we label the value of the [Fe/H] as a function of the dust-to-gas ratio D G for the model computed in the solar neighborhood at the Galactic time of 9.5 Gyr (i.e. model solar value). Fig. 5 . 5. 2 we show the evolution of < P GGP/F GK > IMF and < P GGP/M > IMF probabilities as the function of the [Fe/H] abundance ratio computed at different Galactocentric distances using our chemical evolution models. Because of the inside-out formation, the inner regions exhibit a faster and more efficient chemical enrichment. In fact, the model computed at 4 kpc reaches, at the present time, [Fe/H] value of 0.55 dex, instead of at 20 kpc the maximum [Fe/H] is Upper panels: The probability <P GHZ > IMF to find terrestrial habitable planets but not gas giant around FGK stars (Panel A) and M stars (Panel B) as the function of the Galactocentric radius. In this case we do not consider the destructive effects from nearby SN explosions. The color code indicates different Galactic times. Lower panels: The probability P GHZ to find terrestrial habitable planets but not gas giant around FGK stars (Panel C) and M stars (Panel D) as the function of the Galactocentric radius taking into account the destructive effects from nearby SN explosions. The color code indicates different Galactic times. equal to -0.1 dex. We see that the two probabilities become to be substantially different for over-solar values in the inner regions. For instance, at the present time, i.e. at the maximum values of the [Fe/H] abundance in Panel C ofFig. 1, at 4 kpc the probabilities < P GGP/F GK > IMF and < P GGP/M > IMF show the values of 0.43 and 0.04, respectively. , we derive that at this age the mean Galactic [Fe/H] is -0.5 dex. At this metallicity, as 3 [Fe/H]=−1.48 + 1.05 10 4 D NUMBER OF FGK STARS WITH HABITABLE PLANETSFig. 6. The total number of FGK stars (Upper panel) and M stars (Lower panel) having habitable terrestrial planets but not gas giant ones as functions of the Galactocentric distance and the Galactic time (the N ⋆ lif e quantity in eq. 6) where SN destructive effects are taken into account. The number of stars are computed within concentric rings, 2 kpc wide.G − 4.91 10 5 D G 2 + 1.1410 8 D G 3 − 1.32 10 10 D G 4 + 7.57 10 11 D G 5 − 1.69 10 13 D G 6 0 13 6.66e+07 10.4 1.332e+08 1.998e+08 7.8 2.664e+08 t [Gyr] 3.33e+08 5.2 2.6 4 0 8 R [kpc] 12 16 20 NUMBER OF M STARS WITH HABITABLE PLANETS 0 13 7.06e+08 10.4 1.412e+09 2.118e+09 7.8 2.824e+09 t [Gyr] 3.53e+09 5.2 2.6 4 8 R [kpc] 0 12 16 20 M number F numberGK number Scalo IMF = 0.45 M⊙ 0.08 M⊙ m −2.35 dm 1.4 M⊙ 0.45 M⊙ m −2.35 dm = 11.85. (19) Finally, the predicted local surface mass density of M stars hosting habitable planets predicted by our model is 5.446 M ⊙ pc −2 , and the value for FGK stars is 2.40 M ⊙ pc −2 . The critical metallicity is the metallicity at which the contribution of dust accretion overtakes the dust production from stars(Asano et al. 2013). AcknowledgmentsWe thank the anonymous referee for the suggestions that improved the paper. 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[ "Multi-target Range and Angle detection for MIMO-FMCW radar with limited antennas", "Multi-target Range and Angle detection for MIMO-FMCW radar with limited antennas" ]
[ "Himali Singh \nElectrical Engineering Department\nIndian Institute of Technology Delhi\nIndia\n", "Arpan Chattopadhyay [email protected]. \nElectrical Engineering Department\nIndian Institute of Technology Delhi\nIndia\n" ]
[ "Electrical Engineering Department\nIndian Institute of Technology Delhi\nIndia", "Electrical Engineering Department\nIndian Institute of Technology Delhi\nIndia" ]
[]
Multiple-input multiple-output (MIMO) radar has several advantages with respect to the traditional radar array systems in terms of performance and flexibility. However, in order to achieve high angular resolution, a MIMO radar requires a large number of transmit/ receive antennas, which increases hardware design and computational complexities. Although spatial compressive sensing (CS) has been recently considered for a pulsed-waveform MIMO radar with sparse random arrays, such methods for the frequency-modulated continuous wave (FMCW) radar remain largely unexplored. In this context, we propose a novel multi-target localization algorithm in the range-angle domain for a MIMO FMCW radar with a sparse array of randomly placed transmit/ receive elements. In particular, we first obtain the targets' range-delays using a discrete Fourier transform (DFT)-based focusing operation. The target angles are then recovered at each detected range using CS-based techniques exploiting the sparsity of the target scene. Our simulation results demonstrate the effectiveness of the proposed algorithm over the classical methods in detecting multiple targets with a sparse array.
null
[ "https://export.arxiv.org/pdf/2302.14327v1.pdf" ]
257,232,626
2302.14327
6256541de34b3e9dec0a237ce3f8fa9cad9e3a8b
Multi-target Range and Angle detection for MIMO-FMCW radar with limited antennas Himali Singh Electrical Engineering Department Indian Institute of Technology Delhi India Arpan Chattopadhyay [email protected]. Electrical Engineering Department Indian Institute of Technology Delhi India Multi-target Range and Angle detection for MIMO-FMCW radar with limited antennas Index Terms-Compressive sensingFMCW radarMIMO radarrandom arraysrange-angle estimationsparse arrays Multiple-input multiple-output (MIMO) radar has several advantages with respect to the traditional radar array systems in terms of performance and flexibility. However, in order to achieve high angular resolution, a MIMO radar requires a large number of transmit/ receive antennas, which increases hardware design and computational complexities. Although spatial compressive sensing (CS) has been recently considered for a pulsed-waveform MIMO radar with sparse random arrays, such methods for the frequency-modulated continuous wave (FMCW) radar remain largely unexplored. In this context, we propose a novel multi-target localization algorithm in the range-angle domain for a MIMO FMCW radar with a sparse array of randomly placed transmit/ receive elements. In particular, we first obtain the targets' range-delays using a discrete Fourier transform (DFT)-based focusing operation. The target angles are then recovered at each detected range using CS-based techniques exploiting the sparsity of the target scene. Our simulation results demonstrate the effectiveness of the proposed algorithm over the classical methods in detecting multiple targets with a sparse array. I. INTRODUCTION Frequency-modulated continuous wave (FMCW) radars have become a popular choice for short-range applications like automotive radars [1,2], human vital sign monitoring [3], synthetic aperture radars (SARs) [4], and surveillance systems [5]. The main advantages of FMCW radar are portability, low cost, and high resolution. An FMCW radar transmits a finite train of (piece-wise) linear frequencymodulated (LFM) chirps in each coherent processing interval (CPI). At the receiver, the target returns are mixed with the transmitted signal to obtain a complex sinusoidal beat or intermediate frequency (IF) signal. The targets' locations (and velocities if moving) information can be extracted from the frequencies of this IF signal. To this end, fast Fourier transforms (FFTs) have traditionally been used to estimate the IF signal frequencies [1]. However, to localize targets in the angular domain, multiple transmit and receive antennas are required. In MIMO radars, multiple orthogonal waveforms are transmitted simultaneously with the target returns processed jointly by the multiple receive antennas. The MIMO radar achieves a better angular resolution than conventional radar by exploiting a large number of degrees of freedom of a virtual array synthesized with a small number of physical antenna elements. In this work, we focus on multi-target range-angle detection using MIMO FMCW radars. Conventionally, two-dimensional frequency estimation algorithms are used to estimate both targets' ranges and angles of arrival (AOAs) from the received signal. Other frequency estimation algorithms considered for MIMO FMCW radar include 2D-FFT [6], 2D-MUSIC [7], and ESPRIT [8]. From the array processing theory, it is known that a high angular resolution requires a large array aperture [9]. Further, increasing the aperture without a parallel increase in antenna elements leads to A. C. acknowledges support via the faculty seed grant, professional development fund and professional development allowance from IIT Delhi, and grant no. RP04215G from I-Hub Foundation for Cobotics. H. S. acknowledges support via Prime Minister Research Fellowship. ambiguities in angle estimation. Although MIMO technology helps to achieve higher resolution, the cost of synthesizing a large virtual array with the half-wavelength element spacing (spatial Nyquist sampling rate) can be very high. In this context, sparse linear arrays (SLAs) have been proposed recently for both pulsed-waveform and continuous-wave radars [6,10,11]. Optimal sparse array design was considered in [12] while [6] designed a non-uniform SLA and applied digital beamforming techniques for AOA estimation after interpolating for the missing measurements in the synthesized SLA. On the other hand, [11] suggested matrix completion techniques to complete the corresponding linear array for angle detection. Compressed sensing (CS) addresses sparse signal recovery with fewer measurements [13]. The sparse array setup enables spatial compressive sensing such that the CS recovery naturally suits our target localization problem. Note that the target scene is sparse since only a small number of targets are present in the scene. The CS-recoverybased localization has recently been applied for angle estimation for pulsed-MIMO radar [10]. In [14], CS-based algorithms were used to process measurements from a traditional full array. Besides, spatial compression, CS techniques have also been considered in radars for reduced sampling rate [15,16], interference mitigation [17], and multi-target shadowing effect mitigation in constant false-alarm rate (CFAR) detection [18]. Contributions: In this paper, we present a novel multi-target localization algorithm to detect targets' ranges and AOAs using a random SLA. Prior methods employing CS-based techniques (e.g. [10]) often address only angle detection at a prior known range bin. Here, we consider both range and angle detection in a MIMO FMCW radar. For range detection, we exploit a discrete Fourier transform (DFT)-based focusing operation followed by binary integration [9] of measurements across pulses and virtual array channels, trading off range resolution for higher detection probability. For angle recovery, we use CS-based techniques, which relax the dependence of the angular resolution on the number of antenna elements. Finally, we illustrate the proposed method's performance through numerical simulations, comparing it with classical-FFT processing. The rest of the paper is organized as follows. In the next section, we describe the FMCW radar system model with the random sparse MIMO array setup. In Section III, we present the proposed range and angle detection algorithm. The simulation results are discussed in Section IV, followed by conclusins in Section V. II. RADAR SYSTEM MODEL Consider a colocated MIMO radar system, as shown in Fig. 1, composed of NT transmitters and NR receivers forming a (possibly overlapping) array of total aperture ZT and ZR, respectively, and define Z . = ZT + ZR. The n-th transmitter's and m-th receiver's locations along the x-axis are Zαn/2 and Zβm/2, respectively, where αn ∈ [−ZT /Z, ZT /Z] and βm ∈ [−ZR/Z, ZR/Z]. Note that αn and βm are randomly drawn from appropriate uniform distributions [10]. The transmitters transmit LFM chirps, orthogonal across transmitters. Consider fc as the carrier frequency and γ as the chirp rate of the LFM chirp with chirp duration T . The FMCW radar's transmitted chirp is modeled as s(t) = exp j2π fct + γ 2 t 2 , 0 ≤ t ≤ T, with t as the continuous-time index. A total of P chirps is transmitted in each CPI. Different orthogonal waveform designs for MIMO-FMCW radar transmitters have been proposed in [19,20]. For simplicity, we consider time-domain multiplexing, where the transmitters transmit the same signal with relative time shifts. In our proposed detection algorithm, we process each transmitted chirp independently and use binary integration [9] after detection across pulses (in a CPI) to obtain the estimated ranges. On the contrary, classical-FFT processing considers coherent or non-coherent integration of the pulses to average out the interference and noise before detection [9]. In Section IV, we discuss how binary integration improves the detection probability over classical processing. Similarly, the orthogonality of the transmitted signals allows the corresponding received signal components to be separated at each receiver. Hence, we first focus on the received signal component at the m-th receiver due to the single chirp transmitted from the n-th transmitter. We assume a target scene of K stationary, far-field, non-fluctuating point targets (Swerling Case 0 model [9]). We denote the k-th target's range and angle of arrival (AOA) as R k and θ k , respectively. Denote τ m,n,k as the total time-delay in the k-th target's return at the m-th receiver from the n-th transmitted signal such that the received signal component is given as rm,n(t) = K k=1 a k s(t − τ m,n,k ), where a k is the complex amplitude proportional to the k-th target's radar cross-section (RCS). The time delay τ m,n,k consists of the range delay τ R k and angular delay τ θ m,n,k as τ m,n,k = τ R k + τ θ m,n,k ,(1) where τ R k = 2R k /c and τ θ m,n,k = Z(αn + βm) sin (θ k )/2c with constant c denoting the speed of light. Note that the far-field assumption leads to a constant AOA across the array. After mixing the m-th received signal with the n-th transmitted signal, the FMCW radar's IF signal ym,n(t) is represented as ym,n(t) = K k=1 a * k exp j2π γτ m,n,k t + fcτ m,n,k − γ 2 τ 2 m,n,k + wm,n(t), where (·) * represents the conjugate operation and wm,n(t) is the interference plus-noise term. Each IF signal ym,n(t) is sampled at sampling frequency fs as ym,n[t] = K k=1 a * k exp j2π γτ m,n,k t fs + fcτ m,n,k − γ 2 τ 2 m,n,k + wm,n[t], for 0 ≤ t ≤ N −1, where N = fsT is the total number of samples in a single pulse and wm,n[t] is the sampled noise. Here, we represent the discrete-time index by t. For the NT transmitters and NR receivers MIMO setup, we obtain 'NT NR' sampled measurements {ym,n[t]} 1≤m≤N R ,1≤n≤N T for all P pulses. III. SPARSE ARRAY RECOVERY ALGORITHM In this section, we describe the proposed range-angle detection algorithm. The spatial compressive sensing framework proposed in [10] for pulsed MIMO radar assumes an independent range-Doppler processing and focus only on targets in a given range-Doppler bin for AOA estimation. On the contrary, here, we consider both range and AOA detection. In Section III-A, we adopt a DFT-focusing operation to estimate the targets' ranges and separate the range and AOA information. Finally, in Section III-B, the CS-based recovery provides the AOA estimates at each detected range bin. A. Range detection Consider the N -point DFT of the sampled IF signal ym,n[t] as Ym,n[l] = N −1 t=0 ym,n[t] exp (−j2πlt/N ), = K k=1 a * k exp j2π fcτ m,n,k − γ 2 τ 2 m,n,k × N −1 t=0 exp j2π γτ m,n,k fs − l N t + Wm,n[l],(2)for 0 ≤ l ≤ N − 1, where Wm,n[l] = N −1 t=0 wm,n[t] exp (−j2πlt/N ) represents the noise term. Replacing N = fsT , we first analyze the sum of exponents (2). Consider the sum of M exponents g(x|x) = M −1 q=0 e j(x−x)qω for given constants x and ω. We can approximate |g(x|x)| as N −1 t=0 exp j( 2πγ fs ) τ m,n,k − l γT t in|g(x|x)| = M, |x − x| ≤ π/M ω 0, |x − x| > π/M ω . The approximation implies that in the focus zone |x − x| ≤ π/M ω, the M exponents are coherently integrated while the signal outside the focus zone is severely attenuated. In [15], this focusing approximation was introduced as Doppler focusing across pulses in a CPI to reduce the joint delay-Doppler estimation problem to delay only estimation at a particular Doppler frequency. In our case, the sum of exponents appears naturally in the DFT of ym,n[t]. Using the focusing approximation for the sum of N exponents (indexed by t) in (2), we have Ym,n[l] ≈ K k =1 a * k N exp j2π fcτ m,n,k − γ 2 τ 2 m,n,k + Wm,n[l],(3) where {a k , τ m,n,k } 1≤k ≤K represents the subset of targets which satisfy |τ m,n,k − l/(γT )| ≤ 1/(2γT ) for the given l-th DFT bin. Assuming τ R k τ θ m,n,k for all targets, we have τ m,n,k ≈ τ R k such that the received signal from targets at ranges satisfying |τ R k − l/(γT )| ≤ 1/(2γT ) are coherently integrated, resulting in a (magnitude) peak at the l-th DFT bin. Furthermore, the practical values of γ and T for an FMCW radar ensures that the value 1/(2γT ) is small enough and τ R k ≈ l/(γT ). Hence, using threshold detection to identify the peaks in Ym,n[l] (corrupted by noise), we obtain the range estimates. The estimated range R corresponding to a DFT peak at l -th bin is computed as R = cl 2γT . These range estimates are computed independently for all P pulses and for all NT NR measurements {ym,n[t]} 1≤m≤N R ,1≤n≤N T . The detected ranges are first filtered for false alarms across the P pulses using binary integration, i.e., only the ranges detected in a sufficient number of pulses are considered valid target ranges. Similarly, the detected ranges are also filtered across the NT NR measurements which further reduces the false alarm probability. The classical-FFT range processing also involves threshold detection for peaks in the DFT of the sampled IF signal. However, in classical processing, all the pulses are processed together noncoherently to compute the DFT, which increases the range resolution by increasing the frequency resolution of the computed DFT. On the other hand, by processing each pulse independently, we trade off range resolution for reduced missed detection probability. In particular, in the case of close-range targets, the classical processing often suffers from false peaks dominating the actual target peaks. Using binary integration across pulses and then across NT NR virtual array channels, the detection probability is enhanced with a constant false alarm probability. This performance improvement with binary integration is further discussed in Section IV-A with a simulated example of three close-range targets. B. Angle detection Consider a detected range bin at the l -th DFT point. Substituting + K k =1 a * k N exp j2π fcτ R k − γ 2 (τ R k ) 2 exp (j2π(fc − γτ R k )τ θ m,n,k ), using (τ R k ) 2 (τ θ m,n,k ) 2 . For practical FMCW radars, carrier frequency fc (in GHz), chirp rate γ (in MHz/µs) and short-range delay τ R k (a few µs) are such that the term γτ R k is negligible and Ym,n[l ] = K k =1 a * k N exp j2π fcτ R k − γ 2 (τ R k ) 2 exp (j2πfcτ θ m,n,k ) + Wm,n[l ].(4) Note that the exponential terms with the range and angle delays are now separated in Ym,n[l ] . Denote x k . = a * k N exp j2π fcτ R k − γ 2 (τ R k ) 2where the NT NR × K matrix C(θ) = [c(θ1), . . . , c(θ K )](5) with each column c(θ) = [exp (jπfcZ(α 1 + β 1 )sin(θ)), . . . , exp (jπfcZ(α N T + β N R )sin(θ))] T , known as the virtual array steering vector [10] parameterized by the AOA θ. Here, W represents the NT NR × P noise matrix obtained from similarly stacking Wm,n[l ] from all pulses. We need to recover θ and X from Y with a small number of antenna elements. To this end, we use a sparse localization framework. Assume a grid of G points φ 1≤g≤G of the possible target AOAs θ with G K and negligible discretization errors. Each grid element φg parameterizes a column of C(θ). Hence, we can define a NT NR × G dictionary matrix C = [c(φ1), . . . , c(φG)]. From (5), the measurements Y are then expressed as Y = CX + W,(6) where the unknown G × P matrix X contains the target AOAs and complex amplitudes (x k ). A non-zero row of X represents a target present at the corresponding grid point. Hence, the system (6) is sparse since X has only K G non-zero rows for a particular detected range bin. Given the measurements Y and matrix C, AOA estimation reduces to determining the support (non-zero rows) of X. Note that the matrix C and hence, the recovery guarantees depend on the choice of grid points φ 1≤g≤G as well as the number and (random) positions of the transmitters and receivers ({αn} 1≤n≤N T and {βm} 1≤m≤N R ). In [10], authors also discuss the sufficient conditions on the grid and the random array for recovery of X with high probability. For the recovery of sparse matrix X with limited antenna elements, we consider CS-based algorithms. CS problems can be classified as single measurement vector (SMV) models for P = 1 where Y reduces to a single vector, or multiple measurement vector (MMV) models for P ≥ 1. Our problem (6) is an MMV setting. However, we first consider the SMV setting with P = 1 such that Y = y, X = x and W = w in (6). Recovering a sparse x from NT NR measurements y involves solving the non-convex combinatorial l0-norm problem minx x 0 s.t. y − Cx 2 ≤ ν,(7) where parameter ν is chosen based on the noise level w 2 or the sparsity of x. Solution of (7) requires an exhaustive search of exponential complexity [13]. However, an approximate solution can be obtained using a variety of polynomial complexity algorithms. Matching pursuit (MP) is one such family of methods, which iteratively refines the provisional support by adding one dictionary element at a time. Orthogonal MP (OMP) [21], orthogonal least squares (OLS) [22], and compressive sampling MP (CoSaMP) [23] are some popular MP algorithms for the SMV setting. For the general MMV setting, simultaneous OMP (SOMP) [24] extends the OMP algorithm to matrix measurements. Another class of recovery algorithms is the Basis pursuit (BP) which relaxes the l0-norm in (7) with l1-norm, resulting in a convex problem whose global solution can be found in polynomial time [25]. In Section IV-B, we consider OMP and SOMP, respectively, for the sparse recovery in SMV and MMV settings. IV. SIMULATION RESULTS We now demonstrate the performance of the proposed method in comparison to the classical FFT-processing. In Section IV-A, we first investigate the effect of binary integration for range processing discussed in Section III-A. The simulation results for a sparse target scene are provided in Section IV-B. We considered a MIMO-FMCW radar system transmitting at carrier frequency fc = 9.4 GHz. The transmitted bandwidth was chosen as B = 250 MHz with chirp duration T = 363µs (chirp rate γ = B/T ) and sampling frequency fs = 1.4 MHz such that the range resolution was 0.6 m. One CPI consisted of P = 10 MIMO sweeps. For the sparse array, 3 transmitters and 3 receivers (total 6 antenna elements) were placed uniformly over the array apertures ZT = ZR = 6λ, where λ is the wavelength of the transmitted signal. Note that in this case αn, βm ∈ [−0.5, 0.5] for 1 ≤ n, m ≤ 3. For the full array, we considered 4 transmitters and 8 receivers arranged as in [7]. In particular, two transmitters were placed on either side of the array with an inter-element spacing of λ. The receivers were placed in the middle with an inter-element spacing of 0.5λ and 0.25λ spacing between the closest transmitter-receiver elements. This arrangement results in a virtual array of 20 unique element locations with 0.5λ uniform separation. The target gains were generated as a k = exp (jψ k ) with ψ k drawn from i.i.d. uniform distribution over [0, 2π). The noise term wm,n[t] is modeled as i.i.d. zero-mean complex circular Gaussian noise CN (0, σ 2 I), mutually independent across pulses and virtual array channels. The signal-to-noise ratio (SNR) is then defined as −10 log 10 (σ 2 ) [10]. A. DFT processing: classical and proposed method Consider three close-range targets with ranges R1 = 20.6 m, R2 = 20.0 m and R3 = 19.4 m at AOAs θ1 = θ2 = θ3 = 0 • . Considering the noise-free case, Fig. 2a shows the range-FFT computed in the classical-FFT processing. Fig. 2b shows the DFT computed in the proposed method for three different pulses from measurement y1,1[t], which are then used to estimate the target ranges using binary integration as detailed in Section III-A. We observe that non-coherent processing of the pulses in the classical method provides a refined spectrum as compared to the proposed method of processing one pulse at a time. However, the classical range-FFT suffers from sidelobe effect which results in a false peak of the same order of the third target (R3) peak. Hence, reducing the false alarms (increasing the threshold) results in a missed detection. On the other hand, in the proposed binary integration method, the missed targets in one pulse can be detected at other pulses (or some other ym,n[t] measurement). Hence, binary integration can enhance the detection probability for a constant false alarm rate by trading off range resolution. B. Performance analysis We considered K = 5 targets with target delays and AOAs chosen uniformly at random with ranges in [10m, 40m] and AOAs in [−15 • , 15 • ]. This ensures the target scene has close-range targets as well as multiple targets at the same range. For the proposed CS-based angle recovery, we considered OMP with the vector measurement y as the sum across 10 pulses, and SOMP for matrix measurement Y. In [10], authors assumed a known sparsity level and used the prior information of the actual number of targets K in the CS algorithms. On the contrary, here, we assumed a sparsity level of Kmax for OMP and SOMP algorithms. The target AOAs were then obtained using threshold detection on the recovered signal. Hence, we do not require a prior estimate of K. We consider hit rate and root-mean-squared error (RMSE) of the recovered targets as the performance metrics. A 'hit' is defined as a range-angle estimate within 0.6 m in range and 1 • in angle of the true target. The recovery error is computed for the estimates classified as hits. The target estimates not classified as hits are the false alarms. We vary the thresholds of the threshold detectors to maintain a constant false-alarm rate at different SNRs. The hit rate and false alarm rate for different SNRs, averaged over 300 independent simulations, are shown in Fig. 3 for the proposed method and classical-FFT processing considering both full and sparse arrays. The corresponding range and angle recovery errors are shown in Fig. 4. From Fig. 3, we observe that for high SNRs, the proposed method with OMP-based recovery achieves the same hit rate as the classical processing for the full array with same false alarm rates. However, the full array consists of 12 (4 Tx+ 8 Rx) antenna elements, while the sparse array requires only half of these elements. On the other hand, reducing the transmitter and receiver elements drastically degrades the detection ability of classical-FFT processing. Interestingly, at lower SNRs, the hit rate of the classical processing (full array) reduces due to the side-lobe effect discussed earlier. Note that at high SNRs, the false peaks from the side-lobes are not prominent compared to the actual target peaks. On the contrary, the proposed method maintains the same hit rate with varying noise levels. The SOMPbased angle recovery in the proposed method further helps to increase the detection probability with a reduced false alarm rate, compared to OMP-based recovery. SOMP improves the detection ability by exploiting the correlation among the measurements across different pulses to recover the true target AOAs. In Fig. 4a, we observe that the classical method has a lower range recovery error for both full and sparse arrays, because of the refined range FFT computed in the classical method. The proposed method achieves a slightly higher range error of about 0.15 m. Similarly, in Fig. 4b, the classical method slightly outperforms the proposed method in terms of angle recovery error. However, the classical method's angular resolution (hence, the error) depends on the array aperture. A higher angular resolution requires an increase in the array aperture and hence, the number of antenna elements. On the other hand, the proposed method's angular resolution is determined by the number of grid points G. Hence, the angle recovery error of the proposed method can be reduced with a finer grid φ 1≤g≤G . However, the number and locations of the antenna elements still affect the dictionary matrix C, which in turn, determines the recovery probability of the CS-based algorithms. V. SUMMARY We have proposed a novel sparse-recovery-based multi-target detection algorithm in the range-angle domain for MIMO FMCW radar. The proposed method enables a random array MIMO system to localize multiple targets in a sparse scene with reduced antenna elements compared to the traditional full array system. For range detection, we considered a DFT-based focusing operation with binary integration across pulses and virtual array channels. The binary integration in range detection provided a reduced missed detection probability than the classical non-coherent range-FFT processing. Finally, we considered a sparse recovery framework for target AOAs detection using both SMV and MMV-based CS recovery algorithms. Through numerical simulations, we illustrated the proposed method's target recovery compared to classical-FFT processing. Our numerical experiments suggest that the proposed method can achieve the traditional full-array hit rate with limited antenna elements. Furthermore, the MMV-based angle recovery can outperform both SMV-based and classical-FFT methods. Fig. 1 . 1MIMO radar system ( and • denote receivers and transmitters, respectively). ( 1 ) 1in (3) for τ m,n,k , we obtain Ym,n[l ] = Wm,n[l ] as the complex amplitude independent of the AOAs. Further, we denote Y p m,n [l ] as the l -th DFT coefficient computed for the p-th pulse. Stack the measurements Y p m,n [l ] for all (m, n)-pairs in a NT NR × 1 vector yp. Now, define the 'NT NR × P ' matrix Y = [y1, . . . , yp]. Similarly, define the K × P matrix X = [ x1, . . . , xP ] with xp = [x1, . . . , x K ] T . Now, substituting τ θ m,n,k = Z(αn+βm) sin (θ k )/2c in (4) yields Y = C(θ) X + W, Fig. 2 . 2Normalized DFT magnitude for (a) Classical range-FFT; and (b) Three different pulses for the proposed method (arrows indicate the detected peaks). Fig. 3 . 3Average (a) false alarm rate, and (b) hit rate at different SNRs for classical-FFT processing and the proposed method. Fig. 4 . 4Root MSE in (a) range, and (b) angle estimation at different SNRs for classical-FFT processing and the proposed method. We set Kmax = 10 for both OMP and SOMP. The grid φ 1≤g≤G was chosen as 150 uniformly spaced points in the sin(θ) domain in the interval [−0.7071, 0.7071]. Note that the AOA estimates are uniform in the sin (θ) domain. This holds for classical-FFT processing as well where the DFT samples are equally spaced in the sin (θ) domain between [−1, 1] and assume a non-linear distribution in the θ domain. The grid φ 1≤g≤G spans the interval [−45 • , 45 • ] in the AOA domain. 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Aubry, "Nearly orthogonal waveforms for MIMO FMCW radar," IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 3, pp. 1426-1437, 2013. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. Y C Pati, R Rezaiifar, P S Krishnaprasad, Proceedings of 27th Asilomar conference on Signals, Systems and Computers. 27th Asilomar conference on Signals, Systems and ComputersIEEEY. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, "Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition," in Proceedings of 27th Asilomar conference on Signals, Systems and Computers. IEEE, 1993, pp. 40-44. Orthogonal least squares methods and their application to non-linear system identification. S Chen, S A Billings, W Luo, International Journal of Control. 505S. Chen, S. A. Billings, and W. 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[ "On the nature of Tycho Brahe's supernova", "On the nature of Tycho Brahe's supernova" ]
[ "Pilar Ruiz-Lapuente \nInstituto de Física Fundamental\nConsejo Superior de Investigaciones Científicas\nc/. Serrano 121E-28006MadridSpain\n\nInstitute of Cosmos Sciences\nICC (UB-IEEC)\nc/. Martí i Franqués 1E-08028BarcelonaSpain\n", "Pilar Ruiz-Lapuente " ]
[ "Instituto de Física Fundamental\nConsejo Superior de Investigaciones Científicas\nc/. Serrano 121E-28006MadridSpain", "Institute of Cosmos Sciences\nICC (UB-IEEC)\nc/. Martí i Franqués 1E-08028BarcelonaSpain" ]
[]
At the 450 yr anniversary of its observation, the supernova named after Tycho Brahe, SN 1572, can be explained in the terms used nowadays to characterize Type Ia supernovae (SNe Ia). By assembling the records of the observations made in 1572-74 and evaluating their uncertainties, it is possible to recover the light curve and the color evolution of this supernova. It is found that, within the SNe Ia family, the event should have been a SN Ia with a normal rate of decline. Concerning the color evolution of SNe Ia, the most recently recovered records reaffirm previous findings of its being a normal SN Ia. The abundance studies from X-ray spectroscopy of the whole remnant point to a nuclear burning of the kind of a delayed detonation explosion of a Chandrasekhar-mass white dwarf. A tentative single degenerate path to explosion was suggested from the exploration of the stars in the field in SN 1572. Though, the origin in a double degenerate is being considered as well. Tycho Brahe's supernova, being the first supernova studied by astronomers, is still the subject of very intensive debates nowadays.
10.3389/fspas.2023.1112880
[ "https://export.arxiv.org/pdf/2212.00878v2.pdf" ]
254,220,833
2212.00878
67ac96b869a02363ec25f4208a77b95314d2ebe6
On the nature of Tycho Brahe's supernova Pilar Ruiz-Lapuente Instituto de Física Fundamental Consejo Superior de Investigaciones Científicas c/. Serrano 121E-28006MadridSpain Institute of Cosmos Sciences ICC (UB-IEEC) c/. Martí i Franqués 1E-08028BarcelonaSpain Pilar Ruiz-Lapuente On the nature of Tycho Brahe's supernova Correspondence*:cosmologystars: supernovae: generalSN 1572ISM: supernova remnantswhite dwarfs At the 450 yr anniversary of its observation, the supernova named after Tycho Brahe, SN 1572, can be explained in the terms used nowadays to characterize Type Ia supernovae (SNe Ia). By assembling the records of the observations made in 1572-74 and evaluating their uncertainties, it is possible to recover the light curve and the color evolution of this supernova. It is found that, within the SNe Ia family, the event should have been a SN Ia with a normal rate of decline. Concerning the color evolution of SNe Ia, the most recently recovered records reaffirm previous findings of its being a normal SN Ia. The abundance studies from X-ray spectroscopy of the whole remnant point to a nuclear burning of the kind of a delayed detonation explosion of a Chandrasekhar-mass white dwarf. A tentative single degenerate path to explosion was suggested from the exploration of the stars in the field in SN 1572. Though, the origin in a double degenerate is being considered as well. Tycho Brahe's supernova, being the first supernova studied by astronomers, is still the subject of very intensive debates nowadays. INTRODUCTION In this review, we will start with the historical data on the supernova discovered by Tycho Brahe, SN 1572. We give an account of its light curve and cosmological charaterization. We discuss as well the explosion mechanism of the supernova and the binary path leading to the the explosion, taking into account the most recent work. This article aims to address what is known about the nature of Tycho's SN. Though, due to constraints on the length, not all the contributions in relation to this topic can be discussed. We will try to present the most recent overall view. We will begin with an introduction of SNe Ia as calibrated candles. The understanding of SNe Ia as empirical tools in cosmology had an enormous boost at the end of the past century. Pskovskii (1977Pskovskii ( , 1984 first suggested a correlation between absolute magnitude at maximum and rate of decline of the light curve. The faster the decline of the light curve the dimmer the SN Ia and the slower the decline the brighter the SN Ia. The follow-up of SNe Ia with modern digital CCD detectors, as done in the Calan-Tololo search, confirmed that such correlation had a low intrinsic dispersion. Extensive and accurate observations collected by Phillips, Hamuy and collaborators allowed to build up the mathematical expression. Phillips (1993) first quantified it using a small number of SNe Ia. After that, Hamuy et al (1996a,b) enlarged the study and obtained a more significant fit. The relation has the form: M M AX = a + b[∆m 15 (b) − 1.1](1) where M M AX is the absolute magnitude at maximum, a and b are constants, and ∆m 15 is the number of magnitudes increased in 15 days after maximum. As the first results were given by Phillips (1993), this is called the Phillips relationship. The Supernova Cosmology Project used for that maximum brightness-rate of decline of the light curve the so-called stretch factor, s (Perlmutter et al. 1997, Goldhaber et al. 2001, measuring the amount of broadening or narrowing of the light curve up to 60 days after maximum in relation to a template light curve (see Figure 1 from comparison of ∆m 15 and s). In this way, they were able to use the corrected to template light curves for cosmology, m eff B : m eff B ≡ m B + α(s − 1)(2) where α is the best-fit value of the stretch-luminosity slope from the fit to the primary low-extinction subset 1 . On the other hand, the High-z Supernova Team employed the MLCS (Multicolor Light Curve Shapes: Riess et al. 1995Riess et al. , 1996 method. Riess, Press & Kirshner (1995) give an alternative way to express the intrinsic luminosity of Type Ia supernovae as related to the rate of decline of the light curve. They fit with an overall shape parameter the evolution in luminosity before maximum to well past maximum. Their parameterization, and the one used by Hamuy and collaborators (1996a;b) give a comparable scale of magnitudes for specific SNe Ia The most recent version by this method is the MLCSk2 (Jha, Riess & Kirshner 2007). Several SNe Ia had been discovered after maximum, so the peak brightness had to be extrapolated by comparison with light-curve templates. These SNe were at distances corresponding to redshifts 0.01 < z < 0.1. During the 90's, the Supernova Cosmology Project and the High-z Supernova Team, had started to observe SNe Ia at higher redshifts, 0.18 ≤ z ≤ 0.83, to measure the evolution of the Hubble parameter, H(z). From two different samples of high-redshift SNeIa and using these different methods of light-curve fitting, the two groups reached the same conclusion: the expansion of the universe is accelerating (Riess et al 1998;Perlmutter et al. 1999). Tycho SN 1572 looked like a normal SN Ia, i.e, did not show to be neither overluminous nor underluminous if the historical records of the SN were fit with the stretch relationship for SNe Ia (Ruiz-Lapuente 2004, hereafter called R04). The light curve shape has been confirmed and the color evolution from a typical SN Ia has been proved in more recent historical studies (Neuhaeuser 2022). In recent years, new types of explosions akin to SNe Ia have been observed, which do not belong to the "normal" class, they being either overluminous or underluminous and thus not following the Phillips relationship. The historical view of Tycho's SN has been expanded with historical research that will be mentioned in Section 2. The light curve, color and extinction of the supernova are treated in Section 3. Concerning the search for the progenitor and the explosion mechanism, there have been recent revisions and the current state of the question will be addressed in Section 4. HISTORICAL RECORDS SN 1572 was well observed in Europe (as well as in the Middle-East and Far-East) for almost two years. It added a new aspect to the debate at the time over the Aristotelian cosmological views, as it forced to reconsider the immutability of the heavens and the solid nature of the celestial spheres: the "star" gained brightness and lost it during a period of two years, but it showed no detectable parallax. According to prevalent views about the heavens, mutability would only happen in the sublunar region. This was even the place where comets were assumed to originate. The appearence of SN 1572 challenged the order of the celestial spheres. The observers who followed it up 2 took sides with respect to established Aristotelian views (Ruiz-Lapuente 2005). Today we can still use their observations to see whether that supernova would be of use for cosmology in a different way, as a distance indicator if seen by observers billions of lightyears away from us. It is indeed possible to have a clear idea of where SN 1572 stands among its class. ∆m 15 = 1.09 − 0.161x 1 + 0.013x 2 1 − 0.00130x 1 1 3(4) After several centuries of questioning the nature of SN 1572, the identification as a Type I supernova came through the revision of the light curve done by Baade (1945) and based on data taken or quoted by Tycho Brahe (1603a). Before that time, there were still speculations on whether it was a variable star of some kind, a nova (Morgan 1945) and the suggestion of its cometary nature was still considered (Lynn 1883). The cometary idea expressed in 1573 by Jerónimo Muñoz, was based on the fact that the event is aligned with the Milky Way, so its decay in luminosity could be explained if it were a comet born among the stars that would first approach and then move away from us just along the line of sight. Tycho's stella nova lies indeed only 49-98 pc above the Galactic plane. In modern times, its comparison with other SNe allowed its classification as a Type I supernova (Baade 1945). Later on, this class was shown to contain events of very different nature: those identified with the explosion of white dwarfs (Type Ia, SN Ia) and those corresponding to the collapse of massive stars whose envelope had lost its hydrogen content in the interaction with a binary companion (Type Ib or c, SN Ib or SN Ic, respectively). SN 1572 was of the Type Ia class as discussed by de Vaucouleurs (1985) and by van den Bergh (1993). No further doubts that it is a SN Ia can now be held in view of the X-ray spectrum of its remnant, which clearly differentiates SNe Ia events from SNe coming from the collapse of massive stars (Hughes et al. 1995). A first comparison of its luminosity at maximum with the bulk of SNe Ia would have led to think that it was fainter than normal SNe Ia. van den Bergh (1993) considers whether it could be a peculiar, subluminous SN Ia, like SN 1991bg. However, SN 1572 was heavily obscured. It was reddened by E(B-V)=0.6± 0.04 as it corresponds to the reddening of the stars near its position (R04). After taking into account the extinction undergone across the Milky Way, as well as the measurement of its decline rate, it is concluded that SN 1572 was not a 91bg-like event. Neither was it an overluminous SN Ia like SN 1991T or similars, but rather an event in the middle of the SN Ia class (R04). EVIDENCE ON VISUAL, COLOR EVOLUTION AND LATE DECLINE The SN 1572 data are compared in R04 to templates using the stretch factor s for the characterization of the rate of decline (Perlmutter et al. 1999;Goldhaber et al 2001;Nobili et al. 2003). As already mentioned the stretch factor s method was introduced by the Supernova Cosmology Project to quantify the decline rate of the supernova from data extending up to 60 days after maximum. Even in absence of a measurement of the brightness at maximum, the method produces a fine description of the supernova within the family of decline rates. It makes sense to use the stretch original characterization, as we know what is the reddening suffered by the SN Ia. When we compare magnitude data points and magnitude limits by this method, we clearly see that SN 1572 was not a fast decliner. The best fit corresponds to a stretch factor s ∼ 0.9. SN 1572 was an event, for instance, very similar to SN 1996X, with s = 0.889. A comparison is made in Figure 2, with SN 1996X and with the subluminous SN 1991bg (s = 0.62). The SN 1572 points are shown together with a template light curve with s = 0.9. We see that there is a clear discrepancy with the fast-declining subluminous SN 1991bg. A comparison with the broad light curve of the overluminous SN 1991T shows a similar discrepancy, of the opposite sign. The late decline of SN 1572, from 100 to 450 days after maximum, which is slow and similar to that of the bulk of SNe Ia, is an additional proof that it was a normal one. SN 1572 declined by 1.4 mag in 100 days, similar to SN 1990N and other normal SNe Ia, with decline rates of 1.38-1.5 mag, also in 100 days. In contrast, SN 1991bg declined again faster over the same period (see Filippenko et al. 1992;Leibundgut et al, 1993;Ruiz-Lapuente et al. 1993;Turatto et al. 1996). SN 1572 is clearly not a peculiar "91bg"-like SN Ia. Figure 2. The visual light curve of SN 1572 till 60 days. The solid curve is the V light curve of a s = 0.9 SN Ia, which gives the best account for the decline. Such stretch factor is typical of normal SNe Ia. We show for comparison the V light curve of the normal SN Ia SN 1996X, whose stretch factor is s = 0.889 and of the fast-declining SN 1991bg. SN 1572 was significantly slower than SN 1991bg. The light curves plotted in dashed lines are the templates of 91bg-type events and 91T-like events, which depart significantly from the data. Now using the whole light curve, we consider the templates for a normal event with s = 0.9 (like SN 1996X), for a subluminous SN 1991bg type and for an overluminous event like SN 1991T, all of them matching the respective available observations. Up to 60 days after maximum, such templates coincide with those of Hamuy et al. (1996b) and with the SCP templates for the same s. At later times, they follow the available late-time photometry (Schmidt et al. 1993;Salvo et al. 2011). As reported in R04, the fit of the SN 1572 data to the s = 0.9 template has an acceptable χ 2 of 14.44 for 10 degrees of freedom, whereas the fit to the s = 0.62 template of a fast-declining, subluminous SN Ia (SN 1991bg) has a χ 2 of 53.55 for 3 degrees of freedom (the two premaximum points and the last four points having been omitted due to the lack of data for such stages, in those subluminous events), which is exceedingly high. At the opposite end, the fit to the s = 1.2 template, for slow-declining, overluminous SN Ia like SN 1991T, has a χ 2 of 82.99 for 10 degrees of freedom, which is also too high. We show in Figure 3 the total visual light curve of SN 1572, extending to almost 500 days after maximum light, together with the SN 1991bg and SN 1991T templates. The supernova had not yet leveled-off at 480 days past maximum, according to the last upper limit given by Tycho Brahe. It probably leveled-off later, at around 500 days, since there is a light echo of the SN produced by dust clouds nearby, discovered by Krause et al. (2008). The production of echos depends on the distribution of the clouds, both with respect to the SN and to the observer. So, for instance, in SN 1986G, though being heavily reddened, no echo was observed (Schmidt et al. 1993). SN 1991T and SN 1998bu have shown a slowing down in the V magnitude rate of decline at some 400 days after maxmum, that being due to light echos (Schmidt et al. 1993;Cappellaro et al. 2001). The leveling-off took place at 500 days for those SNe Ia, at some 10 mag from maximum. That, in the case of SN 1572, similar to SN 1998bu, would place the leveling at V = 6 mag, approximately at the limit for the naked eye. SN 1998bu is, in fact, a SN Ia very similar to SN 1996X and SN 1572, its reddening being E(B − V ) = 0.32 ± 0.04 mag only (Hernandez et al. 2000;Cappellaro et al. 2001), which is half the reddening of SN 1572. One could say that SN 1998bu is a Tycho Brahe with half the extinction suffered by the historical SN 1572. (See next subsection about the similarity of SN 1572 and SN 1996X and SN 1998bu as shown in the spectrum observed in the echo). In addition, the late light curve of SN 1572 tells us of the energy deposition due to the decay of 56 Co to 56 Fe in the SN ejecta. The decline being slow points to the explosion of a Chandrasekhar-mass white dwarf and the late decline in V is indicative of a significant deposition of energy by the positrons produced in the decay 56 Co → 56 Fe + e + . Indeed, from 200 days after the explosion, the positrons become the main luminosity source. Departure from full positron trapping in the ejecta is always observed, the reason being incomplete confinement of the positrons and/or incomplete thermalization of their energies. Several causes can produce diversity in the late-time SN light curves: differences in the nucleosynthesis yields and the kinematics of the ejecta, degree of mixing and configuration and intensity of the magnetic field (see Ruiz-Lapuente & Spruit (1998). It is found that departures of the order of 10-15% from full trapping of the positrons at 400 days time can be explained by the distribution of radioactive material, but larger departures, of the order of 30-40% or more, require a lack of confinement of the positrons by the magnetic field or even an enhancement of their escape due to a radially combed magnetic field. In the particular case of SN 1991bg, the very fast drop of the light curve indicates not only an ejected mass below the Chandrasekhar mass but also the absence of a tangled magnetic field. Milne et al. (1999) made a comparison of the predictions from several explosion models for SNe Ia with a sizeable sample of bolometric light curves. No bolometric light curve for SN 1572 exists, the data being only for the V band, but we can draw analogies with SNe Ia for which we know the bolometric data. As said above, the late decline of SN 1572 is an important proof, confirming that Tycho's SN falls in no way within the estimated 16% (Li et al. 2001) of intrinsically subluminous SNe Ia. Such low-luminosity class likely arises from a peculiar type of SNe Ia explosions, likely ejecting a smaller amount of mass than normal SNe Ia (Ruiz-Lapuente et al. 1993). Light echo of SNe Ia Sometimes it is possible to know details about a SN Ia long after its light has gone away (see Figure 3), through the study of the echo coming from dusty regions around the supernova. The light traveling to a dusty region and beeing scattered and absorbed and re-emitted can take some hundred years to arrive. This is the case for Tycho's SN. Through the study by Krause et al (2008), we know that the supernova was a normal one (Krause et al. 2008;Usuda et al. 2013). The spectrum taken of the echo reveals the ion absorptions in full detail and it fits perfectly with a normal SN Ia. Moreover, that paper establishes that the spectrum of SN 1572 matches the comparison spectra of four well observed normal SNe Ia (1994D, 1996X, 1998bu, 2005cf) (Krause et al. 2008). That coincides with the conclusions derived from the light curve (R04). In the same way that the light curves of superluminous SNe Ia and subluminous SNe Ia differ, the first ones being brighter and slower in their rise and decline while the second ones being less luminous and faster in decline, they also differ in spectra. Superluminous SNe Ia lack a well defined Si II λ 6355Å absorption feature at maximum light, though the subsequent evolution is similar to normal SNe Ia. On the other hand, subluminous SNe Ia show a characteristic absorption near 4200Å atributed to Ti II, near maximum light. The spectrum coming from the echo represents those of added epochs corresponding to extent of the scattering cloud. So, typically one compares the echo spectrum with the spectra of a SN Ia time averaged over the brightness peak of the light curve. The dominating characteristics are those of the brightness peak, though it is reasonable to make the weighting by brightnesses of the spectra from 0 to 90 days after explosion, as it is done for several SNe Ia spectra derived from echoes. Reddening and color The most direct estimate of the interstellar reddening in the direction of Tycho's SNR is provided by the measurement of the reddening and extinction of the stars close to the centroid of the remnant and at similar distances. The distance to Tycho SN is in the range of 2 to 4 kpc (de Vaucouleurs (1985) reviewed values obtained by various methods and they lie in this range). An estimate from Tian anf Leahy (2011) The mean reddening above gives an extinction A V = 1.86 ± 0.12 mag, adopting a R V of 3.1 (Sneden et al. 1978;Ricke & Lebofsky 1985). The Galactic extinction data from COBE/DIRBE (Drimmel & Spergel 2001) give A V = 1.77 mag in that direction, the maximum Galactic extinction there being A V = 1.90 mag. Therefore, the extinction measured at the distance of SN 1572 does agree with the COBE/DIRBE values. Once the apparent brightness is corrected for extinction, it confirms Tycho's SN as a normal SN Ia and not a subluminous one. The historical records on the color evolution of SN 1572 can equally be corrected for reddening and obtain the intrinsic color evolution. Two months after maximum light, all SNeIa show a similar evolution in color, and there is a well established law, with low intrinsic dispersion, valid for the period froom 30 to 90 days, studied by Phillips et al. (1999) (Ph99) and based on the work of Lira (1995): (B − V ) 0 = 0.725(±0.05) − 0.0118(t V − 60)(5) where t V is the time from visual maximum. Two months after discovery, the color of SN 1572 was reported to be similar to that of Mars and Aldebaran, which means that B − V was in the 1.36-1.54 mag range. Other color estimates, previous and subsequent to this one, are given in Table 2 of R04. New historical records have been recently added by Neuhaeuser (2022). Based on the observations of Adam Ursinus, Georg Busch and Cyprian Leowitz, this author asserts that by the end of November the supernova appeared withish or silver, becoming yellowish at the end. These new three points confirm that the SN could not be of the SN 1991bg type, since it should have been reddish in that case. Instead, the points fall where they are expected for a normal SN Ia. After correction of the observed colors for our measured reddening of E(B − V ) = 0.6 ± 0.04 mag, the intrinsic color at 55 ± 10 days becomes (B − V ) 0 = 0.76 ± 0.24 mag, in very good agreement with the expected (B − V ) 0 = 0.78 ± 0.15 mag for the given epoch, thus fitting very well in the Ph99 law above. Nobili et al. (2002) have shown that the color evolution of the bulk of the SNe Ia does follow the law, with only a low dispersion of 0.1 mag in the tail. Before maximum, the corrected color of SN 1572 would have been (B − V ) 0 = 0.22 ± 0.29 mag, which is consistent with the fact that normal SNe Ia have (B − V ) 0 ∼ 0 mag. In contrast, SN 1991bg, as well as other subluminous SNe Ia, clearly deviate from the standard color evolution at early epochs also, they being intrinsically redder at maximum, with (B − V ) 0 = 0.6 mag (Leibundgut et al. 1993;Ph99). Additionally, the color evolution for a s = 0.9 SN Ia agrees well with the SN 1572 data. In the nebular phase, at 175 days after maximum from Tycho Brahe's records, the SN went back to a white color. Such behaviour, once the correction for extinction is made, is consistent, once more, with that observed in normal SNe Ia. This takes into account uncertainties in the color estimates and includes the new records above. In R04 the visual absolute magnitude of Tycho's SN is estimated as: M V = -17.72 -5 log(d/3.5 kpc) -A V mag. If corrected for A V = 1.86 ± 0.12 mag, that gives: M V = -19.58 -5 log(d/3.5 kpc) ± 0.42 mag. De Vaucouleurs (1985) gives as the most likely estimate of the distance to Tycho Brahe's SN: d = 3.2 ± 0.3 kpc. Adopting 3.2 kpc for the distance, we have an absolute visual magnitude M V = -19.38 ± 0.42 mag. More recently, however, new discussions of the distance sets it to d = 2.7 ± 1 kpc. This estimate comes from the growth of extinction towards higher distance in the field. as measured from stars in the Gaia DR2 which have measured stellar parameters and colours. From those stars one can track the reddening versus distance towards the direction of SN 1572. It is consistent with other estimates mentioned later on. The absolute magnitude of SN 1572 at maximum should be M V = -19.02 -5 log(d/2.7) ± 0.42 mag, which compares well with M V = -19.12 ± 0.26 mag, the mean magnitude from the Calan Tololo sample (Hamuy et al. 1996a). ON THE EXPLOSION MECHANISM AND PROGENITOR OF TYCHO'S SN After Baade (1938Baade ( , 1945a identified the "nova" B Cas, based on the light-curve data in Tycho Brahe's Progymnasmata, as a Type I supernova, he remarked (1945a,b) that no expanding shell had been detected at the position where the supernova flared up. Exploration of the sky at radio wavelengths, about a decade later (Hanbury Brown & Hazard 1952;Shakeshaft et al. 1955;Baldwin & Edge 1957) did locate such nebula at positions compatible with that given by Baade, within the observational errors. Studies of Tycho's SNR at all wavengths have followed, now covering the full range from radio waves to γ-rays. Especially significant have been the X-ray observations made with the ROSAT (Hughes 2000), XMM-Newton (Decourchelle et al. 2001) and Chandra (Hwang et al. 2002) satellites. Namely, from the XMM-Newton data (Decourchelle et al. 2001), it was found that the chemical abundances and their distribution inside the SNR were those expected from a SN Ia. From the good correlation between the images in the Si XIII K line and the Fe XVII L line, they deduced that some fraction of the inner iron layer had been well mixed with the outer silicon layer. Based on the increasingly detailed data, at all wavelengths, on the morphology, dynamics and chemical composition of Tycho's SNR, there has been physical modeling to infer the explosion mechanism and the nature of the progenitor system of the SN. Badenes et al. (2006), used X-ray observations from XM M − N ewton and Chandra to test different explosion models and found that the fundamental properties of the X-ray emission in Tycho were well reproduced by a one-dimensional delayed detonation model of a Chandrasekhar-mass white dwarf (see Figure 5), interacting with an ambient medium of density ρ AM = 2 × 10 −24 g cm −3 . There is stratification of the chemical composition, which points to a supersonic burning front. , from hydrodynamical calculations of the X-ray spectrum of Tycho's SNR and comparison of the model with Chandra high-resolution images, also conclude that a delayed-detonation model expanding into a uniform interstellar medium is the preferred one for this SNR. The fit of this model gives as best distance 2.8 ± 0.4 kpc. Earlier, Williams et al. (2017), from 3D measurements of the velocities of various ejecta knots using Chandra X-ray observations over a 12 yr baseline, had equally found that delayed-detonation models are favored. Badenes et al. (2007), by comparing their models for the evolution of the SNRs with the observations of several remnants of the Ia type, Tycho's SNR among them, find incompatibility with the pre-supernova models where there is emission of strong, optically thick winds from the progenitor system of the supernova. Such winds would excavate large, low-density cavities around the progenitors and that would be incompatible with the dynamics and the X-ray emission of these SNRs. Chiotellis et al. (2013) find that an uniform ambient density cannot simultaneously reproduce the dynamical and X-ray emission properties of Tycho. A better fit is provided by models in which the remnant was evolving within a dense but small wind bubble. The wind bubble might have different origins, including a sequence of nova explosions or a double-degenerate origin. Sato et al (2019) analyse the clumpy structure of Tycho's SN remnant. Their genus statistic analysis supports a scenario in which the observed structure of the SN Ia remnant arises from initial clumpiness in the explosion (see Figure 6). At present, the cause of the initial ejecta clumping in SNe Ia is found by these authors to be still theoretically unclear and they expect it to gradually become more clear with a 3D simulation covering from the explosion to the remnant phase. The most recent 3D study of the velocity of 59 clumpy, metal-rich ejecta knots, together with the proper motions, estimate a new expansion center of the SNR, at ∼6 arcsecs from the geometrical center (Millard et al. 2022). These authors also find that the southeast quadrant expands faster than the rest of the SNR, therefore confirming some degree of asymmetry in the expansion of the ejecta. Yamaguchi et al. (2017), adopting again a delayed detonation model for a Chandrasekhar-mass white dwarf, explain an iron-rich knot located along the eastern rim of the remnant, surprisingly with no emission from Cr, as originating from a region having reached peak temperatures of (5.3-5.7) ×10 9 K only, with a neutron excess < ∼ 2.0 × 10 −3 , which corresponds either to incomplete Si burning or to an α-rich freeze-out regime, which excludes the dense core of the white dwarf as its origin and points to a region near the boundary of the core as the site of production of the knot. As we have seen, the delayed detonation mechanism in a Chandrasekhar-mass white dwarf appears to be the one that best explains the overall characteristics of Tycho's SNR in the analysis done by various authors. In the delayed detonation models, C is ignited close to the center of the white dwarf when, due to the accretion of mass from a close binary companion, the central density reaches 2 × 10 9 g cm −3 and the temperature rises to ∼ 10 9 K. Burning then propagates subsonically (a deflagration) outwards, causing the layers ahead of the burning front to expand. Hydrodynamical instabilities make the burning front turbulent. When some fraction of the star has already been burned and the front reaches regions having densities below some critical value, the front becomes supersonic (a detonation), sweeping the rest of the star up to the surface. Growth of a C+O white dwarf up to the Chandrasekhar mass by accretion of material from a companion star in a binary system results from a comparatively slow mass transfer (capture of material from the stellar wind of the companion or from a stream due to Roche-lobe overflow by the mass-donor star). Faster mass transfer would happen in the merging with another C+O white dwarf, although it should not be too fast to avoid C ignition close to the surface of the accreting white dwarf before the Chandrasekhar mass is reached. If the explosion would immediately follow the merging, it would hardly have the characteristics of most SNe Ia. If there were a sufficient delay between merging and explosion (MED, see Socker 2019a,b; 2022), however, the exploding object might have becomed a Chandrasekhar-mass white dwarf and then undergo a delayed detonation. Recent work (Neopane et al. 2022) has shown that double-degenerate mergings can actually produce highly magnetized, uniformly rotating white dwarfs, a fraction of them with masses close to the Chandrasekhar mass, which should then explode via the delayed-detonation mechanism. Then, although delayed detonation models seem consistent with a single degenerate path to explosion, they can occur as well, if a sufficient delay between the merging of two white dwarfs and the explosion takes place, in the double degenerate scenario. In fact this DD-MED mechanism is the equivalent in the DD case to the spin up/spin down from Di Stefano et al (2011) applied with the accretion from a non degenerate star (SD path). The SD spin up/spin down models can leave a very faint companion. Such faint companion would be too faint to be catalogued in the Gaia data releases or in the Hubble Space Telescope images taken of the field thus far. The search for a surviving companion of SN 1572 has been a survey of the stars located not too far from the present geometrical centroid of the remnant, at distances within the estimates for that of the SNR, looking for unusually high tangential and radial velocities, photometric and spectroscopic peculiarities and possible excess of Fe-peak elements at the surface (Ruiz-Lapuente 1997; RL04). In the single degenerate channel, the surviving companions of the explosion can, in principle, be at any stage of thermonuclear evolution: main sequence, subgiant, giant or supergiant stars (see Wang & Han 2012;Maoz et al. 2014; Ruiz-Lapuente 2014, 2019, Soker 2019a for reviews). They could also be hot subdwarfs (Meng & Li 2019;Meng & Luo 2021) or fainter objects as mentioned above (Di Stefano et al. 2011). Hydrodynamic simulations of the impact of the SNe Ia ejecta on a non-degenerate companion of any type predict that such stars will survive the explosion after being stripped of some of their mass, heated, and their surfaces possibly contaminated by the slowest moving SN ejecta. The binary system being disrupted, the companions should be ejected at their orbital velocities, plus some kick from the impact of the SN material. In the double degenerate scenario there will be no surviving companion. The first survey (RL04) (see Figure 7) included a number of stars smaller than the ones made later on. Spectra and photometry of all the stars were obtained with several telescopes. Images were taken with the Hubble Space Telescope. Radial velocities were measured from the spectra, and proper motions from the HST images. Spectral types and luminosity classes were determined by modeling the spectra. Comparison with the photometry then gave the distances to the sampled stars. One star, a subgiant, was a 3σ outlier in radial velocity, as compared with the stars at the same distance and position on the sky. The star, labelled Tycho G (see Fgure 7), was an outlier in proper motion as well, and its metallicity showed that it was not a halo star. It was therefore proposed, based on its kinematics, as the likely surviving companion of SN 1572. High-resolution spectra of the star above were later obtained with the HIRES spectrometer on the Keck I telescope. That allowed an accurate determination of the chemical abundances, in addition to a refinement of the stellar atmosphere parameters and a more precise measurement of the radial velocity. Low-resolution spectra of other stars in the Tycho field were also analyzed to determine their spectral types, with good agreement with the results of RL04. Kerzendorf et al. (2009) had remarked that if a companion star were rotating synchronously with the orbital motion (rotation period equal to the orbital period), it should be rotating faster than the proposed candidate. González Hernández et al. (2009) had already argued that the interaction with the SN ejecta can slow down the rotation. Liu et al. (2013) and Pan et al. (2014) later showed, from hydrodynamical simulations, that the rotational velocity can indeed be much reduced by the collision. A number of works have been devoted to the possible companions of Tycho's SN remnant. In , attention was paid to a fast-rotating star of the spectral A type, located close to the geometrical center of the SNR, labelled Tycho B (see Figure 7). Kerzendorf et al. (2018), from UV spectra obtained with the Hubble Space Telescope, concluded that it rather is a foreground star. The idea was that if there were enough Fe II in the SNR and the star were inside it, the spectrum would show blueshifted absorption lines. If it were behind the SNR, in the background, the lines would be both blueshifted and redshifted, as it is the case with the Schweizer-Middleditch star, in the background of SN 1006. No Fe II absorption lines are seen in the spectrum of star B, which by having a surface temperature T eff ∼ 10,000 K, should have significant UV emission. From that, Kerzendorf et al. (2018) conclude that either star B is in the foreground or there is not enough Fe II in the SNR, the material being more highly ionized. Gaia DR3 parallaxes place Tycho B well in the range of distance of the SNR. Ihara et al. (2007) suggested that Tycho E could be the companion of SN 1572 on grounds of absorption seen in the spectra. But this star clearly seems behind the Tycho Brahe's SN (see the Appendix). As we mention later, both Tycho B and Tycho E orbits do not look perturbed by an impact. Bedin et al. (2014) measured the proper motions from HST images of a large sample of stars around the center of Tycho's SNR. The chemical abundances of the candidate star Tycho G were calculated in González Hernández et al (2009), Kerzendorf et al. (2013) and Bedin et al, (2014). The Ni/Fe ratio would point, at most, at moderate or low pollution of the companion of the SN. However, that is what is predicted by the models of Pan et al. (2014): the captured material would be much diluted in the convective envelope of a subgiant star. The advent of the Gaia DR2 opened a new space for the definitive exploration of the candidate stars to companion of Tycho Brahe's SN. Ruiz-Lapuente et al. (2019) used the parallaxes from the Gaia DR2 to reassess the distances to the stars in the Tycho field. The orbits of the stars were also calculated. No orbital peculiarity is seen in Tycho B nor in other candidates stars that had been proposed. The star in RL04 is the most dynamically peculiar in the sample, but there is no real proof that it is the surviving companion it has been looked for. In order to do a full update of this research, in this review the same analysis using now the most recent Gaia data is done, those from the Gaia DR3. The results, which are shown in the Appendix, do not change the conclusions from the 2019 paper using Gaia DR2. The proper motions of the stars are similar (though not identical) to those from Gaia DR2. The parallaxes are better determined. Neither Tycho B nor Tycho E have a peculiar orbit. The only star somehow eccentric and with a proper motion and V R higher than the rest is Tycho G. The peculiarity of a star, Tycho G, comes from three facts: a larger proper motion mostly in declination than the other stars, a larger eccentricity and larger radial velocity in the LSR V R (other star, Tycho U, has a similar proper motion in declination than Tycho G, but no eccentricity and a small V R ). No evidence for a high chemical pollution from the SN Ia is found in this only relatively peculiar star. It might be that this is what occurs for an impact on a main sequence or subgiant companion. It should be addressed as well if a star close to the explosion, but not being the companion, could be perturbed in the way Tycho G appears to have been. The single degenerate path to explosion has not been fully excluded for Tycho Brahe's supernova, from all the abovementioned factors. From the theoretical work done on SN impacts on subgiants and on main sequence stars (see reviews in the previous paragraphs), the velocity of the companion and its pollution might not be very high. In the case of a donor star orbiting very close to the WD, though, one would expect an outlier at many more than 3σ of the proper motion distribution. There have been some other observations pointing indirectly to a single degenerate origin rather than to a double degenerate origin for SN 1572. Zhou et al. (2016) have found that Tycho's SNR is surrounded by a clumpy molecular bubble, expanding at ∼ 60 km s −1 . The bubble is massive and there is morphological correspondence with the SNR. The authors suggests that the origin of the expanding bubble is a fast outflow coming fron the vecinity of the mass-accreting WD that gave rise to SN 1572. To complete this examination, we would like to see whether there are hypervelocity stars in the field of Tycho's SN remnant. The generous assumption of a 1 kpc distance to the remnant, 3000 km s −1 and an age of 450 yr, gives a search radius of 4.7 arcmin. No hypervelocity star is found. But, the whole field is totally empty of hypervelocity stars, even if we amplify the search to a 1 degree around the geomerical center of the remnant. For such radius and star distances between 1.7 and 3.7 kpc, the sample contains 58,691 stars. Thus we can exclude as an explosion mechanism for Tycho Brahe's SN the dynamically driven double-degenerate, double-detonation scenario D 6 mechanism (Shen et al. 2018). These explosions are triggered by the detonation of a surface layer made of He, accreted by the exploding WD from a less massive WD companion. The outburst might happen when the mass-donor has not yet been tidally disrupted. Due to its very high orbital velocity, the WD companion should be ejected as a hypervelocity star (v > 1000 km s −1 ). As we have seen, this is not the case for SN 1572. Figure 8. Histogram of the distribution of tangential velocities of the stars within 1 degree of the geometrical center of Tycho's SNR in the range of distance compatible with SN 1572 (1.7 < d < 3.7 kpc). The data are obtained from Gaia DR3. It can be seen that there are no hypervelocity stars in a very large area around the Tycho's SN remnant. In conclusion of this section, the explosion mechanism favored by the analysis of various authors suggests that a delayed detonation explains better the data than other possible alternatives. The single degenerate and the double degenerate scenario with a delay between merging and explosion fit into this picture. The new scrutiny using Gaia DR3 of the stars in the field of SN 1572 gives a similar conclusion to that reached with the Gaia DR2. A few interesting notes and Figures can be found in he Appendix. In the next decade, we expect to probe the nature of SNe Ia and quantify those coming from mergings of WDs with the new generation of gravitational wave missions in the deci-hertz range (see Yoshida (2020), for instance). We would be able to quantify the rates of SNe Ia coming through this path versus other origins. In the meanwhile, we have a battery of tests that are not so robust as a direct detection. In the conclusions, I summarize the results gathered on the origin of the eariest SN Ia observed by astronomers. CONCLUSIONS We found in R04 that SN 1572 was a supernova very close to the template with a stretch factor s ∼ 0.9. The light curve grows in precision towards the late times, being highly uncertain around maximum brightness. An overall agreement between early, late decline and color with the expected evolution of normal SNeIa supports our conclusion. Type Ia supernovae with stretch factors between 0.9 and 1.1 make the vast majority of the observed population. They are not only those most frequently found in nearby searches, but also the bulk of discoveries in cosmological searches at high-z, as can be seen in the sample of SNe at z > 0.3 found by the Supernova Cosmology Project (Perlmutter et al. 1999). Among SNe Ia of s ∼ 0.9 in nearby galaxies for which very late-time data are available, we have found a close resemblance to SN 1996X in rate of decline. SN 1572 likely has a slightly slower rate. However, whereas SN 1996X was not heavily reddened, the reddening in SN 1572 is E(B-V)=0.60 ± 0.05. The echo of SN 1572 discovered in 2008, reaffirms that SN 1572 is a normal SN Ia. X-ray observations of Tycho's SNR have provided an increasingly detailed picture of the remnant and its surroundings. Although one-dimensional models do fit the overall charateristics of the SNR, threedimensional simulations of its evolution, from the explosion to its current state, are being developed to account for the detailed structure. Concerning the mechanism of explosion, a delayed detonation model seems to give a better account of the nucleosynthesis, as derived by various authors. The possibility for a surviving companion has been throughly reexamined here with Gaia DR3 data. The conclusion is similar to the one reached with the Gaia DR2. Something seems clear from the exploration of the field of Tycho Brahe's supernova: the explosion is not triggered by the detonation of a surface layer made of He, accreted by the exploding WD from a less massive WD companion. Thus, it does not come from the so-called dynamically driven double-degenerate, double-detonation scenario. There are no hypervelocity stars in the field of SN 1572. Finally, the merging of two WDs with a lapse between merging and explosion presents a possible path to this normal SN Ia: hydrodynamical calculations, which show that a delayed detonation in a near Chandrasekhar mass WD can occur in a double degenerate scenario where the explosion occurs with a delay after merging, change the view on the DD scenario. The delayed detonation of a Chandrasekhar WD had been for long investigated through the growth of the WD by accretion from a non-degenerate companion. But both paths seem possible for normal SNe Ia. Further exploration is needed to clarify the case of SN 1572. Several aspects on Tycho Brahe's supernova are still unsolved 450 yrs after its visual detection. Being a normal SN Ia what can be learnt from this explosion, can help to understand the wide majority of supernovae of this type. Acknowledgements. I thank Ralph Neuhaeuser for sharing his findings presented at the EAS in 2022 with me. I would like to thank Jack Hughes, Carles Badenes and Tomonori Usuda for their kind permission to show Figures from their Tycho's SN papers. I would like to thank Bob Fisher for comments on Chandrasekhar explosions through the double degenerate channel and two anonymous referees for pointing out relevant aspects to Tycho Brahe's SN understanding. This work is supported by the project PID2021-123528NB-I00, from the Ministerio de Ciencia e Innovación of Spain. This work has made extensive use of the Gaia DR3. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https: //www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia. In the upper panel, we see that star U reaches the largest distance from the Galactic plane, followed by star G, while stars B and F scarcely depart from the plane. The behaviour of the latter stars is typical of the rest of the sample considered here. In the bottom panel, we see that the orbit of star G, on the Galactic plane, is quite eccentrical, which corresponds to the high value of the proper motion and V R compared to the sample, while the other stars (including star U) have orbits close to circular. Also here, the behaviour of stars B and F is representative of the whole sample. Star E has a similar orbit than stars B and F. Figure 10. Histogram of the distribution in µ δ (in mas yr −1 ) of the stars within 1 degree of the geometrical center of Tycho's SNR in the range of distance compatible with SN 1572 (1.7 < d < 3.7 kpc). The data are obtained from Gaia DR2. The red vertical line shows the µ δ of star G. Figure 11. Distances and distance ranges inferred from the parallaxes in the Gaia DR3 and their uncertainties, together with their proper motions in declination. The dashed vertical lines mark the conservative limits of 2.7 ± 1 kpc on the distance to Tycho's SNR. Solid (blue) error bars correspond to stars from Figure 1 . 1m B magnitudes versus the best-fit stretch factor, s, for high-redshift supernovae. The upper axis gives the equivalent values of ∆m 15 = 1.96(s −1 − 1) + 1.07. Figure from Perlmutter et al. (Supernova Cosmology Project) in Thermonuclear Supernovae (ed. P. Ruiz-Lapuente et al. 1997). © Kluwer Academic Publishers. Reproduced with permission. Figure in R04. © AAS. Reproduced wih permission. Figure 3 . 3The visual light curve of SN 1572 till 500 days. Its late rate of decline is the one of normal SNe Ia. It is very similar to the decline of the s = 0.889 SN 1996X. The visual data of SN1991bg and the template light curves of this SN Ia and SN 1991T are shown for comparison. Figure in R04. © AAS. Reproduced with permission. Figure 4 . 4Spectrum of the light echo of SN 1572 compared with other normal SNe Ia. The spectra are weighted averages around maximum light. Figure from Krause et al (2008). Courtesy of T. Usuda. © Springer Nature. Reproduced with Permission. based on the modeling of the kinematics of the region gives 2.5-3.0 kpc. The stars near the centroid have average reddenings of E(B − V ) = 0.6± 0.04 mag. The quoted value comes from the program to search the companion stars of Galactic SNeIa (Ruiz-Lapuente et al. 2004, hereafer RL04). The candidate stars for SN 1572 are within an angular distance from the centroid of the remnant including the uncertainty on the site of the explosion plus the shift in position corresponding to traveling perpendicularly to the line of sight for 431 years, at the velocities expected for the fastest moving companions. The stars were modeled to obtain the stellar atmosphere parameters T eff , log g and [Fe/H], plus the distance and E(B − V ). Radial velocities were measured from the spectra. The program stars cover 35% of the radius of the remnant, and their reddenings span from E(B − V ) = 0.50 mag to E(B − V ) = 0.8 mag, the values increasing with the distance. Figure 5 . 5Comparison between the emission of the ejecta from a delayed detonation model(Figure 7ofBadenes et al. 2006) and the spatially integrated spectrum of region B. Courtesy of C. Badenes. © AAS. Reproduced wih permission. Figure 6 . 6Figure 2 in Sato et al. (2019). Flux image (1.76-4.2 keV) of Tycho's SNR observed in 2009 by Chandra. The central region (inner white circle) of the remnant has been used for the genus statistics. The southeast ejecta protusions can be seen as well in the Figure. Courtesy of J.P. Hughes. © AAS. Reproduced with permission. Figure 7 . 7Image from the Auxiliary Port at the William Herschel Telescope of the center of the field of SN 1572. It reveals that it is far from being a crowded field. The initial search area in RL04 covers a radius of 0.65 arcmin around RA = 00 h 25 min 19.9 s, Dec. = 64 o 08' 18.2" (J2000) (the Chandra geometrical centre of X-ray emission). From the author's personal archive. Mn or Ni (which implies mass ratios M Cr /M F e < 0.023, M M n /M F e < 0.012 and M sN i /M F e < 0.029) Figure 9 . 9These Figures show the orbits of the stars with the new Gaia DR3 proper motion data. There are no substantial changes compared to our 2019 exploration where we used the Gaia DR2. The orbits of stars B (green), G (red), F ((blue), and U (gray), projected on the Galactic meridian plane (up) and on the Galactic plane (bottom), computed forward on time for the next 500 Myr. Table 1 . 1Proper motions of stars G, B, F and U, in Gaia DR2 and DR3 417±0.191 -4.253±0.093 -4.164±0.143 -4.202±0.097 B -4.505±0.063 -4.201±0.030 -0.507±0.049 -0.518±0.031 F -5.739±0.130 -5.860±0.054 -0.292±0.097 -0.273±0.058 U -1.877±0.113 -1.658±0.054 -5.096±0.083 -4.904±0.057Star µ * α (DR2) µ * α (DR3) µ δ (DR2) µ δ (DR3) (mas yr −1 ) (mas yr −1 ) (mas yr −1 ) (mas yr −1 ) G -4. Table 1 1that, within reasonable uncertainties, might be inside the SNR. See for more stars Ruiz-Lapuente et al. (2019), since results have not changed significantly. This is applied after correcting from extinction as implied by the reddening of the SN Ia. In a more recent version, color changes due to reddening or intrinsic color of SNe are treated equally in the SALT2 standarization. The coefficients for stretch, x1, and color, c, go respectively in a α × x 1 − β × c correction to the observed magnitude. (Some more recent parametrizations add additional terms in the equation). There are relations between x1, the stretch s from the Supernova Cosmology Project and the ∆m 15 : Tycho Brahe's supernova The observers who mostly contributed to measure the position and luminosity, Tycho Brahe, Thomas Digges, Thaddeus Hagecius, Michael Mästlin, Jerónimo Muñoz, Caspar Peucer & Johannes Prätorius held very different views on the meaning of SN 1572. A comparison of their measurements and an account of their views is given elsewhere. Recently Neuhaeuser (2022) has provided more data from German, Italian and Czech astronomers of the epoch such as Adam Ursinus (also called Adam Bär), Francesco Maurolyco, Cyprian Leowitz and Georg Busch. Frontiers AppendixThe most recent survey for a companion in Tycho Brahe's SN has been done so far using the Gaia DR2 (Ruiz-Lapuente et al. 2019). With the recent Gaia DR3, we have examined whether those results are still valid. This is in fact the case: there are minor differences in the proper motions and parallaxes to the stars, but they do not affect the qualitative results. InTable 1, we display the differences for the stars that were used for the orbital calculations. InFigure 9, we present the new orbits of the stars. InFigure 10we show where Tycho G stands in proper motion in the field of stars 1 degree around the geometrical center of the SNR. It is very similar to what it was found with the Gaia DR2. InFigure 11we show how the sample stars used in previous Figures are located now in the Gaia DR3. Some stars have improved their parallax determination. But the overall picture is the same. The absolute photographic magnitude of supernovae. W Baade, Contr. Mnt. Wilson Obs./Carnegie Inst. Was. 600Baade, W. (1938). The absolute photographic magnitude of supernovae. Contr. Mnt. Wilson Obs./Carnegie Inst. Was., 600, 1-20 B Cassiopeiae as a supernova of Type I. W Baade, Contr. Mnt. Wilson Obs./Carnegie Inst. 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[ "Uniform-in-time propagation of chaos for mean field Langevin dynamics", "Uniform-in-time propagation of chaos for mean field Langevin dynamics" ]
[ "Fan Chen \nSchool of Mathematical Sciences\nShanghai Jiao Tong University Shanghai\nChina\n", "Zhenjie Ren \nCEREMADE\nUniversité Paris-Dauphine\nPSL Research University\n75016ParisFrance\n", "Songbo Wang \nCMAP\nCNRS\nÉcole polytechnique\n\nInstitut Polytechnique de Paris\n91120PalaiseauFrance\n" ]
[ "School of Mathematical Sciences\nShanghai Jiao Tong University Shanghai\nChina", "CEREMADE\nUniversité Paris-Dauphine\nPSL Research University\n75016ParisFrance", "CMAP\nCNRS\nÉcole polytechnique", "Institut Polytechnique de Paris\n91120PalaiseauFrance" ]
[]
We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the L p -convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the L 2 -Wasserstein metric and relative entropy.RésuméNous étudions la dynamique de Langevin à champ moyen et le système de particules correspondant. En supposant la convexité fonctionnelle de l'énergie, nous obtenons la convergence dans L p des distributions marginales vers l'unique mesure invariante pour la dynamique à champ moyen. De plus, nous montrons la propagation du chaos uniforme en temps à la fois dans la métrique de Wasserstein d'ordre 2 et dans l'entropie relative. MSC2020 Subject Classifications: 60J60, 60K35 (primary) 35B40, 35Q83, 35Q84 (secondary)
null
[ "https://export.arxiv.org/pdf/2212.03050v2.pdf" ]
254,274,909
2212.03050
eb3e9e4d06c923b23575ac8b7d3e0253879226d1
Uniform-in-time propagation of chaos for mean field Langevin dynamics May 25, 2023 Fan Chen School of Mathematical Sciences Shanghai Jiao Tong University Shanghai China Zhenjie Ren CEREMADE Université Paris-Dauphine PSL Research University 75016ParisFrance Songbo Wang CMAP CNRS École polytechnique Institut Polytechnique de Paris 91120PalaiseauFrance Uniform-in-time propagation of chaos for mean field Langevin dynamics May 25, 2023Langevin diffusionFokker-Planck equationmean field interactionconvergence to equilibriumuniform-in-time propagation of chaoslogarithmic Sobolev inequalityhypercontractivityWasserstein distancerelative entropy We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the L p -convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the L 2 -Wasserstein metric and relative entropy.RésuméNous étudions la dynamique de Langevin à champ moyen et le système de particules correspondant. En supposant la convexité fonctionnelle de l'énergie, nous obtenons la convergence dans L p des distributions marginales vers l'unique mesure invariante pour la dynamique à champ moyen. De plus, nous montrons la propagation du chaos uniforme en temps à la fois dans la métrique de Wasserstein d'ordre 2 et dans l'entropie relative. MSC2020 Subject Classifications: 60J60, 60K35 (primary) 35B40, 35Q83, 35Q84 (secondary) Introduction Recently there has been an increasing interest in modeling the training of neural networks as a convex mean field optimization problem (see e.g. [33,14,44,42,25] and also our Section 3 for explanations.) With some exceptions ( [14,36,38,12]), the majority of the works focus on the entropy-regularized mean field optimization problem and the corresponding mean field Langevin (MFL) dynamics [33,25,13,37,15]. It is first proved in [25] that under a convexity assumption the marginal distributions of the MFL dynamics converge towards its unique invariant measure, which is also the unique minimizer of the mean field optimization problem. Then it is shown in [37,13] that given a uniform logarithmic Sobolev inequality (LSI), the convergence is exponentially quick. The present paper continues to explore this aspect of the MFL dynamics and we prove that under no additional assumptions the MFL dynamics exhibits uniform-in-time propagation of chaos. This property provides upper bounds on the distance between the finite-particle and the mean field dynamics, which offer a theoretical guarantee on the correctness of the finite-particle approximation. Preview of main results Let F : P 2 (R d ) → R be a mean field functional and D m F be its intrinsic derivative. We study the long-time behavior of the following mean field Langevin dynamics: dX t = −D m F (m t , X t )dt + √ 2dW t , where m t = Law(X t ), and the corresponding dynamics of N particles: dX i t = −D m F (µ Xt , X i t )dt + √ 2dW i t , i = 1, . . . , N, where µ Xt = 1 N N i=1 δ X i t . Here W t , W i t are independent d-dimensional standard Brownian motions. We suppose the functional F is such that • the mapping m → F (m) is convex in the functional sense (instead of in the optimal transport sense); • for every x ∈ R d , the mapping m → D m F (m, x) is M F mm -Lipschitz continuous with respect to the L 1 -Wasserstein metric, • for every m ∈ P 2 (R d ), the measure on R d with density proportional to exp − δF δm (m, x) satisfies the ρ-logarithmic Sobolev inequality (LSI) with ρ > 0, Recall m t = Law(X t ), define m N t = Law(X 1 t , . . . , X N t ) and denote by m ∞ the unique invariant measure of the mean field dynamics. Our main results are • if m 0 /m ∞ ∈ L p0 (m ∞ ) for some p 0 > 1, then for every p ∈ R, the norm ∥m t /m ∞ ∥ L p (m∞) → 1 exponentially quickly when t → ∞; • the scaled L 2 -Wasserstein distance and the relative entropy 1 N W 2 2 (m N t , m ⊗N ∞ ), 1 N H(m N t |m ⊗N t ) converge exponentially to a neighborhood of O(N −1 ) size when t → ∞; • the scaled L 2 -Wasserstein distance 1 N W 2 2 (m N t , m ⊗N t ) converges exponentially to a small neighborhood of 0 when t → ∞; if the assumption of the first claim holds, then the scaled relative entropy 1 N H(m N t |m ⊗N t ) has the same convergence; • moreover the exponential rates in the previous claims can be independent of N . Related works Long-time behavior of McKean-Vlasov dynamics. Propagation of chaos in finite time for the stochastic McKean-Vlasov dynamics dX t = b(m t , X t )dt + √ 2dW t , where m t = Law(X t ) is relatively easy to show using synchronous coupling, given that b is a jointly Lipschitz function of the measure and the space variables in the sense of Wasserstein. The bound obtained by this method, however, generally tends to infinity when the time interval extends to infinity. Moreover, the dynamics may possess multiple invariant measures, so uniform-in-time convergence can not be expected without additional assumptions, or without defining it in a more general sense (e.g. convergence modulo symmetries). The research on the long-time behavior of McKean-Vlasov dynamics is active in recent years and we here introduce a setting that has appeared in many previous works. Let U, V : R d → R and consider the special kind of drift b(m, x) = −∇U (x) − ∇V (x −x)m(dx). In this case U is called the external potential and V is called the interaction potential. Now we give a far from exhaustive review of uniform-in-time propagation of chaos (POC) for McKean-Vlasov dynamics. First in the work [32] of Malrieu in 2001, uniform POC is shown for overdamped dynamics under the assumption that both U, V are convex, using a synchronous coupling approach. In an alternative way, Carrillo, McCann and Villani set up the mean field gradient flow framework in their work [11] which the current paper also relies on and showed the exponential convergence of the overdamped mean field system under the assumption that U + 2V is strongly convex. In [34], Monmarché showed uniform POC for kinetic Langevin under the same convexity assumption on U + 2V and this condition is weakened in his follow-up work [22] with Guillin based on the uniform log-Sobolev inequality shown in [21]. In [19] Durmus, Eberle, Guillin and Zimmer showed uniform POC for overdamped Langevin under the assumption that the confining potential U is only weakly convex and V is small enough, using reflection coupling. The reflection coupling technique is then used by Schuh in [43] to show uniform POC for kinetic Langevin, though in this setting the form of the confining potential is more restricted than in the overdamped case. The uniform-in-time convergence is also shown for the overdamped dynamics on a torus in [17] by Delarue and Tse under various settings, the first of which assumes the smallness of interaction without relying on the form of the interaction by a master equation analysis. Lacker and Le Flem showed in [28] the sharp POC rate for the overdamped dynamics in terms of relative entropy, uniformly in time, by studying the relative entropy growth with the help of a uniform log-Sobolev inequality for the mean field flow. We now comment the assumptions and methods of these works. Apart from the second and third settings of [17] and that of [28], the aforementioned works all rely on the smallness or the (semi-, weak) convexity of the interaction potential. This smallness or convexity is then used to control the error between the coupled processes, or to derive a uniform-in-N log-Sobolev inequality for the N -particle system's invariant measure (see [21]). Our setting is different from these works. First, we rely on a functional convexity of the mean field energy functional which is different (and even exclusive in some cases) from the convexity of the interaction potential. We explain this alternative convexity assumption in the following paragraph. Second, our approach does not rely on a uniform-in-N log-Sobolev for the N -particle invariant measure. Finally, we remark that translation-invariant models have also been studied in the last setting of [17] and also in [18], for which there exists a continuum of invariant measures. POC is then obtained modulo the translational symmetry. We also mention that recently Guillin, Le Bris and Monmarché studied in [20] vortex models and extended uniform POC results to the non-gradient and singular interaction. Linear functional convexity. One key assumption of ours is the (linear functional) convexity of the mean field functional F , whose formal definition is given in (2.1) in the following. Except in [45,17] this assumption has not been explicitly exploited to study the long-time behavior of the McKean-Vlasov dynamics. This convexity should be distinguished from the displacement convexity appearing often in optimal transport literature, defined in e.g. [46,Definition 16.1]. We explain in Remark 3.1 that for continuous two-body interaction potentials, Bochner's theorem implies the two notions are even mutually exclusive except for the trivial case. This convexity has appeared somewhat implicitly in [17], where the author studied "H-stable" interaction potentials and showed uniform-in-time propagation of chaos. That is to say the McKean-Vlasov drift is of form b(m, x) = − ∇V (x −x)m(dx) and all Fourier coefficients of the interaction potential V are nonnegative. This implies the corresponding energy F (m) = 1 2 V (x − x)m(dx)m(dx) is functionally convex. Although our results are obtained for dynamics in R d , we can then expect the extension of our method to the torus would also allow obtaining similar results. The main motivation for the new setting is to study the training of two-layer (or one-hidden-layer) neural networks, which we explain in Examples 2 and 4. This form of drift corresponds to gradient descent of the free energy F = F + H in L 2 -Wasserstein space, where H(m) = m(x) log m(x)dx is the absolute entropy of the measure. We refer the readers to [26] for detailed discussions about the linear case F (m) = V (x)m(dx), [2] for a general gradient flow framework in L p -Wasserstein space and [25] for the gradient flow of the free energy F. The first observation in [25] is the convexity F entails the uniqueness of mean field invariant measure m ∞ , The authors then proved the following free energy descent formula central to gradient flow: dF(m t ) dt = − |D m F (m t , x) + ∇ log m t (x)| 2 m t (dx), and, using this formula and LaSalle's invariance principle, showed the Wasserstein convergence of the MFL. In this paper, we prove the same formula under weaker assumptions on the regularity of x → D m F (m, x), thanks to the general framework developed in [2]. Main contributions L p convergence and hypercontractivity of MFL. The exponential convergence of relative entropy for the MFL with convex F has been obtained in [13,37] via log-Sobolev inequalities, extending the classical result [39] where the F is linear in measure. The current paper shows the stronger L p -convergence in Theorem 2.2. For this new convergence to hold we require the initial condition to lie in L p0 for some p 0 > 1. This is different from the situation of relative entropy, where elliptic regularization ensures the relative entropy is finite at all positive time (see Proposition 4.6). Our method of proof is based on the L 2 -convergence and the hypercontractivity, which ports the L 2 -convergence to L p for all p ∈ R. Our key observations are the growth of L p -norm formula (4.8) and the hypercontractive inequalities (4.12) and (4.13) for the mean field flow. Recently the hypercontractivity has also been used in [16] to show the L p -convergence of MFL with Riesz interactions (though on a torus). Finally, we mention in the proof of our propagation of chaos result (Theorem 2.4), we need the L p -convergence for p negative. For this we find the reverse hypercontractivity of the MFL. This property follows from the same formal computations as for the direct hypercontractivity, under the assumption that the invariant measure satisfies an LSI. Convergence of particle system. Working under the mean field setting of [13,37], we show in Theorem 2.3 that the particle system's free energy converges to the N -tensorized invariant measure of the mean field system exponentially modulo an error that is O(N −1 ) per particle. Our proof relies on a decomposition of relative Fisher information and a componentwise application of the log-Sobolev inequality, which introduces the O(N −1 ) error per particle. Our result is different from that of [21], where the particle system's precise convergence to its invariant measure is obtained by the uniform-in-N log-Sobolev inequality. One advantage of our method is that we allow applications with potentially large interactions, including the training of neural networks (see Examples 2 and 4.) Propagation of chaos. By combining the two previous results, i.e. the L pconvergence of the MFL and the entropic convergence of the particle system, we are able to control the distance between the particle system m N t and Ntensorized mean field flow m ⊗N t , in terms of Wasserstein distance and relative entropy. The bound on Wasserstein is a direct consequence of Talagrand's T 2 transport inequality. To control the relative entropy we make use of a classical duality formula (4.15) to link H(m N t |m ⊗N t ) to the −p norm ∥m t /m ∞ ∥ −p where p > 0, whose exponential convergence is guaranteed by Theorem 2.2. As a side result, we also obtain the uniform-in-time concentration of measure of the mean field flow (Theorem 4.18) using this observation. We now compare this method to those of [28,45]. In [28] the authors assume the mean field flow satisfies a uniform LSI and use an entropy growth formula similar to our L p -growth formula to bound the relative entropy. As is remarked in [45] this uniform LSI is difficult to verify in the mean field setting. In particular if one wishes to apply the Holley-Stroock perturbation lemma to the invariant measure m ∞ , the mean field flow needs to satisfy log m t /m ∞ ∈ L ∞ uniformly. Suzuki, Nitanda and Wu assume in [45] that the confining potential is of super-quadratic growth so that this boundedness follows from the ultracontractivity via super LSI. This confining potential is stronger than the quadratic one in our setting and the constants obtained by ultracontractivity depends on the spatial dimension. Another approach, which has appeared in the previous version of the current paper, requires the initial condition to verify log m 0 /m ∞ ∈ L ∞ . Using our current method, we stay in the quadratic growth regime and the only additional requirement, as explained before, is that the initial condition satisfies m 0 /m ∞ ∈ L p0 for some p 0 > 1. Notations Let d be a positive integer and x be an element of R d . We denote the Euclidean norm of x by |x|. Let p ≥ 1. Define P p (R d ) to be the space of probabilities on R d of finite p-moment, i.e. P p (R d ) = {m ∈ P(R d ) : |x| p m(dx) < +∞}. We denote the L p -Wasserstein metric by W p and refer readers to [2,Chapter 7] for its definition and elementary properties. Let F : P 2 (R d ) → R be a mean field functional. We denote by δF δm : P 2 (R d )× R d → R its linear functional derivative and by D m F = ∇ δF δm : P 2 (R d )×R d → R d its intrinsic derivative, if they exist. The definition of linear functional derivative on P 2 (R d ) can be found in [10,Definition 5.43]. Let X, Y be random variables. We denote the distribution of X by Law(X) and say X ∼ m if m = Law(X). We also say X d = Y if Law(X) = Law(Y ). The set of couplings between probabilities µ, ν is denoted by Π(µ, ν). Let N ≥ 2 be integer. The bold letter x N = (x 1 , . . . , x N ) will denote an N -tuple of the elements in R d . We omit the subscript N when there are no ambiguities. Given x N = (x 1 , . . . , x N ) ∈ R dN , we denote the corresponding empirical measure by µ x N = 1 N N i=1 δ x i . For i = 1, . . . , N , we define −i = {1, . . . , N }\{i}, that is, the complementary index set, and we denote the empirical measures of points {x j } j̸ =i by µ x −i N = 1 N − 1 N j=1,j̸ =i δ x j . For a R dN -valued random variable X N = (X i ) N i=1 , we can form the random empirical measures µ X N , µ X −i N . Let I ⊂ {1, . . . , N } and J = {1, . . . , N }\I be the complementary index set. For a probability m N = Law(X) ∈ P(R dN ), we denote its marginal and the (regular) conditional distributions by m N,I = Law(X i ) i∈I , m N,I|J (x J ) = Law((X i ) i∈I |X j t = x j , j ∈ J), where the latter is defined m N,J -almost surely and x J denotes the tuple (x j ) j∈J . We identify i with the singleton {i} when working with indices. Whenever a measure m ∈ P(R d ) has a density with respect to the ddimensional Lebesgue measure, we denote its density function by m equally. Let γ be a positive and σ-finite measure on R d . We define the relative entropy H(m|γ) = log dm dγ (x)m(dx) and the relative Fisher information I(m|γ) = ∇ log dm dγ 2 m(dx) whenever the corresponding integrals are well defined. In the contrary case we set H, I = +∞ respectively. When γ = L d is the Lebesgue measure on R d we omit the dependence on γ and define the absolute entropy and For nonnegative functions f : R d → [0, +∞) we also define its entropy Ent m f = E m [f log f ] − E m [f ] log E m [f ], which is well defined in [0, +∞] by Jensen's inequality. We define c d to be the volume of the d-dimensional unit ball. Organization of paper. In Section 2, we introduce our assumptions, define the mean field Langevin and the particle system and state our main results. In Section 3, we provide examples of MFL systems to which our theorems apply and include numerical experiments on the training of two-layer neural networks. We gives proofs for the mean field and the particle system in Sections 4 and 5 respectively and the most technical ones are postponed to Appendix A. Readers only interested in the proof of Theorem 2.3 can skip Section 4.3. Finally we study the expressiveness of truncated neural networks in Appendix B. Assumptions and main results Assumptions. Let F : P 2 (R d ) → R be a mean field functional. We suppose F is convex in the sense that for all t ∈ [0, 1] and all m, m ′ ∈ P 2 (R d ), F ((1 − t)m + tm ′ ) ≤ (1 − t)F (m) + tF (m ′ ). (2.1) Suppose also its intrinsic derivative D m F : P 2 (R d )×R d → R d exists and satisfies ∀x ∈ R d , ∀m, m ′ ∈ P 2 (R d ), |D m F (m, x) − D m F (m ′ , x)| ≤ M F mm W 1 (m, m ′ ) (2.2) for some constant M F mm ≥ 0. For each m ∈ P 2 (R d ) we define a probability measurem by its densitym (x) ∝ exp − δF δm (m, x) and supposem satisfies the ρ-logarithmic Sobolev inequality (LSI) uniformly in m for some ρ > 0, that is, for every m ∈ P 2 (R d ), ∀f ∈ C 1 b (R d ), ρ Entm(f 2 ) ≤ Em[|∇f | 2 ]. (2.3) We suppose as well sup m∈P2(R d ) sup x∈R d |∇D m F (m, x)| < +∞. (2.4) Finally, for some of the results we additionally suppose that x → D m F (m, x) belongs to C 3 with the bounds sup m∈P2(R d ) sup x∈R d |∇ k D m F (m, x)| < +∞, k = 2, 3. (2.5) Remark 2.1 (Definition ofm). Given m ∈ P 2 (R d ), we definem by its densitŷ m(x) = 1 Z(m) exp − δF δm (m, x) , Z(m) = exp − δF δm (m, x) dx. (2.6) It is assumed implicitly in (2.3) that the normalization constant Z(m) < +∞ for all m ∈ P 2 (R d ) so that the probability measurem is well defined. We will prove in Proposition 4.2 that for this to hold the following is sufficient: the condition (2.2) holds and there exists at least one measure m 0 such that Z(m 0 ) is finite and m 0 satisfies the LSI (2.3). Remark 2.2 (Functional inequalities). We can relax the restriction f ∈ C 1 b in (2.3) by an approximation argument and the inequality holds for functions f whose generalize derivatives satisfies Em[|∇f | 2 ] < +∞. It is well known that the LSI (2.3) implies the Poincaré inequality: ∀f ∈ C 1 b (R d ), 2ρ Varm(f ) ≤ Em[|∇f | 2 ]. (2.7) The restriction f ∈ C 1 b can be analogously relaxed. The LSI (2.3) also implies Talagrand's T 2 -transport inequality: ∀µ ∈ P 2 (R d ), ρW 2 2 (µ,m) ≤ H(µ|m). (2.8) See the original work of Otto and Villani [39] for a proof. We also sketch their argument in the proof of Lemma 4.1. All the three inequalities, namely (2.3), (2.7) and (2.8), are stable under tensorization: if one replaces for some N ≥ 2 the measurem by its tensor productm ⊗N , which is a measure on R dN , and the function f by functions on R dN , then the inequalities hold with the same constant ρ. Mean field and particle system. We study the mean field Langevin dynamics, that is, the following McKean-Vlasov SDE dX t = −D m F (m t , X t )dt + √ 2dW t , where Law(X t ) = m t . (2.9) Let N ≥ 2. The corresponding N -particle system is defined by dX i t = −D m F (µ Xt , X i t )dt + √ 2dW i t , i = 1, . . . , N, where µ Xt = 1 N N i=1 δ X i t . (2.10) Here W, W i are standard Brownian motions in R d and (W i ) N i=1 are independent. Their marginal distributions m t = Law(X t ), m N t = Law(X t ) = Law(X 1 t , . . . , X N t ) solve the Fokker-Planck equations ∂ t m = ∆m + ∇ · (D m F (m t , ·)m), (2.11) ∂ t m N = N i=1 ∆ i m N + ∇ i · (D m F (µ x , x i )m N ). (2.12) The mean field equation (2.11) is non-linear while the N -particle system equation (2.12) is linear. We show in Proposition 4.6 the wellposedness and the regularity of the mean field dynamics (2.11) with initial conditions of finite second moment. Remark 2.3. We have fixed the volatility (diffusion) constant to be √ 2 to simplify the computations. In order to apply our results to the MFL defined by dX t = −D m F (m t , X t )dt + σdW t , where Law(X t ) = m t , with some σ > 0, we apply the rescaling:t = σ 2 2 t,F = 2 σ 2 F andXt = X t . In this way, the new diffusion processt →Xt satisfies the SDE (2.9) with the new mean field functionalF . The same scaling transform applies equally to the particle system. Free energy and invariant measure. We focus on the long-term behavior of the MFL (2.11) and of the corresponding particle system (2.12), where there invariant measures play a key role. Define the mean field free energy F : P 2 (R d ) → (−∞, +∞] by F(m) = F (m) + H(m). (2.13) Given the assumptions (2.1) to (2.4), we can show the existence of a unique minimizer of F, denoted by m ∞ . Furthermore, this measure m ∞ satisfies the first-order condition: m ∞ (dx) =m ∞ (dx) = 1 Z(m ∞ ) exp − δF δm (m ∞ , x) dx. (2. 14) The precise statement and proof is given in Proposition 4.4. Differentiating both sides of the first-order condition, we obtain ∆m ∞ + ∇ · (D m F (m ∞ , x)m ∞ ) = 0, which implies m ∞ is an invariant measure to mean field Fokker-Planck equation (2.11). Conversely, we will show in Corollary 4.8 that under our conditions every invariant measure satisfies the first-order condition and, therefore, there exists the unique one. The N -particle system (2.10) is a classical Langevin dynamics as the equation (2.12) is linear. We define the N -particle free energy F N : P 2 (R dN ) → (−∞, +∞] by F N (m N ) = N F (µ x )m N (dx) + H(m N ). (2.15) We will prove in Proposition 4.3 that under our assumptions (2.1) to (2.3) the minimizer m N ∞ of F N exists, has the density 16) and is invariant to the N -particle Fokker-Planck equation (2.12). By the definition of free energy we have F N (m N ) = H(m N |m N ∞ ) + constant so m N ∞ also minimizes the N -particle free energy F N . m N ∞ (dx) ∝ exp (−N F (µ x )) dx,(2.L p + space for all p ∈ R. We investigate the convergence of the marginal distributions of the mean field dynamics in the L p (m ∞ )-norm for all p ∈ R and take special care when p < 1. Let µ be a probability measure on R d and h : R d → [0, +∞] be a measurable function. For p ̸ = 0 define ∥h∥ L p (µ) = h p dµ 1/p , and for p = 0 define ∥h∥ L 0 (µ) = exp log hdµ . We say h ∈ L p + (µ) if ∥h∥ L p (µ)      < +∞, when p > 0, ∈ (0, +∞), when p = 0, > 0, when p < 0. It is well-known that p → ∥h∥ p is increasing, which ensures that the 0-norm is well defined once ∥h∥ p < +∞ for some p > 0 or ∥h∥ q > 0 for some q < 0. In this paper we will only work with µ equal to m ∞ , the mean field invariant measure. In this case we write ∥h∥ p = ∥h∥ L p (m∞) for simplicity. We also say h ∈ L 1+ (m ∞ ) or h is L 1+ -integrable if there exists a number p 0 > 1 such that h ∈ L p0 (m ∞ ). Main results. Recall that m t and m N t are the respective marginal distributions of the mean field and the N -particle system (2.9) and (2.10). We first improve the exponential entropic convergence result for the MFL dynamics (2.9). Theorem 2.1 (Entropic convergence of MFL). Assume F satisfies (2.1) to (2.4). If m t0 has finite entropy and finite second moment for some t 0 ≥ 0, then for every t ≥ t 0 , Define h t (x) = m t (x)/m ∞ (x). Our first major contribution is the following theorem on the L p -convergence of the MFL dynamics. F(m t ) − F(m ∞ ) ≤ (F(m t0 ) − F(m ∞ ))e −4ρ(t−t0) .(2. Theorem 2.2 (L p -convergence of MFL). Assume F satisfies (2.1) to (2.5). If h 0 ∈ L p0 (m ∞ ) for some p 0 > 1, then for all ρ ′ ∈ (0, ρ) and p ∈ R there exist constants C p = C p (ρ, ρ ′ , M F mm , p 0 , ∥h 0 ∥ p0 ) ≥ 0, τ p = τ p (ρ, p 0 ) ≥ 0 such that for t ≥ τ p , h t belongs to L p (m ∞ ) and satisfies | log ∥h t ∥ p | ≤ C p e −4ρ ′ t . (2.18) Our second major contribution is the uniform-in-N exponential entropic convergence of the particle systems. Theorem 2.3 (Entropic convergence of particle systems). Assume F satisfies (2.1) to (2.4). If m N t0 belongs to P 2 (R dN ) and has finite entropy for some N ≥ 2 and t 0 ≥ 0, then for all ρ ′ ∈ (0, ρ) there exist constants C 1 = C 1 (ρ, ρ ′ , M F mm ), C 2 = C 2 (ρ, ρ ′ , M F mm , d) such that if N > C1 4ρ ′ , then for every t ≥ t 0 , F N (m N t ) − N F(m ∞ ) ≤ (F N (m N t0 ) − N F(m ∞ ))e −(4ρ ′ − C 1 N )(t−t0) + C 2 4ρ ′ − C1 N . (2.19) Remark 2.5. Strictly speaking, the result (2.19) does not imply that the particle systems converge uniformly. We only show 1 N F N (m N t ) approaches the mean field minimum F(m ∞ ) uniformly quickly until they are O(N −1 )-close to each other. Remark 2.6. Theorems 2.1 and 2.3 state results concerning the convergence of the respective free energies, which we will also call "convergence of entropy" or "entropic convergence", since in both cases the differences of free energies are related to relative entropies, as shown in Lemmas 4.9 and 5.2. We now present the main theorem, which establishes the uniform-in-time propagation of chaos in both the Wasserstein distance and the relative entropy. The Wasserstein result follows directly from Theorems 2.1 and 2.3. We also show the entropic propagation of chaos by combining the entropic convergence of the particle system in Theorem 2.3 and the L p -convergence of the mean field dynamics established in Theorem 2.2. Theorem 2.4 (Wasserstein and entropic propagation of chaos). Assume F satisfies (2.1) to (2.4). If m 0 belongs to P 2 (R d ), m N 0 belongs to P 2 (R dN ) and they both have finite entropy for some N ≥ 2, then for all ρ ′ ∈ (0, ρ) the constants C 1 , C 2 in Theorem 2.3 are such that if N > C1 4ρ ′ , then for every t ≥ 0, ρW 2 2 (m N t , m ⊗N t ) ≤ 2N (F(m 0 ) − F(m ∞ ))e −4ρt + 2(F N (m N 0 ) − N F(m ∞ ))e −(4ρ ′ − C 1 N )t + 2C 2 4ρ ′ − C1 N . (2.20) If additionally (2.5) holds and h 0 ∈ L p0 (m ∞ ) for some p 0 > 1, then there exists C 3 , τ ≥ 0 depending on ρ, ρ ′ , M F mm , p 0 and ∥h 0 ∥ L p 0 (m∞) such that if N > C1 4ρ ′ , then for every t ≥ τ , H(m N t |m ⊗N t ) ≤ N C 3 e −4ρ ′ t + 2(F N (m N 0 ) − N F(m ∞ ))e −(4ρ ′ − C 1 N )t + 2C 2 4ρ ′ − C1 N . (2.21) Comments on the assumptions. The conditions (2.2) and (2.4) ensure that the drift is jointly Lipschitz continuous in measure and space, which guarantees the wellposedness of the mean field and the particle system dynamics (2.9) and (2.10). This also implies that the flow is AC 2 in L 2 -Wasserstein space (refer to Definition 4.5), which is the type of curves studied in [2,Chapter 8]. In particular the "chain rule" applies, yielding immediately the energy decrease (4.5) and (5.3). We could weaken the condition (2.2) by replacing the L 1 -Wasserstein metric by the L 2 one. However we would need to additionally suppose x → D m F (µ x , x ′ ) is Lipschitz continuous to ensure the wellposedness of the particle system (2.10). The assumptions (2.1) and (2.3), appearing in the previous works [13,37], are key to the exponential convergence of relative entropy of the MFL. They are also used in this work, along with (2.2), to show the exponential entropic convergence of the particle system in Theorem 2.3. The conditions (2.4) and (2.5) are technical in that they do not contribute to any constants in our results, suggesting that the results may still hold under weaker versions of them. For example, we can expect most results to hold under a one-sided Lipschitz condition on D m F in place of (2.4). The final condition (2.5) allows us to obtain a simple "standard algebra" of the time-dependent semigroup induced by the MFL and to justify easily the computations in L p spaces needed to prove Theorem 2.2. It is possible that our results can also be obtained without the higher-order bounds (for example, by an approximation argument). We, however, choose to work in this setting to avoid excessive technicalities. Applications Sufficient conditions for functional convexity We propose two criteria for the convexity of mean field functionals. The first criterion treats translationally invariant two-body interactions and is a rephrased version of Bochner's theorem. P(R d ) → R defined by F Int (m) = 1 2 V (x − y)m(dx)m(dy) (3.1) is convex if and only if V is the Fourier transform of a positive and finite measure on R d . Proof. Suppose m → F Int (m) is convex. Let µ = N i=1 c i δ x i . Then we have F Int (µ) = 1 2 N i,j=1 c i c j V (x i − x j ). Define ∆ = {(c 1 , . . . , c n ) ∈ R d : c i ≥ 0, N i=1 c i = 1}. The mapping (c i ) N i=1 → F (µ) defined on ∆ is then convex since straight lines in ∆ corresponds to linear interpolations in P(R d ). Hence (V (x i − x j )) N i,j=1 is a positive-definite matrix and V is a positive-definite function. We then apply Bochner's theorem (e.g. [40,Theorem IX.9]). For the reverse implication, supposeV is a positive and finite measure. We have V (x) = (2π) −d/2 e ik·xV (dk). Let µ be arbitrary signed measure on R d . Then by Fubini's theorem we have 1 2 V (x − y)µ(dx)µ(dy) = 1 2(2π) d/2 e ik·(x−y)V (dk)µ(dx)µ(dy) = (2π) d/2 2 μ(−k)μ(k)V (dk) ≥ 0. Example 1 (Regularized Coulomb). It is well-known that in dimension d ≥ 3 the Coulomb potential V C (x) = 1 d(d−2)c d |x| d−2 is the fundamental solution to Laplace's equation, that is to say, ∆V C = −δ 0 . (3.2) Hence its Fourier transformV C verifiesV C (k) = (2π) −d/2 |k| −2 ≥ 0. However V C ̸ ∈ L 1 (R d ) and Theorem 3.1 does not apply (which is consistent with the singularity of V C at 0). To remedy this we propose the regularization V RC (k) = e −r0|k| (2π) d/2 |k| 2 for some r 0 > 0. Its Fourier inverse V RC : R d → R is then indeed a bounded continuous function and has the explicit expression for d = 3: V RC (x) = e −r0|k| e ik·x (2π) 3 |k| 2 d 3 k = arctan(|x|/r0) 2π 2 |x| , when x ̸ = 0, 1 2π 2 r0 , when x = 0. Note that when r 0 → 0, we have V RC (x) → V C (x) for every x ∈ R d . The functional F RC (m) = 1 2 V RC (x − y)m(dx)m(dy) = 1 2 1 2π 2 arctan |x−y| r0 |x − y| m(dx)m(dy) (3. 3) is well defined and convex on P(R 3 ) by Theorem 3.1. Remark 3.1. If the functional F Int satisfies the conditions of Theorem 3.1, then its interaction potential V satisfies ∀x ∈ R d , |V (x)| = 1 (2π) d/2 e ik·xV (dk) ≤ 1 (2π) d/2 V (dk) = V (0). So V attains its global maximum at 0. If V is not constant, then there exists x 0 such that V (x 0 ) < V (0) and by the evenness of V , we have V (−x 0 ) = V (x 0 ) < V (0). In particular, V is not convex and the functional F Int cannot be geodesically convex. In other words, the only functionals of form (3.1) with continuous V that are both functionally and geodesically convex are constant functionals. Remark 3.2. Other regularizations preserving the positivity of the Coulomb potential are also possible. For example we can convolute Laplace's equation (3.2) with a heat kernel ρ ε : x → (2πε) −d/2 exp(−(2ε) −1 x 2 ) to obtain ∆V ′ RC = ∆(V C ⋆ ρ ε ) = −ρ ε . The Fourier transform of V ′ RC readŝ V ′ RC (k) =ρ ε (k) |k| 2 = e −2π 2 ε|k| 2 (2π) d/2 |k| 2 , which is positive and L 1 -integrable. The main reason for our choice of the regularization in Example 1 is that it allows the simple expression (3.3) in three dimensions. The second criterion is an analogue of the property of convex functions under composition. Proposition 3.2. Let X be a Banach space. If V : R d → X is a function of quadratic growth and g : X → R is convex, then the functional F : P 2 (R d ) → R defined by F (m) = g V (x)m(dx) is convex. Proof. Immediate. Example 2 (L 2 -loss of two-layer neural networks). We first explain the structure of two-layer neural networks and then introduce the mean field model for it. Consider an activation function φ : R → R satisfying φ is continuous and non-decreasing, ∃x 0 , x 1 ∈ R, φ(x 0 ) = φ(−∞) = 0 and φ(x 1 ) = φ(+∞) = 1. (3.4) Define S = R × R d × R, where the neurons take values. For each neuron θ = (c, a, b) ∈ S we define the feature map: R d ∋ z → Φ(θ; z) := ℓ(c)φ(a · z + b) ∈ R, (3.5) where ℓ : R → [−L, L] is a truncation function with the truncation threshold L ∈ (0, +∞]. Such truncation has been considered in recent papers [25,37]. The twolayer neural network is nothing but the aggregated feature map parameterized by N neurons θ 1 , . . . , θ N ∈ S: R d ∋ z → Φ N (θ 1 , . . . , θ N ; z) = 1 N N i=1 Φ(θ i ; z) ∈ R. (3.6) The training of neural network aims to minimize the distance between the aggregated output (3.6) and a (only empirically known) label function f : R d → R, i.e. inf (θ 1 ,...,θ N )∈S N d(f, Φ N (θ 1 , . . . , θ N ; ·)) (3.7) for some loss functional d. In this paper, we use the L 2 (µ)-norm as the loss functional where µ ∈ P(R d ) represents the feature distribution. In this way, the objective function of the minimization reads F N NNet (θ 1 , . . . , θ N ) = N 2 |f (z) − Φ N (θ 1 , . . . , θ N ; z)| 2 µ(dz). (3.8) To fit the problem to our theoretical framework, we assume the feature map Φ : S × R d → R satisfies ∀θ ∈ S, Φ(θ; ·) ∈ L 2 (µ), ∃C > 0, ∀θ ∈ S, ∥Φ(θ; ·)∥ L 2 (µ) ≤ C(1 + |θ| 2 ). Now we present the mean field formulation of two-layer neural networks. Let P 2 (S) be the space of probabilities on S of finite second moment and define the class of functions representable by the mean field neural network: N φ,ℓ = {h : R d → R : ∃m ∈ P 2 (S), ∀x ∈ R d , h(x) = E Θ∼m [Φ(Θ; x)]}. (3.9) In particular the N -neuron output functions defined in (3.6) belong to this class since Φ N (θ 1 , . . . , θ N ; ·) = E Θ∼ 1 N N i=1 δ θ i [Φ(Θ; ·)]. Instead of the finite-dimensional optimization (3.7), we consider the following mean field optimization: inf P2(S) F NNet (m), where F NNet (m) := d(f, E Θ∼m [Φ(Θ; ·)]) = 1 2 |f (z) − E Θ∼m [Φ(Θ; z)]| 2 µ(dz). (3.10) The functional F NNet is convex by Proposition 3.2 since F NNet (m) = g V (θ)m(dθ) with V : S ∋ θ → (z → Φ(θ; z)) ∈ L 2 (µ) and g : L 2 (µ) ∋ h → ∥f − h∥ 2 L 2 (µ) ∈ R. Remark 3. 3 (Motivation of mean field formulation). The N -neuron problem (3.8) is non-convex due to the non-linear activation function φ. Inspired by the fact that the width N of two-layer neural networks is usually large in practice, the authors of [33,14,42,25] consider the mean field formulation of neural networks which convexifies the original problem. Remark 3.4 (Absence of geodesic convexity). We highlight here that if F NNet is geodesically convex and regular enough, then the N -neuron problem F N NNet is convex, which is not true. Hence by contradiction F NNet has no geodesic convexity. Indeed, suppose F NNet is geodesically convex. Note that t → 1 N N i=1 δ θ i +tv i is a geodesic in (P 2 , W 2 ) in a neighborhood of t = 0 if θ i are distinct from each other (as the pairing (θ i , θ i + tv i ), i = 1, . . . , N verifies cyclical monotonicity for t small enough). By the geodesic convexity of F NNet and the rela- .10) is zero if µ is compactly supported and no truncation is present (that is, L = +∞ and ℓ is the identity function). However, if a truncation with L < +∞ is applied, all functions h ∈ N φ,ℓ satisfy the bound ∥h∥ ∞ ≤ L and therefore cannot approximate well functions that exceed L. We provide in Proposition B.1 a crude bound on the best approximation error of L-truncated neural networks, which goes to zero as L → +∞. tion F N NNet (θ 1 , . . . , θ N ) = N F NNet ( 1 N N i=1 δ θ i ), we obtain the local convexity of F N NNet on the set S N \∆ N := S N \{(θ 1 , . . . , θ N ) ∈ S N : ∃i ̸ = j, θ i = θ j }. If F N NNet is additionally C 2 , the local convexity implies ∇ 2 F N NNet ≥ 0 on S N \∆ N and by density ∇ 2 F N NNet ≥ 0 everywhere. Therefore F N NNet is convex on S N . Examples of MFL dynamics We construct MFL dynamics for the two examples discussed earlier and demonstrate that our theorems are applicable in both cases. To verify the LSI condition (2.3) we will use the following results. Proposition 3.3. Let µ(dx) = e −V (x) dx be a probability measure in R d for some V ∈ C 2 (R d ). • (Bakry-Émery [4]) If ∇ 2 V ≥ κ then µ satisfies a κ/2-LSI. • (Holley-Stroock [23]) If V = V 1 + V 2 , e −V1 is the density of a probability satisfying an ρ-LSI and V 2 is bounded with oscillation osc V 2 , then µ satisfies a ρ exp(− osc V 2 )-LSI. • (Aida-Shigekawa [1]) If V 2 in the previous statement is Lipschitz-continuous instead of bounded, then µ satisfies an LSI as well. Example 3 (MFL for regularized Coulomb system). Let λ > 0. Define F Ext (m) = λ 2 |x| 2 m(dx). (3.11) We consider the functional F = F RC + F Ext where F RC is defined in (3.3) . By the discussions in Example 1 the functional F satisfies the convexity condition (2.1). Its linear functional derivative reads δF δm (m, x) = V RC (x − y)m(dy) + 1 2 λ|x| 2 and its intrinsic derivative reads D m F (m, x) = ∇V RC (x − y)m(dy) + λx. The conditions (2.2), (2.4) and (2.5) are satisfied because ∥∇ n V RC ∥ ∞ ≤ 1 (2π) d/2 |k| nV RC (dk) = |k| n e −r0|k| (2π) d |k| 2 d d k < +∞ for all n ≥ 0 (and d ≥ 3). In particular, the bound in (2.2) is verified by M F mm = ∥∇ 2 V RC ∥ ∞ . For the uniform LSI, we can apply Holley-Stroock or Aida-Shigakawa, since the first term in δF δm is uniformly bounded and uniformly Lipschitz and the second term verifies the Bakry-Émery condition. The LSI constant given by Holley-Stroock has the simple expression in three dimensions ρ = λ 2 exp(− osc V RC ) = λ 2 exp − 1 2π 2 r0 . The L 1+ -integrability of the initial condition, needed by Theorem 2.2 and the second part of Theorem 2.4, is verified once we have ∃C, ε > 0, ∀x ∈ R, m 0 (x) ≤ Ce −ε|x| 2 . (3.12) However, as the regularization parameter r 0 approaches 0, we observe ρ → 0 and M F mm → +∞, suggesting our method is not suitable for the unregularized Coulomb interaction. We refer readers to [9,8,41,16] for recent developments on the noised gradient flow of Coulomb (and more generally, Riesz) particle systems, where the modulated free energy is used to tackle the singularity in the interactions. Example 4 (MFL for two-layer neural networks). Recall the mean field two-layer neural networks in Example 2. Suppose • the truncation L is finite; • the activation and truncation functions φ, ℓ have bounded derivatives of up to fourth order; • the feature distribution µ has finite second moment; • the label function f belongs to L 2 (µ). On top of the mean field optimization problems (3.10), we add the quadratic regularizer F Ext in (3.11) to the loss, as for the Coulomb system. Then the function and the functional to optimize read F N (θ 1 , . . . , θ N ) = N 2 f (z) − 1 N N i=1 Φ(θ i ; z) 2 µ(dz) + λ 2 N i=1 |θ i | 2 , F (m) = 1 2 |f (z) − E Θ∼m [Φ(Θ; z)]| 2 µ(dz) + λ 2 |θ| 2 m(dθ). The N -neuron loss can be recover from the mean field loss by F N (θ 1 , . . . , θ N ) = N F 1 N N i=1 δ θ i . We verify the assumptions of our theorems one by one. The functional convexity of F = F NNet + F Ext is already proved in Example 2. The linear functional derivative of F reads δF δm (m, θ) = − (f (z) − E Θ∼m [Φ(Θ; z)])Φ(θ; z)µ(dz) + λ 2 |θ| 2 . The first term on the right hand side is uniformly bounded: for every m ∈ P 2 (S) and every θ ∈ S, (f (z) − E Θ∼m [Φ(Θ; z)])Φ(θ; z)µ(dz) ≤ (∥f ∥ L 1 (µ) + ∥ℓ∥ ∞ )∥ℓ∥ ∞ . Hence by Holley-Stroock the uniform LSI condition (2.3) is satisfied with the constant ρ = λ 2 exp(−2(∥f ∥ L 1 (µ) + ∥ℓ∥ ∞ )∥ℓ∥ ∞ ). The intrinsic derivative of F reads D m F (m, θ) = − (f (z) − E Θ∼m [Φ(Θ; z)]) ∂Φ ∂θ (θ; z)µ(dz) + λθ, where the partial derivative of the feature map Φ, defined in (3.5), reads ∂Φ ∂c (θ; z) = ℓ ′ (c)φ(a·z+b), ∂Φ ∂a (θ; z) = ℓ(c)φ ′ (a·z+b)z, ∂Φ ∂b (θ; z) = ℓ(c)φ ′ (a·z+b) for θ = (c, a, b) ∈ S. Similarly we obtain the second order intrinsic derivative: D 2 m F (m, θ, θ ′ ) = ∂Φ ∂θ (θ; z) ⊗ ∂Φ ∂θ (θ ′ ; z)µ(dz). Its 2-norm satisfies the bound |D 2 m F (m, θ, θ ′ )| 2 2 ≤ (∥ℓ ′ ∥ 2 ∞ + ∥ℓ∥ 2 ∞ ∥φ ′ ∥ 2 ∞ (1 + M 2 (µ))), where M 2 (µ) = |z| 2 µ(dz) is the second moment of µ. Thanks to the Kantorovich duality and the Cauchy-Schwarz inequality, the W 1 -Lipschitz constant of m → D m F (m, x) can be given by M F mm = (∥ℓ ′ ∥ 2 ∞ + ∥ℓ∥ 2 ∞ ∥φ ′ ∥ 2 ∞ (1 + M 2 (µ))) 1/2 . So D m F satisfies the condition (2. 2). Since ℓ, φ have bounded derivatives of up to fourth order, the derivatives ∇ k D m F (m, θ) for k = 1, 2, 3 are also uniformly bounded. Thus the technical conditions (2.4) and (2.5) are also satisfied. Finally, the L 1+ -integrability of the initial value m 0 is verified once we require the pointwise Gaussian bound (3.12) on the density of m 0 . Remark 3.6 (Link to practice). In the training of neural networks, the measure µ is an empirical measure 1 K K k=1 δ z k and on the feature points {z k } K k=1 the labels are known f (z k ) = y k . This collection of pairs {z k , y k } K k=1 are the available training data. In practice, instead of the mean field dynamics, we can only simulate the corresponding N -particle system. In other words, we calculate the N -neuron SDE dΘ i t = 1 K K k=1 (y k − Φ N (Θ 1 t , . . . , Θ N t ; z k )) ∂Φ ∂θ (Θ i Remark 3.7 (Noised data). In the previous remark we suppose the data available {z k , y k } N k=1 are precise: y k = f (z k ), while in practice they may be subject to errors: y ′ k = f (z k )+ε k . The new collection of points {z k , y ′ k } N k=1 induces another mean field functional F ′ NNet defined by F ′ NNet (m) = 1 2K K k=1 (y ′ k − E Θ∼m [Φ(Θ; z k )]) 2 . From the triangle inequality for the L 2 -distance we deduce |F ′ NNet (m) − F NNet (m)| ≤ 1 K K k=1 ε 2 k 1/2 F NNet (m) 1/2 + 1 2K K k=1 ε 2 k . The actual N -neuron training process is therefore the noised gradient descent for the functional F ′ := F ′ NNet + F Ext and approximately converges to (m ′ ∞ ) ⊗N where m ′ ∞ minimizes F ′ = F ′ + σ 2 2 H. The difference between respective minima can be bounded as follows: F ′ (m ′ ∞ ) − F(m ∞ ) ≤ F ′ (m ∞ ) − F(m ∞ ) = F ′ NNet (m ∞ ) − F NNet (m ∞ ) ≤ 1 K K k=1 ε 2 k 1/2 F NNet (m ∞ ) 1/2 + 1 2K K k=1 ε 2 k . Hence the additional error converges to zero as the noise in the data (ε k ) K k=1 tends to zero. Remark 3.8 (Advantages over other approaches). Our Theorems 2.3 and 2.4 establish the exponential convergence of the N -neurons training process (3.13) without supposing the truncation satisfies bounds of type ∥∇ k ℓ∥ ∞ < c for some small constant c. This stands in contrast to many previous studies on uniform-in-time propagation of chaos relying on the smallness of the mean field interaction (e.g. [19] and the first setting of [17]). Yet the smallness approach does not apply to general neural networks: in our setting, the smallness requires the Lipschitz constants M F mm to be smaller than a constant times ρ, which we denote by M F mm ≲ ρ, and the relation is difficult to verify. Indeed, using the constants M F mm , ρ obtained in Example 4, we need (∥ℓ ′ ∥ 2 ∞ + ∥ℓ∥ 2 ∞ ∥φ ′ ∥ 2 ∞ (1 + M 2 (µ))) 1/2 ≲ λ 2 exp(−2(∥f ∥ L 1 (µ) + ∥ℓ∥ ∞ )∥ℓ∥ ∞ ). This forces either the regularization λ to be very large or the truncation ∥ℓ∥ ∞ to be very small. In conclusion, our approach based on the functional convexity offers the advantage of obtaining the exponential convergence, albeit at a very slow rate, without such restrictions on λ or ℓ. Numerical experiments As explained in Examples 2 and 4, the MFL dynamics for training two-layer neural networks verifies all the conditions of our theorems, so its particle sys- Parameters Value Setup. We aim to train a neural network to approximate the elementary function z → f (z) = sin 2πz 1 + cos 2πz 2 on [0, 1] 2 . We draw uniformly K points {z i } K k=1 from [0, 1] 2 and calculate the corresponding labels y k = f (z k ) to prepare our training data {z k , y k } K k=1 . These points are plotted in Figure 1. We fix the truncation function ℓ by ℓ(x) = (x ∧ 100) ∨ −100 and the sigmoid activation function φ by φ(x) = 1/(1 + exp(−x)). The Brownian noise has volatility σ (one needs to apply the scaling transform in Remark 2.3 before comparing to the theoretical results), and the quadratic regularization constant λ is fixed in experiments. The initial values ∆t 0.2 T 4000 K 1000 m 0 N (0, 5 2 ) σ 1 λ 10 −5(Θ i 0 ) N i=1 = (c i 0 , a i 0 , b i 0 ) N i=1 of the N neurons are sampled independently from a normal distribution m 0 in four dimensions. The training process (3.13) is discretized with time step ∆t and terminated at time T . The values of the hyperparameters K, σ, m 0 , ∆t, T are listed in Table 1 and the training algorithm is shown in Algorithm 1. We take the number of neurons N to be 2 P for P = 6, . . . , 10 and repeat the training 10 times for each N . Results. We compute the sum of the N −1 -scaled loss 1 N F N NNet (Θ 1 t , . . . , Θ N t ) at each time t and plot its evolution in Figure 2. We observe the value of 1 N F N NNet first decreases exponentially and then decreases more slowly or even stabilizes. To explore the relationship between this residual error and the number of neurons, for each value of N we calculate the average value of 1 N F N NNet during the last 500 training steps and take the average of these values over the 10 independent runs. The results are plotted in Figure 3. Discussions. Our truncation function ℓ does not have bounded derivatives of up to fourth order as required in Example 4 and we can work around this by tak-Algorithm 1: Noised gradient descent for training a two-layer neural network Input: number of particles N , activation φ, truncation ℓ, data set (z k , y k ) K k=1 , noise σ, initial distribution m 0 , time step ∆t, time horizon T Output: ing a sequence of regular ℓ n approximating ℓ since the constants M F mm , ρ depends only on ∥ℓ∥ ∞ , ∥ℓ ′ ∥ ∞ . In our experiment we also ignore the time-discretization error and the difference between training and validation data sets. As shown in Figure 2 the losses first decrease exponentially at a uniform rate for different numbers of neurons, N . This is consistent with the convergence rate ρ ′ − C1 N predicted by Theorems 2.3 and 2.4. However, the LSI constant obtained in Example 4 by Holley-Stroock is excessively small and fails to predict the actual convergence rate. Given that the Holley-Stroock method relies solely on the boundedness of neural networks, this phenomenon suggests the internal structure of neural networks allows for a faster convergence rate that is not captured by the perturbation lemma. (Θ i T ) N i=1 generate i.i.d. Θ i 0 = (A i 0 , B i 0 , C i 0 ) ∼ m 0 , i = 1, . . . , N ; for t = 0, ∆t, 2∆t, . . . , T − ∆t do generate i.i.d. N i t ∼ N (0, 1), i = 1, . . . , N ; // update particles according to discretized Langevin for i = 1, . . . , N do Θ i t+∆t ← Θ i t − (D m F NNet ( 1 N N j=1 δ Θ j t , Θ i t ) + λΘ i t )∆t + σ √ ∆tN i t ; /* where D m F NNet ( N j=1 δ Θ j t , Θ i t ) = 1 K K k=1 (y k − Φ N (Θ 1 t , . . . , Θ N t ; z k )) ∂Φ ∂θ (Θ i t ; z k ) */ We fit the residual losses with the curve α N + β in Figure 3. We choose this parametrization for two reasons: the first term α N corresponds to the error term in the convergence result (2.19) of the free energy 1 N F N (m N t ); the second term β accounts for the facts that F(m ∞ ) ̸ = 0 and that the free energy differs from the neural network's loss by 1 N F N (m N t ) − 1 N F N NNet (m N t ) = λ 2N |θ| 2 m N t (dθ) + σ 2 2N H(m N t ). In particular the relative entropy H(m N t ) can not be directly calculated. Mean field system 4.1 Existence of the measuresm, m ∞ , m N ∞ Our assumptions differ from those in the earlier works, such as [25]. Specifically, we do not require the coercivity condition of type ∀m ∈ P 2 (R d ), ∀x ∈ R d , D m F (m, x) · x ≥ C(|x| 2 − 1). Instead we only assume the condition (2.4) on D m F (m, x). As a result, the existence of the measuresm, m ∞ , m N ∞ , introduced in Section 2, is not obvious. In this subsection we show that thanks to the conditions (2.1) to (2.3) these measures are indeed well defined. First we sketch a proof that regular enough measures satisfying an LSI in R d have finite moments. 3) holds whenm is replaced by µ for some ρ > 0, then µ ∈ ∩ p≥1 P p (R d ) and e α|x| µ(dx) < +∞ for all α ≥ 0. Proof. Here we repeat the argument of Otto and Villani in [39]. Suppose µ satisfies a ρ-LSI (but we do not suppose µ ∈ P 2 (R d ) a priori). For every measure ν ∈ P 2 (R d ) of finite entropy (e.g. the Gaussians), the heat flow ∂ t ν t = ∆ν t + ∇ · (ν t ∇Ψ), ν 0 = νW 2 (ν, ν t ) ≤ 1 √ ρ H(ν|µ) − H(ν t |µ) . (4.1) The sequence ν t are tight in the weak topology of P since we have ρW 2 (ν, ν t ) 2 ≤ H(ν|µ) = (log ν + Ψ)ν < +∞ (recall that Ψ is of quadratic growth). By the lower-semicontinuity of H(·|µ) we must have ν t → µ in P weakly when t → ∞. Then we take lim inf t→∞ on both side of (4.1) and use the lower-semicontinuity of W 2 with respect to the weak topology of P to obtain Talagrand's inequality ρW 2 2 (ν, µ) ≤ H(ν|µ). Hence µ ∈ P 2 . Finiteness of higher moments and exponential moments then follows from concentration inequalities via Herbst's argument (see e.g. the proof of [7, Theorem 5.5]). We give a sufficient condition to the existence ofm for every m ∈ P 2 (R d ) so that the condition (2.3) makes sense. Proof. By definition we have Z(m) = exp − δF δm (m, x) dx = Z(m 0 ) exp δF δm (m 0 , x) − δF δm (m, x) m 0 (dx), where the term on the exponential is of linear growth since its derivative is uniformly bounded: |∇( δF δm (m 0 , x) − δF δm (m, x))| = |D m F (m 0 , x) − D m F (m, x)| ≤ M F mm W 1 (m 0 , m) . But by Lemma 4.1, all exponential moments ofm 0 are finite. Thus Z(m) < +∞ andm is well defined. We now show that the N -particle invariant measure is also well defined. Proof. Fix m 0 ∈ P 2 (R d ). Using convexity we obtain N F (µ x ) ≥ N F (m 0 ) + N δF δm (m 0 , y)(µ x − m 0 )(dy) = N F (m 0 ) − N δF δm (m 0 , y)m 0 (dy) + N i=1 δF δm (m 0 , x i ). The integral δF δm (m 0 , y)m 0 (dy) is finite thanks to Lemma 4.1. Hence e −N F (µx) dx ≤ C e − N i=1 δF δm (m0,x i ) dx = C(Z(m 0 )) N < +∞. Apply the same argument to e α N i=1 |x i | e −N F (µx) dx we obtain the finiteness of exponential moments. H(m) = log m(x)m(x)dx = log m(x) (2π) −d/2 e −x 2 /2 m(x)dx + log(2π) −d/2 − x 2 2 m(x)dx. (4.2) The first term, which is the relative entropy between m and a normalized Gaussian, is always nonnegative and the second term is finite. Moreover the free energy F satisfies for all m, m 0 ∈ P 2 such that m 0 has finite entropy. Since the LSI (2.3) implies the T 2 inequality (2.8), the functional F has P 2 -coercivity: ρW 2 2 (m,m 0 ) ≤ H(m|m 0 ) ≤ F(m) − logm 0 (x)m 0 (dx) − F (m 0 ). The conditions (2.1) and (2.4) imply also the P 2 -lower-continuity of F : if (m n ) n∈N is a sequence convergent to m in the weak topology of P 2 , then we have lim inf n F (m n ) − F (m) ≥ lim inf n δF δm (m, x)(m n − m)(dx) = lim inf n δF δm (m, x) − δF δm (m, 0) (m n − m)(dx) ≥ lim inf n D m F (m, 0) · x − M F mx 2 |x| 2 (m n − m)(dx) = 0. Here the second inequality follows from Taylor's formula and M F mx denotes the constant in the condition (2.4). The entropy H is also P 2 -lower-semicontinuous by the previous decomposition (4.2). The free energy F is then lower-bounded, coercive, lower-semicontinuous and convex, so there exists unique minimizer in P 2 which we denote by m ∞ . Now we show the equivalence between the minimizing property of the free energy F and the first-order condition (2.14). If m 0 satisfies (2.14) thenm 0 = m 0 and from (4.3) we deduce F(m) ≥ F(m 0 ) for all m ∈ P 2 , i.e. m 0 is the minimizer of F. For the reverse implication we refer readers to the necessary part of the proof of [25, Proposition 2.5]. Finally since m ∞ satisfies (2.14) we have ∆m ∞ + ∇ · (D m F (m ∞ , x)m ∞ ) = ∇ · m ∞ ∇ δF δm (m ∞ , x) + log m ∞ = 0, and m ∞ is invariant to (2.11). Remark 4.1. We will establish the uniqueness of the invariant measure of the MFL in Corollary 4.8 after deriving the free energy decrease formula (4.5). Convergence in entropy First we recall the definition of AC 2 curves in [2]. We say x ∈ AC 2 ([a, b]; (X, d)) if additionally m ∈ L 2 ([a, b]). For a globally defined curve x : [t 0 , +∞) → X we say x belongs to the class AC 2 loc and denote x ∈ AC 2 loc ([t 0 , +∞; (X, d)) if x ∈ AC 2 ([t 0 , T ]; (X, d)) for every T ∈ (t 0 , +∞). Now we state the wellposedness and regularity result. 2. moreover, this solution has density and finite entropy for positive time: ∀t > 0, | log m t (x)|m t (x)dx < +∞; 3. if additionally m t0 has finite entropy for some t 0 ≥ 0, then the integral t t0 |∇m s (x)| 2 m s (x) dxds (4.4) is finite for every t ≥ t 0 ; therefore (m t ) t≥t0 ∈ AC 2 loc ([t 0 , +∞); (P 2 , W 2 )) and has tangent vector v t (x) = −D m F (m t , x) − ∇ log m t (x) for t ≥ t 0 a.e. in the sense of [2,Proposition 8.4.5]. Due to the technical nature of this proposition its proof is postponed to Appendix A. Using the results of Proposition 4.6 and applying the formalism of [2], we establish the free energy decrease formula, which is central to our studies of dynamics of gradient flow type. 6. Hence the free energy F = F + H is −λ-geodesically-convex and has differential D m F (m t , ·) + ∇ log m t at m t . For the flow of measures m t we have already obtained its AC 2 -regularity in the previous proposition and its tangent vector reads v t = −D m F (m t , ·) − ∇ log m t at m t for t ≥ t 0 a.e. Then we can apply the chain rule to obtain the absolute continuity of t → F(m t ) and dF(m t ) dt = − |D m F (m t , x) + ∇ log m t (x)| 2 m t (dx), for t ≥ t 0 a.e.∀T > t 0 , F(m T ) − F(m t0 ) = T t0 (D m F (m t , x) + ∇ log m t (x)) · v t (x)m t (dx)dt which is the desired result. Proof. The existence part is already shown in Proposition 4.4. Let m * ∈ P 2 (R d ) be an invariant measure. We let the initial condition m 0 be equal to m * and construct according to Proposition 4.6 the MFL solution (m t ) t≥0 . By the invariance of m * we have m t = m * for all t ≥ 0, so m * must have density and finite entropy. We then apply the energy decrease formula (4.5) and obtain for x ∈ R d a.e., D m F (m * , x) + ∇ log m * (x) = 0. Integrating this equation, we obtain m * solves the first-order condition (2.14) which has unique solution by Proposition 4.4. Now we show the close relation between the free energy and the relative entropies. Lemma 4.9 (Entropy sandwich). Assume F satisfies (2.1) to (2.4). Then for every m ∈ P 2 (R d ) we have H(m|m ∞ ) ≤ F(m) − F(m ∞ ) ≤ H(m|m) ≤ 1 + M F mm ρ + (M F mm ) 2 2ρ 2 H(m|m ∞ ). (4.6) Proof. The first two inequalities are proved in [13,Lemma 3.4]. We show the rightmost one. Recall that Z(m) is the normalization constant defined in (2.6). We have H(m|m) − H(m|m ∞ ) = log m m − log m m ∞ m = log m ∞ m m = δF δm (m, x) − δF δm (m ∞ , x) m(x)dx + log Z(m) − log Z(m ∞ ). The difference between δ := log Z(m) − log Z(m ∞ ) satisfies by Jensen's inequality δ = log Z(m) − log exp − δF δm (m ∞ , x) dx = log Z(m) − log exp − δF δm (m ∞ , x) − logm(x) m(x)dx ≤ log Z(m) + δF δm (m ∞ , x) + logm(x) m(x)dx ≤ log Z(m) + δF δm (m ∞ , x) − δF δm (m, x) − log Z(m) m(x)dx = δF δm (m ∞ , x) − δF δm (m, x) m(x)dx. Then we have by Kantorovich duality and W 1 -Lipschitzianity in (2.2) H(m|m) − H(m|m ∞ ) ≤ δF δm (m, x) − δF δm (m ∞ , x) (m(x) −m(x))dx ≤ ∥D m F (m, x) − D m F (m ∞ , x)∥ ∞ W 1 (m,m) ≤ M F mm W 1 (m, m ∞ )W 1 (m,m) ≤ M F mm W 1 (m, m ∞ )(W 1 (m, m ∞ ) + W 1 (m, m ∞ )). To conclude we note W 1 (m, m ∞ ) ≤ W 2 (m, m ∞ ) ≤ ρ −1 H(m|m ∞ ) by the T 2 inequality (2.8) and W 2 1 (m, m ∞ ) ≤ W 2 2 (m, m ∞ ) ≤ 1 ρ H(m|m ∞ ) ≤ 1 4ρ 2 ∇ logm m ∞ 2m = 1 4ρ 2 |D m F (m, x) − D m F (m ∞ , x)| 2m (x)dx ≤ (M F mm ) 2 4ρ 2 W 2 1 (m, m ∞ ) ≤ (M F mm ) 2 4ρ 3 H(m|m ∞ ), by the T 2 and log-Sobolev inequalities (2.8) and (2.3). The proof of Theorem 2.1, which we reproduce there, is nothing but combining the previous two results. Proof of Theorem 2.1. By Proposition 4.7 we have dF(m t ) dt = − |D m F (m t , x) + ∇ log m t (x)| 2 m t (dx) = −I(m t |m t ) ≤ −4ρH(m t |m t ) ≤ −4ρ(F(m t ) − F(m ∞ )), for t ≥ t 0 a.e. The first inequality is due to the uniform log-Sobolev inequality (2.3) and the second to the entropy sandwich (4.6). The desired result is then given by Grönwall's lemma. L p -convergence and hypercontractivity We now investigate the convergence of the marginal distributions of the MFL dynamics in L p (m ∞ ) for all p ∈ R. ∂ t h = ∆h + (2b ∞ − b t ) · ∇h − [∇ · (b t − b ∞ ) + (b t − b ∞ ) · b ∞ ]h. (4.7) In this subsection we will fix the flow of measures m t to be that constructed in Proposition 4.6 and let h change independently from m t . We will also only consider solutions in L ∞ ([t 0 , T ]; L 1 (m ∞ )) with initial value h t0 ∈ L 1 (m ∞ ) (in the sense of [6, (6.1. 3)]) to the evolution equation (4.7). The solution is then unique according to [6, Theorem 9.6.3] applied to hm ∞ . We will work on a set of regular functions which is both dense and stable under the evolution (4.7). Definition 4.10 (Standard algebra). We define the standard algebra A + to be the set of positive and C 2 functions h : R d → (0, ∞) such that • there exists C > 0 such that for every x ∈ R d , | log h(x)| ≤ C(1 + |x|); • for k = 1, 2 there exists C k > 0 such that for every x ∈ R d , |∇ k h(x)| ≤ exp(C k (1 + |x|)). For a collection of functions (h i ) i∈I we say that h i ∈ A + uniformly for i ∈ I or (h i ) i∈I ⊂ A + uniformly, if there exist constants C, C 1 , C 2 such that the previous bounds holds for all h i , i ∈ I. Remark 4.2. The word "standard algebra" is the terminology in [3]. Readers may have noticed A + is not an algebra in the usual sense, as it contains only positive functions and is not closed under scalar multiplication by −1. To remedy this we can define A = A + − A + and A is truely an algebra. We introduce this unusual set of functions in order to do L p -computations for p < 1. We state the density and stability of A + in the following two propositions respectively. Their proofs are given in Appendix A. Proposition 4.11 (Density of A + ). Let p ≥ 1, q < 1, h : R d → [0, +∞] be a measurable function and µ be a probability distribution on R d having a density with respect to the Lebesgue measure. If h ∈ L p (µ), then there exists a sequence (h n ) n∈N in A + such that h n → h in L p (µ); if h ∈ L q (µ), then there exists a sequence (h n ) n∈N in A + such that ∥h n ∥ q → ∥h∥ q ; and if h ∈ L p ∩ L q (µ), then the sequence in A + can be chosen such that both convergences hold. ) t∈K ⊂ A + uniformly if K ⊂ [t 0 , +∞) is compact. By working in A + we obtain the following L p -norm growth formula. d dt h p m ∞ = p(p − 1) − h p−2 |∇h| 2 m ∞ + h p−1 ∇h · (b t − b ∞ )m ∞ (4.8) for t ∈ [a, b] a.e. Proof. In this proof we suppose t → h(t, x) is C 1 instead of only absolutely continuous. The proof for the general case is similar but more tedious. Notice that the evolution equation (4.7) of h can be rewritten as ∂ t h = (∆ + b ∞ · ∇)h − (b t − b ∞ ) · ∇h − ∇ · (m ∞ (b t − b ∞ )) m ∞ h, where the first term corresponds to the symmetric operator ∆ + b ∞ · ∇ in L 2 (m ∞ ). We then have d dt h p m ∞ = p h p−1 (∆ + b ∞ · ∇)hm ∞ + p h p−1 (−(b t − b ∞ ) · ∇hm ∞ − ∇ · (m ∞ (b t − b ∞ ))h) = −p(p − 1) h p−2 |∇h| 2 m ∞ + p (−h p−1 (b t − b ∞ )∇h + ∇h p · (b t − b ∞ ))m ∞ = p(p − 1) − h p−2 |∇h| 2 m ∞ + h p−1 ∇h · (b t − b ∞ )m ∞ . We can justify the first equality by the dominated convergence theorem and the two integrations by parts in the second one by an approximating sequence of functions, thanks to the fact that h t ∈ A + locally uniformly. Remark 4.3. By dividing (4.8) by p − 1 and taking the limit p → 1, one formally obtains d dt h log hm ∞ = − |∇h| 2 h m ∞ + ∇h · (b t − b ∞ )m ∞ . (4.9) This entropy growth formula is rigorously proved and plays an important role for proving the uniform-in-time propagation of chaos in [28]. The L p -norm growth formula implies the existence of a strongly continuous semigroup in L p (m ∞ ) for all p ∈ [1, +∞). Proof. For h s ∈ A + define h t = h(t, ·) ∈ A + where h is the unique solution of (4.7) in A + . The mapping h s → h t is linear (when the multiplying scalar is positive). For p ̸ = 0, the growth of L p -norm satisfies d du h p u m ∞ ≤ p(p − 1) 4 h p u |b u − b ∞ | 2 m ∞ ≤ p(p − 1) 4 (M F mm ) 2 W 2 1 (m u , m ∞ ) h p u m ∞ for u ∈ [s, t] a.e., by Proposition 4.13 and by Cauchy-Schwarz inequality The existence of the stated constant C s,t,p then follows from an application of Grönwall's lemma. For p ≥ 1, the mapping h s → h t =: P t s h s extends uniquely to a continuous linear one by the density of A + in L p + (m ∞ ). By the dominated convergence theorem we have lim t→s |h t − h s | p m ∞ = 0 when h s ∈ A + , using the fact that (h u ) u∈[s,t] ⊂ A + uniformly. This property extends to general h s ∈ L p + (m ∞ ) by the density in Proposition 4.11. Hence P t s is a strongly continuous semigroup on L p + (m ∞ ). To recover the usual definition of strongly continuous semigroup we note that L p = L p + − L p + and define P t s h : = P t s h + − P t s h − for h ∈ L p (m ∞ ). Specializing to p = 2, we obtain the L 2 -convergence by applying the Poincaré inequality and the result of Theorem 2.1. Proposition 4.15 (L 2 -convergence). Assume F satisfies (2.1) to (2.5). Let m t ∈ C([0, +∞); (P 2 , W 2 )) be a solution to (2.11). If h t0 ∈ L 2 (m ∞ ), then h t ∈ L 2 (m ∞ ) for all t ≥ t 0 . Moreover, for all ρ ′ ∈ (0, ρ), there exists a constant C = C(ρ, ρ ′ , M F mm , ∥h t0 ∥ 2 ) ≥ 0 such that ∀t ≥ t 0 , ∥h t − 1∥ 2 ≤ Ce −2ρ ′ (t−t0) . (4.10) Proof. First suppose h t0 ∈ A + . Thanks to Proposition 4.13 with p = 2, we have d dt h 2 t m ∞ = −2 |∇h t | 2 m ∞ + 2 h t ∇h t · (b t − b ∞ )m ∞ ≤ −2(1 − ε) |∇h t | 2 m ∞ + 1 2ε h 2 t |b t − b ∞ | 2 m ∞ ≤ −4(1 − ε)ρ h 2 t m ∞ − 1 + (M F mm ) 2 2ε W 2 1 (m t , m ∞ )∥h t ∥ 2 2 = −4(1 − ε)ρ∥h t − 1∥ 2 2 + (M F mm ) 2 2ε W 2 1 (m t , m ∞ )∥h t ∥ 2 2 , where we first use the Cauchy-Schwarz inequality before applying the Poincaré inequality (2.7) satisfied by m ∞ and the Lipschitz bound on |b t −b ∞ | = |D m F (m t , x)− D m F (m ∞ , x)|. By the T 2 inequality (2.8) we have W 2 1 (m t , m ∞ ) ≤ W 2 2 (m t , m ∞ ) ≤ ρ −1 H(m t |m ∞ ). Thanks to Lemma 4.9 and Theorem 2.1 we have H(m t |m ∞ ) ≤ F(m t ) − F (m ∞ ) ≤ e −4ρ(t−t0) (F(m t0 ) − F(m ∞ )) ≤ 1 + M F mm ρ + (M F mm ) 2 2ρ 2 e −4ρ(t−t0) H(m t0 |m ∞ ). Finally note that the relative entropy satisfies H(m t0 |m ∞ ) ≤ log ∥h t0 ∥ 2 by Jensen's inequality exp log h 2 t0 m t0 ≤ h 2 t0 m t0 . Chaining up the previous three inequalities we obtain (M F mm ) 2 2ε W 2 1 (m t , m ∞ ) ≤ ρα 2 2ε 1 + α + α 2 2 log ∥h t0 ∥ 2 e −4ρ(t−t0) =: δ(t),(4.11) where we define α := M F mm /ρ. The decrease of L 2 -norm then satisfies d dt ∥h t ∥ 2 2 ≤ −(4ρ ′ − δ(t))∥h t − 1∥ 2 2 + δ(t) with ρ ′ := (1−ε)ρ. Thanks to Grönwall's lemma and the fact that +∞ s δ(u)du ≤ δ(s)/4ρ we obtain ∥h t − 1∥ 2 2 ≤ e −4ρ ′ (t−t0)+ t t 0 δ(s)ds ∥h t0 − 1∥ 2 2 + t t0 e −4ρ ′ (t−s)+ t s δ(u)du δ(s)ds ≤ e δ(t0)/4ρ e −4ρ ′ (t−t0) ∥h t0 − 1∥ 2 2 + t t0 e −4ρ ′ (t−s) δ(s)ds ≤ e δ(t0)/4ρ e −4ρ ′ (t−t0) ∥h t0 − 1∥ 2 2 + δ(t 0 ) t t0 e −4ρ ′ (t−s) e −4ρ(s−t0) ds ≤ e δ(t0)/4ρ e −4ρ ′ (t−t0) ∥h t0 − 1∥ 2 2 + δ(t 0 ) 4(ρ − ρ ′ ) e −4ρ ′ (t−t0) − e −4ρ(t−t0) ≤ e δ(t0)/4ρ ∥h t0 − 1∥ 2 2 + δ(t 0 ) 4ερ e −4ρ ′ (t−t0) . For general h t0 ∈ L 2 (m ∞ ), we take an approximating sequence (h n t0 ) n∈N in A + such that h n t0 → h t0 in L 2 (m ∞ ) according to Proposition 4.12. We have proved that ∥h n t − 1∥ 2 ≤ Ce −γt where h n t = P t t0 h n t0 . By the continuity shown in Corollary 4.14, we have h n t → h t in L 2 (m ∞ ) and therefore (4.10) holds for general h t0 ∈ L 2 (m ∞ ). We recover as well the hypercontractivity of the time-dependent semigroup. Let h be a positive solution to the semigroup (4.7) satisfying h t0 ∈ L q0 (m ∞ ) ∩ L 1 (m ∞ ) for some q 0 ̸ = 1. Let ε ∈ (0, 1] and q(t) solve the ODEq = 4(1 − ε)ρ(q − 1) with the initial condition q(t 0 ) = q 0 . Then h t ∈ L q(t) (m ∞ ) for t ≥ t 0 . Moreover, we have for q 0 > 1, log ∥h t ∥ q(t) ≤ log ∥h t0 ∥ q0 + t t0 δ(s)ds,(4.12) and for q 0 < 1, log ∥h t ∥ q(t) ≥ log ∥h t0 ∥ q0 + t t0 δ(s)ds, (4.13) where δ(t) = 1 4ε (q(t) − 1)(M F mm ) 2 W 2 1 (m t , m ∞ ). Proof. First assume h t0 ∈ A + so that h t ∈ A + for all t ≥ t 0 and that h t ∈ A + uniformly on compact sets of [t 0 , +∞) thanks to Proposition 4.12. Define the function φ(t) = log ∥h t ∥ q(t) . In particular, if q(t) = 0, then φ(t) = log h t m ∞ . By the definition of the stable algebra A + we know φ(t) is well defined for t ≥ t 0 . Moreover, it follows from Fubini's theorem that t → φ(t) is absolutely continuous for t ≥ t 0 and its weak derivative readṡ φ(t) =q (t) q(t) 2 h q(t) t m ∞ h q(t) t log h q(t) t m ∞ − h q(t) t m ∞ log h q(t) t m ∞ + q(t) − 1 h q(t) t m ∞ − h q(t)−2 t |∇h t | 2 m ∞ + h q(t)−1 t ∇h t · (b t − b ∞ )m ∞ . We recognize the term on the first line as the entropy h q(t) t log h q(t) t m ∞ − h q(t) t m ∞ log h q(t) t m ∞ = Ent m∞ (h q(t) t ) which by LSI (2.3) has upper bound Ent m∞ (h q(t) t ) ≤ 1 ρ E m∞ [|∇h q(t)/2 | 2 ] ≤ q(t) 2 4ρ h q(t)−2 t |∇h t | 2 m ∞ . By Cauchy-Schwarz, the second term on the second line satisfies h q(t)−1 t ∇h t ·(b t −b ∞ )m ∞ ≤ ε h q(t)−2 t |∇h t | 2 m ∞ + 1 4ε h q(t) t m ∞ ∥b t −b ∞ ∥ 2 ∞ ≤ ε h q(t)−2 t |∇h t | 2 m ∞ + 1 4ε h q(t) t m ∞ (M F mm ) 2 W 2 1 (m t , m ∞ ). Therefore, for q 0 > 1 (so that q(t) > 1,q(t) > 0), we haveφ(t) ≤ δ(t) while for q 0 < 1 (so that q(t) < 1,q(t) < 0) we haveφ(t) ≥ δ(t). To deal with the case q(t) = 0 we use the continuity of t → φ(t). We have thus shown (4.12) and (4.13) for h t0 ∈ A + . Now consider general h t0 ∈ L q0 + (m ∞ ). In the case q 0 > 1, we use the density of A + (Proposition 4.11) to find a sequence (h n t0 ) n∈N in A + with h n t0 → h t0 in L q0 . To each h n t0 there exists a flow t → h n t in A + satisfying (4.12). For t ≥ t 0 , we also have h n t → h t in L q0 by the semigroup property in Corollary 4.14 so that along a subsequence h n t → h t a.e. By Fatou's lemma we obtain log h q(t) t m ∞ 1/q(t) ≤ lim inf n→∞ (h n t ) q(t) m ∞ 1/q(t) ≤ lim inf n→∞ log ∥h n t0 ∥ q0 + t t0 δ(s)ds = log ∥h t0 ∥ q0 + t t0 δ(s)ds. So (4.12) is proved for general h t0 ∈ L q0 . In the case q 0 < 1, we choose again by Proposition 4.11 a sequence (h n t0 ) n∈N in A + such that h n t0 → h t0 in L 1 and lim n→∞ ∥h n t0 ∥ q0 = ∥h t0 ∥ q0 . By the L 1 -continuity, h n t → h t in L 1 so that along a subsequence h n t → h t pointwise m ∞ -a.e. For q(t) > 0 we have by Fatou's lemma lim inf n→∞ (|h n t | + 1 − |h n t | q(t) )m ∞ ≥ (|h t | + 1 − |h t | q(t) )m ∞ . Thus lim sup n→∞ |h n t | q(t) m ∞ ≤ |h t | q(t) m ∞ . So taking lim sup on both sides of log ∥h n t ∥ q(t) ≥ log ∥h n t0 ∥ q0 + gives us (4.13). For q(t) < 0 we have directly by Fatou lim inf n→∞ (h n t ) q(t) m ∞ ≥ h q(t) t m ∞ so that log ∥h t ∥ q(t) ≥ lim sup n→∞ log ∥h n t ∥ q(t) ≥ lim sup n→∞ log ∥h n t0 ∥ q0 + t t0 δ(s)ds = log ∥h t0 ∥ q0 + t t0 δ(s)ds. To conclude we treat q(t) = 0 by a continuity argument. Take ε ′ ∈ (0, ε) and let q ′ be the solution toq ′ = 4(1 − ε ′ )ρ(q ′ − 1) with q ′ (t 0 ) = q(t 0 ) = q 0 < 1 and δ ′ (t) = 1 4ε ′ (q ′ (t) − 1)(M F mm ) 2 W 2 1 (m t , m ∞ ). We have q ′ (t) < q(t) = 0 so that by previous discussions log ∥h t ∥ q ′ (t) ≥ log ∥h t0 ∥ q0 + t t0 δ ′ (s)ds, whereas log ∥h t ∥ q(t) ≥ log ∥h t ∥ q ′ (t) by the monotonicity of p-norm. We take the limit ε ′ → ε to obtain (4.13). Remark 4.4. The computations are similar to that for the hypercontractivity of a diffusion process whose invariant measure m satisfies a defective LSI, i.e. for some c, δ ≥ 0, ∀f ∈ C 1 b (R d ), Ent m (f 2 ) ≤ c E m [|∇f | 2 ] + δ E m [|f | 2 ]. See [5,Chapter 5] and [3,Chapter 2] for the link between defective LSI and hypercontractivity. After showing the L 2 -convergence and the hypercontractivity, we are finally ready to give the proof of Theorem 2.2. Proof of Theorem 2.2. We will first use Proposition 4.16 to show that after a finite time h lies in L 2 (m ∞ ), then use Proposition 4.15 to let its L 2 (m ∞ )-norm diminish exponentially and finally apply Proposition 4.16 again to port this result to all L p . To this end, pick ε ∈ (0, 1) and defineq 1 (t) = 4(1 − ε)ρ(q 1 (t) − 1) with q 1 (0) = p 0 . Since p 0 > 1, q 1 is exponentially increasing. If p 0 ∈ (1, 2) we set t 1 = (4(1 − ε)ρ) −1 log 1 p0−1 and this definition yields q 1 (t 1 ) = 2. Otherwise p 0 ≥ 2 we simply set t 1 = 0. By the hypercontractivity (4.12) in Proposition 4.16, we have in both cases ∥h t1 ∥ 2 ≤ exp t1 0 δ 1 (s)ds ∥h 0 ∥ p0 , where δ 1 (s) = 1 4ε (q 1 (s) − 1)(M F mm ) 2 W 2 1 (m s , m ∞ ). We have δ 1 (s) ≤ C 1 e −4ερs with C 1 = C 1 (ρ, M F mm , p 0 , ∥h∥ p0 , ε) thanks to the bound on W 2 1 (m t , m ∞ ) similar to (4.11). Hence ∥h t1 ∥ 2 ≤ exp(C 1 /4ερ)∥h 0 ∥ p0 . By Proposition 4.15 we know that for all ρ ′ ∈ (0, ρ ′ ) and all t ∈ [t 1 , +∞), ∥h t ∥ 2 2 ≤ C 2 e −4ρ ′ (t−t1) + 1, where C 2 is a constant depending on both ρ ′ and these that C 1 depends on. Now we define τ p by τ p =      t 1 + 1 4(1−ε)ρ log((p − 1) ∨ 1) if p > 1, t 1 if p ∈ (0, 1), t 1 + 1 4(1−ε)ρ log(2(1 − p)) if p ≤ 0. In the case p > 1, for t ≥ τ p we set t 2 = t − (4(1 − ε)ρ) −1 log((p − 1) ∨ 1) ≥ t 1 and let q 2 solvesq 2 (t) = 4(1 − ε)ρ(q 2 (t) − 1) with q 2 (t 2 ) = 2. Our choice ensures q 2 (t) ≥ p. By the hypercontractivity (4.12) we have ∥h t ∥ q2(t) ≤ exp t t2 δ 2 (s)ds ∥h t2 ∥ 2 , where δ 2 satisfies δ 2 (s) ≤ C 3 e −4ρt2 e −4ερ(s−t2) for all s ≥ t 2 with a constant C 3 depending on ρ, M F mm , p 0 , ∥h∥ p0 and ε. The p-norm then satisfies log ∥h t ∥ p ≤ log ∥h t ∥ q2(t) ≤ log ∥h t2 ∥ 2 + C 3 4ερ e −4ρt2 ≤ 1 2 (∥h t2 ∥ 2 2 − 1) + C 3 4ερ e −4ρt2 ≤ C 2 e −4ρ ′ (t2−t1) + C 3 4ερ e −4ρt2 = C 2 e −4ρ ′ (t−τp) + C 3 4ερ e (1−ε) −1 log(p−1)∨1 e −4ρt . For p ∈ (0, 1), we observe Hölder's inequality h p m ∞ 1 2−p h 2 m ∞ 1−p 2−p ≥ hm ∞ = 1, so that for t ≥ τ p = t 1 we have log ∥h t ∥ p ≥ − 2(1−p) p log ∥h t ∥ 2 . Finally we treat p ≤ 0. Given t ≥ τ p , set t 3 = t − (4(1 − ε)ρ) −1 log(2(1 − p)) ≥ t 1 and let q 3 solvesq 3 (t) = 4(1 − ε)ρ(q 3 (t) − 1) with q 3 (t 3 ) = 1 2 . Our choice ensures q 3 (t) = p. Define δ 3 (s) = 1 4ε (q 3 (s) − 1)(M F mm ) 2 W 2 1 (m s , m ∞ ) . It satisfies the bound δ 3 (s) ≥ −C 4 e −4ρt3 e −4ερ(s−t3) for s ≥ t 3 for some C 4 having the same dependence as C 1 . We obtain by the reverse hypercontractivity (4.13) log ∥h t ∥ p ≥ log ∥h t3 ∥ 1 2 + t t3 δ 3 (s)ds ≥ −2 log ∥h t3 ∥ 2 − C 4 4ερ e −4ρt3 = − log(1 + ∥h t3 − 1∥ 2 2 ) − C 4 4ερ e −4ρt3 ≥ −C 2 e −4ρ ′ (t3−t1) − C 4 4ερ e −4ρt3 ≥ −C 2 e −4ρ ′ (t−τp) − C 3 4ερ e (1−ε) −1 log 2(1−p) e −4ρt . Remark 4.5. We here explain why it is necessary to assume m 0 ∈ L p0 (m ∞ ) in Theorem 2.2. Let m 0 (dx) ∝ exp(− d ν=1 |x ν |), i.e. the d-tensorized exponential distribution and F (m) = 1 2 |x| 2 m(dx). The Langevin dynamics (2.9) is nothing but Ornstein-Uhlenbeck: dX t = −X t dt + √ 2dW t . The SDE is solved explicitly by X t = e −t X 0 + √ 2 t 0 e −(t−s) dW s d = e −t X 0 + 1 − e −2t N , where N ∼ N (0, 1) is a standard normal independent from X 0 . The Langevin has unique invariant measure m ∞ ∝ exp(−|x| 2 /2), i.e. the standard normal distribution in R d . The initial condition m 0 lies in all P p for all p ≥ 1 but m 0 /m ∞ does not belong to L p0 for any p 0 > 1. And so is m t . Indeed, for all ε > 0, E[exp(ε|X t | 2 )] = E[exp(ε(e −t |X 0 | + 1 − e −2t N ) 2 )] ≥ E exp ε 2 (e −2t |X 0 | 2 − 2(1 − e −2t )N 2 ) = E exp ε 2 e −2t |X 0 | 2 E[exp(−ε(1 − e −2t )N 2 )] = +∞. Here we used (a + b) 2 ≥ 1 2 a 2 − b 2 and the independence between X 0 and N . This implies m t m −ε ∞ = +∞ for all ε > 0. Let p > 1. By Hölder's inequality we have To conclude the discussions about the mean field dynamics we show a lemma which uses L p -norms to bound "cross entropy"-like quantities and use it to obtain a uniform-in-time concentration of measure result. The lemma will also be used in the proof of Theorem 2.4. Lemma 4.17. Let µ, ν ∈ P(R d ) and h : R d → (0, +∞) be a measurable function. Then for all p > 0, − 1 p H(ν|µ) + log ∥h∥ L −p (µ) ≤ log hν ≤ 1 p H(ν|µ) + log ∥h∥ L p (µ) . (4.14) Proof. Let X be a measurable space, µ, ν be probabilities on X and U : X → R be a random variable. We have the convex duality inequality (see e.g. [7,Corollary 4.14]) E ν [U ] ≤ H(ν|µ) + log E µ [e U ].(4.15) The right hand side of the inequality is always well defined in (−∞, +∞]. Plug in U = p log h t . For p > 0 we obtain log hν ≤ 1 p H(ν|µ) + 1 p log e p log h µ = 1 p H(ν|µ) + log ∥h∥ L p (µ) , and for p < 0 we obtain log hν ≥ 1 p H(ν|µ) + log ∥h∥ L p (µ) .C ρ ′ = C ρ ′ (ρ, M F mm , p 0 , ∥h 0 ∥ p0 ), τ ρ ′ = τ ρ ′ (ρ, p 0 ) such that for every 1-Lipschitz function f : R d → R, every t ≥ τ ρ ′ and every r ≥ 0, m t [|f − E mt f | ≥ r] ≤ 2 exp −ρ ′ r 2 + C ρ ′ e −4ρ ′ t (r + 1) . (4.16) Proof. Let f : R d → R be 1-Lipschitz continuous and define for t ≥ 0 the moment-generating function ψ t,f (λ) = log E mt e λ(f −Em t f ) . The equality in (4.15) can be attained and therefore we have (see also [7,Corollary 4.14]) ψ t,f (λ) = sup µ≪mt λ(E µ f − E mt f ) − H(µ|m t ). For each µ ≪ m t , the first term satisfies E µ f − E mt f ≤ W 1 (µ, m t ) ≤ W 1 (µ, m ∞ ) + W 1 (m t , m ∞ ) ≤ 1 ρ H(µ|m ∞ ) + W 1 (m t , m ∞ ) by Talagrand's transport inequality (2.8) for m ∞ . The second term satisfies H(µ|m t ) = log µ m t µ = log µ m ∞ − log h t µ = H(µ|m ∞ ) − log h t µ ≥ H(µ|m ∞ ) − 1 p H(µ|m ∞ ) − log ∥h t ∥ p for p > 1 by the previous Lemma 4. 17. Hence for λ ≥ 0 the moment-generating function ψ t,f satisfies ψ t,f (λ) ≤ sup µ≪mt λ 1 ρ H(µ|m ∞ ) + λW 1 (m t , m ∞ ) − 1 − 1 p H(µ|m ∞ ) + log ∥h t ∥ p ≤ λ 2 4(1 − 1 p )ρ + λW 1 (m t , m ∞ ) + log ∥h t ∥ p . For r, λ ≥ 0 we have by Markov's inequality m t [f − E f ≥ r] ≤ e −λr E mt e λ(f −Em t f ) ≤ exp −λr + λ 2 4(1 − 1 p )ρ + λW 1 (m t , m ∞ ) + log ∥h t ∥ p . Take λ = 2(1 − 1 p )ρ. We obtain m t [f −E f ≥ r] ≤ exp − 1 − 1 p ρr 2 + 2 1 − 1 p ρW 1 (m t , m ∞ )r + log ∥h t ∥ p . The bound on m t [f − E f ≤ −r] is obtained by applying the previous inequality to −f . Given ρ ′ ∈ (0, ρ), find p > 1 such that (1 − 1 p )ρ = ρ ′ . The desired result follows from Theorems 2.1 and 2.2. Remark 4.6. Our proof is based on the standard transport method for concentration inequalities and we refer readers to [29,Chapter 6] and [7,Chapter 8] for an introduction to it. In fact, our method allows us to prove a more general perturbative result: if m satisfies a T 1 inequality, h ∈ L p + (m) for p > 1 and hm = 1, then hm also has Gaussian concentration (albeit with a weaker constant). Particle system Convergence in entropy Before giving the proof of Theorem 2.3 we first show two lemmas on entropies. Lemma 5.1 (Information inequalities). Let X 1 , . . . , X N be measurable spaces, µ be a probability on the product space X = X 1 × · · · × X N and ν = ν 1 ⊗ · · · ⊗ ν N be a σ-finite measure. Then N i=1 H(µ i |ν i ) ≤ H(µ|ν) ≤ N i=1 H(µ i|−i (·|x −i )|ν i )µ −i (dx −i ). (5.1) Here we set the rightmost term to +∞ if the conditional distribution µ i|−i does not exist µ −i -a.e. Proof. The inequality is non-trivial only if µ ≪ ν and in this case we denote the relative density by f = dµ/dν. For I ⊂ {1, . . . , N }, we define the conditional densities by f I|−I (x I |x −I ) =    f (x I , x −I ) f (x I , x −I )ν −I (dx −I ) if f (x I , x −I )ν −I (dx −I ) > 0, 0 otherwise. The conditional measures are defined via densities µ I|−I (dx I ) = f I|−I (x I |x −I )ν I (dx I ). In particular, we do not need the regularity of the underlying spaces X 1 , . . . , X N in order to apply disintegration theorems. Define I i = {1, . . . , i} for i = 1, . . . , N . The relative entropy admits the decomposition H(µ|ν) = N i=1 H µ i|Ii−1 (·|x Ii−1 ) ν i µ Ii−1 (dx Ii−1 ). We conclude by applying Jensen's inequality to the convex mappings λ i → H(λ i |ν i ). Lemma 5.2. Assume that F satisfies (2.1) and there exists a measure m ∞ ∈ P 2 (R d ) verifying (2.14). Then for all m N ∈ P 2 (R dN ) of finite entropy, we have H(m N |m ⊗N ∞ ) ≤ F N (m N ) − N F(m ∞ ). (5.2) Proof. Let X be a random variable distributed as m N . By the convexity of F we have F N (m N ) − N F(m ∞ ) = E[N F (µ X ) − N F (m ∞ )] + H(m N ) − N H(m ∞ ) ≥ E N δF δm (m ∞ , x)(µ X − m ∞ )(dx) + H(m N ) − N H(m ∞ ) = − E N log m ∞ (x)(µ X − m ∞ )(dx) + H(m N ) − N H(m ∞ ) = − E N log m ∞ (x)µ X (dx) + H(m N ) = − N i=1 log m ∞ (x i )m N (dx) + H(m N ) = H(m N |m ⊗N ∞ ). Proof of Theorem 2.3. Let t 0 ≥ 0 be such that m t0 has finite entropy and finite second moment. Since ∇ i N F (µ x ) = D m F (µ x , x i ) corresponds to the drift of (2.10), we recognize the particle system flow of measure m N t as a linear Langevin flow with the invariant measure m N ∞ , defined in (2.16). In particular, Proposition 4.7 applied to this dynamics yields dF(m N t ) dt = −I(m N t |m N ∞ ) (5.3) for t ≥ t 0 a.e. In the following we establish a lower bound of the relative Fisher information I t := I(m N t |m N ∞ ) in order to obtain the desired result. (x −i )dx −i = 1, m N,−i t (x −i ) < +∞ everywhere. We are therefore able to define the conditional probability density m N,i|−i t (x i |x −i ) = m N t (x) m N,−i t (x −i ) = m N t (x) m N t (x)dx i which has generalized derivative in x i and is strictly positive everywhere. Decomposing Fisher componentwise. Using the conditional distributions, we can decompose the relative Fisher information by I t = ∇ log m N t (x) m N ∞ (x) 2 m N t (dx) = E ∇ log m N t (X t ) m N ∞ (X t ) 2 = N i=1 E ∇ x i log m N,i|−i t (X i t |X −i t )m N,−i t (X −i t ) m N ∞ (X t ) 2 = N i=1 E ∇ x i log m N,i|−i t (X i t |X −i t ) m N ∞ (X t ) 2 = N i=1 E ∇ x i log m N,i|−i t (X i t |X −i t ) + D m F (µ Xt , X i t ) 2 . Change of empirical measure and componentwise LSI. We replace the empirical measure µ x in D m F by µ x −i . Define δ i 1 (x; y) = D m F (µ x , y) − D m F (µ x −i , y) . Take ε ∈ (0, 1). The Fisher information satisfies I t = N i=1 E ∇ x i log m N,i|−i t (X i t |X −i t ) + D m F (µ X −i t , X i t ) + δ i 1 (X t ; X i t ) 2 ≥ N i=1 E (1 − ε) ∇ x i log m N,i|−i t (X i t |X −i t ) + D m F (µ X −i t , X i t ) 2 − (ε −1 − 1)|δ i 1 (X t ; X i t )| 2 = (1 − ε) N i=1 E I m N,i|−i t (·|X −i t ) μ X −i t − (ε −1 − 1) N i=1 E[|δ i 1 (X t ; X i t )| 2 ], where we used the elementary inequality (2.6). Define the first error (a + b) 2 ≥ (1 − ε)|a| 2 − (ε −1 − 1)|b| 2 andμ x −i is the image of µ x −i under mapping m →m defined in∆ 1 := N i=1 E[|δ i 1 (X t ; X i t )| 2 ] := N i=1 E[|D m F (µ Xt , X i t )−D m F (µ X −i t , X i t )| 2 ]. (5.4) The previous inequality writes I t ≥ (1 − ε) N i=1 E I m N,i|−i t (·|X −i t )|μ X −i t − (ε −1 − 1)∆ 1 . (5.5) We apply the uniform log-Sobolev inequality forμ X i t and obtain 1 4ρ I m N,i|−i t (·|X −i t ) μ X −i t ≥ H m N,i|−i t (·|X −i t ) μ X −i t = log m N,i|−i t (x i |X −i t ) + δF δm (µ X −i t , x i ) m N,i|−i t (dx i |X −i t )+log Z(μ X −i t ). Then we apply Jensen's inequality to log Z(μ x −i ) to obtain log Z(μ X −i t ) ≥ − δF δm (µ X −i t , x i )m ∞ (dx i ) − m ∞ (x i ) log m ∞ (x i )dx i . Chaining the previous two inequalities and summing over i, we have 1 4ρ N i=1 I m N,i|−i t (·|X −i t ) μ X −i t ≥ N i=1 δF δm (µ X −i t , x i ) m N,i|−i t (dx i |X −i t ) − m ∞ (dx i ) + H m N,i|−i t (·|X −i t ) − H(m ∞ ) . (5.6) Another change of empirical measure. We wish to change back µ x −i → µ x in (5.6). Define δ i 2 (x; y) := δF δm (µ x −i , y) − δF δm (µ x , y) and the second error ∆ 2 := N i=1 δ i 2 (x; x i )m N t (dx) − N i=1 δ i 2 (x; x ′ )m ∞ (dx ′ )m N t (dx). (5.7) Then we obtain by taking expectations on both sides of (5.6) 1 4ρ N i=1 E I m N,i|−i t (·|X −i t ) μ X −i t ≥ N E δF δm (µ Xt , y)(µ Xt − m ∞ )(dy) + N i=1 E H m N,i|−i t (·|X −i t ) − N H(m ∞ ) + ∆ 2 . (5.8) Thanks to the convexity of F , the first term satisfies the tangent inequality N E δF δm (µ Xt , y)(µ Xt − m ∞ )(dy) ≥ N E F (µ Xt ) − F (m ∞ ) = F N (m N t ) − N F (m ∞ ). (5.9) For the second term we apply the information inequality (5.1) to obtain N i=1 E −i H m N,i|−i t (·|X −i t ) ≥ H(m N t ). Hence N i=1 E I m N,i|−i t (·|X −i t ) μ X −i t ≥ 4ρ(F N (m N t ) − N F (m ∞ ) + H(m N t ) − N H(m ∞ ) + ∆ 2 ). Using ( I t = I(m N t |m N ∞ ) ≥ 4ρ(1−ε) F N (m N t ) − N F(m ∞ ) + ∆ 2 −(ε −1 −1)∆ 1 . (5.10) Bounding the errors ∆ 1 , ∆ 2 . The transport plan between µ x and µ x −i π i = 1 N j̸ =i δ (x j ,x j ) + 1 N (N − 1) j̸ =i δ (x j ,x i ) (5.11) gives the bound W 1 (µ x , µ x −i ) ≤ 1 N (N −1) j̸ =i |x j − x i |. We use this transport plan to bound the errors ∆ 1 , ∆ 2 . Let us treat the first error ∆ 1 . Since m → D m F (m, x) is M F mm -Lipschitz continuous in W 1 metric, we have |δ i 1 (x; y)| ≤ M F mm W 1 (µ x , µ x −i ) ≤ M F mm N (N − 1) N j=1,j̸ =i |x j − x i |. Under the L 2 -optimal transport plan Law(( X i t ) N i=1 , (X i ∞ ) N i=1 ) ∈ Π(m N t , m ⊗N ∞ ) we have ∆ 1 = N i=1 E[|δ i 1 (X t ; X i t )| 2 ] ≤ (M F mm ) 2 N i=1 E[W 2 1 (µ Xt , µ X −i t )] ≤ (M F mm ) 2 N (N − 1) E 1≤i,j≤N i̸ =j |X j t − X i t | 2 ≤ 3(M F mm ) 2 N (N − 1) E 1≤i,j≤N i̸ =j |X i t −X i ∞ | 2 + |X i ∞ −X j ∞ | 2 + |X j t −X j ∞ | 2 ≤ 3(M F mm ) 2 N (N − 1) 2(N − 1) E N i=1 |X i t −X i ∞ | 2 + N (N − 1) E[|X 1 ∞ −X 2 ∞ | 2 ] . The first term E[ N i=1 |X i t −X i ∞ | 2 ] is the Wasserstein distance W 2 2 (m N t , m ⊗N ∞ ), while the second E[|X 1 ∞ −X 2 ∞ | 2 ] equals 2 Var m ∞ . Hence the first error satisfies the bound ∆ 1 ≤ 6(M F mm ) 2 1 N W 2 2 (m N t , m ⊗N ∞ ) + Var m ∞ . (5.12) Now treat the second error ∆ 2 . The Lipschitz constant of y → δ i 2 (x; y) = δF δm (µ x −i , y) − δF δm (µ x , y) is controlled by |∇ y δ i 2 (x; y)| = |D m F (µ x , y) − D m F (µ x −i , y)| ≤ M F mm W 1 (µ x , µ x −i ) ≤ M F mm N (N − 1) j̸ =i |x j − x i |. Hence |δ i 2 (x; y)−δ i 2 (x; y ′ )| ≤ M F mm N (N −1) j̸ =i |x j −x i ||y −y ′ |. Use Fubini's theorem to first integrate x ′ in the definition of the second error (5.7) and letX ′ ∞ be independent from X t . We obtain |∆ 2 | ≤ N i=1 |δ i 2 (x; x i ) − δ i 2 (x; x ′ )|m ∞ (dx ′ ) m N t (dx) ≤ N i=1 M F mm N (N − 1) N j̸ =i |x j − x i ||x ′ − x i |m ∞ (dx ′ )m N t (dx) = M F mm N (N − 1) N i,j=1 i̸ =j E[|X j t − X i t ||X i t −X ′ ∞ |] ≤ M F mm N (N − 1) 1 2 N i,j=1 i̸ =j E |X i t − X j t | 2 + N − 1 2 N i=1 E |X i t −X ′ ∞ | 2 . Using the same method we used for ∆ 1 , we control the first term by N i,j=1 i̸ =j E |X i t − X j t | 2 ≤ 6N (N − 1) 1 N W 2 2 (m N t , m ⊗N ∞ ) + Var m ∞ . For the second term we work again under the L 2 -optimal plan Law(( X i t ) N i=1 , (X i ∞ ) N i=1 ) ∈ Π(m N t , m ⊗N ∞ ) and letX ′ ∞ remain independent from the other variables. We have N i=1 E |X i t −X ′ ∞ | 2 ≤ 2 N i=1 E |X i t −X i ∞ | 2 + |X i ∞ −X ′ ∞ | 2 = 2N 1 N W 2 2 (m N t , m ⊗N ∞ ) + 2 Var m ∞ . As a result, |∆ 2 | ≤ 4M F mm 1 N W 2 2 (m N t , m ⊗N ∞ ) + 2 Var m ∞ . (5.13) Conclusion. Inserting the bounds on the errors (5.12) and (5.13) to the lower bound of Fisher information (5.10), we obtain I(m N t |m N ∞ ) ≥ 4ρ(1 − ε)(F N (m N t ) − N F(m ∞ )) − (16ρM F mm + 6(ε −1 − 1)(M F mm ) 2 ) 1 N W 2 2 (m N t , m ⊗N ∞ ) − (32ρM F mm + 6(ε −1 − 1)(M F mm ) 2 ) Var m ∞ . Thanks to the Poincaré inequality (2.7) for m ∞ =m ∞ , its variance satisfies 2ρ Var m∞ (x i ) ≤ E m∞ |∇x i | 2 = 1. So Var m ∞ = d i=1 Var m∞ (x i ) ≤ d/2ρ . Using the T 2 -transport inequality (2.8) for m ⊗N ∞ and the entropy sandwich Lemma 5.2 we bound the transport cost by W 2 2 (m N t , m ⊗N ∞ ) ≤ 1 ρ H(m N t |m ⊗N ∞ ) ≤ 1 ρ (F N (m N t ) − N F(m t )). In the end we obtain dF N (m N t ) dt = −I(m N t |m N ∞ ) ≤ − 4(1 − ε)ρ − M F mm N 16 + 6(ε −1 − 1) M F mm ρ F N (m N t ) − N F(m ∞ ) + dM F mm 16 + 3(ε −1 − 1) M F mm ρ . We conclude by applying Grönwall's lemma to the differential inequality. Remark 5.1. If the initial condition m N 0 of the particle system is a tensor product (m 0 ) ⊗N , one may expect the (non-uniform) convergence of the free energy 1 N F(m N t ) → F(m t ) for all t ≥ 0. If this is true, one can take the limit N → ∞ to recover the result of Theorem 2.1. However, while the convergence of the regular part 1 N F (m N t ) → F (m t ) can be expected from the finite-time Wasserstein convergence 1 N sup t∈[0,T ] W 1 (m N t , m ⊗N t ) → 0, the convergence of entropy H(m N t ) → H(m ⊗N t ) is more difficult to obtain. Remark 5.2. We used the convexity of F to achieve two things in the proof: (i) the existence of mean field invariant measure m ∞ ; and (ii) to derive (5.2) and (5.9). Under mild assumptions (i) can also be obtained by a Schauder-type fixed point theorem for the mapping m →m, or by finding stationary points of the mean field free energy F. For (ii), if F is only −κ-semi-convex around m ∞ , in the sense that F (m) − F (m ∞ ) ≥ δF δm (m ∞ , x)(m − m ∞ )(dx) − κ 2 2 W 2 2 (m, m ∞ ), we can expect our method to apply as long as κ is sufficiently small. Remark 5.3. The transport plan (5.11) between µ x , µ x −i is not optimal and this might leave room for improvement of N −1/d -order. has unique global solution defined for t ∈ [0, +∞). By construction the marginal law m t = Law(X t ) is in C([0, +∞); P 2 (R d )), proving the existence of solution. Any solution to the Fokker-Planck equation admits equally this probabilistic representation, then the uniqueness in short time follows from Cauchy-Lipschitz bounds. We extend this uniqueness to the infinity by sewing up the short time intervals, finishing the proof of the first claim. Let ρ t (x) be the density of Gaussian N (0, 2t). The solution m t satisfies Duhamel's formula in the sense of distributions m t = ρ t ⋆ m 0 + t 0 ρ t−s ⋆ ∇ · (m s D m F (m s , ·))ds = ρ t ⋆ m 0 + d i=1 t 0 ∇ i ρ t−s ⋆ (m s D m F i (m s , ·))ds. Note that ∥∇ρ t ∥ L p (R d ) ≤ C d,p t − 1 2 + d 2 ( 1 p −1) , which is integrable around 0+ when p < d d−1 . In this case apply Young's convolution inequality to obtain ∥m t ∥ L p (R d ) ≤ ∥ρ t ∥ L p (R d ) ∥m 0 ∥ TV + d i=1 t 0 ∥∇ i ρ t−s ∥ L p (R d ) ∥m s D m F i (m s , ·)∥ TV ds, where sup s∈[0,t] ∥m s D m F i (m s , ·)∥ TV ≤ sup s∈[0,t] C (1 + |x|)m s (dx) < +∞. Hence ∥m t ∥ L p (R d ) < +∞ for all t > 0. This and the second moment bound |x| 2 m t (dx) < +∞ are sufficient for the finiteness of entropy, i.e. the integral | log m t (x)|m t (x)dx is finite, which is our second claim. Indeed for the lower bound on entropy we use the decomposition in (4.2), while the upper bounds follows from m log m ≤ m p −m p−1 for all p > 1. The drift D m F (m t , x) has uniform linear growth in x: |D m F (m s , x)| ≤ M F mx |x| + sup s∈[t0,t] |D m F (m s , 0)|, where M F mx is the constant in (2.4) and the second term is finite by the compactness of set (m s ) s∈[t0,t] in P 2 . As a result, t t0 |D m F (m s , x)| 2 m s (dx)dt < +∞. We then apply [6, Theorem 7.4.1] to obtain the finiteness of (4.4). Especially, ∇m ∈ L 1 loc ((0, +∞); L 1 (R d )). Rewrite the Fokker-Planck equations as a conti- nuity equation ∂ t m + ∇ · (m t v t ) = 0 where v t (x) = −D m F (m t , x) − ∇ log m t (x). We have t t0 |v s (x)| 2 m s (dx)ds ≤ 2 t t0 |D m F (m s , x)| 2 m s (dx)ds + t t0 |∇m s (x)| 2 m s (x) dxds < +∞. Hence by [2,Theorem 8.3.1] the flow m t is locally AC 2 in (P 2 , W 2 ). The vector field v t (x) = −D m F (m t , x) − ∇ log m t (x) solves the continuity equation ∂ t m t + ∇ · (m t v t ) = 0 (A.1) in the sense of distributions and v t writes in the gradient form v t = −∇( δF δm (m t , x)+ log m t (x)) = −∇φ t . We finally verify v t is indeed a tangent vector of m t according to [2, Definition 8.4.1], i.e. v t ∈ Tan mt P 2 (R d ) = {∇φ : φ ∈ C ∞ c (R d )} L 2 (mt) . Let η R : R d → [0, 1] be a smooth function supported on B(2R), has the constant value 1 on B(R) and satisfies |∇η(x)| ≤ 2/R for all x. We have |∇φ t − ∇(φ t η R )| 2 m t ≤ 2 B(2R)\B(R) (|φ t | 2 |∇η R | 2 + |∇φ t | 2 |1 − η R | 2 )m t . The second term tends to 0 when R → ∞ while the first satisfies B(2R)\B(R) |φ t | 2 |∇η R | 2 m t ≤ 2 R 2 B(2R)\B(R) δF δm (m t , x) 2 + | log m t (x)| 2 m t ≤ 2C R 2 B(2R)\B(R) (1 + |x| 4 )m t (dx) + 2 R 2 B(2R)\B(R) | log m t | 2 m t ≤ 2C R 2 B(2R)\B(R) (1 + 4R 2 |x| 2 )m t (dx) + 2 R 2 B(2R)\B(R) | log m t | 2 m t . Here the first term tends to 0 since m t ∈ P 2 , while the second term tends to 0 by the integrability of | log m t | 2 m t , which follows from the elementary inequality m| log m| 2 ≤ C p m p 1 m≥1 + 2(|x| 2 m + sup t∈[0, 1] t(log t) 2 e −|x| )1 m<1 for p > 1 and x ∈ R d . Hence ∇(φ t η R ) → ∇φ t in L 2 (m t ). It then suffices to approximate the (essentially) compactly supported function φ t η R by C ∞ c functions in the L 2 (m t )-norm. We can do this by taking a sequence of compacted supported mollifiers ρ n and applying them to obtain ∇(φ t η R ) ⋆ ρ n → ∇(φ t η R ) in L 2 (m t ) when n → ∞. Proof of Proposition 4.11. Let h be a positive function. Define the functions k n = 1 B(n) (h ∧ n) ∨ 1/n and k n,m = ρ m ⋆ h n , where (ρ m ) m∈N is a sequence of C ∞ mollifiers. They satisfy ∀x ∈ R d , 1 n ≤ k n (x), k n,m (x) ≤ n and |∇ ℓ k n,m (x)| ≤ n∥∇ ℓ ρ m ∥ ∞ < +∞. In particular k n,m ∈ A + . We have k n → h in L p (µ) whenever h ∈ L p (µ) for p ≥ 1 and ∥k n ∥ q → ∥h∥ q whenever h ∈ L q (µ) for q ≤ 1 by the dominated convergence theorem. Since for all n ∈ N the function k n ∈ L 1 (R d ), we have k n,m → k n in L 1 (R d ) when m → ∞. Hence k n,m → k n a.e. when m → ∞ along a subsequence. Then we can apply again the dominated convergence to obtain k n,m → k n in L p (µ) for all p ≥ 1 and ∥k n,m ∥ q → ∥k n ∥ q for all q < 1. We can thus taking a subsequence of (n, m) → (+∞, +∞) so that k n,m → h in the desired ways. Proof of Proposition 4.12. Fix T > t 0 . We denote by C a positive constant that depends on max k=1,2,3 sup m,x |∇ k D m F (m, x)| and on the initial condition h ′ ∈ A + ; and by C Q a positive constant that depends additionally on the quantity Q. The constants C, C Q may change from line to line. Define g(t, x) = ∇ · (b t − b ∞ ) + (b t − b ∞ ) · b ∞ . It satisfies |g(t, x)| ≤ C(1 + |x|) for all (t, x) ∈ [t 0 , T ] × R d as ∥∇ k (b t − b ∞ )∥ ∞ ≤ C for k = 0, 1 and t ∈ [t 0 , T ]. Fix t ∈ [t 0 , T ]. Let (X t,x s ) s∈[0,t−t0] be the stochastic process solving dX t,x s = (2b ∞ − b t−s )ds + √ 2dW s (A.2) with X t,x 0 = x and define as well its extremal process M t,x s = sup 0≤u≤s |X u | for s ∈ [0, t − t 0 ]. Since the drift satisfies (2b ∞ − b t ) · x ≤ C T |x| 2 + C T for all (t, x) ∈ [t 0 , T ] × R d , we obtain the Gaussian moment bound E exp(C −1 T |M t,x t−t0 | 2 ) ≤ C T exp(C T |x| 2 ) by Itō's formula and Doob's maximal inequality. As a consequence the exponential moments are finite: ∀α ≥ 0, E exp(α|M t,x t−t0 |) ≤ C T,α exp(C T,α |x|). Set h(t 0 , ·) = h ′ . We construct the solution by the Feynman-Kac formula for (4.7) h(t, x) := E exp − t−t0 0 g(t − s, X t,x s )ds h(t 0 , X t,x t−t0 ) . It is standard that the h constructed above solves (4.7) in the sense of distributions. We verify h t ∈ A + for all t ∈ [t 0 , T ]. For the upper bound we apply the Cauchy-Schwarz inequality to obtain h(t, x) ≤ E exp −2 t−t0 0 g(t − s, X t,x s )ds 1/2 E[h(t 0 , X t,x t−t0 ) 2 ] 1/2 ≤ E[exp(C T (1 + |M t, x t−t0 |))] 1/2 E[exp(C T (1 + |X t,x t−t0 |))] 1/2 ≤ E[exp(C T (1 + |M t,x t−t0 |))] ≤ exp(C T (1 + |x|)). We applied the bound on g and h in the second inequality and used the exponential moment bound on M t−t0 in the last. For the lower bound we use Cauchy-Schwarz from the other direction: h(t, x) ≥ E exp t−t0 0 g(t − s, X t,x s )ds −1 E[h(t 0 , X t,x t−t0 ) 1/2 ] 2 ≥ C −1 T E[exp(C T |M t,x t−t0 |)] −1 E[exp(−C T |X t,x t−t0 |)] 2 ≥ C −1 T E[exp(C T |M t,x t−t0 |)] −1 E[exp(C T |X t,x t−t0 |)] −2 ≥ C −1 T E[exp(C T |M t,x t−t0 |)] −3 ≥ C −1 T exp(−C T |x|). Again we applied the bound on g and h on the second inequality and used the exponential moment bound on M t−t0 on the last line. So we have proved the bound of both sides | log h(t, x)| ≤ C T (1 + |x|), that is, the "zeroth-order" condition of A + . Now derive the continuity of x → h(t, x). Let the stochastic processes (X t,x · ) x∈R d be coupled by sharing the same Brownian motion in their defining SDEs (A.2). The mapping x → X t,x s is continuous almost surely as its matrix-valued partial derivative ∂X t,x · ∂x solves the SDE d ∂X t,x s ∂x = ∇(2b ∞ (X t,x s ) − b t−s (X t,x s )) ∂X t,x s ∂x ds whose wellposedness is guaranteed by the bound |∇ 2 (2b ∞ − b t−s )(x)| ≤ 3 sup m∈P2(R d ) sup x∈R |∇ 2 D m F (m, x)| ≤ C.|M t,x t−t0 | 2 )] ≤ C T exp(C T |x 0 | 2 ) for all x 0 ∈ R d . We obtain h(t, x) → h(t, x 0 ) when x → x 0 by applying the dominated convergence theorem to the Feynman-Kac formula. We sketch the part for verifying the conditions on derivatives. Differentiate the evolution equation (4.7). We obtain for k = 1, 2, ∂ t ∇ k h = ∆∇ k h + (2b ∞ − b t ) · ∇∇ k h + k i=2 k i ∇ i (2b ∞ − b t ) · ∇∇ k−i h + k i=1 k i ∇ i g(t, x)∇ k−i h + [∇(2b ∞ − b t ) · ∇∇ k−1 h + g(t, x)∇ k h]. We then write the Feynman-Kac formula for ∇ k h, k = 1, 2. The first two terms on the right hand side of the equation corresponds to the same stochastic process, to which the Gaussian moment bound applies. The third and fourth term are lower-order derivatives, continuous in space and bounded by |∇ k−i h(t, x)| ≤ exp(C T (1 + |x|)) by the induction hypothesis. The last term corresponds to the exponential in the Feynman-Kac formula, whose growth in x remains linear. So we can argue as before to derive |∇ k h(t, x)| ≤ exp(C T (1+|x|)) for all (t, x) ∈ [t 0 , T ] × R d . The continuity of x → ∇ k h(t, x) for k = 1, 2 follows analogously. Since x → h(t, x) are twice-differentiable the generalized derivative ∂ t h exists by the evolution equation (4.7). Finally all the constants in the bounds depend only additionally on T , so (h t ) t∈[t0,T ] ⊂ A + uniformly. B Truncated neural networks We study the expressiveness of truncated neural networks in this section. First recall that the activation function φ satisfies (3.4) and we have applied a finite truncation threshold L. We obtain the following crude bound on the best approximation error for truncated neural networks. Proof. Suppose first ∥f ∥ 1 = 1. By the property (3.4) we can find a 0 > 0, b 0 ∈ R such that φ(−a 0 + b 0 ) = 0 and φ(b 0 ) = 1. Hence the function φ 0 defined by φ 0 (x) = L 2 φ(a 0 x + b 0 ) − L 2 φ(a 0 (x + 1) + b 0 ) = ℓ(c + ) 2 φ(a 0 x + b) + ℓ(c − ) 2 φ(a 0 (x + 1) + b 0 ) is nonnegative and supported in [−1, 1], has the value L 2 at 0 and belongs to N φ,ℓ . Let L S d−1 = L be the Lebesgue measure on the sphere S d−1 . Its mass L(S d−1 ) is then equal to dc d , where c d is the volume of the d-dimensional unit ball. We define a function Φ ′ 0 : R d → R by Φ ′ 0 (x) = 1 dc d ω∈S d−1 φ 0 (ω · x)L(dω). The function Φ ′ 0 is then continuous, nonnegative, supported in B(0, 1) with Φ ′ 0 (0) = L 2 and belongs to N φ,ℓ . We rescale Φ ′ 0 by defining Φ 0 (x) = Φ ′ 0 (ax) for some a > 0 so that R d Φ 0 = 1. In this case Φ 0 is supported in B(C/L 1/d ) for some C = C(φ, d). Define the function Φ y (·) = Φ 0 (· − y) for y ∈ R d . By definition the functions Φ y , −Φ y belongs to N φ,ℓ . We let Φ(x) = f (x − y)Φ 0 (y)dy = f (y)Φ y (x)dy = |f (y)| sgn f (y)Φ y (x)dy. Since |f | = 1, we have Φ ∈ N φ,ℓ . The error bound then follows from |f (x) − Φ(x)| ≤ |f (x − y) − f (x)|Φ 0 (y)dy ≤ B(C/L 1/d ) |f (x − y) − f (x)|Φ 0 (y)dy ≤ C L 1/d ∥∇f ∥ ∞ . For general f we apply a homothety to define the new function g(x) = f (∥f ∥ 1/d 1 x) and apply the approximation result to g. Remark B.1. The function Φ 0 can be made arbitrarily close (in appropriate senses) to L 2 1 B((2/(Lc d )) 1/d ) . In this way the constant C in the bound (B.1) can be made independent from φ. The price we pay is that the minimum is not necessarily attained and we only have the bound inf Φ∈N φ,ℓ ∥f − Φ∥ ∞ ≤ 2 Lc d 1/d ∥∇f ∥ ∞ ∥f ∥ 1/d 1 . Gradient descent. Our dynamics is a special case of McKean-Vlasov with gradient-type drift: b(m, x) = −D m F (m, x) = −∇ δF δm (m, x). Fisher information by H(m) := H(m|L d ), I(m) := I(m|L d ). Theorem 3 . 1 ( 31Bochner). Let V : R d → R be a bounded, continuous and even function. The functional F : Remark 3.5 (Expressiveness of truncated networks). It is well known that twolayer neural networks are universal approximators, that is, they can approximate any continuous function on R d arbitrarily well with respect to the compact-open topology ([24, Theorem 2.4]). This implies that the infimum in (3 Figure 1 : 1Data samples {z k , y k } K k=1 (schematic). Figure 2 : 2Individual (shadowed) and 10-averaged (bold) losses versus time steps. Figure 3 : 3Average losses of last 500 steps for individual trainings (shadowed) and its 10-average (bold). Lemma 4. 1 . 1Let µ(dx) = e −Ψ dx be a probability in R d where Ψ is twice differentiable with the bound |∇ 2 Ψ| ≤ C. If µ satisfies an LSI, i.e. (2. Proposition 4. 2 . 2Assume F satisfies (2.2). If there exists a measure m 0 ∈ P 2 (R d ) such thatm 0 is well defined (i.e. Z(m 0 ) < +∞) and m 0 satisfies LSI (2.3), thenm are well defined (i.e. Z(m) < +∞) for all m ∈ P 2 (R d ). Proposition 4. 3 . 3Assume F satisfies (2.1) and (2.3). Then the measure m N ∞ in (2.16) is well defined and has finite exponential moments for all N ≥ 2. Proposition 4 . 4 . 44Assume F satisfies (2.1) to(2.4). Then the mean field free energy F, defined in (2.13), has unique minimizer m ∞ . The minimizer m ∞ is also the unique solution to the first-order equation (2.14) and an invariant measure to the MFL dynamics (2.11).Proof. Recall that F(m) = F (m) + H(m) where the absolute entropy H(m) is well defined for m ∈ P 2 and has value in (−∞, +∞] thanks to the decomposition F (m) − F (m 0 ) ≥ δF δm (m 0 , x)(m − m 0 )(dx) + H(m) = − logm 0 (x)(m − m 0 )(dx) + H(m) = H(m|m 0 ) + logm 0 (x)m 0 (dx) Definition 4. 5 . 5Let (X, d) be a complete metric space and x : [a, b] → X be a continuous mapping. We say x is absolutely continuous (a.c., x ∈ AC([a, b]; (X, d))) if there exists m ∈ L 1 ([a, b]) such that ∀a ≤ s < t ≤ b, Proposition 4. 6 ( 6Existence, uniqueness and regularity of MFL). Assume F satisfies (2.2) and (2.4). Then 1. for all m 0 ∈ P 2 (R d ) there exists a unique continuous flow m : [0, +∞) → P 2 (R d ) solving weakly the Fokker-Planck equation (2.11); Proposition 4 . 7 ( 47Energy decrease). Assume F satisfies (2.2) and(2.4). If m t0 is a measure of finite entropy and finite second moment for some t 0 ≥ 0, then the free energy F, defined in (2.13), is absolutely continuous along the flow (m t ) t≥t0 constructed in Proposition 4.6. Moreover it has derivative ( 4 . 5 ) 45Proof. We will apply the chain rule result of[2, Proposition 10.3.18] and we verify its conditions, namely, the differentiability of the free energy F = F + H and of the flow of measures m t . Firstly under the conditions (2.2) and (2.4) we can apply the argument of [13, Lemma A.2] to show that F : P 2 (R d ) → R is −λ-geodesically-convex for some λ > 0 and it has differential D m F (m t , ·) at m t . Secondly the entropy H : P 2 (R d ) → (−∞, +∞] is also 0-geodesically-convex by the result of [2, Proposition 9.3.9] and for t ≥ t 0 a.e. has subdifferential ∇ log m t at m t by[2, Theorem 10.4.6], thanks to the regularity bounds in the previous Proposition 4. Corollary 4. 8 ( 8Uniqueness of invariant measure). Under (2.1) to (2.4) there exists a unique invariant measure in P 2 (R d ) to the mean field dynamics (2.11). For notational simplicity define b t (x) := −D m F (m t , x), b ∞ (x) := −D m F (m ∞ , x) and h t (x) = m t (x)/m ∞ (x). The relative density h t solves Proposition 4.12 (Stability of A + under flow). Assume F satisfies (2.1) to(2.5). For every t 0 ≥ 0 and h ′ ∈ A + , there exists a solution h : [t 0 , +∞) → A + to (4.7) with initial value h(t 0 , ·) = h ′ . Moreover the temporal weak derivative ∂ t h exists and h t belongs to A + locally uniformly, i.e. (h t Proposition 4 . 413 (L p -norm growth). Assume F satisfies (2.1) to (2.5). Let p ̸ = 0 and h : [a, b] → A + be a solution to the evolution (4.7). Then the growth of p-norm t → h p t m ∞ is absolutely continuous and has derivative Corollary 4 . 414 (L p -continuity of flow). Under the hypotheses of Proposition 4.13, for every p ≥ 1 and every a ≤ s ≤ t ≤ b there exists a constant C s,t,p > 0 such that h p t m ∞ ≤ C s,t,p h p s m ∞ holds for every solutions to (4.7) in A + . Therefore the evolution equation (4.7) determines a strongly continuous (and positive) semigroup (P t s ) s≤t in L p + (m ∞ ) for p ∈ [1, +∞). Proposition 4 . 16 ( 416Hypercontractivity). Assume F satisfies (2.1) to (2.5). = +∞. Similarly, Nelson's theorem [3, Théorème 2.3.1] shows the optimality of the exponent's growth in Proposition 4.16. Theorem 4 . 18 ( 418Uniform-in-time concentration of measure). Under the hypotheses of Theorem 2.2, for all ρ ′ ∈ (0, ρ) there exist constants Regularity of conditional distribution. By the elliptic positivity (see e.g. [6, Theorem 8.2.1]), we know that for all t > t 0 andx ∈ R dN , m N t (x) > 0 with explicit lower bound. Let i ∈ {1, . . . , N }. Define marginal density m N,−i t (x −i ) = m N t (x)dx i .It is strictly positive everywhere by the positivity of m N t and is lower semicontinuous (in x −i ) thanks to the continuity of x → m N t (x) and Fatou's lemma. Since Fubini gives m N,−i t 5.5) and recalling the definition of free energies F(m) = F (m) + H(m), F N (m N ) = F N (m N ) + H(m N ), we obtain Proposition B. 1 . 1If there exist c + , c − ∈ R such that ℓ(c + ) = L and ℓ(c − ) = −L, then there exists some constant C = C(φ, d) such that for all f ∈ C 1 c (R d ; R), there exist a function Φ ∈ N φ,ℓ satisfying ∥f − Φ∥ ∞ ≤ C L 1/d ∥∇f ∥ ∞ ∥f ∥ Table 1 : 1Hyperparameters of neural network training.tem satisfies the exponential convergence bound (2.19). We now present our numerical experiments. The norm of∀s ∈ [0, t − t 0 ], ∀x ∈ R d , ∂X t,x s ∂x ≤ C T a.s.by Grönwall's lemma. Therefore we haveE[exp(C −1∂X t,x s ∂x satisfies T sup x:|x−x0|≤1 t ; z k )dt − λΘ i t dt + σdW i t ,(3.13) for i = 1, . . . , N . The first drift term of the diffusion is the gradient ∇ θ i F N (Θ 1 t , . . . , Θ N t ), so the time-discretization of this diffusion is nothing but the noisy gradient descent (NGD) algorithm for training neural networks. We refer readers to[47,49,30,48,35] for its applications. The second drift term −λΘ i t , coming from our quadratic regularization, is called weight decay in the field of machine learning. It is believed to lead to better generalizations of the trained neural network (see[27,31]). t t0 δ(s)ds Acknowledgements. We would like to thank an anonymous reviewer whose comments motivated us to improve the quality of this paper.Funding. The research of Zhenjie Ren was supported by the FIME Research Initiative.Propagation of chaosProof of Theorem 2.4. The triangle inequality for the L 2 -Wasserstein distance gives us W2 2). By Talagrand's inequality (2.8) for m ⊗N ∞ we bound the Wasserstein distances bywhere we applied Lemmas 4.9 and 5.2. We then apply Theorems 2.1 and 2.3 to obtain (2.20). Now suppose additionally(2.5)andwhere m N,i t is the i-th marginal of m N t . 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[ "Emission line star catalogues post-Gaia DR3 A validation of Gaia DR3 data using LAMOST OBA emission catalogue", "Emission line star catalogues post-Gaia DR3 A validation of Gaia DR3 data using LAMOST OBA emission catalogue", "Emission line star catalogues post-Gaia DR3 A validation of Gaia DR3 data using LAMOST OBA emission catalogue", "Emission line star catalogues post-Gaia DR3 A validation of Gaia DR3 data using LAMOST OBA emission catalogue" ]
[ "B Shridharan \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "Blesson Mathew \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "Suman Bhattacharyya \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "T Robin \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "R Arun \nIndian Institute of Astrophysics\nKoramangalaBangaloreIndia\n", "Sreeja S Kartha \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "P Manoj \nTata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia\n", "S Nidhi \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "G Maheshwar \nIndian Institute of Astrophysics\nKoramangalaBangaloreIndia\n", "K T Paul \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "Mayank Narang \nTata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia\n", "T Himanshu \nTata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia\n", "B Shridharan \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "Blesson Mathew \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "Suman Bhattacharyya \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "T Robin \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "R Arun \nIndian Institute of Astrophysics\nKoramangalaBangaloreIndia\n", "Sreeja S Kartha \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "P Manoj \nTata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia\n", "S Nidhi \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "G Maheshwar \nIndian Institute of Astrophysics\nKoramangalaBangaloreIndia\n", "K T Paul \nDepartment of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia\n", "Mayank Narang \nTata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia\n", "T Himanshu \nTata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia\n" ]
[ "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Indian Institute of Astrophysics\nKoramangalaBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Tata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Indian Institute of Astrophysics\nKoramangalaBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Tata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia", "Tata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Indian Institute of Astrophysics\nKoramangalaBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Tata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Indian Institute of Astrophysics\nKoramangalaBangaloreIndia", "Department of Physics and Electronics\nCHRIST (Deemed to be University)\nHosur Main RoadBangaloreIndia", "Tata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia", "Tata Institute of Fundamental Research\nHomi Bhabha RoadMumbaiIndia" ]
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Aims. Gaia DR3 and further releases have the potential to identify and categorise new emission-line stars in the Galaxy. We perform a comprehensive validation of astrophysical parameters from Gaia DR3 with the spectroscopically estimated emission-line star parameters from LAMOST OBA emission catalogue. Methods. We compare different astrophysical parameters provided by Gaia DR3 with those estimated using LAMOST spectra. By using a larger sample of emission-line stars, we perform a global polynomial and piece-wise linear fit to update the empirical relation to convert Gaia DR3 pseudo-equivalent width to observed equivalent width, after removing the weak emitters from the analysis. Results. We find that the emission-line source classifications given by DR3 is in reasonable agreement with the classification from LAMOST OBA emission catalogue. The astrophysical parameters estimated by esphs module from Gaia DR3 provides a better estimate when compared to gspphot and gspspec. A second degree polynomial relation is provided along with piece-wise linear fit parameters for the equivalent width conversion. We notice that the LAMOST stars with weak Hα emission are not identified to be in emission from BP/RP spectra. This suggests that emission-line sources identified by Gaia DR3 is incomplete. In addition, Gaia DR3 provides valuable information about the binary and variable nature of a sample of emission-line stars.
10.1051/0004-6361/202244353
[ "https://export.arxiv.org/pdf/2209.13221v1.pdf" ]
252,544,870
2209.13221
c32c618f9ebccb096a7abf954338e5ddd6590913
Emission line star catalogues post-Gaia DR3 A validation of Gaia DR3 data using LAMOST OBA emission catalogue September 28, 2022 B Shridharan Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia Blesson Mathew Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia Suman Bhattacharyya Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia T Robin Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia R Arun Indian Institute of Astrophysics KoramangalaBangaloreIndia Sreeja S Kartha Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia P Manoj Tata Institute of Fundamental Research Homi Bhabha RoadMumbaiIndia S Nidhi Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia G Maheshwar Indian Institute of Astrophysics KoramangalaBangaloreIndia K T Paul Department of Physics and Electronics CHRIST (Deemed to be University) Hosur Main RoadBangaloreIndia Mayank Narang Tata Institute of Fundamental Research Homi Bhabha RoadMumbaiIndia T Himanshu Tata Institute of Fundamental Research Homi Bhabha RoadMumbaiIndia Emission line star catalogues post-Gaia DR3 A validation of Gaia DR3 data using LAMOST OBA emission catalogue September 28, 2022Received xxxx; accepted xxxxAstronomy & Astrophysics manuscript no. aandacatalogues -stars: emission-line, Be -stars: variables: Herbig Ae/Be -methods: data analysis -techniques: spectro- scopic Aims. Gaia DR3 and further releases have the potential to identify and categorise new emission-line stars in the Galaxy. We perform a comprehensive validation of astrophysical parameters from Gaia DR3 with the spectroscopically estimated emission-line star parameters from LAMOST OBA emission catalogue. Methods. We compare different astrophysical parameters provided by Gaia DR3 with those estimated using LAMOST spectra. By using a larger sample of emission-line stars, we perform a global polynomial and piece-wise linear fit to update the empirical relation to convert Gaia DR3 pseudo-equivalent width to observed equivalent width, after removing the weak emitters from the analysis. Results. We find that the emission-line source classifications given by DR3 is in reasonable agreement with the classification from LAMOST OBA emission catalogue. The astrophysical parameters estimated by esphs module from Gaia DR3 provides a better estimate when compared to gspphot and gspspec. A second degree polynomial relation is provided along with piece-wise linear fit parameters for the equivalent width conversion. We notice that the LAMOST stars with weak Hα emission are not identified to be in emission from BP/RP spectra. This suggests that emission-line sources identified by Gaia DR3 is incomplete. In addition, Gaia DR3 provides valuable information about the binary and variable nature of a sample of emission-line stars. Introduction Emission-line stars (ELS) are a class of objects with emission lines, particularly Hα, at 6563 Å in the spectrum. They also exhibit physical processes such as stellar winds, jets or outflows, and/or mass accretion through the circumstellar disc. The hot ELS are classified mainly into main-sequence Classical Ae/Be (CAe/CBe; Rivinius et al. 2013) and pre main-sequence (PMS) Herbig Ae/Be (HAeBe; Waters & Waelkens 1998) based on its evolutionary stage. Many large sky surveys such as 2MASS (Cutri et al. 2003), WISE (Cutri et al. 2012), IPHAS (Drew et al. 2005), etc., have improved the ELS research by providing precise photometric measurements which are used to classify the ELS into various categories (Koenig & Leisawitz 2014;Witham et al. 2008). The Gaia Data Release 3 (Gaia DR3) catalogue represents a substantial advance in Galactic stellar astronomy. Gaia DR3 (Gaia Collaboration et al. 2021) builds on previous releases by improving the quality of previously released data and introducing entirely new data products, such as mean dispersed BP/RP spectra from spectro-photometry and radial velocity spectra (RVS), in addition to their integrated photometry in G BP , G RP , and the white light G-band published in Gaia EDR3 (De Angeli et al. 2022). Gaia BP/RP and/or RVS spectra is now available for sources with G < 19 mag, and astrophysical parameters for sources with G < 17.6 mag. e-mail: [email protected] The previous Gaia releases played a pivotal role in identifying and studying new populations of ELS in the Galaxy. Some notable examples are the selection of 11,000 high confidence PMS from Sco OB2 association (Damiani et al. 2019), and understanding the dynamics of young stellar objects (YSOs) in the Vela OB association (Cantat-Gaudin et al. 2019). The Spitzer/IRAC Candidate YSO (SPICY) catalogue was compiled from the YSO candidates identified using the high-quality astrometric data from Gaia EDR3 along the Galactic midplane (Kuhn et al. 2022). More homogeneous studies on the stellar parameters of YSOs were carried out by Arun et al. (2019) and Wichittanakom et al. (2020) using Gaia DR2, andGuzmán-Díaz et al. (2021) and Vioque et al. (2022) using Gaia EDR3. Even though Gaia has extensively improved stellar parameters of the previously known ELS in the Milky Way, the unavailability of Hα emission measurements for the Gaia sources hindered the classification of more ELS. The Large sky Area Multi-Object fibre Spectroscopic Telescope (LAMOST) has observed and catalogued 10,431,197 spectra of astronomical sources in their latest DR7 data release. Due to the availability of such a large database of spectra, the number of newly identified ELS has improved. Hou et al. (2016) identified 10,436 early-type ELS using LAMOST DR2 and studied various Hα profiles. Shridharan et al. (2021, hereafter CAe, and 56 HAeBe stars based on the analysis of optical/IR magnitudes and colours. This makes it one of the largest homogeneous ELS catalogue with a thorough classification using spectroscopy and available photometry. More recently, Zhang et al. (2022) identified 25,886 early-type ELS from LAMOST DR7. Even though the number of ELS objects increased with such large spectroscopic surveys, they cannot be classified accurately unless astrometric and photometric data are available. Hence, the field of ELS improves when the large spectroscopic surveys and all-sky astrometric surveys progress in tandem. This is achieved by the recently released Gaia DR3 data which provides astrometric, photometric, and spectroscopic parameters for more than 200 million objects. There is no doubt that the DR4 and further releases will greatly improve the ELS research. As the first step in this direction, we compare the new dataset released by Gaia DR3 with a previously existing, wellcharacterised spectroscopic catalogue. In this work, we aim to provide an external validation for the astrophysical parameters and to improve our ELS catalogue with newly available data from Gaia DR3. Data analysis and Results We use the 3339 ELS from LEMC and queried various DR3 tables using the source identifier from EDR3. The query was made using ADQL facility in the Gaia archive 1 . We explore the different datasets that Gaia provides with its new release. Classification and astrophysical parameters from Gaia DR3 The gaiadr3.astrophysical_parameters table provides plenty of information using the BP/RP spectra, the details of which can be found in Fouesneau et al. (2022, hereafter APSIS-II). The comparison between the sub-classification of ELS reported in LEMC with the classification done using the 'Extended Stellar Parametrizer for Emission-Line stars (ESP-ELS)' module of Gaia DR3 (mentioned as classlabel_espels), for a sample of 506 stars, is shown as a heatmap in the top panel of Figure 1. The bottom panel of Figure 1 shows the heatmap of the spectral type comparison between 3109 ELS from LEMC with those estimated from the 'Extended Stellar Parametrizer for Hot Stars (ESP-HS)' module in Gaia DR3 (denoted as spectraltype_esphs). From the figure (Figure 1; top panel), we see that the classification provided by LEMC and classlabel_espels DR3 matches well. Of 315 CBe stars with Gaia DR3 estimates, 303 (96 %) stars are classified as 'BeStar', 11 (4 %) stars as 'HerbigStar'and 1 star as 'wN 'by Gaia DR3. The quality of classlabel_espels classification is given by the classlabel_espels_flag, where classlabel_espels_flag <= 2 denotes a probability larger than 50%. Interestingly, the 11 stars which are classified as Herbig stars are having quality flag classlabel_espels_flag >= 4. For the 303 stars classified as 'BeStar', 155 stars have classlabel_espels_flag <= 2 and 148 have classlabel_espels_flag > 2. The sub-sample of 89 stars with unclear classifications in LEMC (Be**, Be*, Em*) 2 can now be classified as 'Be Star'(83) and 'Herbig Star'(6). 1 https://gea.esac.esa.int/archive/ 2 Be** = LEMC B-type star but no detection in Gaia EDR3 Be* = LEMC B-type star with Gaia EDR3 detection but not in 2MASS Em* = Hα emission object for which spectral type could not be calcu- The bottom panel of Figure 1 shows the comparison between the spectral type given in LEMC and those estimated by Gaia DR3, spectraltype_esphs. It can be seen that stars with spectraltype_esphs='B', the spectral type estimated are reasonably matching with LEMC spectral types ranging from O (<1%), B0-B5 (25%), B5-B9 (57%) to A0-A5 (17%). However, the problem with spectraltype_esphs can be seen clearly when we consider the stars with LEMC spectral type B8 (767 stars). Of the 767 stars, 255 (33%) stars are classified by Gaia DR3 to be spectraltype_esphs=F/G/K. This is a very significant deviation from the accurate spectral type given in LEMC, which was performed through a semi-automated template matching technique. The deviation of 33% towards later spectral types should be kept in mind before using the spectraltype_esphs in future studies. A possible explanation for the observed deviation can be the line-of-sight extinction. The 33% of the B8 stars misclassified by Gaia DR3 as -Gaia DR3 O8 O9 B0 B1 B2 B3 B5 B6 B7 B8 B9 A0 A1 A2 A3 A4 A5 A6 A7 A9 F0 F1 F2 F3 F4 F5 Spectral type from LAMOST OBA catalog T eff (K) Teff teff_gspphot teff_esphs teff_gspspec O8 O9 B0 B1 B2 B3 B5 B6 B7 B8 B9 A0 A1 A2 A3 A4 A5 A6 A7 A9 F0 F1 F2 F3 F4 F5 Spectral type from LAMOST OBA catalog , whereas the extinction value for the 59% of B8 stars classified to be B spectral type, is within 0-1 mag. Thus, higher the observed extinction value, higher the chances of Gaia DR3 spectral type estimation being different from the spectral type in LEMC. Gaia DR3 provides several astrophysical parameters such as T e f f , logg, Vsini, mass, radius and luminosity based on the BP/RP spectrum. For hot stars and ELS, they have used special modules to estimate these parameters. We compare all the different T e f f estimates with our spectral type to identify the best value for hot ELS. It should be noted that spectral type estimates from LEMC, although performed meticulously, have errors of about ±2 subtypes. Figure 2 shows the distribution of various T e f f and logg estimates of ELS available from Gaia DR3 with spectral type estimated in LEMC. It is very evident from Figure 2 (top) that for B-type stars, T e f f is significantly underestimated using RVS spectra (teff_gspspec). Two different modules were used to estimate T e f f using BP/RP spectra i.e., teff_gspphot and teff_esphs. Figure 2 (top) reveals that the teff_esphs value matches better when compared to T e f f from Pecaut & Mamajek (2013) calibration table and also, it has significantly lower inter-quartile range (IQR) when compared to teff_gspphot. We notice a large number of outliers in the teff_gspphot boxplot for each spectral type, which questions its validity. Hence it is clear from our analysis that teff_esphs provides an better T e f f estimate for B-type stars. In addition, there are other T e f f estimates available from modules such as teff_gspphot_marcs, teff_gspphot_ob and teff_gspphot_a in the gaiadr3.astrophysical_parameters_supp table. An appropriate model selection can be done based on the object of interest. Similarly, Figure 2 (bottom) shows the distribution of logg values for a subsample of LEMC stars. The logg estimate from RVS spectra (logg_gspspec) shows a large scatter when compared to the logg estiamtes from BP/RP spectra i.e., logg_gspphot and logg_esphs, which are distributed in the range 3-4 dex. Since the LEMC sample contains mainly CBe and HAeBe stars, it is fair to expect logg to be within 3-5. Hence we conclude that, when compared to other modules used in Gaia DR3, the ESP-HS module provides accurate astrophysical parameters and can be used for the analysis of OBA stars. According to Frémat et al. (2022), the Vsini estimations from the vbroad module degrades noticeably at T e f f > 7500 K and G RVS > 10. Therefore, the Vsini would be highly inaccurate for our sample of hot ELS stars. Consequently, we did not include Vsini analysis in the present study. Comparison of Gaia DR3 pEW with EW from LAMOST spectra Gaia DR3 made available pseudo-equivalent width (pEW) measurements of Hα for about 235 million sources, which are given in Gaia DR3 astrophysical_parameters table as ew_espels_halpha parameter. The classification and the ELS catalogue provided by Gaia DR3 are dependent on this pEW calculation. However, due to the low resolution of BP/RP spectra, using pEW solely may not provide a complete list of ELS which can be identified from Gaia DR3. Hence it is important to calibrate pEW values with actual EW measurements carefully. APSIS-II provides an empirical relation between pEW and the EW values available from various ELS catalogues in the literature ( Figure 21 and Table 3 of APSIS-II). They estimated the slope of the linear fit to be in the range of 2.26 and 2.83, which can be used to convert pEW to actual Hα EW. We improve upon this analysis by performing a second degree polynomial fit to a large sample of 1088 CBe stars from LEMC. Even though we have a bigger sample of 3339 ELS, we do not attempt to make a fit with other classes to avoid problems like emission inside the absorption core (CAe stars; Anusha et al. 2021), the low number statistics (HAeBe) and the contamination from [NII] forbidden lines. We use the sample of 1088 CBe stars from LEMC for which the EW were measured homogeneously using IRAF (Anusha et al., in prep). Stars showing Hα emission peak inside the absorption core are shown (light blue diamonds) in Figure 3 and were not used in the analysis. We emphasise here that, Gaia DR3 identifies the Hα to be in emission only if the emission peak is above the local continuum. Thus, for B-type stars, Gaia DR3 can identify sources as ELS only if the observed EW is greater than 0.5 nm. For A-type stars, the threshold value will only increase, since the Hα absorption peaks at A0 spectral type (Gray & Corbally 2009). Hence the catalogue of ELS provided by Gaia DR3 may not be complete with weak emitters, specifically those with emission peak inside the absorption core. This is a known caveat owing to the very low resolution of BP/RP spectra (Martayan et al. 2008). The second degree polynomial relation is shown in Equation 1. EW LEMC (nm) = −0.54 + 1.60 × pEW Gaia − 0.49 ×pEW 2 Gaia (nm)(1) Since we have larger sample of CBe stars when compared to Silaj et al. (2010) and Raddi et al. (2015), we also perform piecewise linear fit in intervals of 0.5 nm. The slope and intercept of the linear fit along with IQR (EW LEMC ) and median absolute deviation (MAD) along EW LEMC axis as a representative of the scatter is given for each interval range. A global polynomial fit and a piece-wise fit for different intervals of pEW values are shown in Figure 3. As seen from piece-wise linear fit values, the slope gets steeper as we move towards intense emitters. We suggest using the respective slopes and intercepts for calculating observed EWs from each pEW range for hot ELS ( Figure 3). However, the users should be aware that the LAMOST and Gaia have obtained the spectra at different epochs; the scatter and deviation of some points can be attributed to intrinsic variability of some CBe stars that range in the orders of days to years (Mathew & Subramaniam 2011;Cochetti et al. 2021). The addition of pEW measurement in DR3 will improve the sample of ELS and can serve as target list for future Hα ELS surveys. Synthetic photometry from BP/RP spectra For 686 stars in the LEMC, we could not estimate the spectral type due to the low SNR in the bluer region of the LAMOST spectra. Due to the observation strategy of LAMOST DR5, the majority of our sample is towards the galactic anti-center direction (Figure 1 of Shridharan et al. 2021). Limited photometric survey footprints towards this region restricted us from studying these stars photometrically or using SEDs to estimate their stellar parameters. For our sample of 3339 ELS, 2872 stars have continuous BP/RP spectra. The gaiaxpy package enables the user to calculate the synthetic magnitudes based on the continuous BP/RP spectra from DR3. We used the gaiaxpy package to generate Johnson, SDSS, PanSTARRS, and IPHAS photometric magnitudes for our sample of 2872 stars. To show the improvement of creating SED using BP/RP spectra, we present a representative SED which compares the data before and after incorporating Gaia DR3 data, in Figure 4. The SED is fitted with a Python routine used in Arun et al. (2021) and Bhattacharyya et al. (2022). We suggest that for sources with bad quality photometric measurements, synthetic photometry from BP/RP spectra can be used to improve the SED studies. Non-single stars and variable stars One of the major improvements in Gaia DR3 is the classification of 813,687 stars as non-single stars with orbital binary solutions for 356,132 stars. Massive stars are known to have a binary companion or clustering around it (Chini et al. 2013). Hence, we use non-single stars catalogue to find the binary stars in our sample. Among our sample of ELS, only 10 have solutions in gaiadr3.nss_two_body_orbit which gives the parameters for spectroscopic and eclipsing binaries. They also provide mass_ratio, eccentricity, inclination, teff_ratio, which can be used to characterise binaries. The LAMOST ID along with the parameters from gaiadr3.nss_two_body_orbit is shown in Table 1. Gaia DR3 classified a sample of its sources into different variable categories based on multi-epoch photometry. From LEMC, 363 stars are classified as variable stars in Gaia DR3. Epoch photometry of variable stars with good quality classification (best_class_score > 0.6) are shown in Figure 5. The cause of variability can also be related to the evolving nature of Hα emission region and hence, a detailed analysis of these stars will be taken up in a future work. Summary The newly released Gaia DR3 data will accelerate the field of astronomy as it provides astrophysical parameters for 470,759,263 sources using the mean BP/RP spectra. In that, 2,382,015 sources are classified as hot stars which can increase the number of known CBe and HAeBe stars. As a first step towards achieving this, we compared the astrophysical parameters provided by DR3 with carefully classified OBA-type ELS identified from LAMOST DR5. We see that the ELS classification provided by Gaia DR3 as classlabel_elseps matches reasonably well with our LEMC catalogue. Gaia DR3 also provides new classification and spectral type estimate for stars classified as 'Em'and 'Em[e]'in LEMC catalogue. The mismatch between the spectral types provided by Gaia DR3 (spectraltype_esphs) and LEMC was evident on comparison. The spectraltype_esphs estimates should be used with caution along with quality flag provided. Gaia DR3 also provides T e f f from 3 different modules using both BP/RP spectra and RVS spectra. Based on our comparison of T e f f values with spectral types from LEMC catalogue, we see that teff_esphs values matches well with the theoretical values. The teff_gspspec values are severely underestimated for early B-type stars. Similarly, teff_gspphot estimate may not be reliable because of the scatter and high number of outliers. We conclude that, teff_esphs should be used as the T e f f estimate for early-type ELS. We used the sample of 1088 CBe stars from LEMC to perform a global polynomial fit and piece-wise fit analysis to obtain a relation to convert pEW to the actual Hα EW. In cases where one needs a more accurate estimate of actual Hα EW for a specific range of pEW, the piece-wise slope and intercept values can be used. It should be noted that the weak emitters (with emission peak inside the absorption core) in LEMC have positive pEW values in Gaia DR3. This directly implies the incompleteness of ELS catalogue provided by Gaia DR3. We also checked for non-single stars and variable stars present in LEMC catalogue. Among our sample, 10 non-single stars with 7 of them classified as spectroscopic binaries for which various parameters are provided. From LEMC, 363 stars are classified as variables. These Hα emitting binaries and variable ELS will be studied in a future work. To summarise, this work provides an account of how the data provided by the recent Gaia DR3 can improve the study of ELS. Along with photometry and astrometric measurements, the availability of BP/RP spectra for a large number of sources will increase the number of already known ELS. The astrophysical parameters estimated from the BP/RP and RVS spectra will help to study a large number of ELS with ease. Fig. 1 . 1(Top) A heatmap representation of comparison between LEMC classification and spectraltype_esphs provided by Gaia DR3. (Bottom) A comparison between spectral type from LEMC and spectraltype_esphs provided by Gaia DR3. The number statistics for each category is provided inside each cell and a color bar is given for the reference. Fig. 2 . 2(Top) The distribution between different T e f f values provided by Gaia DR3 and the spectral type estimated in LEMC is shown as a boxplot. The red dashed line represent the T e f f vs spectral type calibration relation fromPecaut & Mamajek (2013). (Bottom) The distribution between logg values of stars from Gaia DR3 and the spectral type estimated in LEMC is shown as a boxplot. Fig. 3 . 3The figure shows a scatter plot between pEW provided by Gaia DR3 and EW estimated for LEMC CBe stars. The red crosses shows the outliers based on a normal distribution analysis. The grey filled circles show the stars with Hα peak above the continuum and light blue diamonds with Hα peak below the continuum.A global second degree polynomial fit is shown in black dashed lines along with the equation at the top left of the plot . Further, the piecewise fit within each interval is shown along with the fit parameters in corresponding colours. Negative EW values denote lines in emission. F/G/K have higher extinction values in both Green's 3D dustmap (Green et al. 2019) and Gaia DR3 (AG-DR3) λ F λ in y-axis and wavelength in xaxis. Red triangles denote the photometric flux available pre-Gaia DR3. Black hollow circles represent the synthetic photometry calculated using Gaia BP/RP continuous spectrum. Gaia DR3 mean sampled BP/RP spectrum are shown in green solid line. The best fit BT-NextGen spectrum are shown in grey. 5 Fig. 5 . 55averaged G band observing time JD-2455197.G-band multi-epoch photometry of stars classified as variables. The different subplots show the different variable classes as provided by Gaia DR3 with the class specification shown in red letters. The y-axis is limited to a range of 2 mag to visualize the variability in each class. The description on each variable class is provided in , called as LEMC) compiled a catalogue of 3339 hot ELS from 451,695 O, B, and A-type spectra from the LAMOST DR5 release. After careful spectral type re-estimation, they reported 1088 CBe, 233 Article number, page 1 of 6 arXiv:2209.13221v1 [astro-ph.GA] 27 Sep 2022 A[e] Ae Ae** B[e] Be Be* Be** Em Em[e]HAe HBe HFe? Oe* Oe**classification BeStar HerbigStar TTauri wN classlabel_espels 3 21 303 7 42 47 14 11 4 1 1 1 2 9 11 3 6 4 9 3 1 2 1 O8 O9 B0 B1 B2 B3 B5 B6 B7 B8 B9 A0 A1 A2 A3 A4 A5 A6 A7 A9 F0 F1 F2 F3 F4 F5 Spectral type from OBA catalog EW LEMC = -0.49×pEW 2 Gaia + 1.60×pEW Gaia -0.54 (2 nd degree polynomial fit)1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50 pEW Gaia (nm) 6 5 4 3 2 1 0 1 2 EW LEMC (nm) Slope=2.379 Intercept=-0.35 IQR=0.6 MAD=0.47 Slope=2.31 Intercept=-0.3 IQR=0.61 MAD=0.466 Slope=1.936 Intercept=-0.5 IQR=0.5 MAD=0.364 Slope=1.175 Intercept=-0.51 IQR=0.21 MAD=0.159 H peak above continuum H peak below continuum Table 2 of 2Eyer et al. Table 1 . 1Table containing the non-single stars data from Gaia DR3 for a subset of LEMC stars. SB1 = Spectroscopic BinaryDR3Name LAMOST ID Classification nss_solution_type Period Inclination Eccentricity Center_of_mass_velocity (days) (degrees) (km/s) 440218930776209664 J030719.47+523014.2 Be*-onlynir SB1 47.53 - 0.182 -57.457 2685047840736856448 J212547.34-022251.2 Ae*-onlynir SB1 195.252 - 0.326 -143.58 4549079418323779712 J173714.72+160334.7 F[e]? SB1 8.55 - 0.4349 -15.50 760463232938062464 J112045.14+362535.6 Ae SB1 6.93 - 0.053 1.405 948585824160038912 J072441.51+404013.1 Be SB1 8.20 - 0.077 9.46 666842467830419200 J074942.34+153117.3 Em SB1 11.47 - 0.25765 -13.58 277055356579399296 J043023.15+550408.8 Be*-onlynir SB1 57.082 - 0.01306 -20.97 2081810716132810368 J200645.38+435107.9 Em[e] Orbital 573.77 - 0.469 - 3441613167517590400 J053943.43+265316.2 Em[e] EclipsingBinary 1.48 77.59 0.0 - 3340108762301888256 J054241.85+114343.3 A[e] EclipsingBinary 0.876 72.08 0.0 - Acknowledgements. We would like to thank the Science & Engineering Research Board (SERB), a statutory body of the Department of Science & Technology (DST), Government of India, for funding our research under grant number CRG/2019/005380. We thank the Center for Research, CHRIST (Deemed to be University), Bangalore, India, for funding our research under the grant number MRP DSC-1932. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https: //www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. 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[]
[ "LMCODEC: A LOW BITRATE SPEECH CODEC WITH CAUSAL TRANSFORMER MODELS", "LMCODEC: A LOW BITRATE SPEECH CODEC WITH CAUSAL TRANSFORMER MODELS" ]
[ "Teerapat Jenrungrot \nUniversity of Washington\nSeattle\n", "Michael Chinen \nGoogle\n\n", "W Bastiaan Kleijn \nGoogle\n\n\nSchool of Engineering and Computer Science\nVictoria University of Wellington\n\n", "Jan Skoglund \nGoogle\n\n", "Zalán Borsos \nGoogle\n\n", "Neil Zeghidour \nGoogle\n\n", "Marco Tagliasacchi \nGoogle\n\n" ]
[ "University of Washington\nSeattle", "Google\n", "Google\n", "School of Engineering and Computer Science\nVictoria University of Wellington\n", "Google\n", "Google\n", "Google\n", "Google\n" ]
[]
We introduce LMCodec, a causal neural speech codec that provides high quality audio at very low bitrates. The backbone of the system is a causal convolutional codec that encodes audio into a hierarchy of coarse-to-fine tokens using residual vector quantization. LMCodec trains a Transformer language model to predict the fine tokens from the coarse ones in a generative fashion, allowing for the transmission of fewer codes. A second Transformer predicts the uncertainty of the next codes given the past transmitted codes, and is used to perform conditional entropy coding. A MUSHRA subjective test was conducted and shows that the quality is comparable to reference codecs at higher bitrates. Example audio is available at https://mjenru ngrot.github.io/chrome-media-audio-papers/publi cations/lmcodec.
10.1109/icassp49357.2023.10095442
[ "https://export.arxiv.org/pdf/2303.12984v1.pdf" ]
257,687,322
2303.12984
a871ef076dabe4c640be005129e05dedfd0bfb14
LMCODEC: A LOW BITRATE SPEECH CODEC WITH CAUSAL TRANSFORMER MODELS Teerapat Jenrungrot University of Washington Seattle Michael Chinen Google W Bastiaan Kleijn Google School of Engineering and Computer Science Victoria University of Wellington Jan Skoglund Google Zalán Borsos Google Neil Zeghidour Google Marco Tagliasacchi Google LMCODEC: A LOW BITRATE SPEECH CODEC WITH CAUSAL TRANSFORMER MODELS Index Terms-speech codingTransformersself-supervised learninggenerative adversarial networks We introduce LMCodec, a causal neural speech codec that provides high quality audio at very low bitrates. The backbone of the system is a causal convolutional codec that encodes audio into a hierarchy of coarse-to-fine tokens using residual vector quantization. LMCodec trains a Transformer language model to predict the fine tokens from the coarse ones in a generative fashion, allowing for the transmission of fewer codes. A second Transformer predicts the uncertainty of the next codes given the past transmitted codes, and is used to perform conditional entropy coding. A MUSHRA subjective test was conducted and shows that the quality is comparable to reference codecs at higher bitrates. Example audio is available at https://mjenru ngrot.github.io/chrome-media-audio-papers/publi cations/lmcodec. INTRODUCTION Speech coding, which consists of compressing speech signals to a limited number of bits with minimal distortion, is at the core of communication technologies such as mobile telephony or Voice over IP (VoIP). Opus [1] and EVS [2] are state-of-the-art speech coding techniques that combine traditional coding tools, such as Linear Predictive Coding (LPC), Code Excited Linear Prediction (CELP), and Modified Discrete Cosine Transformation (MDCT) to achieve high coding efficiency over different content types and bitrates. These waveform and parametric codecs rely on psychoacoustics expertise to design signal processing pipelines with maximal coding efficiency. Yet, while fast and interpretable, such handcrafted pipelines only represent a fraction of the potential models for a speech codec. This has motivated data-driven approaches to train neural networks to perform speech coding. These networks leverage large amounts of training data while relaxing the assumptions made on the type of transformations applied by the system [3][4][5][6][7][8][9][10]. In particular, the SoundStream neural codec combines a causal convolutional architecture with a residual vector quantizer. This quantization method produces a hierarchy of coarse-to-fine codes, and allows for efficient compression while providing bitrate scalability. As a result, Sound-Stream at 3 kbps matches the quality Opus at 12 kbps. However, the quality of most codecs, be they handcrafted or trained, degrades significantly at bitrates lower than 3 kbps. In this work, we introduce LMCodec, a low bitrate speech codec that combines recent advances in neural audio coding and audio generative modeling. LMCodec uses autoregressive Transformers [11] This work was done during a research internship at Google. on SoundStream tokens to (i) model the entropy of the distribution of coarse tokens and (ii) predict fine tokens from the coarse ones. At inference, LMCodec extracts the codes of a SoundStream model from the input waveform. However, instead of sending all codes to the receiver like a SoundStream codec would do, LMCodec only transmits entropy-coded coarse tokens. On the receiver side, a generative language model is used to predict fine tokens from the coarse ones, and a SoundStream decoder then reconstructs audio from the complete token sequence. LMCodec takes inspiration from the AudioLM [12] generative model, which also predicts fine SoundStream tokens from coarse ones. However, unlike AudioLM, LMCodec does low bitrate compression rather than generative modeling, and to do so leverages AudioLM both as a generative model and an entropy model. Other Transformer-based models for low bitrate coding have been proposed [7,13]. The codec in [13] enriches SoundStream with embeddings extracted from a self-supervised speech representation model [14] and achieves speech compression at a rate of 600 bps. [7] synthesizes speech from a combination of phonetic, pitch and speaker representations to achieve 365 bps. Unlike these models, LMCodec is a fully causal model, which is thus amenable to online encoding and decoding. Our primary contribution is the design of a new neural speech codec, which achieves state-of-the-art results outperforming many previous codecs operating at three to four times the rates according to subjective human evaluation metrics. Subjective evaluations demonstrate how LMCodec allows for low bitrate speech coding with minimal distortion, with LMCodec at approximately 1-1.5 kbps matching the performance of Opus at 12 kbps. We furthermore analyze the failure modes of our system, as well as the discrepancies in bit allocations between speech and non-speech sections of an audio signal. PROPOSED MODEL In this section, we describe our proposed speech codec consisting of four components: an encoder, a residual quantizer, an AudioLM block, and a decoder. The encoder, residual quantizer, and decoder follow similar structures from SoundStream. At the very high level, the encoder takes raw speech in the time domain as an input and extracts low-rate features that contain sufficient information to reconstruct the speech. The residual quantizer finds discrete representations of the inherently continuous encoded features. AudioLM poses the modeling of the quantized discrete representation as a language modeling problem and estimates the probability distribution of the next discrete audio token given previous audio tokens. Finally, the decoder reconstructs the input speech signal from the discrete encoded features. SoundStream We now briefly describe the SoundStream model [10] that we used for creating high-quality audio tokens. Encoder Given a raw speech signal x ∈ [−1, 1] T of length T , the encoder E : [−1, 1] T → R Te×Ne creates a sequence of embeddings of length Te T , each with dimension Ne. In our proposed model, the encoder takes raw waveform speech at T = 16 kHz as input and generates Ne = 128 dimensional speech features with a frame rate of 50 Hz. The architecture of the encoder is fully convolutional based on causal 1D convolutions. Hence, the algorithmic delay is determined by the overall striding factor (i.e., T /Te = 320 samples or 20 ms). Residual Vector Quantizer (RVQ) Transmission of continuous speech features over low-bandwidth channels is achieved via vector quantizers (VQs) [10], where the features are turned into discrete representations while introducing minimal distortion. Given the encoded features e ∈ R Te×Ne , the residual quantizer Q : R Te×Ne → {0, . . . , 2 log Nc -1} Te×Nq computes the corresponding binary representation of e and its inversion, where Nq is the number of quantizers and Nc is the codebook size of a single quantizer. In our proposed model, we always use the codebook of size Nc = 2 10 and vary the number of layers in the residual VQs: Nq ∈ {3, 4, 6, 12, 24}. Decoder The decoder D : R Te×Ne → [−1, 1] T synthesizes the original speech signal from the post-quantized embeddings. In our work, [15]. Numbers next to the markers refer to the number of coarse-level codes NC. we adopt the CNN-based decoder method trained with adversarial loss in addition to losses on waveform and spectral domains. The architecture of the decoder is similar to that of the encoder, with a transposed convolutional layer to upsample the output. The adversarial training framework relies on two types of discriminators: waveform domain and short time Fourier Transform (STFT) domain discriminators. AudioLM In this subsection, we describe the problem of language modeling of SoundStream tokens. Adding a language model in the bottleneck enables interesting modeling tasks, including modeling the distribution of future SoundStream tokens (Section 2.2.1) or tokens at different VQ layers (Section 2.2.2). For the rest of this paper, let NC and NF denote the number of quantizers for the coarse-level and fine-level AudioLMs, respectively. Figure 1 shows the overall architecture of our proposed model, in which we use NC = 4 and NF = 8. In our experiment, we use various combination of (NC, NF ) ranging from NC +NF = 3 to NC+NF = 24. Additionally, let c (n) k denote the SoundStream token at frame n and VQ layer k. Coarse-level AudioLM The goal of the coarse-level AudioLM is to model the distribution of the next coarse SoundStream tokens. Specifically, we are interested in modeling the conditional distribution of the next SoundStream tokens given the past information pC c (n) k c (n) k−1 , . . . , c (n) 1 coarse-level current frame , c (n−1) N C , . . . , c (1) 1 past information (1) for k ∈ {1, . . . , NC}. Given the distribution of the future SoundStream tokens, we build a codec by using lossless Entropy Coding (Section 2.3). More specifically, the discrete probability distribution of SoundStream tokens can be estimated both at the sender and the receiver sides, and we use this to drive an entropy codec. Note that in our proposed method, we only need to transmit NC tokens per single audio frame. The remaining NF tokens are generated at the receiver side only as described in the next section. (2) for k ∈ {NC + 1, . . . , NC + NF }. Note that our model is causal, in contrast to AudioLM. Since we only transmit the coarse-level tokens, we model the distribution of the fine-level tokens by assuming that we have access to ground-truth coarse-level SoundStream tokens. We note that, while [12] also proposes a similar fine-level AudioLM stage, our contribution here is the causal formulation of the task, which makes our approach more suitable and amenable to online decoding. Entropy Coding (EC) Given the distribution of coarse-level SoundStream tokens, we transmit data by using entropy coding, a lossless data compression technique. In this work, we provide experimental results using Huffman coding, in addition to the estimated entropy rate. We treat each code from the residual VQs separately and do not perform any grouping to reduce the upper bound on the bitrate. We first note that our proposed codec requires only sending coarse-level SoundStream tokens using entropy coding. Specifically, given raw audio, LMCodec first encodes audio into SoundStream tokens and models the probability distribution of the next Sound-Stream tokens, driving the entropy codec. Note that the discrete probability distribution of SoundStream tokens can be estimated both at the sender and the receiver sides, so the receiver can losslessly reconstruct the coarse tokens. To generate audio output from only coarse-level tokens, we use a fine-level AudioLM to synthesize fine-level tokens from the transmitted coarse-level tokens and then generate audio from both coarse-level and fine-level tokens using SoundStream decoder. Training Strategy We adopt a 2-stage training paradigm. First, we train only the encoder, quantizer, and decoder. Then, we freeze the weights of these components and train only the AudioLM components. We train the coarse-level and fine-level AudioLM models separately. Loss Functions We trained the SoundStream model using the standard adversarial loss, feature matching loss, reconstruction loss, and quantization loss according to [10]. In training AudioLM models, we use the standard cross-entropy loss for language modeling over the vocabulary space. Training configurations To create our codec modules, we adapted the architectures of the encoder, quantizer, generator, and discriminators used in Sound-Stream [10] and AudioLM from T5X. Both AudioLM models are the decoder-only models based on the base model of t5.1.1 (with approximately 250 million parameters). The SoundStream model is trained on 16 kHz audio from the LibriVox dataset [16] for 1M steps. Both coarse-level and fine-level AudioLM models are trained on 16 kHz audio from the Libri-Light dataset [17] for 1M steps with a batch size of 32 and sequence length of 1024 SoundStream tokens with Adafactor optimizer [18] with a decay rate of 0.8. We trained multiple coarse-level and fine-level AudioLM models to achieve varieties of bitrates. The bitrates are calculated based on the entropy coding of codes from coarse-level AudioLM. EVALUATION To demonstrate the performance of our proposed method, we evaluate LMCodec using both objective and subjective evaluations. For objective evaluation, we report the accuracy of LMCodec future token prediction and objective metrics including ViSQOL [19], WARP-Q [20], SSL-MOS [15], WER, and CER together with bitrate based on the test split from the clean LibriSpeech dataset [21]. For subjective evaluation, we perform two MUSHRA-like [22] subjective tests to compare the audio quality with standard state-ofthe-art speech codecs at medium bitrate (i.e., 1 kbps to 12 kbps) and low rate (i.e., 0.5 kbps to 1.5 kbps). The tests were conducted respectively on 91 and 94 crowd-sourced raters using headphones over 32 clean utterances from VCTK dataset [23]. Raters who did not score the reference above 80 at least 80% of the time were discarded, as were raters who rated more than 75% of non-reference samples 80 or above. 40 raters for the medium rate test and 33 raters for the low rate test met this requirement. As shown in Figure 2, the raters found that LMCodec-4/6 with 4 quantizers at 1.1 kbps perform significantly better than 12 kbps Opus. LMCodec-8/12 with 8 quantizers at 2.6 kbps has comparable performancce to SoundStream at 6 kbps. The low-rate MUSHRA test compares recent transformer neural codecs and lower bitrate SoundStream models. The raters preferred LMCodec to the transformer models from [13] and SoundStream at the same rate. Table 1 shows the accuracy of the future token prediction and the bitrate performance of LMCodec from the test split of the clean LibriSpeech [21]. For accuracy, we note that perfect accuracy means the model knows perfectly what the next tokens are. In the context of fine-level AudioLM, this suggests that the model does not necessarily need to synthesize the correct code to produce reasonable audio output. The bitrates are computed based on the future token's distributions obtained from LMCodec. For Huffman coding, we use the ground truth tokens encoded with the Huffman algorithm. Additionally, we note that the distributions of future tokens are updated every timestep based on the model, different from how other entropy codecs that may have fixed distributions operate. So, the Huffman bitrate may sometimes be lower than the bitrate derived from the entropy. Discussion In this section, we additionally discuss some of the interesting audio effects from LMCodec. We suggest that readers listen to some of the audio samples from our model. In particular, our model with only one quantizer is able to produce reasonable human voice with some babbling effects. The amount of babbling is reduced as the number of quantizers used in the codec increases. This suggests that there are some underlying hierarchical structure in SoundStream tokens, and the proposed codec can potentially be operating at very low bitrate, given that the coarse-to-fine prediction is accurate. In Figure 4, we visualize the distribution of code prediction from the AudioLM model when the input is at the middle of a phoneme and between phonemes. We also found that the model is very confident if the audio input is the middle of the phonemes, as the language model network is able to learn underlying linguistic behavior of the utterances. On the other hand, the model has lower confidence in predicting the next token when reaching silence sections, suggesting that our proposed causal model is unable to predict future word really well. This confirms the babbling effect that we observed in the audio output from our proposed codec, which increases as we restrict the amount of information to describe each frame (e.g., by transmitting fewer codes or dropping frames). Figure 2 shows the comparison of LMCodec with low-rate and medium-rate audio codecs. In particular, we find that LMCodec-4/6 performs better than SoundStream with 3 quantizers at 1.5 kbps but slightly worse than SoundStream with 12 quantizers at 6 kbps which is on par with LMCodec-8/12. We note that LMCodec-4/6 and LMCodec-8/12 are based on SoundStream with 6 and 12 quantizers respectively. Our results suggest that LMCodec effectively takes advantages from entropy coding and synthesizing reasonable fine-level codes from coarse-level codes. When comparing with SoundStream at similar rate, LMCodec essentially outperforms. Voice Activity Detection (VAD) In this section, we show the performance of LMCodec applied only on audio regions with voice activity. We use an open-source RNNoise model [24], which uses Mel-Frequency Cepstral Coefficients (MFCC) and outputs the probability of voice activity every 10 ms frame size. Since the frame size of SoundStream tokens is 20 ms, we run RNNoise on 2 consecutive 10-ms frames and define that the 20-ms SoundStream frame has a voice activity if and only if the probability that 2 consecutive frames have voice is over 0.8. Table 2 shows the bitrate of LMCodec on two scenarios: (i) transmitting only voices and (ii) transmitting entire speech signals but using zero bits for non-voices. We report the bitrate derived from the entropy and the bitrate based on Huffman coding. We note the first scenario has slightly lower bitrates as compared to bitrates from Table 1 because the entropy for non-speech signals is usually higher than the entropy for speech signals. Additionally, the second scenario provides the lower bound estimate of bitrates when transmitting very low bits for non-voice signals similar to Opus with variable bitrate scheme. Objective Evaluation We present an objective evaluation on the audio examples from VCTK dataset [23] in Figure 3. First, we demonstrate that the word error rate (WER) and character error rate (CER) are decreasing as the number of quantizers used in the LMCodec increases until around Table 1: Accuracy and bitrates. Bitrate without entropy coding is equivalent to 500 bps per quantizer (i.e., 6 kbps for 12 quantizers). Given the space limit, we only present the numerical results for LM-Codec with 12 RVQ layers and LMCodec models shown in Figure 2. 4-6 quantizers, suggesting that the semantic content is stored in the coarse tokens. To evaluate WER and CER, we use two ASR models from AWS Transcribe service and Conformer model [14] trained on LibriSpeech [21]. Second, ViSQOL [19] and WARP-Q [20], metrics designed for neural speech codecs, increases and decreases respectively, implying that the fine tokens are responsible for finegrained acoustic details. Third, SSL-MOS [15] shows that the overall speech quality improves by increasing the number of quantizers. Despite neural speech codecs metrics ViSQOL and WARP-Q indicating worse performance at about 4-6 quantizers, our listening test shows very high quality audio results with small number of quantizers. This suggests that the language model of LMCodec is able to model the distribution of the fine tokens given the coarse tokens reasonably well even if the synthesized fine tokens are different from the ground truth ones. This drives metrics like ViSQOL and WARP-Q down as they primarily rely on the comparison between synthesized audio and its corresponding ground truth reference audio. (N C , N F ) When comparing LMCodec with different total number of quantizers, we first note that the upper bound performance of LM-Codec with 6 quantizers is lower than the upper bound performance of LMCodec with 12 or 24 quantizers. However, LMCodec with a lower total number of quantizers reaches better performance faster than LMCodec with a higher total number of quantizers. CONCLUSION Our experiments show that the proposed codec significantly outperforms the original neural speech codec with respect to the quality of synthesized speech when operating in the ultra-low bitrate regime. In addition, the subjective experiments indicate comparable to or better perceptual speech quality compared to conventional codecs operating at higher rates. Fig. 1 : 1Overall pipeline of the proposed codec. Fig. 2 : 2MUHSRA-like subjective evaluation from state-of-the-art codecs with medium and low bitrates. LMCodec-x/y refers to our model with NC = x and NC + NF = y. wav2vec[13] is a recent neural codec based on SoundStream and Transformer. Fig. 3 : 3Objective evaluation of different LMCodec models. (left) LMCodec with a fixed number of RVQ layers (i.e., NC + NF = 12) on various standard metrics. (right) LMCodec with NC + NF ∈ {6, 12, 24} on SSL-MOS Fig. 4 :N 4Distribution of codes prediction for inputs from the non-voice section and inputs from the middle of phonemes2.2.2. Fine-level AudioLMSimilar to the coarse-level AudioLM, the fine-level AudioLM predicts the top VQ layers given the information about bottom VQ layers in addition to the past information. Specifically, we are interested in modeling the distribution of the fine-level SoundStream tokens conditioned on the coarse-level tokens and the past information: C +N F , . . . , c 1 past information (N C , N F )Accuracy Entropy Huffman (2, 1) 15.5% 534.0 bps 542.5 bps (3, 1) 14.3% 837.1 bps 845.7 bps (4, 2) 13.1% 1163.9 bps 1173.5 bps (1, 11) 16.1% 262.8 bps 262.6 bps (2, 10) 15.7% 533.5 bps 540.7 bps (3, 9) 14.9% 844.6 bps 847.4 bps (4, 8) 13.4% 1154.2 bps 1174.3 bps (6, 6) 11.9% 1853.7 bps 1861.2 bps (8, 4) 10.6% 2561.8 bps 2577.6 bps (10, 2) 9.7% 3300.0 bps 3324.8 bps (12, 0) 8.9% 4094.5 bps 4092.1 bps Transmitting only voices Transmitting non-voices with zero bitsEntropy Huffman Entropy Huffman (2, 1) 545.6 bps 554.1 bps 303.1 bps 307.9 bps (3, 1) 850.5 bps 858.6 bps 472.1 bps 476.6 bps (4, 2) 1165.6 bps 1173.7 bps 647.2 bps 651.7 bps (1, 11) 268.3 bps 268.7 bps 149.3 bps 149.5 bps (2, 10) 523.7 bps 530.0 bps 290.8 bps 294.3 bps (3, 9) 816.5 bps 819.1 bps 453.2 bps 454.7 bps (4, 8) 1108.5 bps 1129.7 bps 615.2 bps 627.0 bps (6, 6) 1775.2 bps 1783.6 bps 985.5 bps 990.2 bps (8, 4) 2457.3 bps 2471.5 bps 1363.5 bps 1371.4 bps (10, 2) 3170.9 bps 3196.7 bps 1763.4 bps 1777.8 bps (12, 0) 3958.2 bps 3951.5 bps 2207.6 bps 2203.9 bps Table 2 : 2Coding performance of LMCodec with VAD. 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[]
[ "Constraint-Based Qualitative Simulation", "Constraint-Based Qualitative Simulation" ]
[ "Krzysztof R Apt [email protected] ", "Sebastian Brand [email protected] ", "\nNational University of Singapore\nSingapore\n", "\nCWI and UvA\nAmsterdamThe Netherlands\n", "\nNational University of Singapore\nSingapore\n" ]
[ "National University of Singapore\nSingapore", "CWI and UvA\nAmsterdamThe Netherlands", "National University of Singapore\nSingapore" ]
[]
We consider qualitative simulation involving a finite set of qualitative relations in presence of complete knowledge about their interrelationship. We show how it can be naturally captured by means of constraints expressed in temporal logic and constraint satisfaction problems. The constraints relate at each stage the 'past' of a simulation with its 'future'. The benefit of this approach is that it readily leads to an implementation based on constraint technology that can be used to generate simulations and to answer queries about them.To support this claim, we implemented this approach in the generic constraint programming system ECL i PS e[22]and discuss here several case studies.
10.1109/time.2005.19
[ "https://arxiv.org/pdf/cs/0504024v1.pdf" ]
5,985,017
cs/0504024
f4b517ce0d243468432a78612b009aaab82d4ee8
Constraint-Based Qualitative Simulation 7 Apr 2005 Krzysztof R Apt [email protected] Sebastian Brand [email protected] National University of Singapore Singapore CWI and UvA AmsterdamThe Netherlands National University of Singapore Singapore Constraint-Based Qualitative Simulation 7 Apr 2005arXiv:cs/0504024v1 [cs.AI] We consider qualitative simulation involving a finite set of qualitative relations in presence of complete knowledge about their interrelationship. We show how it can be naturally captured by means of constraints expressed in temporal logic and constraint satisfaction problems. The constraints relate at each stage the 'past' of a simulation with its 'future'. The benefit of this approach is that it readily leads to an implementation based on constraint technology that can be used to generate simulations and to answer queries about them.To support this claim, we implemented this approach in the generic constraint programming system ECL i PS e[22]and discuss here several case studies. Introduction Qualitative reasoning was introduced in AI to abstract from numeric quantities, such as the precise time of an event or the location or trajectory of an object in space, and to reason instead on the level of appropriate abstractions. Two different forms of qualitative reasoning were studied in the literature. The first one is concerned with reasoning about continuous change in physical systems, monitoring streams of observations and simulating behaviours, to name a few applications. The main techniques used are qualitative differential equations, constraint propagation and discrete state graphs. For a thorough introduction see [14]. The second form of qualitative aims at reasoning about contingencies such as time, space, shape, size, directions, through an abstraction of the quantitative information into a finite set of qualitative relations. One then relies on complete knowledge about the interrelationship of these qualitative relations. This approach is exemplified by temporal reasoning due to [1], spatial reasoning introduced in [10] and [20], reasoning about cardinal directions (such as North, Northwest), see, e. g., [16], etc. For a recent overview of this approach to spatial reasoning, see [8]. Qualitative simulation deals with reasoning about possible evolutions in time of models capturing qualitative information. One assumes that time is discrete and that only changes adhering to some desired format occur at each stage. [15] discusses qualitative simulation in the first framework, while qualitative spatial simulation is considered in [9]. Our aim here is to show how qualitative simulation in the second approach to qualitative reasoning (exemplified by qualitative temporal and spatial reasoning) can be naturally captured by means of temporal logic and constraint satisfaction problems. The resulting framework allows us to concisely describe various complex forms of behaviour, such as a simulation of a naval navigation problem or a solution to a version of a piano movers problem. The domain knowledge is formulated using a variant of linear temporal logic with both past and future temporal operators. Such temporal formulas are then translated into constraints. The usual constraint-oriented representation of the second approach to qualitative reasoning is based on modelling qualitative relations as constraints. See, for example, [11] for an application of this modelling approach. In contrast, we represent qualitative relations as variables. This way of modelling has important advantages. In particular, it is more declarative since model and solver are kept separate; see the study of the relation variable model in [6]. In our case it allows us to express all domain knowledge on the same conceptual level, namely as constraints on the relation variables. Standard techniques of constraint programming can then be used to generate the simulations and to answer queries about them. Simulation Constraints Constraint Satisfaction Problems We begin by briefly introducing Constraint Programming. Consider a sequence X = x 1 , . . . , x m of variables with respective domains D 1 , . . . , D m . By a constraint C on X, written C(X), we mean a subset of D 1 × · · · × D m . A constraint satisfaction problem (CSP) consists of a finite sequence of variables X with respective domains and a finite set C of constraints, each on a subsequence of X. A solution to a CSP is an assignment to its variables respecting their domains and constraints. We study here CSPs with finite domains. They can be solved by a top-down search interleaved with constraint propagation. The top-down search is determined by a branching strategy that controls the splitting of a given CSP into two or more CSPs, the 'union' of which is equivalent to (i. e., has the same solutions as) the initial CSP. In turn, constraint propagation transforms a given CSP into one that is equivalent but simpler. We use here heuristicscontrolled domain partitioning as the branching strategy and hyper-arc consistency of [19] as the constraint propagation. Hyper-arc consistency is enforced by removing from each variable domain the elements not used in a constraint. Intra-state Constraints To describe qualitative simulations formally, we define first intra-state and inter-state constraints. A qualitative simulation corresponds then to a CSP consisting of stages that all satisfy the intra-state constraints. Moreover, this CSP satisfies the inter-state constraints that link the variables appearing in various stages. For presentational reasons, we restrict ourselves here to binary qualitative relations (e. g., topology, relative size). This is no fundamental limitation; our approach extends directly to higher-arity relations (e. g., ternary orientation). We assume that we have at our disposal • a finite set of qualitative relations Q, with a special element denoting the relation of an object to itself; • consistency conditions on Q-scenarios; we assume the usual case that they can be expressed as relations over Q, specifically as a binary converse relation conv and a ternary composition relation comp, • a conceptual neighbourhood relation between the elements of Q that describes which atomic changes in the qualitative relations are admissible. Example. Take the qualitative spatial reasoning with topology introduced in [10] and [20]. The set of qualitative relations is the set RCC8, i. e., In what follows we represent each stage t of a simulation by a CSP P t uniquely determined by a qualitative array Q t . Here t is a variable ranging over the set of natural numbers that represents discrete time. Instead of Q t [A, B] we also write Q[A, B, t], reflecting that, in fact, we deal with a single ternary array. Q = {disjoint, Inter-state Constraints To describe the inter-state constraints, we use as atomic formulas statements of the form Q[A, B] ? q where ? ∈ {=, =} and q ∈ Q, or 'true', and employ a temporal logic with four temporal operators, (eventually), (next time), (from now on), U (until), and their 'past' counterparts, −1 , −1 , −1 , and S (since). While it is known that past time operators can be eliminated, their use results in more succinct (and in our case more intuitive) specifications; see, e. g., [18]. Inter-state constraints are formulas that have the form φ → ψ. Both φ and ψ are built out of atomic formulas using propositional connectives, but φ contains only past time temporal operators and ψ uses only future time operators. Intuitively, at each time instance t, each inter-state con- straint φ → ψ links the 'past' CSP t i=0 P i with the 'fu- ture' CSP tmax i=t+1 P i . So we interpret φ in the interval [0..t], and ψ in the interval [t + 1 .. t max ]. We now explain the meaning of a past or future temporal formula φ with respect to the underlying qualitative array Q in an interval [s..t], for which we stipulate s t. We write |= [s..t] φ to express that φ holds in the interval [s..t]. Propositional connectives. These are defined as expected, in particular independently of the 'past' or 'future' aspect of the formula. For example, |= [s..t] ¬φ if not |= [s..t] φ, |= [s..t] φ 1 ∨ φ 2 if |= [s..t] φ 1 or |= [s..t] φ 2 . Conjunction φ 1 ∧ φ 2 and implication φ 1 → φ 2 are defined analogously. Future formulas. Intuitively, the evaluation starts at the lower bound of the time interval and moves only forward in time. |= [s..t] Q[A, B] ? c if Q[A, B, s] ? c where ? ∈ {=, =}; |= [s..t] φ if |= [r. .t] φ and r = s + 1 and r t; |= [s..t] φ if |= [r..t] φ for all r ∈ [s..t]; |= [s..t] φ if |= [r..t] φ for some r ∈ [s..t]; |= [s..t] χ U φ if |= [r..t] φ for some r ∈ [s..t] and |= [u..t] χ for all u ∈ [s .. r − 1].|= [s..t] Q[A, B] ? c if Q[A, B, t] ? c where ? ∈ {=, =}; |= [s..t] −1 φ if |= [s..r] φ and r = t − 1 and s r; |= [s..t] −1 φ if |= [s..r] φ for all r ∈ [s..t]; |= [s..t] −1 φ if |= [s..r] φ for some r ∈ [s..t]; |= [s..t] χ S φ if |= [s..r] φ for some r ∈ [s..t] and |= [u..t] χ for all u ∈ [r + 1 .. t]. Furthermore, we write Q[A, B] ∈ {q 1 , . . . , q k } as an ab- breviation of (Q[A, B] = q 1 ) ∨ . . . ∨ (Q[A, B] = q k ). The meaning of Q[A, B] / ∈ {q 1 , . . . , q k } is analogous. The bounded quantification ∃A ∈ {o 1 , . . . , o k }. φ(A) represents the disjunction φ(o 1 ) ∨ . . . ∨ φ(o k ). Universal quantification ∀A ∈ {o 1 , . . . , o k }. φ(A) is interpreted analogously. As usual, A in φ(A) denotes a placeholder (free variable), and φ(o i ) is obtained by replacing A in all its occurrences by o i . with the obvious meaning (EQ is the identity relation). Ligozat [16] provides the composition table for this form of qualitative reasoning and shows that it captures consistency. An Example: Navigation The buoy positions are given by the following global intra-state constraints: Q[buoy a , buoy c ] = NW, Q[buoy a , buoy b ] = SW, Q[buoy b , buoy c ] = NW. All objects occupy different positions: ∀A, B ∈ O. A = B → Q[A, B] = EQ. The initial position of the ship is south of buoy c, so we have Q[ship, buoy c ] = S. The ship is required to follow a path around the buoys. In Fig. 2, the positions required to be visited are marked with bold circles. We stipulate (Q[ship, buoy a ] = W ∧ (Q[ship, buoy b ] = N ∧ (Q[ship, buoy c ] = E ∧ (Q[ship, buoy c ] = S )))) to hold in the interval [0 .. t max ]. A tour of 13 steps exists (and is found by our program); it is indicated in Fig. 2. Temporal Formulas as Constraints We explain now how a temporal formula (an inter-state constraint) is imposed on the sequence of CSPs representing the spatial arrays at consecutive times. Such a formula is reduced to a sequence of constraints by eliminating the temporal operators. We provide two alternative translations. The first simply unfolds the temporal operators into primitive constraints, while the second retains more structure and avoids duplication of subformulas by relying on array constraints. Consider a temporal formula φ → ψ where φ uses only 'past' time operators and ψ uses only 'future' time operators. Given a CSP t i=s P i , we show how the past temporal logic formula φ translates to a constraint cons − ([s..t], φ) and how the future temporal logic formula ψ translates to a constraint cons + ([s..t], ψ), both on the variables of t i=s P i . We assume that the target constraint language has Boolean constraints and reified versions of simple comparison and arithmetic constraints. Reifying a constraint means associating a Boolean variable with it that reflects the truth of the constraint. For example, (x = y) ≡ b is a reified equality constraint: b is a Boolean variable reflecting the truth of the constraint x = y. We denote by cons([s..t], φ) ≡ b the sequence of constraints representing the fact that the formula φ has the truth value b in the interval [s..t]. The 'past' or 'future' aspect of a formula is indicated by a marker − or + , resp., when relevant. The translation of φ proceeds by induction and is initiated with cons([s..t], φ) ≡ 1 (where s t). Unfolding Translation We translate the propositional connectives into appropriate Boolean constraints. The temporal operators are unfolded over the simulation stages. For example, the 'future' formula (Q[A, B] = q) in the interval [1..3] translates to (Q[A, B, 1] = q) ≡ b 1 , (Q[A, B, 2] = q) ≡ b 2 , (Q[A, B, 3] = q) ≡ b 3 , and b 1 ∨ b 2 ∨ b 3 = 1, with fresh Boolean variables b 1 , b 2 , b 3 . Translation for 'future' formulas. cons + ([s..t], true) ≡ b is b = 1; cons + ([s..t], ¬φ) ≡ b is b ′ = ¬b, cons + ([s..t], φ) ≡ b ′ ; cons + ([s..t], φ 1 ∨ φ 2 ) ≡ b is (b 1 ∨ b 2 ) ≡ b, cons + ([s..t], φ 1 ) ≡ b 1 , cons + ([s..t], φ 2 ) ≡ b 2 ; cons + ([s..t], Q[A, B] ? c) ≡ b is (Q[A, B, s] ? c) ≡ b where ? ∈ {=, =}; cons + ([s..t], φ) ≡ b is (b 1 ∧ b 2 ) ≡ b, cons + ([r..t], φ) ≡ b 2 , (s + 1 t) ≡ b 1 , (s + 1 = r) ≡ b 1 ; cons + ([s..t], φ) ≡ b is ( r∈s..t b r ) ≡ b, cons + ([r..t], φ) ≡ b r for all r ∈ [s..t]; cons + ([s..t], φ) ≡ b is ( r∈s..t b r ) ≡ b, cons + ([r..t], φ) ≡ b r for all r ∈ [s..t]; cons + ([s..t], χ U φ) ≡ b is cons + ([s..t], φ ∨ χ ∧ (χ U φ)) ≡ b. Translation for 'past' formulas. This case is symmetric to the 'future' case except for the 'backward' perspective. So we have cons − ([s..t], Q[A, B] ? c) ≡ b is (Q[A, B, t] ? c) ≡ b where ? ∈ {=, =}, for example. The remaining cases are defined analogously. Observe that the interval bounds s, t in cons([s..t], φ) are treated as constants such that s t. Array Translation This alternative translation avoids the potentially large disjunctive constraints caused by unfolding the and U operators. The idea is to push disjunctive information inside variable domains, with the help of array constraints. Reconsider the formula (Q[A, B] = q) in the interval [1..3]. It is translated into a single array constraint, with the help of a fresh variable x ranging over time points: Q[A, B, x] = q, 1 x, x 3. Array constraints generalise the better-known element constraint. Constraint propagation for array constraints is studied in [5] and used in our implementation. When negation occurs in the formula, a complication arises with this translation approach, however. Just negating the associated truth value, as in the unfolding translation, is now incorrect. We therefore first transform a formula into negation normal form (NNF). The array translation of NNF formulas follows. We give it only for 'future' formulas and where different from the unfolding translation. The case of negation does not apply anymore. cons + ([s..t], φ) ≡ b is cons + ([s..t], φ ∧ ( true → φ) ≡ b; cons + ([s..t], φ) ≡ b is s r, r t, cons + ([r..t], φ) ≡ b; cons + ([s..t], χ U φ) ≡ b is (b 1 ∧ (b 2 ∨ b 3 )) ≡ b, s r, r t, cons + ([r..t], φ) ≡ b 1 , (s = r) ≡ b 2 , s u, u r, (u = r − 1) ≡ b 3 , cons + ([s..u], χ) ≡ b 3 . The interval end points s, t in cons([s..t], φ) can now be variables with domains, in contrast to the case of the unfolding translation where s, t are constants. We are careful to maintain the invariant s t and state appropriate constraints to this end. Therefore, for example, we unfold φ into a conjunction only step-wise, as the formula φ ∧ ( true → φ). Example. Let us contrast the two alternative translations for a formula from the navigation domain. Consider The array translation results in just two array constraints, namely Q[ship, buoy, r 1 ] = E and Q[ship, buoy, r 2 ] = S, The four ordering constraints 1 r 1 , r 1 n r 1 r 2 , and r 2 n control the fresh variables r 1 , r 2 . φ ≡ (φ 1 ∧ φ 2 ), Simulations By a qualitative simulation we mean a finite or infinite sequence PS = P 0 , P 1 , . . . of CSPs such that for each chosen inter-state constraint φ → ψ we have that the constraint cons([0 .. t 0 ], φ) → cons([t 0 + 1 .. t], ψ) is satisfied by the CSP t i=0 P i , • if PS is finite with u elements, for all t 0 ∈ [0 .. u − 1], t = t max , • if PS is infinite, for all t 0 0, t t 0 + 1. Thus, at each stage of the qualitative simulation, we relate its past (and presence) to its future using the chosen interstate constraints. Consider an initial situation I = P 0 and a final situation F x determined by a qualitative array of the form Q x , where x is a variable ranging over the set of integers (possible time instances). We would like to determine whether a simulation exists that starts in I and reaches F t , where t is the number of steps. If one exists, we may also be interested in computing a shortest one, or in computing all of them. Simulation algorithm. The algorithm given in Figure 3 provides a solution to the first two problems in presence of a non-circularity constraint. The sequence PS of CSPs is initially empty and subsequently step-wise extended; so it remains finite. We view PS as a single CSP, which consists of regular finite domain variables and constraints and which thus fits into the problem format solvable by a standard constraint programming techniques. We employ the auxiliary procedures prop and solve. The call to prop performs constraint propagation of the intrastate and inter-state constraints. In our implementation, the hyper-arc consistency notion is used. As a result, the variable domains are pruned and less backtracks arise when solve is called. If the outcome is an inconsistent CSP, the value false is returned in the failure flag. The call to solve checks if a solution to the CSP corresponding to the given sequence of CSPs exists. If so, a solution and true is returned, otherwise ∅, false . In our implementation, solve is a standard backtrack search (based on variable domain splitting) combined with constraint propagation as in the prop procedure. We use the constant t max equal to the number of different qualitative arrays, i. e., t max = |O| · (|O| − 1) · 2 |Q|−1 . If the desired simulation exists, the above algorithm finds a shortest one and outputs it in the variable solution. Implementation We implemented the simulation algorithm of Fig. 3 and both alternative translations of temporal formulas to constraints in the ECL i PS e constraint programming system [22]. The total program size is roughly 1500 lines of code. Propagation Support for enforcing hyper-arc consistency for Boolean and many reified constraints, as well as for extensionally defined constraints such as conv, comp and the conceptual neighbourhood constraint, is directly available in ECL i PS e (by its fd/ic and propia libraries). For array constraints, we use the ECL i PS e implementation discussed in [5]. The availability of these (generic) implementations of propagation mechanisms explains why we chose hyper-arc consistency. We emphasise, however, that in a relation variable model, constraint propagation is relevant only for efficiency. Search We use the basic backtracking algorithm provided by ECL i PS e , but we control it with the heuristics described in the following section. Various other, advanced search strategies are available in ECL i PS e , for example Limited Discrepancy Search [13]. Although we did not experiment with these techniques, we believe it is worth doing so, and it is not difficult to modify our implementation (the solve procedure) accordingly. Heuristics Our implementation also incorporates the specialised reasoning techniques for RCC8 [21] and the cardinal directions [16]. In these studies, maximal tractable subclasses of the respective calculi are identified, and corresponding polynomial decision procedures are discussed. Our context requires that these techniques are treated as heuristics, due to the presence of side constraints (notably the inter-state constraints). With a relation variable model for qualitative spatial reasoning, these heuristics fall into the customary class of variable and value ordering heuristics for guiding search in constraint programming. In our implementation, the search heuristic splits the relation variable domains appropriately so that one of the new domains belongs to a maximal tractable subclass of the respective calculus. Case Studies We now report on two case studies. In both of them, the solutions were found by our implementation within a few seconds. Piano Movers Problem Consider the following version of the piano movers problem. There are three rooms, the living room (L), the study room (S) and the bedroom (B), and the corridor (C). Inside the study room there is a piano (P) and inside the living room a table (T); see Figure 4. Move the piano to the living room and the table to the study room assuming that none of the rooms and the corridor are large enough to contain the piano and the table at the same time. Additionally, ensure that the piano and the table at no time will touch each other. To formalise this problem, we describe the initial situation by means of the following formulas: φ 0 ≡ Q[B,L] = disjoint ∧ Q[B,S] = disjoint ∧ Q[L,S] = disjoint, φ 1 ≡ Q[C,B] = meet ∧ Q[C,L] = meet ∧ Q[C,S] = meet, φ 2 ≡ Q[P,S] = inside ∧ Q[T,L] = inside. We assume that initially φ 0 , φ 1 , φ 2 hold, i. e., the constraints cons − ([0..0], φ 0 ), cons − ([0..0], φ 1 ) and cons − ([0..0], φ 2 ) are present in the initial situation I. Below, given a formula φ, by an invariant built out of φ we mean the formula φ → φ. Further, we call a room or a corridor a 'space' and abbreviate the subset of objects {B, C, L, S} by S. We now stipulate as the inter-state constraints the invariants built out of the following formulas: • the relations between the rooms, and between the rooms and the corridor, do not change: φ 0 ∧ φ 1 , • at no time do the piano and the table fill completely any space: ∀s ∈ S. (Q[P, s] = equal ∧ Q[T, s] = equal) , • together, the piano and the table do not fit into any space. More precisely, at each time, at most one of these two objects can be within any space: Remarkably, the interaction with our program revealed in the first place that our initial formalisation was incomplete. For example, the program also generated solutions in which the piano is moved not through the corridor but 'through the walls', as it were. To avoid such solutions we added the following intrastate constraints. • each space is too small to be 'touched' (met) or 'overlapped' by the piano and the After these additions, our program generated the shortest solution in the form of a simulation of length 12. In this solution the bedroom is used as a temporary storage for the table. Interestingly, the table is not moved completely into the bedroom: at a certain moment it only overlaps with the bedroom. Phagocytosis The second example deals with a simulation of phagocytosis: an amoeba absorbing a food particle. This problem is discussed in [9]. We quote: "Each amoeba is credited with vacuoles (being fluid spaces) containing either enzymes or food which the animal has digested. The enzymes are used by the amoeba to break down the food into nutrient and waste. This is done by routing the enzymes to the food vacuole. Upon contact the enzyme and food vacuoles fuse together and the enzymes merge into the fluid containing the food. After breaking down the food into nutrient and waste, the nutrient is absorbed into the amoeba's protoplasm, leaving the waste material in the vacuole ready to be expelled. The waste vacuole passes to the exterior of the protozoan's (i. e., amoeba's) body, which opens up, letting the waste material pass out of the amoeba and into its environment." To fit it into our present framework, we slightly simplified the problem representation by not allowing for objects to be added or removed dynamically. In this problem, we have six objects, amoeba, nucleus, enzyme, vacuole, nutrient and waste. The initial situation is described by means of the three following constraints: We model the splitting up of the food into nutrient and waste material by QQ[nutrient, waste] = equal→ (φ 1→ φ 2∨ φ 3 ) ∨ Q[nutrient, waste] = equal; with φ 1 ≡ Q[nutrient, vacuole] = inside ∧ Q[enzyme, nutrient] = overlap ∧ Q[enzyme, waste] = overlap φ 2 ≡ Q[nutrient, waste] = overlap φ 3 ≡ Q[nutrient, waste] = equal The dotted operators express if-then-else, that is, a→ b∨ c ≡ (a → b) ∧ (¬a → c). The final situation is described by means of the constraints Q[amoeba, waste] = disjoint, Q[amoeba, nutrient] ∈ {contains, covers}. Our program generated a simulation consisting of 9 steps. Final Remarks The most common approach to qualitative simulation is the one discussed in [14, chapter 5]. For a recent overview see [15]. It is based on a qualitative differential equation model (QDE) in which one abstracts from the usual differential equations by reasoning about a finite set of symbolic values (called landmark values). The resulting algorithm, called QSIM, constructs the tree of possible evolutions by repeatedly constructing the successor states. During this process, CSPs are generated and solved. This approach is best suited to simulate evolution of physical systems. A standard example is a simulation of the behaviour of a bath tub with an open drain and constant input flow. The resulting constraints are usually equations between the relevant variables and lend themselves naturally to a formalisation using CLP(FD), see [7, chapter 20] and [3]. The limited expressiveness of this approach was overcome in [4], where branching time temporal logic was used to describe the relevant constraints on the possible evolutions (called 'trajectories' there). This leads to a modified version of the QSIM algorithm in which model checking is repeatedly used. Our approach is inspired by the qualitative spatial simulation studied in [9], the main features of which are captured by the composition table and the neighbourhood relation discussed in Example 2.2. The distinction between the intra-state and inter-state constraints is introduced there, however the latter only link the consecutive states in the simulation. The simulation algorithm of [9] generates a complete tree of all 'evolutions', usually called an envisionment. In contrast to [9], our approach is constraint-based. This allows us to repeatedly use constraint propagation to prune the search space in the simulation algorithm. Further, by using more complex inter-state constraints, defined by means of temporal logic, we can express substantially more sophisticated forms of behaviour. While the prevalent approach to constraint-based modelling of qualitative spatial knowledge maps qualitative relations to constraints, we use variables to express qualitative relations. The relation variable approach is much more declarative, separating the model from the solver. The advantage of a relation variable model for qualitative simulations is that the knowledge of the spatial domain as well as of the application domain can be expressed on the same conceptual level, by intra-state and inter-state constraints. This leads to a model that can easily be realised within a typical constraint programming system using generic propagation and search techniques, and is also immediately open to advances in these systems. Simulation in our approach subsumes a form of planning. In this context, we mention the related work [17] in the area of planning which shows the benefits of encoding planning problems as CSPs and the potential with respect to solving efficiency. Also related is the TLPLAN system where planning domain knowledge is described in temporal logic [2]. The planning system is based on incremental forward-search, so temporal formulas are just unfolded one step at a time, in contrast to the translation into constraints in our constraint-based system. Finally, [12] discusses how a qualitative version of the piano movers problem can be solved using an approach to qualitative reasoning based on topological inference and graph-theoretic algorithms. Our approach is substantially simpler in that it does not rely on any results on topology apart of a justification of the composition table. Figure 1 . 1The eight RCC8 relations Figure 2 . 2Navigation path Past formulas. Here the evaluation starts at the upper bound and moves backward. A ship navigates around three buoys along a specified course. The position of the buoys is fixed; seeFig. 2. We reason qualitatively about the cardinal directions Q = {N, NE, . . . , W, NW, EQ} φ 1 ≡Figure 3 . 13(Q[ship, buoy] = E) and φ 2 ≡ (Q[ship, buoy] = S), in the interval [1..n] for a constant n, as a 'future' formula. So we consider the sequence of constraints cons + ([1..n], φ) for each translation. The unfolding translation generates many reified equality constraints of the form (Q[ship, buoy, k] = D) ≡ b i,k , where D is E or S. More specifically, n + n i=1 i = n(n + 3)/2 such constraints and as many new Boolean variables are created. Many of the constraints are variants of each other differing only in their Boolean variable b i,k . Simulate spatial array Q, state constraints, t max −→ solution PS := ; t := 0 while t < t max do P t := create CSP from Q t and impose intra-state constraints PS := append P t to PS and impose inter-state constraints PS, failure := prop(PS) if not failure then PS ′ := PS with final state constraints imposed on P t solution, success := solve(PS ′ ) if success then return solution t := t + 1 return failure The simulation algorithm Figure 4 . 4A piano movers problem ∀s ∈ S. ¬(Q[P, s] ∈ {inside, coveredby} ∧ Q[T, s] ∈ {inside, coveredby}), • at no time instance do the piano and the table touch each other: Q[P, T] = disjoint. The final situation is captured by the constraints Q[P, L] = inside and Q[T, S] = inside. We fix now a sequence O of objects of interest. By a qualitative array we mean a two-dimensional array Q on O × O such that Each qualitative array determines a unique CSP. Its variables are Q[A, B], with A and B ranging over the sequence of the assumed objects O. The domains of these variables are appropriate subsets of Q. An instantiation of the variables to elements of Q corresponds to a consistent Q-scenario.meet, equal, covers, coveredby, contains, inside, overlap}; see Fig. 1, which also shows the neighbourhood relation between these relations. • for each pair of objects A, B ∈ O, the expression Q[A, B] is a variable denoting the (basic) relation be- tween A, B. So its initial domain is a subset of Q. • the consistency conditions hold on Q, so for each triple of objects A, B, C the following intra-state constraints are satisfied: reflexivity: Q[A, A] = equal, converse: conv(Q[A, B], Q[B, A]), composition: comp(Q[A, B], Q[B, C], Q[A, C]). [amoeba, nutrient] = disjoint, Q[amoeba, waste] = disjoint, Q[nutrient, waste] = equal. We have the intra-state constraints Q[enzyme, amoeba] = inside, Q[vacuole, amoeba] ∈ {inside, coveredby}, Q[vacuole, enzyme] ∈ {disjoint, meet, overlap, covers}, and, concerning the nucleus, Q[nucleus, vacuole] ∈ {disjoint, meet}, Q[nucleus, enzyme] ∈ {disjoint, meet}, Q[nucleus, amoeba] = inside. The inter-state constraints are Q[nutrient, amoeba] = meet → Q[nutrient, amoeba] = overlap, Q[nutr., amoeba] ∈ {inside, coveredby, overlap → Q[nutr., amoeba] ∈ {inside, coveredby}. c 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Maintaining knowledge about temporal intervals. J F Allen, Communications of the ACM. 2611J. F. Allen. Maintaining knowledge about temporal intervals. 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[ "Lithium fluoride (LiF) target preparation for nuclear physics experiment", "Lithium fluoride (LiF) target preparation for nuclear physics experiment" ]
[ "Lalit Kumar Sahoo \nSaha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia\n\nHomi Bhaba National Institute Anushaktinagar\n400094MumbaiIndia\n", "Ashok Kumar Mondal \nSaha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia\n\nHomi Bhaba National Institute Anushaktinagar\n400094MumbaiIndia\n", "Dipali Basak \nSaha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia\n\nHomi Bhaba National Institute Anushaktinagar\n400094MumbaiIndia\n", "Chinmay Basu \nSaha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia\n", "Suraj Kumar \nSaha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia\n", "Karan " ]
[ "Saha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia", "Homi Bhaba National Institute Anushaktinagar\n400094MumbaiIndia", "Saha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia", "Homi Bhaba National Institute Anushaktinagar\n400094MumbaiIndia", "Saha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia", "Homi Bhaba National Institute Anushaktinagar\n400094MumbaiIndia", "Saha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia", "Saha Institute of Nuclear Physics Bidhan nagar\n700064KolkataIndia" ]
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The LiF target preparation on self-supporting Ag backing (LiF/Ag) is discussed in a detailed manner using vacuum evaporation process. The target thickness is measured using the energy loss of three line alpha source. 183.74 µg/cm 2 thickness of LiF is achieved through the evaporation process. Good uniformity of targets is observed. Non-uniformity in targets is found within 6 %. The XPS analysis confirms the presence of both the F and Li atoms on sample surface.
10.1016/j.vacuum.2023.112055
[ "https://export.arxiv.org/pdf/2208.03425v1.pdf" ]
251,402,501
2208.03425
63ed8b4a54a8ca5b68bf64b297f1563bcc032363
Lithium fluoride (LiF) target preparation for nuclear physics experiment Lalit Kumar Sahoo Saha Institute of Nuclear Physics Bidhan nagar 700064KolkataIndia Homi Bhaba National Institute Anushaktinagar 400094MumbaiIndia Ashok Kumar Mondal Saha Institute of Nuclear Physics Bidhan nagar 700064KolkataIndia Homi Bhaba National Institute Anushaktinagar 400094MumbaiIndia Dipali Basak Saha Institute of Nuclear Physics Bidhan nagar 700064KolkataIndia Homi Bhaba National Institute Anushaktinagar 400094MumbaiIndia Chinmay Basu Saha Institute of Nuclear Physics Bidhan nagar 700064KolkataIndia Suraj Kumar Saha Institute of Nuclear Physics Bidhan nagar 700064KolkataIndia Karan Lithium fluoride (LiF) target preparation for nuclear physics experiment LiFThree line alphaEvaporationXPSFloating PACS: 8115Ef3450Bw3360Fy The LiF target preparation on self-supporting Ag backing (LiF/Ag) is discussed in a detailed manner using vacuum evaporation process. The target thickness is measured using the energy loss of three line alpha source. 183.74 µg/cm 2 thickness of LiF is achieved through the evaporation process. Good uniformity of targets is observed. Non-uniformity in targets is found within 6 %. The XPS analysis confirms the presence of both the F and Li atoms on sample surface. Introduction Targets in nuclear physics are the systems that are subjected to the bombardment of accelerated charged particles such as protons, neutrons, or heavy ions. Nuclear targets are a key factor in the accomplishment of nuclear reactions.The use of high quality targets is essential for successful nuclear interactions. Cross-sections resulting from the nuclear interaction are highly dependent on the quality of targets used in the experiment. The importance of target quality increases particularly in astrophysical reactions where cross sections are in order of nb or pb. Lithium fluoride (LiF) on self supported Ag foils (LiF/Ag) is one of the crucial target for many reactions like 19 F(p,α), 19 F(p,γ), 7 Li(p,α), 7 Li(p,n), 19 F(p,n) etc [1,2,3,4]. Preparation of a thin target is also an essential requirement for many reactions where energy loss through target would be less. LiF is a good alternative to prepare thin targets as compared to metallic lithium or fluorine. LiF has been preferred for generation of mono-energetic neutrons [2,5] which can be efficiently used as neutron beam. Both the target requirement of Li and F are satisfied with the deposition of LiF compound on Ag backing. However, the detailed and efficient description of LiF target preparation was not discussed in great detail. Smail Damache et al. [6] has described the LiF target preparation in some details . In this paper, we presented a detailed description of the smooth preparation of lithium fluoride (LiF) target on self-supporting silver backing by vacuum evaporation method with analysis of prepared targets. Deposition Set-up The deposition setup is divided into two parts according to the mode of deposition. One part is for thermal evaporation and second one is for e − -beam evaporation. Figure 1 (left) shows the thermal evaporation part. It consists of a copper electrode and a boat that hold the sample. The Copper electrode is used for resistive heating purposes in the application of current. One substrate holder is placed above it as shown in figure 1. In the same chamber, the e − beam setup is placed as shown in figure 1 (right). It consists of the crucible (for placing sample), shutter (initially used to collect the impurities if any are present and to stop them from depositing on the substrate during this process), and substrate holder attached with the rotating plate. The whole setup is covered with aluminium foil to prevent the deposition on the chamber walls. Figure 2 shows the schematic view of setup used for vacuum evaporation. Preparation Method Preparation method of LiF on self supporting silver foil consists of two steps. First step requires initial silver evaporation and second steps requires LiF deposition over deposited silver foil. The following steps describe the whole process efficiently. (i) First, three glass slides of 76mm×25mm are attached to substrate holder of rotating plate as shown in figure 1 (right). Glass substrates are cleaned with ethanol before attaching to the substrate holder. A thin layer of BaCl 2 is deposited through e − beam evaporation which will act as releasing agent. Then, pure silver ( 99.99 %) is deposited over the deposited BaCl 2 using e − beam evaporation. The deposition of Ag is started at 33 mA of current with 0.1 A 0 /s initial deposition rate and gradually goes on increasing with increase of current up to 50 mA with final deposition rate of 2.2 A 0 /s. After deposition of Ag, glass slide is immersed in distilled water. The inter layer of BaCl 2 is dissolved in water and Ag foil floats on the surface of water. Ag foils were carefully picked up by Al frame in order to avoid the impact of surface tension of water. Figure 3 shows the floating process of Ag foil and silver foil mounted on Al frame of 8-10 mm diameter opening. Substrate and source was at a distance of 22.5 cm during whole deposition process. The substrate holder was rotating at speed of 5 rpm to maintain uniformity of deposition. (ii) Then, self supporting Ag foils on Al frame were fixed to another substrate holder (figure 1 left (e)) at 10.5 cm distance from the copper electrode. The reduction in distance of substrate holder from the sample placed in boat can be attributed to low vapour pressure of LiF sample. LiF was deposited successfully by resistive heating method. Ag foils mounted on Al frame is sandwiched with a 15 mm diameter frame for restricting deposition area of LiF. Evaporated LiF on Ag backing is shown in figure 3. Materials used for deposition, melting point, and current used for evaporation are tabulated in the table 1. Analysis of prepared targets Thickness of prepared LiF targets on self supporting Ag backing is measured using three line alpha source ( 239 Pu, 241 Am, and 244 Cm).It is measured in two steps. First, thickness of Ag is measured using alpha energy loss.Then using known thickness of silver, energy loss through Ag can be calculated and energy available after loss through Ag foil is used as initial incident energy for thickness calculation of LiF sample. Figure 4 shows the setup used for thickness measurement using alpha loss. Collimator shown in figure 4 is used to scan different position of samples for confirmation on uniformity of prepared targets. Silicon detector is used for detecting alpha particles after passing the sample. Setup shown in figure 4 was placed within a SS chamber with vacuum maintained at 10 −5 mbar. The three line alpha source was placed inside the groove before collimator. Energy associated with three line alpha source 239 Pu, 241 Am, and 244 Cm are 5155 KeV, 5486 KeV and, 5805 KeV respectively. The detector needs to be calibrated properly to measure the thickness. This can be achieved by taking measurement without placing foil in front of alpha source. After placing the Ag foil in front of alpha source, the detector will detect the alpha particle with reduced energy. . This stopping power is calculated using SRIM code [7]. Thickness of the foil is calculated by x = ∆E − dE dx (2) where, x= thickness of foil in µg/cm 2 , ∆E = Energy loss through foil in KeV and -dE dx is the stopping power measured in KeV/ (µg/cm 2 ). Thickness of one of the Ag foil is 398.62 µg/cm 2 (0.38 µm) using equation 2. Uniformity of foil is checked by scanning the different position of foil through energy loss method with a collimator and variation in thickness of Ag foil is between 5-10 % from the average thickness of foil. However, LiF is evaporated over Ag foil. So the available energy of three lines alpha after loss of energy through Ag foil will be initial energy for LiF samples. So by subtracting the alpha energy loss through silver foil from initial energy of alpha source, the energy available for LiF sample are calculated. Figure 7 (left) shows the calibration curve for three alpha energies with channel number and the relation is found to be Energy(KeV ) = 102.24 + 6.50 × channel number Total energy loss for both silver and LiF samples are determined from the peak shift as shown in figure 7 (right). Thickness measurement of LiF sample for three alpha energy loss are tabulated in table 2. Loss in energies through Ag in table 2 are calculated assuming the average thickness (0.35 µm) of silver foils.Thickness of LiF is calculated using the same relation (equation 2) as used for Ag. Uniformity of LiF sample is calculated using same procedure of collimator at different position. Maximum variation in thickness of LiF sample is with in 6 % from the mean value. The presence of LiF samples on Ag backing are confirmed by X-ray photo electron spectroscopy (XPS). It is a surface sensitive technique used to identify the atoms present in sample surface. The XPS of samples were characterized using non-monochromatic Mg K α x-ray source with energy of 1253.6 eV. The hemispherical e − analyser with radius of 150 mm and 20 eV pass energy were used to observe the ejected photo electrons. The XPS data were analysed by XPSpeak41 software [8]. The background is also subtracted by Shirley method using XPSpeak41. The figure 6 shows the XPS spectra of LiF sample. The spectra shows peaks at 29.58 eV and 684.62 eV denoting F 2s [9] and F 1s [10] respectively. The peak at 55.54 eV denotes Li 1s state [11]. XPS analysis confirms the presence of 47.43 % of Li and 52.56 % of F atoms in the deposited targets. Conclusion In this paper, a smooth and detailed procedure is discussed for fabrication of LiF target on self-supporting Ag backing. Good number of LiF targets are fabricated successfully on self-supporting Ag backing. The thickness of one of the target of LiF is found to be 183.74 µg/cm 2 by observing loss of energy of three line alpha source. However, the thickness can be tuned by monitoring the current and evaporation time. Maximum non-uniformity observed is within 6 %. So good uniformity is achieved during deposition. This targets are intended to be used for mostly alpha and proton induced reaction especially suitable for nuclear astrophysics experiments. The high melting point of LiF is also suitable for use in high current accelerator facility. XPS spectra confirms the presence of Li and F atom on the surface of sample. The atomic concentration observed from XPS analysis is 47.43 % of Li and 52.56 % of F. Acknowledgement Figure 1 : 1(left) Set-up for thermal evaporation (a)Copper electrode for resistive heating (b) Mounted boat for placing sample (c)Substrate holder (d) Quartz. (Right) Setup for e-beam evaporation in same chamber (e) Crucible for placing sample (f) e-beam shutter (g) Substrate holder with rotating plate. Figure 2 : 2Schematic View of Vacuum Chamber Figure 3 :Figure 4 : 34(Left) Floating process of silver foil (Right) Silver foil mount on Al frame (Middle) Deposited LiF on self supporting Ag foils mounted on substrate holder Set-up for thickness measurement Figure 5 ( 5left) shows the calibration curve of three line alpha source. The equation between energy of three line alpha and channel number is found to be Energy(KeV ) = 183.77 + 7.99 × channel number (1) Figure 5 ( 5right) shows the shift in peak of three line alpha energy with and without Ag foil respectively. Energy loss for three line energy can be calculated from the peak shift in figure 5 (right) using equation 1. Energy loss (∆E) for three alpha energies are 136.93 KeV, 133.78 KeV, and 125.32 KeV. Stopping power for three alpha energy on Ag foils are 0.3361 KeV/(µg/cm 2 ), 0.3241 KeV/(µg/cm 2 ), and 0.3137 KeV/(µg/cm 2 ) Figure 5 : 5(left) Calibration curve (right) Shift in peak due to energy loss through Ag foil Figure 6 : 6(left) Calibration curve (right) Shift in peak due to energy loss through Ag+LiF foil Table 1 : 1Materials used and PropertiesMaterials Melting point( 0 C) Current Mode of evaporation BaCl 2 962 7-15 mA e − beam Ag 961.8 33-50 mA e − beam LiF 848.2 60-65 A Resistive heating Table 2 : 2Thickness tabulation of LiF sampleEnergy loss in Ag (KeV) Total en- ergy loss in Ag+LiF (KeV) Energy loss in LiF (KeV) -dE dx of LiF (KeV/ (µg/cm 2 )) Thickness of LiF (µg/cm 2 ) Avg. thickness of LiF (µg/cm 2 ) 123.04 251.42 128.38 0.7152 179.50 118.65 244.85 126.20 0.6845 184.36 183.74 114.84 237.96 123.12 0.6571 187.36 The authors would like thank Prof. Supratic Chakraborty, Prof. Satyazit Hazra, Ms. Mousri Paul, and Mr. Subhankar Mondal of Surface Physics and Material Science division,SINP,Kolkata for allowing us to use XPS facility for material characterization. . I L , N UCLEAR AND P ARTICLE P HYSICS. 40125102I. L. et al., J OURNAL OF P HYSICS G: N UCLEAR AND P ARTICLE P HYSICS 40 (2013) 125102. . F , Physical review. 114201F. et al., Physical review 114 (1959) 201. . H , Nuclear Physics A. 193H. et al., Nuclear Physics A 193 (1972) 497-509. . S E , Z. Phys. A -Hadrons and Nuclei. 342S. E. et al., Z. Phys. A -Hadrons and Nuclei 342 (1992) 471-482. . D , NIM A. 686D. et al., NIM A 686 (2012) 75-81. . S D , NIM B. 308S. D. et al., NIM B 308 (2013) 46-53. . J F , NIM B. 268J. F. et al., NIM B 268 (2010) 1818-1823. . K H , Physica Scripta. 1K. H. et al., Physica Scripta 1 (1970) 277-280. . M G , J. Phys. Chem. Solids. 41785M. G. et al., J. Phys. Chem. Solids 41 (1980) 785. . M W , J. Am. Chem. Soc. 95751M. W. et al., J. Am. Chem. Soc. 95 (1973) 751.
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[ "A Lightweight CNN-Transformer Model for Learning Traveling Salesman Problems", "A Lightweight CNN-Transformer Model for Learning Traveling Salesman Problems" ]
[ "Minseop Jung \nDepartment of Computer Science and Engineering\nIncheon National University\n22012IncheonSouth Korea\n", "Jaeseung Lee \nDepartment of Computer Science and Engineering\nIncheon National University\n22012IncheonSouth Korea\n", "Jibum Kim \nDepartment of Computer Science and Engineering\nIncheon National University\n22012IncheonSouth Korea\n" ]
[ "Department of Computer Science and Engineering\nIncheon National University\n22012IncheonSouth Korea", "Department of Computer Science and Engineering\nIncheon National University\n22012IncheonSouth Korea", "Department of Computer Science and Engineering\nIncheon National University\n22012IncheonSouth Korea" ]
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Article history:Traveling salesman problem, Combinatorial optimization problem, CNN-Transformer, Lightweight model ABSTRACT Transformer-based models show state-of-the-art performance even for large-scale Traveling Salesman Problems (TSPs). However, they are based on fully-connected attention models and suffer from large computational complexity and GPU memory usage. We propose a lightweight CNN-Transformer model based on a CNN embedding layer and partial self-attention. Our CNN-Transformer model is able to better learn spatial features from input data using a CNN embedding layer compared with the standard Transformer models. It also removes considerable redundancy in fully connected attention models using the proposed partial self-attention. Experiments show that the proposed model outperforms other state-of-the-art Transformer-based models in terms of TSP solution quality, GPU memory usage, and inference time. Our model consumes approximately 20% less GPU memory usage and has 45% faster inference time compared with other state-of-the-art Transformer-based models. Our code is publicly available at https://github.com/cm8908/CNN_Transformer3.
10.48550/arxiv.2305.01883
[ "https://export.arxiv.org/pdf/2305.01883v1.pdf" ]
258,461,478
2305.01883
ffd5a2ce4e310476ab6f312ea0dd57b576d3f9ea
A Lightweight CNN-Transformer Model for Learning Traveling Salesman Problems Minseop Jung Department of Computer Science and Engineering Incheon National University 22012IncheonSouth Korea Jaeseung Lee Department of Computer Science and Engineering Incheon National University 22012IncheonSouth Korea Jibum Kim Department of Computer Science and Engineering Incheon National University 22012IncheonSouth Korea A Lightweight CNN-Transformer Model for Learning Traveling Salesman Problems 1 Article history:Traveling salesman problem, Combinatorial optimization problem, CNN-Transformer, Lightweight model ABSTRACT Transformer-based models show state-of-the-art performance even for large-scale Traveling Salesman Problems (TSPs). However, they are based on fully-connected attention models and suffer from large computational complexity and GPU memory usage. We propose a lightweight CNN-Transformer model based on a CNN embedding layer and partial self-attention. Our CNN-Transformer model is able to better learn spatial features from input data using a CNN embedding layer compared with the standard Transformer models. It also removes considerable redundancy in fully connected attention models using the proposed partial self-attention. Experiments show that the proposed model outperforms other state-of-the-art Transformer-based models in terms of TSP solution quality, GPU memory usage, and inference time. Our model consumes approximately 20% less GPU memory usage and has 45% faster inference time compared with other state-of-the-art Transformer-based models. Our code is publicly available at https://github.com/cm8908/CNN_Transformer3. Introduction The Traveling Salesman Problem (TSP) is a classic NP-Hard problem in computer science and operation research that seeks to find the shortest possible route to visit every city exactly once and return to the starting city [1]. Finding an optimal solution for the TSP is computationally expensive when the number of cities is large. Researchers have studied a variety of heuristics and approximation algorithms that can provide high-quality solutions to the problem in a reasonable amount of time. One of the most simple and popular heuristics is a nearest-neighbor heuristics. It starts at a randomly chosen city and repeatedly selects the nearest unvisited city as the next city to visit while there are unvisited cities. Finally, it returns to the starting city to complete the tour. Another famous heuristics for TSP is the Christofides algorithm [2]. It finds an approximate solution using the minimum spanning tree of the graph representing the cities, which is a tree that connects all the cities with the minimum possible total edge weight. It is known to provide a solution that is guaranteed to be within a factor of 3/2 of the optimal solution. Another famous tool for solving TSP using a heuristics approach is to use Google-OR tools [3]. It performs local search and meta-heuristics to find the approximate solutions of a wide range of combinatorial optimization problems such as TSP and vehicle routing problems. However, heuristic approaches trade optimality for computational cost and are expressed in the form of rules [4,5]. For many TSP instances, Concorde is considered as the fastest and most exact TSP solver that produces the optimal solution [6]. Concorde uses an Integer Programming solver with Cutting Planes and Branch-and-Bound. It assumes a symmetric TSP where the distance between two cities is the same in each opposite direction [7]. Gurobi also finds optimal TSP results but Concorde is faster than Gurobi because it is specialized for TSPs [4,8]. Many studies have been conducted to find an approximate solution of TSP based on deep learning. Among these studies, the pioneering work is the Pointer Network [7]. It is a supervised learning-based approach that uses RNN-based encoders and decoders. In the experiment, a planar symmetric TSP is assumed, and they use a beam search decoding procedure to remove invalid tours such as visiting the same city twice or ignoring a destination. Bello et al. updated the learning parameters of the LSTM-based model with a reinforcement learning-based approach that uses the tour length as a negative reward signal [9]. Nazari et al. added an embedding instead of using the RNN encoder of the Pointer Network to reduce the computational complexity without impacting performance [10]. Joshi et al. proposed a method for predicting the edge probability matrix of the entire graph through a graph convolutional neural network model and a supervised learning-based approach [11]. Stohy et al. proposed a hybrid pointer network model for TSP that demonstrated good performance for large-scale TSP instances [12]. However, it suffers from a long inference time towing to a more complex model structure compared to the baseline graph pointer network [13]. Several efforts have been made to introduce a convolutional neural network (CNN) to the TSP. Researchers used 2D convolution for TSP but did not show good performance [14,15]. Sultana et al. introduced a new model that combines 1D-CNN with LSTM but is still an RNN-based model [16]. Recent attention-based transformer models have shown good performance in various research fields [17,18,19,20]. Researchers successfully used a transformer-based model to find approximate solutions for TSP [21,22,4,23,5,24,25]. Deudon et al. proposed a novel approach for solving TSP using deep reinforcement learning. The city coordinates are utilized as inputs, and the model is trained using reinforcement learning to predict a distribution of a city sequence [21]. Kool et al. proposed a transformer-based model consisting of purely attention blocks and trained the model using REINFORCE for solving various routing problems such as TSP and vehicle routing problems [4]. Wu et al. proposed a transformer-based deep reinforcement learning framework that trains an improvement heuristic that iteratively improves an initial solution [22]. Researchers proposed an approach that applied multiple rollout and data augmentation methods to Kool's attention model [23]. Recently, Bresson et al. proposed a TSP Transformer model [5]. It is based on a standard Transformer encoder with multi-head attention and residual connection but uses batch normalization instead of using layer normalization. It uses an auto-regressive decoding approach and introduces a self-attention block in the decoder part. It constructs the query using all cities in the partial tour with a self-attention module [5]. They showed a state-of-the-art (SOTA) performance for various TSP instances and reported performance with an optimal (optimality) gap of 0.0004% for TSP50 and 0.39% for TSP100. Although the TSP Transformer model shows the SOTA for many TSP instances, it has a complex model structure based on a fully-connected attention-based model and also involves a large GPU usage. Moreover, the training and inference time are very long [26]. Recently, various studies have been conducted to reduce the computational complexity of standard transformer models [27,28,29]. For TSP, a recent study has been conducted to make the model lightweight while removing the learnable decode [24]. A similar study to lightweight TSP Transformer model is performed in [25] . Yang et al. proposed a memory-efficient Transformer-based model for TSP. Their model successfully reduces GPU/CPU memory usage compared with the standard Transformer-based models [25]. In this paper, we propose a novel CNN-Transformer model based on partial self-attention by performing attention only on recently visited nodes in the decoder. Linear embedding in the standard Transformer model does not consider local spatial information and has limitations in learning local compositionality. We add a CNN embedding layer to the standard Transformer model to extract the local spatial features of the input data as the CNN is effective in learning the spatial invariance of nodes in the Euclidean space. Second, the standard Transformer model is based on fully-connected attention-based models [26]. Therefore, it suffers from huge computational complexity and memory consumption. Furthermore, the Transformer model structure has a weakness at learning local compositionality owing to its fully-connected topology. For TSP, we improve the attention mechanism by proposing partial self-attention that focuses only on recently visited nodes in the decoder. We observed that it significantly reduces the redundancy in a fully-connected topology in the Transformer model for TSP and also improves the quality of TSP solution by removing excessive attention connections. The main contributions of our paper are summarized as follows: -To the best of our knowledge, we propose the first CNN-Transformer-based model for learning TSP solutions. Our results show that the CNN embedding layer is very effective in learning local spatial features of various TSP instances. -The proposed model is based on partial self-attention that performs attention only on recently visited nodes in the decoder. Therefore, the proposed model is able to better learn local compositionality compared with the standard Transformer model that is based on the fully-connected topology. -By removing the redundant attention connections in the decoder, the proposed model significantly reduces the GPU memory usage and has much lower inference time compared with the standard Transformer model. Proposed Model We propose a CNN-Transformer model that combines a convolutional neural network embedding layer with a standard Transformer model. The proposed CNN-Transformer model has encoders as well as decoders but our model has a CNN embedding layer to extract local spatial information. We improve the attention mechanism based on a partial selfattention, which removes unnecessary attention connections. , where x i ∈ R 2 represents the 2D Cartesian coordinates of the points. The output of the model, denoted as π π π = {π 1 , . . . , π n }, is a sequence that represents the optimal predicted tour where π t is the node index selected at the t th decoding step at the decoder. Let D(x i , x j ) be the distance between nodes x i and x j . Our goal is to minimize the total tour length n−1 t=1 D(x πt , x πt+1 ) + D(x πn , x π1 ) while visiting each node exactly once and then return to the starting node. Encoder The proposed encoder is composed of a CNN embedding layer and L identical encoder layers as illustrated in Figure 2. The CNN embedding layer generates embedding vectors by extracting local spatial information from the input data points, which is passed on to the subsequent encoder layers. Each encoder layer consists of two sublayers: multi-head self-attention (MHSA) sublayer and point-wise feed forward (FF) sublayer. The MHSA sublayer performs multi-head self-attention to capture the dependencies between each node and the point-wise FF performs non-linear activation. The residual connection [30] and batch normalization [31] were incorporated between each sublayer. Similar to previous studies [4,5], we use batch normalization instead of using layer normalization as it can effectively handle a large number of nodes. CNN embedding layer We add the CNN embedding layer in the encoder for extracting spatial information from the input nodes. Let X k-NN i be the concatenation of feature vectors of the i th node and its k-nearest neighboring node feature vectors closest in distance to the i th node. Then, the embedding vector of the i th node in the encoder, x emb i , is computed as the sum of node embedding of i th input nodes, x i W emb and the convolution of X k-NN i , which is denoted as: x emb i = x i W emb + Conv(X k-NN i ) ∈ R d ,(1) where W emb ∈ R 2×d is a learnable parameter for node embedding. A fixed value of kernel size k+1 is used to ensure that Conv(X k-NN i ) is in the same d-dimension space as node embedding, x i W emb . Encoder layer The encoder has L identical encoder layers, and the first encoder layer takes x emb i from CNN embedding layer as input. Each encoder layer has two sublayers: MHSA sublayer and point-wise FF sublayer. MHSA sublayer. The input of the MHSA sublayer of l th encoder layer, E l−1 , is the output of the (l − 1) th encoder layer. The output of the MHSA sublayer of l th encoder layer,Ê l , is obtained by first applying multi-head self-attention (MH l ) to E l−1 , followed by residual connection and batch normalization (BN l ). The function MH l (Q, K, V) takes three inputs Q, K, V, which represent the query, key, and value vectors, respectively, to perform multi-head attention mechanism at the l th encoder layer. The output of the MHSA sublayer,Ê l , is formulated as follows: E l = BN l E l−1 + MH l E l−1 , E l−1 , E l−1 ,(2) where E 0 = {x f , x emb 1 , . . . , x emb n } ∈ R (n+1)×d . Here, E 0 , the input of the first encoder layer, is created by concatenating start token x f with {x emb 1 , . . . , x emb n }. We add x f to create a virtual node feature vector that learns dependencies with other node features, so that the decoding can start at the best possible location [5]. Point-wise FF sublayer. The input of the point-wise FF sublayer in the l th encoder layer, which is composed of two linear projections and a ReLU activation, isÊ l . It performs non-linear activation followed by residual connection and batch normalization and produces output E l , which is denoted as: E l = BN l Ê l + FF l Ê l ,(3) where FF l is a FF sublayer of l th encoder layer. Encoder output. Let e L i be the encoder output of the i th node of L th encoder layer and f be the index of start token, respectively. The final encoder output of L th encoder layer, E L = {e L f , e L 1 , . . . , e L n }, is produced and fed into the decoder. Decoder We perform decoding auto-regressively, one node at a time. The decoder is comprised of four layers, each of which are followed by residual connection and layer normalization. The first layer is multi-head partial self-attention layer (MHPSA), which extracts past information by performing attention with encoder outputs of already visited nodes in previous steps. Unlike previous works, we use fewer reference vectors for performance and computational efficiency. The second layer is a masked multi-head attention layer (MMHA), which performs attention mechanism where the query is an output of MHPSA layer output and reference vectors are encoder outputs of unvisited nodes. The third layer is the point-wise FF layer, which performs linear projection and non-linear activation, similar to an point-wise FF sublayer in the encoder. The pointer layer selects the next node to visit by calculating a probability distribution over the unvisited nodes. MHPSA layer. We are motivated by the fact that recently visited nodes are more relevant to the node to be selected in the current step than nodes that are visited earlier. Based on this intuitive fact, the proposed partial self-attention performs attention only on recently visited nodes. We expect that the proposed model is able to better learn local compositionality compared with previous works based on fully-connected attention. MHPSA performs self-attention using the decoder input of current time step t, h t , as query and decoder inputs of recently visited nodes as reference vectors. The decoder input at time step t, denoted by h t , is calculated as follows: h t = e L πt−1 + PE t ,(4) where π 0 = f and PE t denotes positional encoding at time step t. The proposed partial self-attention uses decoder inputs for only the m last visited nodes as reference vectors. For instance, suppose the current time step is t, then the decoder inputs used at time step t − m to t are used as reference vectors, denoted as H t = {h t−m , . . . , h t } ∈ R m×d . Consequently, memory usage and computation time are significantly reduced. The output of MHPSA layer,ĥ t , is calculated as follows: h t = LN h t + MH L+1 (h t , H t , H t ) ,(5) where LN refers to layer normalization. MMHA layer. Masked multi-head attention layer performs attention mechanism usingĥ t as query and E L as reference vectors. We use the same masked multi-head attention layer as [5]. Let ζ t ∈ R n+1 be the mask where the value is one for unvisited and zero for visited nodes to the attention weight, respectively. Then, the output from the MMHA layer is calculated as follows:h t = LN ĥ t + MMH L+1 ĥ t , E L , E L , ζ t ,(6) where MMH(Q, K, V, ζ) is a modified function of MH(Q, K, V ), which takes an additional input mask ζ and replaces Attention function [17] with MaskedAttention function formulated as follows: MaskedAttention(Q, K, V, ζ) = softmax QK √ d k ζ V, where denotes element-wise product operation. Point-wise FF layer. The input of the point-wise FF layer ish t and the output ish t , which is denoted as: h t = LN h t + FF L+1 h t .(7) Pointer layer. The goal of the pointer layer is to compute a distribution over unvisited nodes. We perform singlehead attention by usingh t as query and E L as reference vectors. We use attention weights as probability distributions, p t , that determine the next node to visit. Masking is used to avoiding already visited nodes. Then, p t can be computed as: p t = softmax c · tanh qK √ d ζ t ,(8) where q and K are query and reference vectors of single head attention and c is a hyperparameter that controls the range of the logits [9]. During the training phase, the decoder considers p t as a categorical distribution for sampling node indices, and the node index with the highest probability is selected during the inference phase. This process is repeated n times, resulting in a node indices sequence π π π = {π 1 , . . . , π n }, which is the final output of the model. Model training based on reinforcement learning We trained our model based on reinforcement learning. The loss function is the average tour length. Let θ be training model parameters. Then, p(π π π; θ) is the probability that the model generates π π π, which can be defined as: p(π π π; θ) = n t=1 p(π t |π 1 , . . . , π t−1 ; θ), where p(π t |π 1 , . . . , π t−1 ; θ) is the probability that π t is chosen from p t at time step t. We use REINFORCE algorithm to update θ. A duplicate version of θ, θ b , is used as a baseline. We denote π π π an index sequence that the model parameterized by θ b generates in a greedy way. Let (π π π) and (π π π ) be the total tour length of node sequences of training and baseline models, respectively. Then, a REINFORCE loss L(θ) is optimized by gradient descent methods using the REINFORCE algorithm: L(θ) = E π π π∼p(π π π;θ) [ (π π π) − (π π π )] . The gradient of L(θ) is computed as: ∇ θ L(θ) ≈ π π π ( (π π π) − (π π π )) ∇ θ log p(π π π; θ). We optimize the training model using ∇ θ L(θ) in one epoch. When one epoch ends, we compare the average tour length of training and baseline models. We copy θ to θ b if the average tour length of the training model is shorter than that of the baseline model. Experiment Datasets Random dataset. We assume a 2D planar symmetric TSP. For model training and validation, we use a randomly generated data from a uniform distribution on the fly in [0, 1] × [0, 1]. We generated 10,000 test instances. We trained and tested on TSP problems of n = 50 (TSP50) and 100 nodes (TSP100). The output tour using Concorde [6] was considered to have exact solutions and labels of test instances. TSPLIB dataset. TSPLIB [32] is a widely-used benchmark dataset for evaluating the performance of TSP solvers on a variety of real-world data with varying distributions. We choose ten 2D-Euclidean problem instances from TSPLIB, which are considered relatively difficult. Let N and A be the number of nodes and the square area covered by the nodes, respectively [16]. Then, we use a critical parameter value, l √ N ·A , to evaluate the difficulty level of TSPLIB where l is the optimal tour length [33,34]. A critical parameter value close to 0.75 indicates a higher difficulty level. We normalize each TSPLIB instance such that all TSPLIB instances are in [0, 1] × [0, 1]. Hyperparameters and decoding strategy Hyperparameters. We did not tune the hyperparameters of the proposed model. For model training, we use an Adam optimizer with a fixed learning rate of 0.0001. A batch size of 512 is used. We train for 100 epochs using training data but further increasing the number of epochs could improve the model's performance. Our experiments were conducted on an AMD EPYC 7513 32-Core Processor and a single Nvidia A6000 GPU. Decoding strategy. At test time, we employ both greedy and beam search for decoding. Beam search [35] is a breadth-first search strategy that considers top-B cases in every decoding time step and chooses the best solution at the end of the decoding. We set the beam width (B) as 2,500 in order to compare our results with other SOTA models [5] that use beam width of 2,500. List of Experiments In our study, we conducted four experiments to evaluate the proposed model's performance. The third experiment used both the random dataset and the TSPLIB dataset, while the other experiments only used the random dataset. For Experiments 1 and 2, we use greedy decoders. Experiment 1. We test whether the proposed partial self-attention in the decoder is more effective in improving the performance of the model compared with the existing fully-connected self-attention in the standard Transformer decoder. We test by decreasing the number of reference vectors (m) from 100 to 5. in extracting local spatial information and produces better output performance by performing an ablation study. Experiment 3. We compare the proposed model with other solvers, including the optimal solver (Concorde [6]), heuristic solver (Google OR-Tools [3]), and other SOTA transformer-based models for TSP [4,5]. Table 1 Metrics The performance of the model was evaluated using the following metrics. Average predicted tour length. Letl TSP i be the predicted tour length of the i th instance. Then, the average tour length is computed as 1 n n i=1l TSP i , where n is the number of total test instances. Here, we set n as 10,000. Optimality gap. The optimality gap is computed as the average percentage of the predicted tour length to the optimal solution, which is computed as 1 n n i=1 l TSP i l TSP i − 1 , where n is the number of total test instances. Here, l TSP i is the optimal solution of the i th instance produced by Concorde [6]. We set n as 10,000. Training time. The training time is measured as the time taken to train 10,000 instances. Inference time. We report the inference time taken to solve the test set of 100 instances of TSP100 using beam search decoding. Beam width (B) was set to 2,500 and batch size was set to one due to the limitation of memory capacity. GPU Memory Usage. Maximal memory usage capacitated by the training process is measured. Table 3 presents the results of the ablation study. Both models use greedy decoders. Our results show that for both TSP50 and TSP100, the proposed model outperforms the proposed model without the CNN embedding layer. We observed that removing the CNN embedding layer degrades the overall performance. Therefore, we can conclude that the CNN embedding layer in our model is effective in extracting spatial features and improving the output performance. Experiment 2 Experiment 3 Random dataset. Table 4 compares the performance of the proposed model with other SOTA Transformer-based models. We present the results by dividing the table into three sections: exact solver, heuristics, and deep learning models. Concorde [6], which is known to produce optimal results, shows the best performance. Among deep learning models, the proposed model has the best performance both for greedy and beam search decoding. The proposed model also outperforms the heuristics (OR-tools [3]) for both TSP50 and TSP100. For TSP50, the proposed model is only behind by a 0.1% optimality gap compared with the exact solver, Concorde. Our model with B=2,500 reduces the optimality gap from 0.11% to 0.10% for TSP50 and 1.26% to 1.11% for TSP100 compared with the TSP Transformer. TSPLIB dataset. Two most difficult TSPLIB instances are the kroC100 and the berlin52 instances. The berlin52 is known to be a hard TSP instance because many nodes are highly constrained in very small regions. For both very hard real-world instances, the proposed model outperforms the TSP Transformer model. Figure 4 displays the output of the proposed model and TSP Transformer on kroC100 and berlin52. For the kroC100, the optimal tour length of Concorde is 20,749. The average tour length of our model is 21,523, while that of the TSP Transformer model is 21,788. The average tour length of our model is 7,610 but the average tour length of the TSP transformer model is 7,637. The main strength of our model lies in its reduced GPU memory usage due to the partial self-attention in the decoder. We remove redundant attention connections by applying the partial self-attention in the decoder, which reduces inference time. Specifically, the proposed model consumes approximately 20% less GPU memory compared with the TSP Transformer model. Owing to its lower memory consumption, our model is able to use a larger batch size and further decrease the training time. Another strength of our model lies in its significantly faster inference time compared to the TSP Transformer. Specifically, the inference time of our model is 45% less than that of the TSP Transformer for TSP100. Discussion Recent studies tried several heuristic search algorithms such as Monte Carlo Tree Search [36,37] or 2-opt search [21,22] to further enhance the quality of TSP solutions. The proposed model uses beam search decoding techniques, but other heuristic search algorithms for TSP can be combined to further enhance the performance. We also observed that shortest tour heuristic proposed in [11] which selects the shortest tour among the set of B complete tours also improve the output performance of the model. Our results show that the proposed model is successful in significantly reducing memory consumption and inference time. Our model is also based on a standard Transformer model with multiple layers of transformer blocks and adds a CNN embedding layer to extract spatial features. We plan to apply various lightweight techniques for Transformer-based models such as proposed in [26,27,28,38,29] and also for CNN [39,40,41,42]. Conclusion In this paper, we proposed a CNN-Transformer model based on a partial self-attention in the decoder. Our model is able to extract and learn spatial features from the input data and produces better TSP solutions compared with the standard Transformer-based models. Our results show that our model is able to better learn local compositionality compared with the standard Transformer model. We also observed that the proposed model significantly reduces the GPU memory usage and inference time by applying partial self-attention in the decoder. The proposed partial self-attention is effective in improving the TSP solution quality by removing the redundant and excessive attention connections in the decoder. The proposed model outperforms the existing SOTA Transformer-based models in terms of solution quality, GPU memory usage, and inference time for random as well as real-world datasets. Fig. 1 . 1Overview of the proposed CNN-Transformer model for TSP. Figure 1 1illustrates the overall structure of the proposed model. The input of the proposed model is a planar point set X = {x 1 , . . . , x n } with n nodes (cities) Fig. 2 . 2Proposed encoder architecture with the CNN embedding layer and L identical encoder layers. Figure 3 Fig. 3 . 33illustrates an example for the decoding process when the current time step (t) is 10 and the number of reference vectors (m) is three. Proposed decoder architecture. This figure shows a decoding process when the current time step t = 10 and the number of reference vectors m = 3. We use the same hyperparameters for all TSP problem sizes. The proposed model has six encoder layers (L = 6) with k = 10. The kernel size of CNN embedding layers is set to 11. The model has 128 hidden dimensions (d = 128) and the hidden dimension of each point-wise FF layer (also sublayer) and is set to 512. The model has eight attention heads. The logit range clipping value c in the decoder is set to 10. summarizes the main difference between the proposed model with other Transformer-based models. Here, 'node' means the hidden feature vector for a particular node generated by the encoder or MHSA layer of each model and 'graph' is the average value of the hidden feature vectors of all nodes. Experiment 4 . 4We compare the computational complexity between the proposed CNN-Transformer model with other Transformer-based models. 2 presents the average tour length and the optimality gap of our model with various m values. The experimental results show that the performance of our model improves as the m value decreases. We also observe that the smallest m value (i.e., m = 5) shows the best output performance. For TSP100, the proposed model using partial self-attention with m = 5 achieved an optimality gap of 2.83%, but the proposed model using the fully-connected attention, i.e., m = 100, produced an optimality gap of 3.10%. The proposed model using partial self-attention, which focuses on recent nodes, is effective in improving the performance compared with the model using the fully-connected attention. We also observed that excessive attention connection can degrade the model performance. From here, we select the number of reference vectors (m) as five because it shows the best model performance among compared m values. Traditional optimization-based solvers like Concorde[6] still outperform neural network models in terms of performance, but neural net models have faster inference time compared with Concorde. For example, Concorde takes approximately 0.16 seconds to solve 10,000 TSP instances of TSP100 whereas the proposed model takes approximately 0.1 seconds. Table 1 . 1Experiment 2.We remove the CNN embedding layer in our CNN-Transformer model and test whether it is effectiveComparison of transformer-based models for TSP: Kool [4] and TSP Transformer [5] Here, MHSA and MMHA denote Multi-Head Self-Attention and Masked Multi-Head Attention, respectively. MHSA was not used in Kool [4] Model Encoder Decoder Embedding MHSA MMHA Query Key Query Key Kool [4] Linear embedding - - First node + last node + graph Every unvisited node TSP Transformer [5] Linear embedding Last node Every visited node Last node Every unvisited node Ours CNN embedding Last node Last-m visited node Last node Every unvisited node Table 2 . 2Average tour length and the optimality gap (%) of our model with various m (number of reference vectors) values on TSP50 and TSP100 using the random dataset. Here, our model with m = 100 indicates the fully-connected attention models and other m values indicate partially-connected attention models. The greedy decoder is used for this experiment. TSP50 TSP100 m Tour length Gap Tour length Gap 100 - - 8.005 3.10% 50 5.750 1.06% 7.993 2.94% 20 5.748 1.01% 7.988 2.87% 5 5.745 0.97% 7.985 2.83% Table 3. Ablation study. Average tour length and the optimality gap (%) of our model without CNN and with CNN on TSP50 and TSP100 using the random dataset. The greedy decoder is used for this experiment. TSP50 TSP100 Tour length Gap Tour length Gap Ours (without CNN) 5.754 1.12% 8.004 3.08% Ours 5.745 0.97% 7.985 2.83% 4. Results 4.1. Experiment 1 Table Table 4 . 4Average tour length and optimality gap of various models and solvers on TSP50 and TSP100 using the random dataset. Here, B and * mean the beam size and the result from other papers, respectively. TSP50 TSP100 Method Type Tour length Gap Tour length Gap Concorde [6] Exact solver 5.689 0.00% 7.764 0.00% OR Tools [3]* Heuristic 5.80 1.83% 7.99 2.90% Kool et al. [4]* Greedy 5.80 1.76% 8.12 4.53% Beam search (B=5000) 5.72 0.47% 7.93 2.18% TSP Transformer [5] Greedy 5.750 1.05% 8.015 3.22% Beam search (B=2500) 5.696 0.11% 7.863 1.26% Ours Greedy 5.745 0.97% 7.985 2.83% Beam search (B=2500) 5.695 0.10% 7.851 1.11% Table 5. Average tour length and optimality gap for the TSPLIB instances. The critical parameters close to 0.75 indicate harder TSP instances. The greedy decoder is used for this experiment. Problem Critical parameter Concorde TSP Transformer Ours Tour length Gap Tour length Gap kroC100 0.75 20,749 21,788 5.01% 21,523 3.73% berlin52 0.74 7,542 7,637 1.26% 7,610 0.90% kroA100 0.77 21,282 21,747 2.18% 21,620 1.59% ch150 0.78 6,528 7,390 13.20% 7,050 8.00% ch130 0.78 6,110 6,569 7.51% 6,552 7.23% rd100 0.81 7,910 8,078 2.12% 8,221 3.93% st70 0.86 675 710 5.19% 676 0.15% eil101 0.98 629 681 8.27% 673 7.00% eil76 1.03 538 565 5.02% 564 4.83% eil51 1.05 426 438 2.82% 429 0.70% Table 5 5presents the output performances for our model and TSP Transformer[5] on various TSPLIB instances. Both models were trained on TSP50. We observe that the proposed model gives consistent results for various TSPLIB instances and outperforms TSP Transformer in most TSPLIB instances. Therefore, the proposed model using partial self-attention is also effective in real-world datasets. The proposed model outperforms TSP Transformer for all TSPLIB instances except one case, i.e., rd100.Fig. 4. Output tours of Concorde[6], TSP Transformer[5], and our model for (a) kroC100 and (b) berlin52 using the TSPLIB dataset.Concorde Tour Length: 20749 TSP Transformer Tour Length: 21788 Ours Tour Length: 21523 (a) kroC100 Concorde Tour Length: 7542 TSP Transformer Tour Length: 7637 Ours Tour Length: 7610 (b) berlin52 Table 6 . 6Comparison of model parameters, GPU memory usage, and training (T) time for 10,000 TSP instances and inference (I) time for 100 TSP instances of TSP100. Here, 'GPU memory' means the GPU memory usage during training process using batch size of 512.Table 6summarizes the overall model complexity and runtimes of TSP Transformer and our model for TSP100. The proposed model has more model parameters compared with the TSP Transformer as we added the CNN embedding layer in our model. Owing to the increased number of parameters, the training time of the TSP Transformer is slightly lower than that of our model when the batch size is 512. 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Xu, Solve traveling salesman problem by monte carlo tree search and deep neural network, arXiv preprint arXiv:2005.06879 (2020). S Mehta, M Ghazvininejad, S Iyer, L Zettlemoyer, H Hajishirzi, arXiv:2008.00623Delight: Deep and light-weight transformer. arXiv preprintS. Mehta, M. Ghazvininejad, S. Iyer, L. Zettlemoyer, H. Hajishirzi, Delight: Deep and light-weight transformer, arXiv preprint arXiv:2008.00623 (2020). A G Howard, M Zhu, B Chen, D Kalenichenko, W Wang, T Weyand, M Andreetto, H Adam, arXiv:1704.04861Mobilenets: Efficient convolutional neural networks for mobile vision applications. arXiv preprintA. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, T. Weyand, M. Andreetto, H. Adam, Mobilenets: Efficient convolutional neural networks for mobile vision applications, arXiv preprint arXiv:1704.04861 (2017). 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[ "https://github.com/cm8908/CNN_Transformer3." ]
[ "Analytically determined topological phase diagram of the proximity- induced gap in diffusive n-terminal Josephson junctions OPEN", "Analytically determined topological phase diagram of the proximity- induced gap in diffusive n-terminal Josephson junctions OPEN" ]
[ "JabirMorten Amundsen \nDepartment of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway\n", "Ali Ouassou \nDepartment of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway\n", "Jacob Linder \nDepartment of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway\n" ]
[ "Department of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway", "Department of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway", "Department of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway" ]
[]
Multiterminal Josephson junctions have recently been proposed as a route to artificially mimic topological matter with the distinct advantage that its properties can be controlled via the superconducting phase difference, giving rise to Weyl points in 4-terminal geometries. A key goal is to accurately determine when the system makes a transition from a gapped to non-gapped state as a function of the phase differences in the system, the latter effectively playing the role of quasiparticle momenta in conventional topological matter. We here determine the proximity gap phase diagram of diffusive n-terminal Josephson junctions ( ∈ N n ), both numerically and analytically, by identifying a class of solutions to the Usadel equation at zero energy in the full proximity effect regime. We present an analytical equation which provides the phase diagram for an arbitrary number of terminals n. After briefly demonstrating the validity of the analytical approach in the previously studied 2-and 3-terminal cases, we focus on the 4-terminal case and map out the regimes where the electronic excitations in the system are gapped and non-gapped, respectively, demonstrating also in this case full agreement between the analytical and numerical approach.
10.1038/srep40578
null
15,024,558
1610.08508
f60f32975cbb721177952da57d5ebd2e21ddb85f
Analytically determined topological phase diagram of the proximity- induced gap in diffusive n-terminal Josephson junctions OPEN Published: 17 January 2017 JabirMorten Amundsen Department of Physics Norwegian University of Science and Technology N-7491TrondheimNorway Ali Ouassou Department of Physics Norwegian University of Science and Technology N-7491TrondheimNorway Jacob Linder Department of Physics Norwegian University of Science and Technology N-7491TrondheimNorway Analytically determined topological phase diagram of the proximity- induced gap in diffusive n-terminal Josephson junctions OPEN Published: 17 January 201710.1038/srep40578received: 26 October 2016 Accepted: 07 December 20161 Scientific RepoRts | 7:40578 | Correspondence and requests for materials should be addressed to M.A. (email: [email protected]) Multiterminal Josephson junctions have recently been proposed as a route to artificially mimic topological matter with the distinct advantage that its properties can be controlled via the superconducting phase difference, giving rise to Weyl points in 4-terminal geometries. A key goal is to accurately determine when the system makes a transition from a gapped to non-gapped state as a function of the phase differences in the system, the latter effectively playing the role of quasiparticle momenta in conventional topological matter. We here determine the proximity gap phase diagram of diffusive n-terminal Josephson junctions ( ∈ N n ), both numerically and analytically, by identifying a class of solutions to the Usadel equation at zero energy in the full proximity effect regime. We present an analytical equation which provides the phase diagram for an arbitrary number of terminals n. After briefly demonstrating the validity of the analytical approach in the previously studied 2-and 3-terminal cases, we focus on the 4-terminal case and map out the regimes where the electronic excitations in the system are gapped and non-gapped, respectively, demonstrating also in this case full agreement between the analytical and numerical approach. The interest in topological quantum phases of matter has grown steadily in recent years, and the fundamental importance of this topic in physics was recently recognized by Thouless, Haldane, and Kosterlitz being awarded the 2016 Nobel prize in physics for their contribution to this field. So far, specific material classes such as telluride-based quantum wells (HgTe, CdTe), bismuth antimony (Bi 1−x Sb x ) and bismuth selenide (Bi 2 Se 3 ) have received the most attention in the pursuit of symmetry-protected topological phases and excitations [1][2][3][4] . However, it was recently proposed 5 that similar physics could be obtained using conventional superconducting materials. More specifically, by using multiterminal Josephson junctions, the authors of ref. 5 showed that it was possible to create an artificial topological material displaying Weyl singularities under appropriate conditions. In multiterminal Josephson junctions hosting well-defined Andreev bound states, the crossing of these states with the Fermi level has been shown to be analogous to Weyl points in 3D solids with the Andreev bound state taking on the role of energy bands and the superconducting phase differences corresponding to quasiparticle momenta. A considerable advantage in utilizing multiterminal Josephson junctions rather than 3D solids to study exotic phenomena such as Weyl singularities and topologically different phases is that the phase differences are much more easily controlled experimentally than the quasiparticle momenta. In order to probe electronic excitations with topological properties, a key goal is to map out the phase diagram of the system in terms of when it is gapped or not. A gapped system here means that there are no available excitations in a finite interval around the Fermi level. The reason for why this is important is that transitions between topologically protected states can occur via gap closing, and so by identifying under which circumstances the system makes such a transition provides information about when the topological nature of the system's quantum state changes. The arguably easiest way to probe such a phase transition is via the readily available density of states measurements, which pick up whether or not the system is gapped at a specific energy. The electronic density of states can be probed via conductance measurements, for instance in the form of tunneling between the system and a small metallic tip using so-called scanning tunneling microscopy. Recent previous works have considered the case of 3-terminal Josephson junctions, both in ballistic 6,7 and diffusive systems 8 , and also the 4-terminal case in the case of chaotic cavities being connected to each other and the superconductors 9 . In particular the 4-terminal case is of interest due to the possibility of creating Weyl singularities 5 . In terms of experimental realization, metallic diffusive systems are of high relevance as the conditions for realizing such systems are far less stringent than, for instance, the discrete Andreev bound states of quantum dots. However, the proximity-gap phase diagram has not yet been studied for the 4-terminal case involving diffusive normal metals. Motivated by this, we here determine the proximity gap phase diagram of diffusive n-terminal Josephson junctions (  ∈ n ), both numerically and analytically, by identifying a class of solutions to the Usadel equation 10 at zero energy in the full proximity effect regime. We present an analytical equation which provides the phase diagram for an arbitrary number of terminals n. After briefly demonstrating the validity of the analytical approach in the previously studied 2-and 3-terminal cases, we focus on the 4-terminal case and map out the regimes where the electronic excitations in the system are gapped and non-gapped, respectively, demonstrating also in this case full agreement between the analytical and numerical approach. Our results may serve as a guideline for exploring the interesting physics of multiterminal devices involving the experimentally prevalent and accessible scenario of diffusive metals connected to superconductors, which has a long history 11 . Theory We will use the quasiclassical theory of superconductivity which is known to yield good agreement with experimental measurements on mesoscopic superconducting devices. As only non-magnetic structures will be considered here, only singlet Cooper pairs exist and it is possible to work in Nambu-space alone due to the spin degeneracy. Using a field operator basis ψ ψ ψ = ↑ ↓ † ( , ), the 2 × 2 quasiclassical Green function matrix g describing the existence of superconductivity in the system via the anomalous correlation function f reads: =       −         g g f f g(1) Here, {g, f} are complex scalars that depend on position r and quasiparticle energy E. In a bulk BCS superconductor with order parameter Δ = Δ 0 e iφ , g takes the form: =      − −       φ φ − g c s s c e e ( 2) BCS i i where c ≡ cosh(θ), s ≡ sinh(θ), and θ = atanh[Δ 0 /(E + iδ)]. Here, δ accounts for inelastic scattering processes and causes a smearing of the spectral density. In writing g BCS , we have used that = c c and = − s s. The above matrix may be Ricatti-parametrized 12 in the same way as one would do in the case of non-degenerate spin (see e.g. ref. 13 for a general Ricatti-parametrization in this case) with two differences: (i) we have to let γ γ → −  , and (ii) treat γ γ  { , } as scalars rather than matrices. More specifically, we write the Green function in the form 1 . The Usadel equation in the normal wires, which governs the behavior of the Green function g, reads: γγ γ γ γ γ =       − − −       ∼ ∼    g N N N N (1 ) 2 2 ( 1 )(3)with γγ = = + ∼ −  N N (1 )τ ∂ ∂ + = D g g E g ( ) i[ , ] 0,(4)x x z where D is the diffusion coefficient, τ z is the third Pauli matrix, and E is the quasiparticle energy. Since we are interested in mapping out the regime where the system is gapped, it suffices to consider the behavior of g at the Fermi level (E = 0). In this case, we have γ γ =  ⁎ , and the Ricatti-parametrized Usadel equation [obtained by inserting Eq. (3) into Eq. (4)] determining γ takes the form γ γ γ γ ∂ − ∂ + = . ⁎ 2( ) 1 0 (5) x x 2 2 2 This equation has the following general and exact solution if  γ ∈ : γ = + . x cx c ( ) tan( )(6) 1 2 Although a purely real γ might seem like a very particular case, this scenario in fact allows us to gain important information about the proximity-gap phase diagram. To see this, consider the expression for the normalized (against its normal-state value) density of states  at zero energy:  γ γ = − + . 1 1(7) 2 2 The solution γ = 0 corresponds to the absence of superconducting correlations, i.e. completely closed gap, in which case the density of states resumes its normal-state value = 1  . The solutions γ = ± 1 correspond to the fully gapped case = 0  where no available quasiparticle excitations exist at the Fermi level. The existence of such points can now be identified analytically by determining c 1 and c 2 in Eq. (6) via the boundary conditions in the N-terminal system. We later proceed to do so explicitly. It is also worth noting that Eq. (5) also has a general solution when γ is purely imaginary [  γ ∈ , Re(γ) = 0]: γ = + . x cx c ( ) i tan( )(8) 1 2 The solution Eq. (6) is of particular relevance in the case where the phase differences between the terminals is nπ, with n = 0, 1, 2, … The reason for this is that in such a scenario, one can choose a gauge where all superconducting order parameters are purely imaginary in the reservoirs (phases φ j = π/2 or 3π/2), which renders the BCS anomalous correlation function = φ f se i j to be purely real at zero energy since s(E = 0) = − i. If one assumes ideal boundary conditions at the superconducting interfaces, meaning that f is continuous, there are no imaginary terms in the boundary conditions or in the equation of motion for γ itself, meaning that the solution γ can be taken as real. From Eq. (7), it is clear that the maximum value of the Fermi-level density of states in the presence of a superconducting proximity effect is = 1 max  . We can thus conclude that the analytical approach presented above is valid whenever the superconducting phase differences between the terminals are nπ. The above class of exact solutions are useful since they are valid at specific phase differences and provide information about whether or not the DOS is gapped there. However, we have identified an additional class of exact solutions which is useful because it is valid at any phase-differences where = 0  , which is precisely the regime of interest. By noting that = 0  only when |γ| = 1, a reasonable ansatz is:  γ = − ∈ . S x ie , ( ) (9) S x i ( ) The prefactor − i is just a convention that simplifies the boundary conditions for S. Insertion into Eq. (5) gives immediately = + S x ax b ( )(10) where a and b are real constants determined by the boundary conditions. Besides allowing us to analytically determine the region in phase-space where the system is gapped, this solution also allows us to analytically compute the topological number associated with the gapped regime defined as 14 : = ∇ ⋅ ∮ m S d r r ( )(11) where S(r) is interpreted as the phase of the superconducting correlations at E = 0. There are several ways to relate the Riccati parameter γ to the physical properties of the system. First of all, it can be related to the physically observable density of states using Eq. (7). Moreover, when the system is fully gapped so that the zero-energy density of states = 0  , γ is in fact just the anomalous Green function f, which quantifies the superconducting correlations in the system. This can be seen by comparing Eqs (1) and (3): in general, the anomalous Green function is given by f = 2Nγ, but since γ = − ie iS(r) for a fully gapped system, we find that N = [1 + e +iS(r) e −iS(r) ] −1 = 1/2 using the definition given above. It is assumed that the Green functions in the superconductors may be approximated by bulk expressions, and that the interfaces to the normal metals are transparent. This leads to the boundary conditions S(r j ) = φ j , where r j are the locations of the terminals in Fig. 1, and φ j are the corresponding phases. This can be deduced by comparing with the anomalous Green function in a bulk superconductor, f BCS = − ie iφ . Although Eq. (9) is exact whenever the system is gapped ( = 0  ), it cannot be used carelessly because one still has to specify for which choices of the phases φ j it is valid. It is clearly valid when all phases are equal in the system, so that the phase-difference between all terminals is zero. As we will later show, it is also valid in large regimes of phase-space, and one needs a criterion for when Eq. (9) can be used. Such a criterion can be obtained Since the wires are assumed to be diffusive, their precise geometrical orientation does not influence the topological properties of the system. For instance, the same 3-terminal topological phase diagram would have been obtained if the leads were connected in a Y-shape rather than a T-shape: only the physical properties of the wires (e.g. their Thouless energies) are of consequence. Scientific RepoRts | 7:40578 | DOI: 10.1038/srep40578 in a convenient way by noting that as soon as S(x) acquires a non-zero imaginary part, the consequence is that ≠ 0  . Identifying the condition for when a complex S(x) becomes a possible solution is thus our strategy for describing analytically the topological phase diagram. By using Eq. (9) with  ∈ S x ( ) and writing S(x) = S r (x) + iS i (x), Eq. (5) becomes ∂ + ∂     − +     = S S i( ) 1 2 1 e 0 (12) x x S 2 2 2 i It is observed that the solution of Eq. (12) reduces to Eq. (10) in the limit S i (x) → 0. This means that by allowing a small S i (x), it is possible to map out regions where Eq. (10) is not valid and the imaginary component begins to matter. To do so, we Taylor expand the square bracket of Eq. (12), and insert the perturbation expansion λ λ = + + +… S x S x S x S x ( ) ( ) i( ( ) ( ) )(13)r i i 1 2 2 where  S x S x ( ) ( ) i r 1 and +  S S ik ik 1 . The expansion parameter λ is a helper variable used to collect different orders of the expansion. This gives λ ∂ = S : 0(14) x r 0 2 λ ∂ + ∂ = S S S : ( ) 0(15) x i x r i 1 2 1 2 1 and similarly for higher orders of λ. It is noticed in particular that Eq. (10) remains a solution for S r (x). The first order correction S i1 (x) is easily solved, giving = + S x C ax C ax ( ) cos( ) s in( ) (16) i1 1 2 In an n-terminal Josephson junction with transparent interface between superconductors and the normal metal, it is clear that |γ| = 1 at the interface regardless of the phase. The proper boundary conditions are therefore that S i1 (x j ) = 0, with x j being the position of superconducting interface j. In addition, current conservation at the intersection between the arms of the multiterminal junction requires continuity of the Green function as well as the following relation between derivatives: ∑ γ ⋅ ∇ =  e 0 (17) j j j where γ j is the solution of the Usadel equation in arm j, and  e j is a unit vector pointing towards the intersection. Using these conditions, it is possible to formulate a criterion for when the purely real solution for S(x) is valid, namely: Any combination of boundary conditions for which the only solution for S i1 (x) possible is one where C 1 = C 2 = 0. The curves where this is not satisfied may be found from the boundary conditions for an n-terminal Josephson junction as ∑ ψ ψ = = tan 0 (18) j n j j 1 with ψ j given as ∑ ψ φ φ φ φ = − = − = n 1 (19) j j j k n k 1 Equations (18) and (19) represent a key analytical result in this manuscript as they provide the phase diagram for the proximity-induced gap for an arbitrary number of terminals n. It is emphasized that the curves satisfying Eq. (18) only determine when a small imaginary contribution to S(x) is possible and hence for which phases a transition between gapped and ungapped regimes in phase space occur. These curves are therefore referred to as transition curves. Higher order terms in the perturbation expansion are required in order to more accurately describe the ungapped regions. This is however not necessary when only interested in the gapped regions. It will be shown that it is possible to distinguish between the two regimes using only the first order correction. To complement our analytical considerations, we also perform a fully numerical determination of the proximity-gap phase diagram by solving the Usadel equation numerically for any phase differences φ j and without assuming ideal boundary conditions. In the following sections, we first provide a brief discussion of the already known 2-terminal and 3-terminal cases in order to prove the correctness of our novel analytical approach. Then, we proceed to discuss the less explored 4-terminal case in more detail. We comment here that multiterminal geometries beyond effective 1D models can also be treated using the recently developed 15 numerical solution of the full Usadel equation in 3D, allowing for the study of non-trivial geometrical effects. Moreover, previous works have considered analytical solutions of the Usadel equation using the so-called θ-parametrization in SN bilayers [16][17][18] and also approximate solutions in the SNS case [19][20][21] , whereas in our work the analytical solution is exact for the key cases of (i) = 0  and (ii) for phase differences nπ between the terminals. Scientific RepoRts | 7:40578 | DOI: 10.1038/srep40578 Results: 2-terminal case Assuming ideal boundary conditions at the superconducting interfaces x = − L/2 and x = L/2 see Fig. 1(a), real solutions of γ must satisfy γ = tan(c 1 x + c 2 ) where: − + = − + = − φ φ c L c c L c tan( /2 ) ie and tan( /2 ) ie (20) 1 2 i 1 2 i L R This restricts the superconducting phases to be φ j = {π/2, 3π/2} in order to ensure  γ ∈ . A number of solutions can be obtained from this. If φ L = π/2 and φ R = 3π/2 or vice versa, the solution is c 2 = 0 which gives a DOS in the center of the wire = = x ( 0) 1  . This is the expected result for a phase difference of π between the superconducting terminals. If instead the phase difference is zero, meaning φ φ π π = = { /2, 3 /2} L R , then the solution is c 2 = ± π/4, providing  = = x ( 0) 0. This is also consistent with the result that the DOS is allowed to be fully gapped when there is no phase difference. These results are in agreement with the condition given in Eq. (18), which identifies φ L − φ R = nπ, n = 1, 2, … as the only configurations for which a non-zero density of states is possible. The phase-dependent minigap in an SNS junction was originally considered in ref. 19. Results: 3-terminal case In the 3-terminal case, we consider the geometry of Fig. 1(b). The regions in phase space where = = = x y ( 0, 0) 0  is mapped out using Eq. (18). Since only phase differences matter physically, we fix the phase of one superconducting terminal, φ D = 0, without loss of generality. Transition curves indicating the transition between gapped and ungapped regions are shown in Fig. 2(a) for the extended phase space [− 2π, 2π] × [− 2π, 2π]. It can be seen that one such curve encircles the origin, with a near-elliptical shape, thereby splitting the plane into two regions. It is known that the origin resides in a gapped region, so that the outer region may be identified as ungapped. There also appears several open curves in the second and fourth quadrant. These curves are considered to be metastable solutions, corresponding to a higher phase-winding of the superconducting correlations in the normal wires, and are not investigated further. Due to the 2π-periodicity of the superconducting phases, the physically relevant transition curves must be translated into [0, 2π] × [0, 2π], as shown in Fig. 2(b). The density of states may also be computed analytically in the select points where the boundary conditions are real. Using Eq. (6), the solutions in the left, down, and right arm are written as γ L = tan(c 1 x + c 2 ), γ R = tan(c 3 x + c 4 ), γ D = tan(c 5 x + c 6 ). For this particular calculation, it is necessary to set φ D = π/2 in order to make γ = − = φ ie 1 D BCS, i D real. At the intersection point (x, y) = (0, 0) continuity of the Green function and its derivative ensure continuity of the current. We assume here for simplicity equal lengths and normal-state conductances of the three normal wires, although the analytical treatment does not require this in general. In this case, we obtain the boundary conditions The values of {φ L , φ R } are restricted to π/2 and 3π/2 in order to ensure the validity of the solution for γ. Since  ∈ c tan 2 , the last boundary condition is equivalent to c 1 + c 5 − c 3 = 0. The above non-linear system of equations may be solved analytically, keeping the physically acceptable solution which gives  > 0. For instance, for (φ L , φ R ) = (3π/2, 3π/2) one finds that = − ± c tan( ) 2 3 2 . The positive solution is the physically acceptable one since it provides > 0  . The Fermi-level DOS in the center of the system (x, y) = (0, 0) is given by = = = − + x y c c ( 0, 0) 1 tan( ) 1 tan( ) ,(22) 2 2 2 2  and we find from the solution of c 2 that: φ φ π π φ φ π π φ φ π π φ φ π π = = =             = . = . = . = x y ( 0, 0) 0, if ( , ) ( /2, /2) 0 866, if ( , ) ( /2, 3 /2) 0 866, if ( , ) (3 /2, /2) 0 866, if ( , ) (3 /2, 3 /2) (23) L R L R L R L R  These solutions may be compared with the numerical solution of the full proximity-gap phase diagram in Fig. 3(a), where it can be seen that the analytically determined transition curves of Fig. 2(b) trace out exactly the regions where the density of states is non-zero. The excellent correspondance is explained by the rapid transition between the two regimes, as shown by the numerical solution. In addition, the four red circles are gauge-equivalent to the above phase-choices (note that in the figure we have set φ D = 0). As seen, the analytical expressions match the numerical result. In order to model a more realistic setting with finite interface transparencies, we provide the phase diagram using the Kupriyanov-Lukcihev boundary conditions 22 in Fig. 3(b). The interface transparency is quantified by the parameter ζ = R R / j B j N j , , where R B,j is the barrier resistance and R N,j is the normal-state resistance of wire j. As seen, the gapped region extends compared to the fully transparent case, in agreement with ref. 8. Results: 4-terminal case We now focus on the 4-terminal case and demonstrate both the robustness of the analytical approach developed above in addition to providing comprehensive numerical results. The transition surface in the, now three dimensional, extended phase space is shown in Fig. 4(a), where φ U has been fixed to zero and metastable solutions have been removed for clarity. It can be seen to have an ellipsoidal shape, which is an expected generalization of the 3-terminal case. Figure 4(b-d) show slices of the surface after translation into the first quadrant for φ D = 0, π 2 and π, respectively. The resulting phase diagram displays a more complicated behavior than in the 3-terminal case. At φ D = 0, the phase diagram is similar to the 3-terminal case, but as φ D is increased toward π/2 one of the gapped regions expands greatly at the expense of the other gapped regions which are separated from each other by a "barrier" of finite DOS  ≠ 0. As φ D is further increased toward π, the phase-diagram morphs into a qualitatively different shape than at φ D = 0, and at φ D = π two of the gapped regions have been almost completely expelled from the phase diagram whereas two gapped "valleys" remain, the latter again separated by a non-gapped region. With purely real boundary conditions, and φ = This non-linear system of equations may be solved analytically. Due to the requirement that  γ ∈ , we restrict our attention to {φ L , φ R , φ U } taking the values π/2 and 3π/2. We provide the solutions in Table 1 which again match the values obtained from a fully numerical solution, thus indicating the correctness of our analytical approach. We now proceed to present numerical results for the 4-terminal case when there exists a finite interface resistance between the superconducting terminals and the normal wires, which is experimentally more realistic. We fix φ U = 0 without loss of generality and plot the evolution of the proximity-gap phase diagram, quantified via the zero-energy DOS  at the intersection point (x, y) = (0, 0), as the remaining superconducting phases {φ D , φ L , φ R } are varied in Fig. 5. Once again, the analytical transition curves correspond well with the regions where the numerically computed density of states differs from zero. In an experimental setting, the phase-differences can be tuned by connecting the superconducting terminals and thus creating loops which a magnetic flux can pass through, the latter controlling φ j . We consider in Fig. 6 the special case where the flux penetrating all loops is the same, meaning that the phase difference between each pair of terminals is equal to φ (except between the up and left terminal, see inset of Fig. 6). We set all wire lengths L j = L and interface resistances to be equal for simplicity, and consider different sizes L. Regardless of L, the superconducting correlations vanish completely at φ = π/2 and φ = π, as indicated by  taking its normal state value (φ L , φ R ) = (π/2, π/2) (φ L , φ R ) = (3π/2, π/2) (φ L , φ R ) = (π/2, 3π/2) (φ L , φ R ) = (3π/2, 3π/2) φ D = π/2 = . 0 00  = . 0 71  = . 0 71  = . 1 00  φ D = 3π/2  = . 0 71 = . 1 00   = . 1 00  = . 0 71 Table 1. Analytically obtained values of  at special points in phase-space. The solution for the zeroenergy DOS  at the intersection point of the wires (x, y) = (0, 0) obtained through analytically solving the non-linear equations for γ j assuming transparent interfaces to the superconducting terminals (in contrast to Figs 5 and 6 where a finite interface resistance is used). We fixed φ U = π/2. At all points (φ U , φ D , φ L , φ R ) shown in the table, the analytically obtained value of  matches the numerically obtained solution. Scientific RepoRts | 7:40578 | DOI: 10.1038/srep40578 ( = 1). The gapped region at 0 < φ < π/2 for small lengths L/ξ ≪ 1 starts to fill up with available electronic excitations as L increases. Conclusion The main new results in this work are the class of analytical solutions of the Ricatti-parametrized Usadel equation at E = 0 in the full proximity effect regime, the equations (18) and (19) providing the transition between the gapped and non-gapped regimes for an arbitrary number of terminals n, and the specific results for the 4-terminal case. An interesting expansion of the present work would be to explore how magnetic interfaces [23][24][25] and spin-orbit coupling would influence the proximity-gap phase diagram and topological properties of multiterminal Josephson junctions, as recent works have demonstrated that in particular the latter of these can induce several novel effects in both diffusive and ballistic superconducting hybrids 13,26-34 . Figure 1 . 1Multiterminal Josephson junctions. The density of states  at zero energy (Fermi level) is measured at the point indicated by a star, i.e. at the intersection of the diffusive normal wires. (a) 2-terminal, (b) 3-terminal, and (c) 4-terminal setups. Figure 2 . 2Analytically calculated transition curves between gapped and ungapped regions in the 3-terminal case. Plot of curves where the first order correction S i1 (x) can have non-zero solutions. (a) Structure of the condition in the extended phase space, showing metastable solutions. (b) Translation of physically relevant curves into [0, 2π] × [0, 2π]. Scientific RepoRts | 7:40578 | DOI: 10.1038/srep40578 π U 2 ,Figure 3 . 23the solutions in the left, down, right, and up arm are written as γ L = tan(c 1 x + c 2 ), γ D = tan(c 3 x + c 4 ), γ R = tan(c 5 x + c 6 ), γ U = tan(c 7 x + c 8 ). As in the previous section, we assume here for simplicity equal lengths and normal-state conductances of the four normal wires. The resulting boundary conditions take the form: Numerically calculated proximity-gap phase diagram for 3-terminal Josephson junctions. Plot of the Fermi level density of states  for a 3-terminal setup as a function of the phases φ L and φ R . For both plots, we set L/ξ = 0.67 and δ/Δ 0 = 5 × 10 −3 . The phase of the 'down' superconducting terminal has been set to φ D = 0. (a) Ideal boundary conditions. (b) Kupriyanov-Lukichev boundary conditions with finite interface resistance. We have set ζ j = 2.5, j = {L, R, D}. Scientific RepoRts | 7:40578 | DOI: 10.1038/srep40578 Figure 4 . 4Analytically calculated transition curves between gapped and ungapped regions in the 4-terminal case. The mapping of three-dimensional phase space was performed using Eq. (18), with φ U = 0. (a) Transition surface in extended phase space. (b-d) Translation of physically relevant curves into the first quadrant for φ D = 0, π 2 and π, respectively. Figure 5 . 5Numerically calculated density of states at E = 0 for a 4-terminal Josephson junction for different phase-configurations. Setting the upper superconducting phase to zero without loss of generality, φ U = 0, we plot the evolution of the proximity-gap phase diagram, quantified via the zero-energy density of states  at the intersection between the wires, as the phases at the other superconducting terminals are varied. We have set the wire lengths equal to L/ξ = 0.67 and the interface contact with the superconductors parametrized by a finite interface resistance ratio to the bulk resistance ζ = 2.5. The blue regions correspond to the gapped regime where  = 0. Figure 6 . 6Numerically calculated density of states at E = 0 for a 4-terminal Josephson junction for equal flux through the loops. 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[ "PSEUDONORMS AND p-ADIC BIRATIONAL TORELLI THEOREM", "PSEUDONORMS AND p-ADIC BIRATIONAL TORELLI THEOREM" ]
[ "Chen-Yu Chi " ]
[]
[]
A p-adic analogue of the pseudonorm version of the birational Torelli type theorem is obtained via a comparison theorem of image closures. Among other results obtained, we have a criterion for existence of rational points of canonically polarized surfaces over finite fields.Spec R. Note that X(K), the set of all K-points of X, is a K-analytic manifold 1 . We let Γ an X(K), K ⊗m X(K) = all K-analytic section of K ⊗m X(K) on X(K) ,where K X(K) = ∧ dim X T * X(K) is the canonical bundle of X(K), and letwhere ω X/R = ω X ⊗ π * ω Spec R . We have a natural homomorphismby viewing scheme-theoretic sections on X as K-analytic sections on X(K). Since all K-analytic sections are Borel measurable sections, we have by Definition 2.4 the following maps1 See [9] 2.4 for the definitions and basic properties. K-analytic manifolds are assumed to be 2nd countable.Finally, if V is a nonzero R-submodule of Γ(X, ω ⊗m X/R ), we have the restricted m-th pluricanonical R-rational map 2l=0 Sym l R V. Now consider two integral schemes X and Y which are proper and smooth over Spec R. The following is the first main result of the current paper.Theorem 1.1. For an m ∈ N and for an isomorphismof R-schemes identifies the image closures of the R-rational maps ϕ |V X | and ϕ |V Y | .We will see later (Proposition 4.2) that for every m ∈ N the pseudonormed space Γ(X, ω ⊗m X/R ), · m is an R-birational invariant of X in the sense that associated to every R-birational map XNote that if the base change X K is of general type, then there exists some m X ∈ N such that ϕ X,m := ϕ Γ(X,ω ⊗m X/R ) is an R-birational map from X onto the image closure of ϕ X,m for every m m X . 3 Consequently, Theorem 1.1 has the following corollary.Corollary 1.2 (p-adic birational Torelli theorem). Suppose that X K and Y K are of general type over K. If X(K) = ∅ = Y (K), if for some m ∈ N there is an R-linear isometry Γ(Y, ω ⊗m Y /R ), · m T / / Γ(X, ω ⊗m X/R ), · m , and if both ϕ X,m and ϕ Y,m maps birationally onto their image closures, then there exists an R-birational map X f / / Y and some c ∈ R × such that T = cf * .Our next main result is related to the condition X(K) = ∅. It is well known that,
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[ "https://export.arxiv.org/pdf/2211.09335v1.pdf" ]
253,581,595
2211.09335
5d8616e44e54d21f86ce1a9ff2f1f2e4733c8628
PSEUDONORMS AND p-ADIC BIRATIONAL TORELLI THEOREM 17 Nov 2022 Chen-Yu Chi PSEUDONORMS AND p-ADIC BIRATIONAL TORELLI THEOREM 17 Nov 2022 A p-adic analogue of the pseudonorm version of the birational Torelli type theorem is obtained via a comparison theorem of image closures. Among other results obtained, we have a criterion for existence of rational points of canonically polarized surfaces over finite fields.Spec R. Note that X(K), the set of all K-points of X, is a K-analytic manifold 1 . We let Γ an X(K), K ⊗m X(K) = all K-analytic section of K ⊗m X(K) on X(K) ,where K X(K) = ∧ dim X T * X(K) is the canonical bundle of X(K), and letwhere ω X/R = ω X ⊗ π * ω Spec R . We have a natural homomorphismby viewing scheme-theoretic sections on X as K-analytic sections on X(K). Since all K-analytic sections are Borel measurable sections, we have by Definition 2.4 the following maps1 See [9] 2.4 for the definitions and basic properties. K-analytic manifolds are assumed to be 2nd countable.Finally, if V is a nonzero R-submodule of Γ(X, ω ⊗m X/R ), we have the restricted m-th pluricanonical R-rational map 2l=0 Sym l R V. Now consider two integral schemes X and Y which are proper and smooth over Spec R. The following is the first main result of the current paper.Theorem 1.1. For an m ∈ N and for an isomorphismof R-schemes identifies the image closures of the R-rational maps ϕ |V X | and ϕ |V Y | .We will see later (Proposition 4.2) that for every m ∈ N the pseudonormed space Γ(X, ω ⊗m X/R ), · m is an R-birational invariant of X in the sense that associated to every R-birational map XNote that if the base change X K is of general type, then there exists some m X ∈ N such that ϕ X,m := ϕ Γ(X,ω ⊗m X/R ) is an R-birational map from X onto the image closure of ϕ X,m for every m m X . 3 Consequently, Theorem 1.1 has the following corollary.Corollary 1.2 (p-adic birational Torelli theorem). Suppose that X K and Y K are of general type over K. If X(K) = ∅ = Y (K), if for some m ∈ N there is an R-linear isometry Γ(Y, ω ⊗m Y /R ), · m T / / Γ(X, ω ⊗m X/R ), · m , and if both ϕ X,m and ϕ Y,m maps birationally onto their image closures, then there exists an R-birational map X f / / Y and some c ∈ R × such that T = cf * .Our next main result is related to the condition X(K) = ∅. It is well known that, Introduction In [6] and [4], Yau and the author have initiated a study of birational equivalence via the pseudonorm functions on pluricanonical spaces of complex projective manifolds and obtained theorems of Torelli type for birational equivalence. In the current paper, we prove a p-adic version of the aforementioned result. In order to state the main result we first set up some notation and terminology. Let K be a completion of an algebraic number field at a prime divisor, R its ring of integers, and F q its residue field. Below we call such K a p-adic field if q is a power of some prime number p. | · | K will be a fixed non-archimedean absolute value on K compatible with the valuation on R. We consider an integral R-scheme X π / / Spec R which is proper and smooth over 1 if X(F q ) = ∅, then X(K) = X(R) = ∅ by Hensel's lemma and the valuative criterion. For a fixed X and a fixed prime number p, clearly X(F q ) = ∅ if q = p r is sufficiently 2 For S-schemes X and Y , a rational map X f / / Y is said to be S-rational if there exists a dense open set U of X which intersects every fibre of X over S such that U f |U / / Y is an S-morphism; f is S-birational if U can be further chosen so that f (U ) is open in Y and f maps U onto f (U ) isomorphically. 3 Here one may extend a birational map over K to an R-birational map by a classical result of Matsusaka and Mumford ( [17] Theorem 1). See also [13] Theorem 5.13 for a rephrasing close to the current situation. large. It is then natural to ask how large q would be sufficient to imply X(F q ) = ∅. Let n = dim X Fq . When X Fq is smooth, in discussions one usually assumes that X Fq is geometrically connected, for otherwise X(F q ) = ∅. According to the solution to Weil's conjecture [7], an estimate of #X(F q ) can be obtained in terms of the Betti numbers. To obtain explicit results following this line, one has to have information of the Betti numbers, and this is mostly achievable for completely intersections. For example, if X Fq is a complete intersection in P n+r Fq of multi-degree (d 1 , . . . , d r ), Deligne [7] has obtained the estimate #X(F q ) − q n − q n−1 − · · · − 1 b ′ n (n + r, d 1 , . . . , d r )q n 2 where b ′ n (n + r, d 1 , . . . , d r )q n 2 is the n-th primitive Betti number of any nonsingular complete intersection of P n+r of dimension n and multi-degree (d 1 , . . . , d r ), which can be written down explicitly in terms of n, r, and (d 1 , . . . , d r ). Among many explicit results along this line, it is known for ecample that (cf. [3] Theorem 1.2) X(F q ) = ∅ if q > 2 2 + r j=1 (d j − 1) 2 (d 1 · · · d r ) 2 . A natural question is then to know what one can say about situations which involve possibly non-complete-intersections. Besides, we look for results on the nontriviality of X(F q ) involving intersection numbers of divisors, instead of involving Betti numbers. We have the following result on existence of rational points over F q involving the intersection numbers of the canonical divisor and a very ample divisor. Theorem 1.3. Let W be a complete smooth geometrically connected scheme over F q of dimension n. Suppose that H is a very ample divisor. Then W (F q ) = ∅ if q > max H •n (H •n − 1) n , K W • H •(n−1) + (n − 1)H •n + 2 2 , where • denotes the numerical intersection product of divisors. One might hope to deduce a result of similar form by applying the Lang-Weil bound [16], which considers varieties with an embeding into an ambient projective space. However, the Lang-Weil bound involves the dimension of the ambient projective space. Theorem 1.3 does not require knowing such information. Now we specialize to the canonically polarized situation, which provides the most basic objects for the birational Torelli-type theorem. By the fundamental work of [8] we obtain the following explicit effective result. Corollary 1.4 (Existence of rational points on canonically polarized surfaces). For any complete smooth geometrically connected surface W over F q with K W ample, W (F q ) = ∅ if q > max 25K •2 W (25K •2 W − 1) 2 , 30K •2 W + 2 2 . 3 This can be derived by simply taking H = K ⊗5 W in Theorem1.3, since K ⊗5 W is very ample according to [8]. It seems that so far there is no established generalization of such effective very ampleness result when the dimension is greater than two. In dimension two. On the other hand, one may also consider the more general situation that W be a smooth surface of general type. Explicit effective result of similar kind may be obtained in this more general situation, but it requires more elaborative treatments of singularities of canonical models, which will appear in [5]. As for the organization of the paper, in Sec. 2 we briefly review the notion of padic integration and introduce the pseudonorms. In Sec. 3 we prove a density theorem (Theorem 3.1) for rational points in the set of geometric points of integral schemes over a complete normed field. Although for later use we only need the density theorem for p-adic fields, which can be more directly obtained by using p-adic measure theory, we think it is worth proving a more general version the proof of which makes no use of integration. In Sec. 4 we show that the pseudonormed pluricanonical spaces for a family of R-birational invariants. In Sec. 5 we prove a p-adic parallel (Theorem 5.2) to an equimeasurability theorem of Rudin. Rudin's theorem is for the field C, and the C-version of the birational Torelli type theorem can be viewed as a special case of Rudin's result, as pointed out to me by S. Antonakoudis. See the survey article [21]. In his proof Rudin quotes Wiener's invariant subspace theorem. We preferred not proceeding detailed examinations of whether Wiener's result has a suitable analogue for K; instead we adopt a setting of a flavour of the Hilbert Nullstellensatz and makes more elementary use of the Fourier transformation. A key step in both Rudin's and our proofs is to construct a non-identically 0 function of particular type. The analysis in the case of C and our case of p-adic are quite different, the latter being simpler. Finally, in Sec. 6 we derived the first main theorem from the p-adic equimeasurability theorem. In contrast with the case of C, we have K = K, and further care of density needs to be taken. Finally, in Sec. 7 we prove Theorem 1.3. The method is to reduce the proof to the case of curves and then apply the Hasse-Weil bound. In the case over finite fields, this requires finding possibly non-generic but smooth hyperplane sections, and fortunately we found [1]. Finally I like to make some notes about the current paper and work of Shuang-Yen Lee. The first version of the current paper only consisted of Theorem 1.1 and the padic birational Torelli theorem, obtained in October, 2020; it did not treat the question about the nontriviality of X(K). Later on April 30, 2021, I happened to see [14], which is a draft of an undergraduate thesis by Lee supervised by Chin-Lung Wang. In [14] Lee obtained the 1-dimensional case of the p-adic birational Torelli theorem, essentially following the arguments of Royden [18], which is independent from the approach we have adopted here, the equimeasurability framework of Rudin. The approach of Royden seems hard to be carried out in higher dimensions to yield the most general birational Torelli theorem, at least for now. On May 8, 2021 I informed Chin-Lung Wang and other collegue that earlier I have obtained Theorem 1.1 and the p-adic birational Torelli theorem, with a copy of my first draft. In July 2021, I reported on Theorem 1.1 and Corollary 1.2, as well as the p-adic analogue of Rudin's equimeasurablity theorem (Theorem 5.2), in the HAYAMA Symposium on Complex Analysis in Several Variables XXII. Very recently I am aware of a preprint by Lee [15], which is a revised extension of [14]. Theorem 0.1 of [15] reads very similar to our Theorem 1.1, except that we have R as the base ring instead of K. Actually, assuming Theorem 5.2 and replacing all appearances of R by K in our proof of Theorem 1.1 (Sec. 6) one obtains a proof of Theorem 0.1 of [15]. Besides, [15] contains essentially a proof of my Theorem 5. Royden's approach taken by [14] and adopted Rudin's equimeasurability framework in order to obtain higher dimensional results. Now come back to [14]. By using the Hasse-Weil bound (cf. [11] Exercise V 1.10), it is mentioned in [14] that if X is a smooth geometrically connected curve over F q of genus g and if q > 4g 2 , then X(F q ) = ∅. This raised my interests in finding similar conditions in higher dimensions, with g replaced by intersection numbers involving K X and polarization divisors, as mentioned above. Part of the outcomes are Theorem 1.3 and Corollary 1.4, which I wrote in the second version of this paper, also sent to several colleagues around May 2021. Acknowledgement The p-adic case of pseudonorm problems was raised to the author by Professor Shing-Tung Yau about ten years ago. The author would like to thank Professor Yau for having shared with him many problems related to pseudonorms and for constant encouragement. The author also like to thank Professor Dinh Tien-Cuong for showing his interest in the current work during the HAYAMA symposium 2021. Integration on K-analytic manifolds and pseudonorms Let (K, |·| K ) denote a p-adic valued field, and let µ K,n be the associated Haar measure on the locally compact Hausdorff abelian group K n normalized by setting µ K,n (B) = 1 for the unit ball B = B 1 (0). We have the following analogue of the fundamental formula for the Lebesgue measure on R n (cf. [9] 7.4): Change-of-variable formula 2.1. Let U ϕ / / V be a K-bianalytic map between open sets U and V of K n . For any C-valued Borel measurable function h on V we have V h dµ K,n = V (h • ϕ) |ϕ ′ | K dµ K,n if one of the two integral exists, where ϕ ′ (u) denotes the Jacobi matrix of ϕ at the point u ∈ U. In the following we let X be a K-analytic manifold of dimension n (always assumed Hausdorff and second countable). Suppose that X is covered by coordinate patches (U i , ϕ i ) (i ∈ I) with coordinate systems u i = (u 1 i , . . . , u n i ). We may and will assume that ϕ i (U i ) is compact, since every open ball in K n is compact. Proof. The question being local completely, we only need to consider the zero set S of a K-analytic function F on a compact open subset U of K n . The case n = 1 is clear since all points of S are isolated. Now we assume n = 2. For every p ∈ U we may apply the Weierstrass preparation theorem, which works for arbitrary field equipped with a complete absolute value. By a linear change of coordinate we may find open discs B ⊆ K n−1 and B ′ ⊆ K such that p ∈ B × B ′ ⊆ U and the projection from S ∩ B × B ′ to B has all its fibres finite. By Fubini's theorem we see that µ K,n S ∩ (B × B ′ ) = 0. Definition 2.2. We say that a subset S ⊆ X is said to be of measure 0 in S if µ K,n ϕ i (S ∩ U i ) = 0 for every i ∈ I. Since U is compact, we have µ K,n (S ∩ U) = 0. Now we let K X denote the canonical bundle of X, the K-analytic line bundle of exterior forms of degree n. To each Borel measurable section ω of K ⊗m X , (m ∈ N) one may associate canonically a regular 4 Borel measure ω 1 m on X. To see this, we first express ω| U i in the form f i (u 1 i , . . . , u n i ) (du 1 i ∧ · · · ∧ du n i ) ⊗m where f i is a continuous function on the open set ϕ i (U i ) of K n . By Riesz's representation theorem, the linear functional C c (U i ) / / C h ✤ / / ϕ i (U i ) (h • ϕ −1 i )|f i | 1 m dµ K,n defines a regular Borel measure µ i on U i . By the above change-of-variable formula we see that µ i | U i ∩U i ′ = µ i ′ | U i ∩U i ′ for every pair (i, i ′ ). Thus µ i patches together to form a regular Borel measure on X, which is the expected ω 1 m . Definition 2.4. We define the m-th pseudonorm ω m := X ω 1 m ∈ [0, ∞] for every Borel measurable section ω of K ⊗m X on X. Density of rational points in the set of geometric points Let W be an integral algebraic scheme defined over a field K. In this section we discuss the density of W (K) in W (K) with respect to the Zariski topology. The density does not hold in general, as shown by the real algebraic set {(x, y) ∈ R 2 | x 2 + y 2 = 0} in the complex one {(x, y) ∈ C 2 | x 2 + y 2 = 0}. Note that (0, 0) is a singular point of the complex algebraic scheme defined by the polynomial X 2 + Y 2 = 0. The following density theorem indicates that this phenomenon cannot happen assuming smoothness. Theorem 3.1. Let K be a nontrivial complete valued field. Suppose that W is an integral algebraic scheme over K with dim W > 0. If x ∈ W (K) is a smooth point 5 of W , then every K-analytic neighborhood of x in W (K) is dense in W (K) with respect to the Zariski topology over K. Proof. Suppose that W is locally defined around x by polynomials F i (X 1 , . . . , X r ) ∈ K[X 1 , . . . , X r ] (i = 1, . . . , s) in an affine space A r K . Then x ∈ W (K) is a smooth point if and only if rk K ∂F i ∂X j (x) = rk K ∂F i ∂X j (x) = r − dim W (the Jacobian criterion). In particular, every open neighborhood of x in W (K) is K-analytically isomorphic to an open set of K n (n = dim W ), by the K-analytic implicit function theorem. When dim W = 1, such a neighborhood is an infinite set, and hence is Zariski dense in the irreducible curve W (K). We then proceed by induction on dim W . Suppose that the 5 If an integral algebraic scheme over K has a smooth K-point, then it is geometrically integral. statement holds when dim W < n (with n > 1) and consider the case dim W = n. Since the question is local in nature, we may assume that W is a closed subscheme of some projective space P N K . Consider any K-analytic coordinate patch U of W (K) around x. We claim that U is Zariski dense in W (K). Were this false, U ⊆ V + (G) W (K) for some nonzero homogeneous polynomial G(Y 0 , . . . , Y N ) with all coefficients in K. For any d ∈ N we let N d := N +d d . For z = (z α 0 ···α N ) α 0 +···+α N =d ∈ K N d \ {0} we let L z (Y 0 , . . . , Y N ) = α 0 +···+α N =d z α Y α + · · · + z N Y N , H z = V + (L z ) be the corresponding hyperplane of P N K , and P x := [z] ∈ P N d −1 (K) | x ∈ H z (K) . Note that P x is itself a projective hyperplane of P N d (K) defined over K, since x is a K-rational point. Consider the sets V 1 := [z] ∈ P x W (K) H z (K) , V 2 := [z] ∈ P x T x W (K) H z (K) , V 3 := [z] ∈ P x W (K) ∩ H z (K) V + (G) , and V 4 := {[z] ∈ P x | W K ∩ (H z ) K is integral } , where T x W (K) is the projective tangent space of W (K) at x, and the intersection W K ∩ (H z ) K is to be understood as scheme theoretic. It is direct to verify that V 1 , V 2 , and V 3 are all nonempty open sets in P r (K). By a strengthened version of Bertini's theorem ([10] Corollary 3.7) we see that, if we choose d sufficiently large, then V 4 contains a nonempty open set in P r (K). Note that U ∩H z (K) is K-analytically smooth at x. To see this, note that around x the set of defining equations of U in P N (K) is the same as that of W (K) in P N (K). Therefore the condition of the Jacobian criterion for the latter (the algebraic case over K) at x implies that for the former (the K-analytic case). Claim. P x ∩ P N d −1 (K) is dense in P x . Were the claim proved, one may find some point [z] ∈ V 1 ∩ V 2 ∩ V 3 ∩ P N d (K). Since [z] ∈ P N d (K), we have a reduced closed subscheme W ′ of W such that W ′ (K) = W (K) ∩ H z (K). (1) Since [z] ∈ V 1 , by Krull's Hauptidealsatz W ′ (K) has pure dimension n − 1 and hence x is not an isolated point of W ′ (K). (2) Since [z] ∈ V 2 , x is a regular point of W ′ (K). (3) Since [z] ∈ V 3 , V + (G) ∩ W ′ (K) is a proper closed subset of W ′ (K). (4) Since [z] ∈ V 4 , W ′ is integral. Therefore, we may apply the induction hypothesis to W ′ . The induction hypothesis implies that U ∩ H z (K) is dense in W ′ (K). However, U ∩ H z (K) is contained in the proper closed subset V + (G) ∩ W ′ (K) of W ′ (K), as contradicts to its density in W ′ (K). It remains to verify the claim. Since P x is a projective hyperplane of P N d −1 (K) defined over K, it suffices proving the following statement: P k (K) is dense in P k (K) with respect to the Zariski topology. Suppose that P k (K) is not dense in P k (K), say, P k (K) ⊆ V + (F ) for some nonzero homogeneous polynomial F (Z 0 , . . . , Z k ) = α F α Z α ∈ K[Z 0 , . . . , Z k ]. Fixing a basis e s (s ∈ S) of K over K, we may write F (Z 0 , . . . , Z k ) = s∈S 0 e s α F s,α Z α with S 0 finite and F s,α ∈ K for all (s, α). The condition P k (K) ⊆ V + (F ) is equivalent to that α F s,α b α • = 0 for every s ∈ S 0 if (b 0 , . . . , b k ) ∈ K k+1 \ {0}. Since K is infinite, this implies that F s,α = 0 for every (s, α), and hence F = 0, a contradiction. Remark 3.2. Theorem 3.1 has a simpler proof using measure theory when K is a p-adic field and W is proper and smooth over its ring of integers R. This is the only situation we need for the rest of the paper. If this is the case, for any closed subset Z W , it is well known that Z(K) = Z(R) is of measure 0 in the K-analytic manifold W (K) = W (R). On the other hand all nonempty open subsets W (K) has strictly positive measure. Then we obtain the expected density by working on affine charts. R-birational equivalence and pseudonorms In this section we suppose that K is a p-adic field and R its ring of integers. Consider an R-birational map X f / / ❴ ❴ ❴ Y between smooth R-schemes. We recall the following easy version of global change-of-variable formula. Lemma 4.1. If f is an R-morphism, we have ω m = f * ω m for any ω ∈ Γ(Y, ω ⊗m Y /R ). Proof. There exists a dense open subset V of Y such that f is an R-isomorphism from f −1 (V ) to V . By the local change-of-variable formula 2.1 we have V (K) ω 1 m = f −1 V (K) f * ω 1 m . Let Z = Y \ V and Z ′ = X \ f −1 (Z) . Then Z(K) and Z ′ (K) are K-analytic subsets of the K-analytic manifolds Y (K) and X(K), respectively. Theorem 3.1 implies that they are nowhere dense, and hence are of measure 0 in their ambient manifolds by Remark 2.5. Therefore, ω m = Y (K) ω 1 m = V (K) ω 1 m = f −1 V (K) f * ω 1 m = X(K) f * ω 1 m = f * ω m 9 The pull-back map f * extends to the R-birational case. Moreover, we have: Theorem 4.2. If both X and Y are proper over Spec R, then for every m ∈ N there is a canonical R-linear isometry Γ(Y, ω ⊗m Y /R ), · m f * / / Γ(X, ω ⊗m X/R ), · m . Proof. By Hironaka's desingularization theorem [12] there exists a common resolution W which is projective over R and is compatible with f : W g~⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ h ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ X f / / ❴ ❴ ❴ ❴ ❴ ❴ ❴ Y. Note that · m takes finite values on Γ(W, ω ⊗m W/R ) since W is projective over R. Thus · m takes finite values on both Γ(X, ω ⊗m X/R ) and · m takes finite values on Γ(Y, ω ⊗m Y /R ) by Lemma 4.1. It suffices to establish the isometries for g and h, i.e., we may assume f an R-morphism. Therefore, there exists an open subset V of Y with codim Y (Y \ V ) 2 such that f is an R-isomorphism from f −1 (V ) to V . We have a unique R-linear maps f * fitting in the following commutative diagram: Γ(Y, ω ⊗m Y /R ) injective (·)| V f * / / Γ(X, ω ⊗m X/R ) ≃ (·)| f −1 (V ) Γ(V, ω ⊗m Y /R ) (f | f −1 (V ) ) * ≃ / / Γ f −1 (V ), ω ⊗m X/R . By lemma 4.1 all the above maps preserves · m . It is direct to see that (f −1 ) * = (f * ) −1 (f −1 being the inverse R-rational map of f ). p-adic analogue of Rudin's equimeasurability theorem Definition 5.1. Let (X, A, µ) and (Y, B, ν) be measured spaces, (Z, C) be a measurable space, and Z X F > > ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ Y G _ _ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ be measurable maps, i.e., F −1 (C) ∈ A and G −1 (C) ∈ B for every C ∈ C. We say that f and g are equimeasurable if F * µ = G * ν, i.e., µ F −1 (C) = µ G −1 (C) for every C ∈ C. The following is a p-adic analogue of a theorem due to Rudin ([19] 1.4 Theorem I). Theorem 5.2. Suppose that (K, | · | K ) is a p-adic field and view K n as the measurable space equipped with the σ-algebra of all Borel subsets. Let r ∈ Q >0 . For any measurable maps K n X F =(f 1 ,...,fn) = = ④ ④ ④ ④ ④ ④ ④ ④ Y G=(g 1 ,...,gn) a a ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ from measured spaces (X, A, µ) and (Y, B, ν) to K n , if f i ∈ L 1 (X, A, µ) and g i ∈ L 1 (Y, B, ν) for i = 1, . . . , n and X 1 + v 1 f 1 (x) + · · · + v 1 f 1 (x) r K dµ = Y 1 + v 1 g 1 (x) + · · · + v 1 g 1 (x) r K dν (5.3) for every (v 1 , . . . , v n ) ∈ K n , then F and G are equimeasurable. Before giving the proof we first make some formal settings. Let (S, M) be a measur- and T * f := f • T for every M-measurable function f. Then S T * f dm = S f d(T * m) if f is integrable with respect to |m|. We have T * |m| = |T * m| if T is further assumed bijective. The following lemma is easily derived from the definitions. T * F ⊆ F ⇒ T * (F ⊥ ) ⊆ F ⊥ resp. T * M ⊆ M ⇒ T * (M ⊥ ) ⊆ M ⊥ . Proof of Theorem 5.2. We let (S, M) be K with its σ-algebra of Borel subsets and F = |at + b| r K a ∈ K × , b ∈ K . Let (u 1 , . . . , u n ) be the coordinate system of K n . For every v = (v 1 , . . . , v n ) ∈ K n we consider "projection map" K n Lv / / K with L v (u 1 , . . . , u n ) = v 1 u 1 + · · · + v n u n . We set m v = (L v ) * (F * µ − G * ν). Then (5.3) says nothing but m v ∈ F ⊥ for every v ∈ K n . (5.5) Now we set F ′ := (F ⊥ ) ⊥ ∩ L 1 (K, ds) (ds being the normalized Haar measure). Since F is invariant under all affine transformations, so is F ⊥ and (F ⊥ ) ⊥ by Lemma 5.4. Since L 1 (K, ds) is also invariant under all affine transformations, 6 so is F ′ Therefore, for any f ∈ F ′ and any m ∈ F ⊥ , we have (f * m)(t) = K f (t − s) dm(s) = 0 for every t ∈ K. Applying the above argument to m = m v and taking the Fourier transform yield that f m v = 0 for every f ∈ F ′ and for every v ∈ K n (5.6) where we identify the character group K with K as topological groups. More precisely, let Λ be any fixed nontrivial element of K, the pairing (cf. [20] Theorem 2.2.1) K × K / / S 1 (τ, t) ✤ / / exp 2πiΛ(τ t) gives an isomorphism K ≃ K. Claim. m v = 0 (and hence m v = 0) for every v ∈ K n . Assume the claim for now. By (5.5) we have m v = 0 for every v ∈ K n . This implies F * µ − G * ν = 0 (and hence F * µ − G * ν = 0) as follows. We have similar identifications K n ≃ K n ≃ K n via the pairing) K n × K n / / S 1 (v • , u • ) ✤ / / exp 2πiΛ − j v j u j F * µ − G * ν(v 1 , . . . , v n ) = K n exp 2πiΛ − j v j u j d(F * µ − G * ν)(u • ) = K exp 2πiΛ(−t) dm v (t) = m v (1) = 0. It remains proving the claim. Note that by (5.3) with (z 1 , . . . , z n ) = (0, . . . , 0) we have m v (0) = K dm v = K n d(F * µ − G * ν) = X dµ − Y dν = 0. By (5.6), it suffices to show that for every τ ∈ K × there exists some f ∈ F ′ with f (τ ) = 0. Actually it will be enough to obtain a function f 0 ∈ F ′ which is not identically zero: for such f 0 we have f 0 nontrivial, say f 0 (τ 0 ) = 0 for some τ 0 ∈ K × . Then 0 = f 0 (τ 0 ) = K exp 2πiΛ(−τ 0 s)f 0 (s) ds = K exp 2πiΛ(−τ s)f 0 (τ τ −1 0 s) |τ τ −1 0 | K ds = f (τ ) where f (s) := f 0 (τ τ −1 0 s) |τ τ −1 0 | K . Since F ′ is invariant under dilation, we see that f ∈ F ′ . To construct such a nontrivial f 0 ∈ F ′ , we consider functions of the form f 0 (s) = N k=1 a k |1 + c k s| r K (5.7) where a 1 , . . . , a N ∈ R and c 1 , . . . , c N ∈ K × are to be chosen. We first try to analyse the behaviour of f 0 (s) when |s| K is sufficiently large. Let b j = r j ∈ Q be the binomial coefficients. Then (1 + σ) r := ∞ j=1 b j σ j converges in K when |σ| K < 1 since |b j | K 1 for all j ∈ N. Moreover, we have |(1 + σ) r | K = |1 + σ| r K when |σ| K < 1. This can be see as follows. Write r = l m with l, m ∈ N. When |σ| K < 1, by multiplying power series we see that (1 + σ) r m = (1 + σ) l , and hence (1 + σ) r K = (1 + σ) r m 1 m K = (1 + σ) r m 1 m K = (1 + σ) l 1 m K = 1 + σ l m K = 1 + σ r K . Therefore, |1 + s| r K = |s| r K 1 + 1 s r K when |s| K > 1. From now on we denote | · | K by | · |. When |s| > max{1, |c 1 |, . . . , |c N |}, we have f 0 (s) = N k=1 a k |c k | r |s| r ∞ j=1 b j c −j k s −j = N k=1 a k |c k | r |s| r max |b j | |c k | −j |s| −j j ∈ N . 13 Note that 1 = |b 0 | |c k | 0 |s| 0 > |b j | |c k | −j |s| −j if and only if |s| K > |b j | 1 j K |c k | . In summary, f 0 (s) = N k=1 a k |c k | r |s| r when |s| > max 1, |c 1 |, . . . , |c N |, max j∈N |b j | 1 j max{ |c k | −1 | k = 1, . . . , N} . If a k and c k (k = 1, . . . , N) are chosen so that N k=1 a k |c k | r = 0, (5.8) then f 0 vanishes outside a sufficiently big ball, and hence is bounded and continuous. In particular, f 0 ∈ L 1 (K, ds). Besides, f 0 ∈ F ⊆ (F ⊥ ) ⊥ , and hence f 0 ∈ F ′ . It remains to make f 0 nontrivial. Since f 0 (0) = N k=1 a k , it suffices to set N = 2 and choose c 1 , c 2 ∈ K × with |c 1 | K = |c 2 | K and a 1 , a 2 satisfying (5.8). Proof of Theorem 1.1 Recall that the R-linear isometry V Y , · m T / / V X , · m induces an R-isomorphism γ := PT * : PV * X ≃ / / PV * Y . Let X ′ denote the image closure of ϕ |V X | , which is defined over R. X ′ is an integral schemes of finite type over R. Similar setting will be adopted for Y . In order to show that PT * identifies X ′ with Y ′ , it suffices to prove the statement over K-points: Claim 1. If γ X ′ (K) = Y ′ (K), then X ′ is identified with Y ′ via γ. γ(X ′ ) and Y ′ are integral closed R-subschemes of the projective scheme PV * Y ≃ P N R . Suppose that γ(X ′ ) and Y ′ are locally defined on an affine open set U ≃ Spec A of P N R by ideals I and J A. Since K is algebraically closed and γ X ′ (K) = Y ′ (K), we have I ⊗ R K = J ⊗ R K. Note that 14 as can be seen by the following commutative diagram with exact rows: I = A ∩ (I ⊗ R K) (all viewed as subsets of A ⊗ R K)0 / / I ⊗ R K / / A ⊗ R K / / (A/I) ⊗ R K / / 0 0 / / I ? O O / / A ? O O / / A/I ? O O / / 0 . Similarly, J = A ∩ (J ⊗ R K) = A ∩ (I ⊗ R K) = I. Claim 2. Y ′ (K) ⊆ γ X ′ (K) . Fix an R-basis of η 0 , . . . , η N ∈ Γ(Y, ω ⊗m Y /R ). Then (T η 0 ) K , . . . , (T η N ) K form a basis of V X,K . The rational maps γ • ϕ |V X | and ϕ |V Y | can be viewed as mapping some nonempty open sets X 0 ⊆ X resp. Y 0 ⊆ Y , respectively, to the affine R-scheme A N R , given by T η 1 T η 0 , . . . , T η N T η 0 and η 1 η 0 , . . . , η N η 0 , which determine maps K N X 0 (K) F K 6 6 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ Y 0 (K) G K h h ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ K N ? O O X 0 (K) ? O O F =(f 1 ,...,f N ) 5 5 • • • • • • • • • • • • • • • • Y 0 (K) ? O O G=(g 1 ,...,g N ) h h | | | | | | | | | | | | | | | | . We let µ and ν be the measures on X and on Y associated to T η 0 1 m and η 0 1 m , respectively. Then for every (v 1 , v 2 , . . . , v N ) ∈ K N we have T η 0 + N i=1 v i η i 1 m = 1 + N j=1 v i f i 1 m K T η 0 1 m and η 0 + N i=1 v i η i 1 m = 1 + N i=1 v i g i 1 m K η 0 1 m . Since X and Y are proper over R, we have X 0 (K) T (η i ) 1 m = X(K) T (η i ) 1 m < ∞ and Y 0 (K) η i 1 m = Y (K) η i 1 m < ∞, or equivalently, f i ∈ L 1 X 0 (K), B X 0 , µ and g i ∈ L 1 Y 0 (K), B Y 0 , ν for i = 1, . . . , N, where B X 0 and B Y 0 are the corresponding σ-algebra of Borel sets. The isometry condition on V Y , · m T / / V X , · m is then equivalent to that X 0 (K) 1 + N j=1 v i f i 1 m K T η 0 1 m = Y 0 (K) 1 + N i=1 v i g i 1 m K η 0 1 m for every (v 1 , v 2 , . . . , v N ) ∈ K N . By Theorem 5.2 we conclude that F and G are equimeasurable (Definition 5.1). Now suppose that Y ′ (K) γ X ′ (K) . We will show that the equimeasurability of F and G is violated. By Chevalley' theorem, there exists a nonempty Zariski open subset U of Y ′ \ γ(X ′ ) such that ϕ −1 |V Y | (U) ⊆ Y 0 and U = ϕ |V Y | ϕ −1 |V Y | (U) = G ϕ −1 |V Y | (U) . We denote ϕ −1 |V Y | (U) by W . Since Y is smoorh and Y (K) is assumed nonempty, Theorem 3.1 implies that W (K) = ∅, and hence it contains a nonempty open subset of the K-analytic manifold Y (K). On the other hand, F −1 (U) = ∅ since U ∩ γ(X ′ ) = ∅. Existence of rational points over finite fields As indicated in the introduction, the nontriviality of X(K) comes from that of X(F q ). In this section we prove Theorem 1.3. First recall the Hasse-Weil bound when W is a smooth geometrically connected complete curve of genus g: 1 + q − #W F q ) 2g √ q (see for example [11] Exercise V 1.10). This implies thatW (F q ) = ∅ if q > 4g 2 . Existence of rational points over finite fields (Theorem 1.3). Let W be a complete smooth geometrically connected scheme over F q of dimension n. Suppose that H is a very ample divisor. Then W (F q ) = ∅ if q > max H •n (H •n − 1) n , K W • H •(n−1) + (n − 1)H •n + 2 2 . Proof. In the following we let (·) denote the operation of base-change induced by a fixed extension F q → F q , and objects without an overline are defined over F q . We may assume that n 2. The very ample divisor H gives an embedding of W into a projective space P N Fq . We let d = H •n be the degree of W in P N Fq . Since q > d(d − 1) n , there exists a hyperplane H 1 in P N Fq such that H 1 intersects W transversally ([1] Theorem 1). Since n = dim W 2, we see that W ∩ H 1 is connected ([11] Corollary 7.9). We let W 1 = W ∩ H 1 . Then W 1 has degree d in P N Fq , too, and q > d(d − 1) n d(d − 1) n−1 . We may repeat the argument to find hyperplane H 1 , . . . , H n−1 in P N Fq such that W j = W j−1 ∩ H j is smooth and geometrically connected. In particular, W n−1 is a complete smooth geometrically connected curve over F q . By the adjunction formula, 2g(W n−1 ) − 2 = (K W + H 1 + · · · + H n−1 ) • H 1 • · · · • H n−1 = K W • H •(n−1) + (n − 1)H •n . By the Hasse-Weil bound, we see that W (F q ) = ∅ since q > K W • H •(n−1) + (n − 1)H •n + 2 2 . This completes the proof. 2. Both proofs of the p-adic equimeasurability result models on Rudin's setting. Rudin's proof of equimeasurability theorem over C uses properties of Fourier transform and Wiener's invariant subspace theorem. For the p-adic analogue, my proof makes the Fourier transform arguments work and bypasses the usage of Wiener's invariant subspace theorem; Lee's proof is more direct in nature, which relies on a decomposition of the unity function via the Schwartz-Bruhat function and bypasses the usage of Fourier transform. According to Lee, he did not know that I had obtained my Theorem 1.1 nor Theorem 5.2; in April 2022, under the suggestion by Chin-Lung Wang, he abandoned Lemma 2 . 3 . 23Every nowhere dense K-analytic subsets of a K-analytic manifolds is of measure 0. F able space and M(S, M) the space of all complex measures on (S, M). For any family F of M-measurable functions on S we let ⊥ ∈ M(S, M) f is integrable with respect to |m| and S f (s) dm(s) = 0 for every f On the other hand, for any family M ⊆ M(S, M) we let M ⊥ :=    f : M-measurable f is integrable with respect to |m| and S f (s) dm(s) = 0 for every m For any M-measurable map S T / / S we let T * m := m • T −1 for every m ∈ B(S, M) Lemma 5. 4 . 4Let S T / / S be a bijective M-measurable map. For any family F of M-measurable functions on S (resp. any M ⊆ M(S, M)), we have Here a (positive) Borel measure µ is said to be regular if (i) µ(K) is finite for every compact set K, (ii) µ(B) = inf{µ(U ) | B ⊆ U and U is open in X} for every Borel set B, and (iii) µ(U ) = sup{µ(K) | K ⊆ U and K is sompact}.6 The invariance of L 1 (K, ds) under dilations comes from the change-of-variable formula 2.1. An effective Bertini theorem over finite fields. E Ballico, Adv. Geom. 34E. Ballico, An effective Bertini theorem over finite fields, Adv. Geom. 3 (2003), no. 4, 361-363. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21. S Bosch, W Lütkebohmert, M Raynaud, Néron models. BerlinSpringer-VerlagS. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21. Springer-Verlag, Berlin, 1990 Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field. Matera Cafure, Privitelli , Finite Fields Appl. 31Cafure, Matera, and Privitelli, Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field, Finite Fields Appl. 31 (2015), 42-83. Pseudonorms and theorems of Torelli type. C.-Y Chi, J. Differential Geom. 1042C.-Y. Chi, Pseudonorms and theorems of Torelli type, J. Differential Geom. 104 (2016), no. 2, 239-273. Rational points of surfaces of general type over finite fields. C.-Y Chi, to appearC.-Y. Chi, Rational points of surfaces of general type over finite fields, to appear. A geometric approach to problems in birational geometry. C.-Y Chi, S.-T Yau, Proc. Natl. Acad. Sci. USA 105. Natl. Acad. Sci. USA 105C.-Y. Chi and S.-T. Yau, A geometric approach to problems in birational geometry, Proc. Natl. Acad. Sci. USA 105 (2008), no. 48, 18696-18701. . P Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. No. 43P. Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. No. 43 (1974), 273-307. Canonical models of surfaces of general type in positive characteristic. T , Inst. Hauteś Etudes Sci. Publ. Math. No. 67T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hauteś Etudes Sci. Publ. Math. No. 67 (1988), 97-144. An introduction to the theory of local zeta functions. J Igusa, AMS/IP Studies in Advanced Mathematics. 14International PressJ. Igusa, An introduction to the theory of local zeta functions. AMS/IP Studies in Advanced Mathematics, 14. American Mathematical Society, Providence, RI; International Press. M Ghosh, A Krishna, arXiv:1912.09076Bertini theorems revisited. M. Ghosh and A. Krishna, Bertini theorems revisited, arXiv:1912.09076. Graduate Texts in Mathematics. R Hartshorne, Springer-Verlag52New York-HeidelbergAlgebraic geometryR. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52. Springer-Verlag, New York-Heidelberg, 1977. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. H Hironaka, Ann. of Math. 2ibid.H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 1964 205-326. Young person's guide to moduli of higher dimensional varieties. Algebraic geometry-Seattle. S Kovács, Proc. Sympos. Pure Math. 22Amer. Math. Soc., ProvidenceS. Kovács, Young person's guide to moduli of higher dimensional varieties. Algebraic geometry- Seattle 2005. Part 2, 711-743, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Provi- dence, RI, 2009. Characterizing algebraic curves using p-adic norms. S.-Y. Lee, unpublishedS.-Y. Lee, Characterizing algebraic curves using p-adic norms, April, 2021 (unpublished). S.-Y. Lee, arXiv:2210.06767Global Igusa zeta function and K-equivalence. S.-Y. Lee, Global Igusa zeta function and K-equivalence, arXiv:2210.06767, Oct. 2022. Number of points of varieties in finite fields. S Lang, A Weil, Amer. J. Math. 76S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819-827. Two fundamental theorems on deformations of polarized varieties. T Matsusaka, D Mumford, Amer. J. Math. 86T. Matsusaka and D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668-684. Automorphisms and isometries of Teichmuller space. H L Royden, Advances in the Theory of Riemann Surfaces. 66H. L. Royden, Automorphisms and isometries of Teichmuller space, 1971 Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies, 66, 369-383. . W Rudin, Equimeasurability , Indiana Univ, Math. J. 253W. Rudin, L p -isometries and equimeasurability, Indiana Univ. Math. J. 25 (1976), no. 3, 215-228. Fourier analysis in number fields, and Hecke's zeta-functions. J Tate, Algebraic Number Theory (Proc. Instructional Conf. BrightonJ. Tate, Fourier analysis in number fields, and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), 305-347. On the pseudonorm project of birational classification of algebraic varieties, Geometry and analysis on manifolds. S.-T Yau, Progr. Math. 308Birkhäuser/SpringerS.-T. Yau, On the pseudonorm project of birational classification of algebraic varieties, Geometry and analysis on manifolds, 327-339, Progr. Math., 308, Birkhäuser/Springer, Cham, 2015. . Chen-Yu Chi, TaipeiDepartment of Mathematics, National Taiwan UniversityTaiwan Email address: [email protected], [email protected] Chi: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan Email address: [email protected], [email protected]
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[ "Resource allocation in open multi-agent systems: an online optimization analysis", "Resource allocation in open multi-agent systems: an online optimization analysis" ]
[ "Renato Vizuete ", "Charles Monnoyer De Galland ", "Julien M Hendrickx ", "Paolo Frasca ", "Elena Panteley " ]
[]
[]
The resource allocation problem consists of the optimal distribution of a budget between agents in a group. We consider such a problem in the context of open systems, where agents can be replaced at some time instances. These replacements lead to variations in both the budget and the total cost function that hinder the overall network's performance. For a simple setting, we analyze the performance of the Random Coordinate Descent algorithm (RCD) using tools similar to those commonly used in online optimization. In particular, we study the accumulated errors that compare solutions issued from the RCD algorithm and the optimal solution or the noncollaborating selfish strategy and we derive some bounds in expectation for these accumulated errors.
10.1109/cdc51059.2022.9993038
[ "https://arxiv.org/pdf/2207.09316v1.pdf" ]
250,644,279
2207.09316
f57ef409a06d42e8e37e202f9f24e01a424b2138
Resource allocation in open multi-agent systems: an online optimization analysis Renato Vizuete Charles Monnoyer De Galland Julien M Hendrickx Paolo Frasca Elena Panteley Resource allocation in open multi-agent systems: an online optimization analysis The resource allocation problem consists of the optimal distribution of a budget between agents in a group. We consider such a problem in the context of open systems, where agents can be replaced at some time instances. These replacements lead to variations in both the budget and the total cost function that hinder the overall network's performance. For a simple setting, we analyze the performance of the Random Coordinate Descent algorithm (RCD) using tools similar to those commonly used in online optimization. In particular, we study the accumulated errors that compare solutions issued from the RCD algorithm and the optimal solution or the noncollaborating selfish strategy and we derive some bounds in expectation for these accumulated errors. I. INTRODUCTION We consider the optimal resource allocation problem, where a fixed amount of resource must be distributed among n agents while minimizing some separable cost function f [1]. Problems of this type can be found in many different fields of research including distributed computer systems [2], games [3], smart grids [4], etc. In some specific formulations like actuator networks [5] or power systems [6], each agent i holds a quantity d i (which we call here the "demand" of agent i), so that the total amount of resource to be distributed is n i=1 d i ; the problem can then be written as (1) where each function f i : R p → R is α-strongly convex and β-smooth, and represents the local cost held by agent i. min x∈R np f (x) = n i=1 f i (x i ) s.t. n i=1 x i = n i=1 d i , Problems of this type have received a lot of attention in the last years and most of them are related to a possible change in the budget due to a variation in the demand of some of the agents [7,8]. However, in these works, only quadratic functions are considered which significantly restrict the set of potential cost functions of the agents and do not correspond [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). to the standard assumptions in the field of convex and smooth optimization [9,10]. In addition to the possible variations of the budget over time in (1), the composition of the system may also change during the whole process due to the arrival, departure or replacement of agents at a time-scale comparable to that of the process, giving rise to open multi-agent systems. Those are motivated by the growing size of the systems that tends to slow down the process as compared to the time-scale of potential changes in the set of agents. More generally, systems naturally allowing agents to join and leave are becoming common, such as e.g. multi-vehicle systems or with the Plug and Play implementation [11,12]. In the case of (1), it results in the system size n t , the local cost functions f t i , and the local demands d t i becoming time-varying. As a consequence, the instantaneous optimum of (1), denoted x * ,t , changes with the time as well, preventing usual convergence. Due to these possible changes in the dimension of the system, most of the related works in the field of open multi-agent systems are focused on the analysis of scalar performance indexes associated with the process, which allow overcoming the problem of time-varying dimensions. For instance, in [13,14] the variance is proposed as a metric for the analysis of a pairwise gossip algorithm, while in [15] the mean squared error is the object of study in randomized interactions. Regarding optimization, problems such as (1) typically imply a minimization process on a long period, and hence the cost is expected to be paid on a regular basis. In such setting, a natural way of measuring the performance of an algorithm is to compute its accumulated error with respect to a given strategy over a finite number of iterations. Similar metrics occur in the context of online optimization [16], where the objective is to minimize the socalled regret, commonly defined as the accumulated error of the estimate x t with respect to x * := arg min x T t=1 f t (x), or sometimes with respect to the time-varying solution of (1) x * ,t := arg min x f t (x) such as e.g., in [17]. Other extensions of the regret include the case of time-varying constraints, where a similar metric is used to measure the violation [18]. In this work, we analyze the performance of the Random Coordinate Descent algorithm (RCD) [19,20] to solve (1) in open systems. We study the loss accumulated by the RCD algorithm with respect to the time-varying optimal solution x * ,t over a finite number of iterations, and its gain with respect to the selfish strategy x s,t , which consists in the absence of collaboration between the agents (i.e., x s,t i = d t i ), by obtaining upper bounds. Finally, we consider the case of quadratic cost functions, for which tighter results are derived. II. PROBLEM FORMULATION The set of real numbers is denoted by R and the set of nonnegative integers by Z ≥0 . For two vectors x, y ∈ R n , x, y = x y = n i=1 x i y i denotes the usual Euclidean inner product and x = √ x x the Euclidean norm. The set of n-dimensional vectors with nonnegative entries is denoted by R n ≥0 . We denote the vector of size n constituted of only zeros by 0 n and of only ones by 1 n . A. Open resource allocation problem We consider the problem (1), where we restrict to nonnegative states x i ∈ R p ≥0 for the agents, and where the local cost functions satisfy the following assumption. Assumption 1 (Local cost function): The local cost func- tion f i : R p ≥0 → R ≥0 of any agent i is • continuously differentiable; • α-strongly convex: f i (x) − α 2 x 2 is convex ∀x; • β-smooth: ∇f i (x) − ∇f i (y) ≤ β x − y , ∀x, y; • satisfies arg min x∈R p ≥0 f i (x) = 0 p and f i (0 p ) = 0. More generally, we use F p α,β to denote the set of functions f : R p ≥0 → R ≥0 satisfying these conditions. Assumption 1 means that the cost paid by an agent is always nonnegative, and is zero only when the agent does not contribute at all to any activity, i.e. x i = 0 p , which is the minimal value taken by x i . It follows from Assumption 1 that the global cost satisfies f (x) = n i=1 f i (x i ) ∈ F np α,β . To problem (1) we associate an undirected graph G = (V, E), so that at random times a pair of agents (i, j) ∈ E is uniformly randomly chosen to interact and exchange information. Moreover, we assume the system is subject to random instantaneous arrivals and departures of agents in the system, respectively resulting in a new agent joining the system with its own local cost function and demand, or in an agent leaving the system and never coming back, with the possibility of sending a last message to their neighbours. We also consider replacements, which consist in the simultaneous occurrence of both an arrival and a departure. Hence, the system size n t , the local cost functions f t i and the demands d t i evolve with time, and consequently the instantaneous solution of (1) is time-varying as well, and is denoted x * ,t . B. Simplifying assumptions and reformulation For this preliminary work, we restrict to the specific case defined by the following assumptions. Assumption 2 (1-D functions): The local cost function of any agent at any time is one-dimensional: f t i : R ≥0 → R ≥0 . Assumption 3 (Homogeneous demand): The demand associated with any agent i at any time t is d t i = 1. Assumption 4: The graph G = (V, E) is complete. Moreover, we restrict to the case where the openness of the system is solely characterized by replacements of agents (i.e., the simultaneous occurrence of an arrival and a departure), so that the system size is fixed, and n t = n for all time t. Hence, the system only evolves at the instantaneous occurrences of either pairwise interactions of agents, resulting in a possible Evolution of the function value f t evaluated with the RCD algorithm x t defined in (8), the optimal solution x * ,t , and the selfish strategy x s,t , in a system subject to replacements of agents (i.e., simultaneous departures and arrivals) on average once every 4 RCD steps. update of their states (denoted U ) or replacements (denoted R). We call these occurrences "events", and we can define the set of all the events that can possibly take place in the system, which we call "event set", as follows: Ξ := R ∪ U = i∈V R i ∪ (i,j)∈E U ij , (2) where R i denotes the replacement of agent i and U ij a pairwise interaction between agents i and j. We assume that two distinct events never occur simultaneously, so that the system evolves in a discrete manner, where each time-step k ∈ Z ≥0 corresponds to the occurrence of an event ξ k ∈ Ξ. Assumption 5: An event ξ k is independent of all other events ξ j , j = k and of the state of the system x t until time k, so that at each time-step either an update (i.e., an event U ) happens with fixed probability p or a replacement (i.e., an event R) with fixed probability 1 − p. Let S n := x ∈ R n ≥0 : 1 x = n be the feasible set, we can now express (1) in our setting under the assumptions of this section: min x∈Sn f t (x) = n i=1 f t i (x i ).(3) C. Performance metrics Natural indexes for measuring the performance of an algorithm in our setting consist in evaluating its accumulated error over a finite number of iterations with respect to a given strategy. We define the two following strategies of interest in the context of the resource allocation problem: • Perfect collaboration: at each time instant t the agents know the optimal solution of (3) denoted x * ,t ; • Selfish players: the agents do not collaborate to minimize f t , and they operate at their individual desired point so that x s,t = 1 n at all t. Hence, for any T , the estimate x t obtained with a welldesigned algorithm is expected to satisfy T t=1 f t (x * ,t ) ≤ T t=1 f t (x t ) ≤ T t=1 f t (x s,t ).(4) The evolution of these strategies, compared with that of a given algorithm, is illustrated in Fig. 1. We define the following performance metrics to analyze the value provided by the RCD algorithm x t with respect to the strategies above: Dynamical Regret: Reg T := T t=1 f t (x t ) − f t (x * ,t ) ; (5) Benefit: Ben T := T t=1 f t (x s,t ) − f t (x t ) ; (6) Potential Benefit: Pot T := T t=1 f t (x s,t ) − f t (x * ,t ) . (7) The "dynamical regret" and "benefit" respectively measure the accumulated error from using a given algorithm with respect to the optimal solution x * ,t and the accumulated gain from using it instead of the selfish strategy x s,t . The "potential benefit" is independent of the algorithm; it represents the accumulated advantage of the optimal strategy with respect to the selfish one, and satisfies Pot T = Ben T + Reg T . Observe that the regret commonly used in online optimization typically compares x t with the overall optimal solution taken over all the iterations, i.e., x * = arg min x∈Sn T t=1 f t (x). In that sense, it differs from the dynamical regret in (5), which compares x t with the time-varying instantaneous optimal solution x * ,t = arg min x∈Sn f t (x) at each iteration, such as e.g., in [17]. D. Random Coordinate Descent algorithm and objective We consider the Random Coordinate Descent algorithm (RCD) introduced in [19], such as whenever a pair of agents (i, j) ∈ E interact, they update their respective estimates as x + i = x i − 1 β (f i (x i ) − f j (x j )) x + j = x j − 1 β (f j (x j ) − f i (x i )).(8) We moreover assume that whenever an agent in joins the system, it initializes its estimate as x in = d in = 1,(9) and whenever an agent out leaves the system, it sends a last message to all its neighbours (i.e., all the other agents in our setting) with its current estimate x out and its demand d out so the agents i = out update their estimates as x + i = x i + x out − x i n = 1 − 1 n x i + 1 n x out .(10) We show in the following proposition that RCD iterations, arrivals and departures as they are defined in (8) to (10) guarantee that as long as the initial estimate x 0 is feasible, then all the estimates remain feasible. Proposition 1 (Well-posedness): The event set (2) guarantees that if x 0 ∈ S n , then x t ∈ S n for all t. Proof: We first consider arrivals: the nonnegativity of x i and preservation of the constraint is a direct consequence of (9). In the case of departures, the nonnegativity of x i is a direct consequence of (10). Moreover, if the constraint is satisfied at iteration k with n k = n, then under the departure of the agent labelled out we have i =out x k+1 i = n − x out + x out n (n − 1) − n − x out n = n − 1. We finally consider iterations of the RCD algorithm. From Assumptions 1 and 2, it follows that for any x i ≥ 0 x i f i (x i ) ≥ α |x i | 2 ≥ 0, so that f i (x i ) ≥ 0. Moreover, since f i is β-smooth, one has f i (x i ) ≤ βx i , and therefore at each update of the RCD algorithm between agents i and j there holds x + i = x i − 1 2β f i (x i ) − f j (x j ) ≥ x i − 1 2β (βx i ) = x i 2 , establishing the nonnegativity of x i . A similar analysis can be used for x j . Due to the symmetry of the update rule, the constraint is always preserved and we conclude the proof. Our goal is to analyze the performance of the RCD algorithm (8) with the arrival and departure rules (9) and (10) in the setting described in Sections II-A and II-B using the metrics defined in Section II-C in expectation. III. UPPER BOUNDS ON THE PERFORMANCE METRICS We now derive upper bounds on the evolution of the Potential Benefit and the Dynamical Regret respectively defined in (7) and (5) in expectation. Whereas the former is only related to the problem itself, the latter actually depends on the algorithm we consider. We first provide the following lemmas, where Lemma 1 directly follows from the equivalence of the norms. Lemma 1: Let x ∈ S n , then n ≤ x 2 ≤ n 2 . Lemma 2: Let f (x) = n i=1 f i (x i ) , where all f i satisfy Assumptions 1 and 2, then for any x ∈ S n there holds α 2 n ≤ f (x) ≤ β 2 n 2 .(11) Proof: From Assumptions 1 and 2, f (0) = f (0) = 0. Hence, since f is β-smooth and using Lemma 1, there holds f (x) ≤ β 2 x 2 ≤ β 2 n 2 which establishes the upper bound. Similarly, since f is α-strongly convex and by using Lemma 1, it follows that f (x) ≥ α 2 x 2 ≥ α 2 n, which establishes the lower bound, and concludes the proof. Lemma 2 provides a global upper bound on the difference between any two solutions x, y ∈ S n : f t (x) − f t (y) ≤ n 2 (nβ − α).(12) This can be used to derive upper bounds on any of the metrics defined in Section II-C, e.g., Ben T ≤ n 2 (nβ − α)T . A. Potential Benefit We first obtain in the following theorem an upper bound on the expected value of the potential benefit, which we remind quantifies the accumulated advantage of using the optimal strategy rather than not collaborating at all. Theorem 1: In the setting of Section II, there holds Pot T ≤ n 2 α (κ − 1) T,(13) and in particular lim T →∞ Pot T T ≤ n 2 α(κ − 1).(14) Proof: Remember that x s,t = 1 n by definition, and that f t (0 n ) = 0 and ∇f t (0 n ) = 0 n from Assumption 1. Hence, there holds from the β-smoothness of f t f t (x s,t ) ≤ ∇f t (0 n ) (x s,t ) + β 2 x s,t 2 = β 2 1 n 2 = β 2 n. Similarly, since f t is α-strongly convex, we get f t (x * ,t ) ≥ ∇f t (0 n ) (x * ,t ) + α 2 x * ,t 2 = α 2 x * ,t 2 ≥ α 2 n, where the last inequality follows from Lemma 1. Hence f t (x s,t ) − f t (x * ,t ) ≤ β 2 n − α 2 n = n 2 (β − α) holds, and injecting it into (7) yields (13). The last result then follows from dividing (13) by T . Notice that since the dynamical regret is nonnegative by definition, the bounds (13) and (14) also hold for the benefit since Ben T = Pot T − Reg T ≤ Pot T . B. Dynamical Regret We now obtain an upper bound on the expected dynamical regret defined in (5), where we remind x t is obtained with the RCD algorithm defined in Section II-D. For that purpose, we first introduce the following intermediate quantities: C t := f t (x t ) − f t (x * ,t ); (15) ∆f t := f t+1 (x t+1 ) − f t (x t ); (16) ∆f * t := f t+1 (x * ,t+1 ) − f t (x * ,t ).(17) Thus, C t corresponds to the instantaneous loss of the RCD algorithm with respect to the optimal solution at iteration t, and ∆f t and ∆f * t respectively stand for the instantaneous variation at one iteration of the total estimated cost and optimal cost, such that C t+1 = C t + ∆f t − ∆f * t . In the following proposition, we study the effect or replacements on ∆f t in order to later characterize C t in expectation, and consequently the expected dynamical regret. Proposition 2: In the setting of Section II the replacement of an agent, denoted R, results in E ∆f t | R ≤ 5 2 β − 3 2 α.(18) Proof: We analyze the effects of arrivals and departures separately. Let g denote the local cost function of the joining agent at an arrival, then f t+1 (x t+1 ) = f t (x t ) + g (1) and ∆f t = f t (x t ) + g(1) − f t (x t ) = g(1) ≤ β 2 ,(19) where the last inequality follows from Assumption 1, and in particular the β-smoothness of g. Consider now a departure, and let denote the label of the leaving agent, such that f t+1 ( (10). From the definition of departures, is uniformly selected among the n agents in the system and by taking the expected value ∆f t over the leaving agent, one gets the following, where we omit the reference to time to lighten the notation: x t+1 ) = i = f t i (x t+1 i ), with x t+1 i = x t i + x t i +x t n fromE [∆f t ] = n =1 1 n   i = f i x i + x − x i n − f (x)   = 1 n n =1 f n − 1 n x + x n 1 n − f (x ) − f (x) = 1 n n =1 f n − 1 n x + x n 1 n − n + 1 n f (x). Since f is β-smooth from Assumption 1, one has f n − 1 n x + x n 1 n ≤ f (x) + 1 n ∇f (x), x 1 n − x + β 2n 2 x 1 n − x 2 , and it follows that E [∆f t ] ≤ 1 n 2 n =1 ∇f (x), x 1 n − x + β 2n 3 n =1 x 1 n − x 2 − 1 n f (x).(20) From Assumption 1, in particular since f is β-smooth and αstrongly convex, it satisfies α x 2 ≤ ∇f (x), x ≤ β x 2 for any x. Hence, reminding that n =1 x = n, the first sum of (20) can be upper bounded by n =1 ∇f (x), x 1 n − x = n ∇f (x), 1 n − x ≤ n(βn − α x 2 ). The second sum of (20) can be expressed as: n =1 x 1 n − x 2 = n =1 n i=1 x 2 − 2x x i + x 2 i = n =1 nx 2 − 2nx + x 2 = 2n x 2 − n . Then (20) is upper bounded by E [∆f t ] ≤ 1 n βn − α x 2 + β n 2 x 2 − n − 1 n f (x). Lemmas 1 and 2 yield x 2 ≤ n 2 and f (x) ≥ α 2 n, so that E ∆f t ≤ 2β − 3 2 α.(21) The conclusion follows from adding (19) and (21). We can now use Proposition 2 to study the evolution of the expected dynamical regret in the following theorem. Theorem 2: In the setting of Section II, there holds EReg T ≤ C 0 T t=1 η t +(1−p) T −1 t=0 η t (M f + (T − t)θ) ,(22) where η = 1 − p κ(n−1) (with p the probability that a given event is an update from Assumption 5), M f = n 2 (βn − α) and θ = 5 2 β − 3 2 α. Proof: Let γ = 1 − 1 κ(n−1) denote the contraction rate of the RCD algorithm as defined in (8) [19]. Remember that there holds C t+1 = C t + ∆f t + ∆f * t for all times t. Hence, at any time-step t one has E C t+1 = E C t + ∆f t − ∆f * t .(23) From Assumption 5 the event at iteration t is an update, denoted U t , with probability p, or a replacement, denoted R t , with probability 1 − p. In the case of an update, we have x * ,t+1 = x * ,t , so that ∆f * t = 0. Hence, we have ∆f t = C t+1 − C t , and since E C t+1 |C t , U t ≤ γC t with the RCD algorithm from [19]: E ∆f t |, U t = E C t+1 − C t |U t ≤ (γ − 1)E C t . (24) In the replacement case, there holds E ∆f t |R t ≤ θ = 5 2 β − 3 2 α,(25) where θ comes from Proposition 2. Injecting (24) and (25) into (23) then yields E C t+1 ≤ E C t +p(γ − 1)E C t +(1 − p) θ − E ∆f * t = ηE C t + (1 − p) θ − E ∆f * t ,(26) where η = 1 + p(γ − 1). E C t ≤ η t C 0 + (1 − p) t−1 j=0 η t−j−1 θ − E ∆f * t .(27) Injecting this last result into (5) then yields EReg T = T t=1 E C t ≤ C 0 T t=1 η t +(1 − p) T t=1   t−1 j=0 η t−j−1 θ − E ∆f * t   . After some term re-organization, it becomes EReg T ≤ C 0 T t=1 η t + (1 − p) T −1 t=0 (T − t)η t θ − (1 − p) T −1 t=0 η t   T −t j=0 E ∆f * j   . Finally, using Lemma 2, one concludes that − T −t j=0 E ∆f * j = −E T −t j=0 ∆f * j = −E f T −t+1 (x * ,T −t+1 ) − f 1 (x * ,1 ) ≤ n 2 (βn − α), and M f = n 2 (βn − α) yields the conclusion. We now analyze the asymptotic behavior of the averaged regret in the following corollary. Corollary 1: Let ρ R := 1−p p . In the same setting as that of Theorem 2, there holds lim T →∞ EReg T T ≤ ρ R (n − 1)β 5κ − 3 2 . (28) Proof: Starting from (22), we have EReg T T ≤ C 0 T t=1 η t T + (1 − p) T −1 t=0 M f η t T + 1 − t T η t θ . Remember that η = 1 − p κ(n−1) ≤ 1, so that T t=1 η t < T , and lim T →∞ T t=1 η t T = 0. Hence lim T →∞ EReg T T ≤ lim T →∞ (1 − p) T −1 t=0 1 − t T η t θ. Moreover, for η < 1, one shows lim T →∞ EReg T T ≤ (1 − p) 1 1 − η θ = 1 − p p κ(n − 1)θ, and the conclusion follows from the definitions of θ in Theorem 2, and from ρ R := 1−p p . The upper bounds for the potential benefit (13) and the dynamical regret (22) linearly scale with T . This behavior is rather natural for the former, which does not depend on the algorithm. For the latter, it is most likely unavoidable due to the introduction at each replacement of perturbations of nondecaying magnitude which no algorithm can instantaneously compensate. Interestingly, this behavior contrasts with standard results in online optimization, where a sublinear growth in T is desired to cancel the asymptotic averaged regret [16, Ch. 1.1]. However, these results usually apply on another definition of the regret, where x t is compared with an overall time-independent strategy x * computed over all T iterations, in opposition with x * ,t which is optimal for each iteration. Moreover, the corresponding asymptotic upper bounds linearly grow with n and α for (14), and with n−1 and β for (28), consistently with their expected behavior. In particular, the scaling of (28) with n − 1 follows from the convergence rate of the RCD algorithm γ = 1− 1 κ(n−1) . Interestingly, (28) is proportional to ρ R (n − 1) = (1 − p) n−1 p , and the bound can thus be seen as the ratio between the probability for a given agent to be involved in a RCD update p n−1 (involved in γ), and the impact of replacements at the system level 1 − p, independently of n. This is consistent with the bound on the impact of replacements in (18) which is independent of n (by contrast, alternative situations such as e.g., if all agents were to be reset at each replacement are expected to generate an impact growing with n). Hence, for small values of ρ R (i.e., rare replacements), the bound guarantees that the asymptotic dynamical regret remains reasonably bounded, and decays to zero when ρ R → 0, i.e., for closed systems. Finally, observe that (28) is proportional to 5κ−3 2 , consistently with the fact that a larger interval for the possible curvature of the cost functions should generate a larger potential error at replacements. This factor is a potential source of conservatism, e.g., with respect to (14) where the scaling is in 1 2 (κ − 1). More generally, it is not clear yet whether other algorithms than the RCD might provide tighter bounds. Remark 1: The proofs of Theorem 2 and Corollary 1 can directly be adapted to any contraction rate γ < 1, and are thus easily generalized to any other algorithm that guarantees linear convergence; in particular lim T →∞ EReg T T ≤ ρ R θ 1−γ . C. The case of quadratic functions The bound on the expected dynamical regret can be refined for the particular case where all local functions are quadratic, i.e. satisfy the following additional assumption. Assumption 6 (Quadratic functions): The local cost function of any agent i at time t is of the form f i (x i ) = φ i x 2 i , φ i ∈ α 2 , β 2 .(29) The parameter φ i is randomly chosen according to a distribution with a finite support determined by the interval α 2 , β 2 . Observe that functions satisfying Assumption 6 necessarily satisfy Assumption 1 as well. Under Assumption 6, we can obtain a tighter bound than that of Proposition 2, presented in the following proposition. Proposition 3: In the setting of Section II, and under Assumption 6, the replacement of an agent R results in E [∆f t | R] ≤ 3n 2 − 3n + 1 2n 2 (β − α) .(30) Proof: The arrival case is treated the same was as in the proof of Proposition 2, resulting in ∆f t ≤ β 2 . The departure case follows the same first steps with f i (x) = φ i x 2 , and E [∆f t ] = n =1 1 n   i = φ i x i + x −xi n 2 − f (x)   = 1 n n =1   i = φ i x i + x −xi n 2 − x 2 i − φ x 2   = 1 n 2 n =1 i =l φ i 1−2n n x 2 i + 2x i x n−1 n + x 2 n − f (x) n . Using the fact that n =1 i = φ i x 2 i = (n − 1)f (x) and that φ i ≤ β 2 for all i, we obtain E ∆f t ≤ 1 n 2 1 − 2n n (n − 1) − 1 n f (x) + β 2n 2 n =1 i = 2x i x n − 1 n + x 2 n . Observe that n =1 i =l x i x = n 2 − x 2 and n =1 i = x 2 = (n − 1) x 2 , so that a few algebraic manipulations yield E ∆f t ≤ − 3n 2 − 3n + 1 n 3 f (x) + β n − 1 2n 3 2n 2 − x 2 . Using x 2 ≥ n (from Lemma 1) and f (x) ≥ α 2 n (from Lemma 2) then yields E ∆f t ≤ − 3n 2 − 3n + 1 2n 3 α + β 2n 2 − 3n + 1 2n 3 , and combining with the arrival case concludes the proof. The result above allows us stating the following theorem, which improves Theorem 2 and Corollary 1 respectively for the case of quadratic functions. Theorem 3: In the setting of Section II, and under Assumption 6, there holds EReg T ≤ C 0 T t=1 η t +(1−p) T −1 t=0 η t (M f + (T − t)θ) ,(31) where η = 1 − p κ(n−1) (with p the probability that a given event is an update from Assumption 5), M f = n 2 (βn − α) and θ = (β − α) 3n 2 −3n+1 2n 2 . In particular, lim T →∞ EReg T T ≤ ρ R (n − 1) 3n 2 −3n+1 2n 2 β (κ − 1) .(32) Proof: The proof follows the exact same steps as those of Theorem 2 and Corollary 1 where Proposition 3 is used instead of Proposition 2. The upper bound for the quadratic case is qualitatively better than that of the general case; it was derived based on an additional information of the cost function, thus resulting in tighter bounds. In this case, the dependence of (32) is in 3 2 (κ − 1), which is consistent with the result derived for the potential benefit (14). In particular, for n becoming large, (32) and (28) become equivalent up to a constant β. Moreover, (32) becomes 0 when κ = 1, consistently with the expected behavior of the RCD algorithm for quadratic functions since all the cost functions would then be the same. D. Numerical Results To illustrate the results of Theorems 1 to 2, we consider a system of 5 agents with κ = 10 and ρ R = 0.0125, the latter implies that on average there is one replacement every 80 events. We consider two possibilities: random replacements (RR) where the local function is randomly uniformly chosen among the set of piecewise quadratic functions satisfying Assumption 1, and adversarial replacements (AR), where these functions are quadratic functions φ i x 2 , with φ i ∈ α 2 , β 2 . The AR setting is expected to be less favorable than the RR setting, since replacements might result in the largest change of local functions. Notice that the bounds (13) and (22) are independent of the distribution from which the local cost function are assigned to the agents when they join the system, so that they hold for any such assignment rule. Fig. 2 compares the results of Theorem 1 and Corollary 1 with simulations for both random and adversarial replacements in the setting described above. Even though the theoretical bounds are conservative, they capture well the qualitative behavior of these metrics. In particular, consistently with Pot T and EReg T that grow linearly with T , the bounds in the figure do not converge to zero, and a remaining asymptotic error is observed. Our bounds are tighter for the (14) and (28), with simulated results, either with random replacements (RR) or adversarial replacements (AR). adversarial replacement case than the random replacement case, this suggests that our bounds might be tight for some particular choice of the joining functions at replacements, especially that on the potential benefit. Fig. 3 compares the results of Theorems 2 and 3 with simulations in the same setting as described previously, with random replacements (RR). The figure shows that bound (32) is tighter for quadratic functions, following the fact that we have access to more information regarding the local cost functions, thus improving the estimation of the effect of replacements on the expected regret. IV. CONCLUSION We analyzed the performance and behavior of the Random Coordinate Descent algorithm (RCD) for solving the optimal resource allocation problem in an open system subject to replacements of agents, resulting in variations of the total cost function and of the total amount of resource to be allocated. We considered a simple preliminary setting where the budget is homogeneous and the graph is complete, and used tools inspired from online optimization to show that it is not possible to achieve convergence to the optimal solution with the RCD algorithm in expectation in open system, but that the error is expected to remain reasonable. We have derived upper bounds on the evolution of the regret and the potential benefit in expectation and showed that due to the random choice of the new local cost function during replacements, an error is expected to be accumulated with time and cannot be compensated. A natural continuation of this work is thus the derivation of the corresponding upper bound for the benefit, and of lower bounds for this quantities in order to validate the observed behavior. More generally, our bounds could be extended to more general settings, and their tightness can be improved to match more accurately the actual performance of the algorithm. Moreover, since our approach is based on the analysis of the effect of arrivals and departures of agents combined into replacements, the next step of this study is to generalize it to the case where the system size changes with the time, i.e., where arrivals and departures are decoupled. Research supported in part by the Agence Nationale de la Recherche (ANR) via grant "Hybrid And Networked Dynamical sYstems" (HANDY), number ANR-18-CE40-0010 and by the "RevealFlight" ARC at UCLouvain, by the Incentive Grant for Scientific Research (MIS) "Learning from Pairwise Data" of the F.R.S.-FNRS. R. Vizuete and C. Monnoyer de Galland equally contributed to this work. R. Vizuete and E. Panteley are with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France. R. Vizuete and P. Frasca are with Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, F-38000 Grenoble, France. C. Monnoyer de Galland and J. M. Hendrickx are with the ICTEAM institute, UCLouvain, Louvain-la-Neuve, Belgium. C. Monnoyer de Galland is a FRIA fellow (F.R.S.-FNRS). (E-mail adresses: Fig. 1. Evolution of the function value f t evaluated with the RCD algorithm x t defined in (8), the optimal solution x * ,t , and the selfish strategy x s,t , in a system subject to replacements of agents (i.e., simultaneous departures and arrivals) on average once every 4 RCD steps. t = 0, and therefore there holds Fig. 2 . 2Evolution of the averaged asymptotic expected Potential Benefit (on the left) and dynamical regret (on the right) in a system of 5 agents with ρ R = 0.0125 and κ = 10. Each plot compares the upper bounds, respectively from Fig. 3 . 3Evolution of the expected averaged regret for a system of 5 agents holding quadratic functions with ρ R = 0.0125 and κ = 10. The plain blue line and the dash-dotted red line respectively correspond to the upper bounds of Theorems 2 and 3 respectively. The dotted yellow line corresponds to simulation where we consider random replacements (RR). Resource Allocation Problems: Algorithmic Approaches. T Ibaraki, N Katoh, MIT pressT. Ibaraki and N. 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[]
[ "EMBEDDABILITY OF MULTIPLE CONES", "EMBEDDABILITY OF MULTIPLE CONES" ]
[ "D Repovš ", "W Rosicki ", "A Zastrow ", "M Željko " ]
[]
[]
The main result of this paper is that if X is a Peano continuum such that its n-th cone C n (X) embeds into R n+2 then X embeds into S 2 . This solves a problem proposed by W. Rosicki.
10.1016/j.topol.2008.02.007
[ "https://arxiv.org/pdf/0706.0557v2.pdf" ]
11,847,569
0706.0557
abcab97ac0e26c08a8717f38cb66ffa5180ec819
EMBEDDABILITY OF MULTIPLE CONES 29 Mar 2008 D Repovš W Rosicki A Zastrow M Željko EMBEDDABILITY OF MULTIPLE CONES 29 Mar 2008arXiv:0706.0557v2 [math.GT] The main result of this paper is that if X is a Peano continuum such that its n-th cone C n (X) embeds into R n+2 then X embeds into S 2 . This solves a problem proposed by W. Rosicki. Introduction The classical Lefschetz-Nöbeling-Pontryagin Embedding Theorem [10] asserts that every compact metric space X of dimension n embedds into R 2n+1 . We are interested in the relationship between the embeddability of X and embeddability of its Cartesian product X × I n with a cube I n (resp. its cone C(X), iterated cone C n (X) = C(. . . (C(X)) . . .), suspension Σ(X)). Clearly, if X embeds in R m , then X × I n and C n (X) embed into R n+m . However, sometimes they embed into lower-dimensional Euclidean space. Such is the case for the spheres S n , where S n , C(S n ) ∼ = B n+1 and S n × I all embed into R n+1 . Let X be a Peano continuum. It was proved in [14] that if the cone C(X) of X embeds into R 3 , then X embeds into S 2 . As a consequence, if the suspension Σ(X) of X embeds into R 3 , then X is planar. Note that for each n ≥ 3, there exists a Peano continuum X n such that X n is not embeddable in S n , whereas the cone C(X n ) of X n is embeddable in R n+1 (see [14]). The main result of this paper is Theorem 1.1 which solves a problem from [14]. Our proof is based on the methods of [4] and [14]. Theorem 1.1. Let X be a Peano continuum. Suppose that for some n ∈ N, C n (X) is embed- dable in R n+2 . Then X is embeddable in S 2 . Let X be a Peano continuum. Claytor [7] proved that X is embeddable in S 2 if and only if X does not contain any of the Kuratowski curves K 1 , K 2 , K 3 , K 4 (see Figure 1). Preliminaries A space X is said to be planar if X is embeddable in R 2 . We say that X is locally planar if for every point x ∈ X there exists a neighbourhood U x of x in X such that U x is embeddable in R 2 . Rosicki [13,Theorem 1.1] proved that if a Peano continuum X is embeddable in R 3 and X is a nontrivial Cartesian product X = Y × Z then one of the factors is either an arc or a simple closed curve. Rosicki [13] also proved that if a Peano continuum X is embeddable in R 3 and is homeomorphic to the product Y × S 1 then the factor Y must be planar. Alternatively, if X = Y × [0, 1] is embeddable in R 3 andȞ 1 (X) =Ȟ 2 (X) = 0 then Y must be planar. Cauty [4], generalizing Rosicki [13], proved that for every n > 3 and every Peano continuum X such that X × I n−2 is embeddable into an n-manifold, it follows that X must be locally planar. This theorem was stated earlier by Stubblefield [15]. However, Burgess [2] found a mistake in his proof. Borsuk [1] constructed an example of a locally connected, locally planar continuum X which is not embeddable into any surface. This continuum contains a sequence (X n ) of subsets K 1 K 2 K 3 K 4 Figure 1. Kuratowski curves K 1 , K 2 , K 3 , K 4 homeomorphic to Kuratowski curve K 1 which converge to an arc. Cauty [4] proved that X×I n−2 is not embeddable into any n-manifold so the converse to his theorem does not hold. Local separation We say that a subset D ⊂ R n locally separates R n at the point x 0 ∈ D into k ∈ N components if there exists ε > 0 such that for all 0 < δ < ε, the set B(x 0 , δ) \ D has exactly k components A 1 , . . . , A k for which x 0 ∈ A i , for all i ∈ {1, . . . , k}. It is easy to prove the following lemma using similar methods as in the proof of Lemma 1 in [14]. Lemma 3.1. A homeomorphic image of any n-disk locally separates R n+1 at any point of its interior into two components. Note that C n (X) = σ n−1 * X = {xt + y(1 − t); x ∈ σ n−1 , y ∈ X, t ∈ [0, 1]}, where σ n−1 is an (n − 1)-simplex. Then σ n−1 * {x} is an n-ball and σ n−1 * I is an (n + 1)-ball. We consider σ n−1 as a subset of σ n−1 * X. Lemma 3.2. Let I i , i ∈ {1, . . . , k}, k > 1 be arcs with common endpoints and pairwise disjoint interiors and C k = C n ( k i=1 I i ) = σ n−1 * ( k i=1 I i ). Let h : C k → R n+2 be an embedding. Then h(C k ) locally separates R n+2 at any point h(x 0 ), where x 0 is an interior point of σ n−1 , into k components (where σ n−1 is considered as a subset of C k ). Proof. The proof is by induction on k. If k = 2, then C 2 = σ n−1 * S 0 * S 0 hence h(C 2 ) locally separates R n+2 at h(x 0 ) into two components, by Lemma 3.1. Assume that Lemma 3.2 holds for k − 1. Choose ε > 0 smaller than the distance between h(x 0 ) and the image of ∂σ n−1 * ( k i=1 I i ). Let δ > 0 be so small that D k = h(C k ∩ B(x 0 , δ)) ⊂ B(h(x 0 ), ε). There exists an open connected set U k ⊂ R n+2 such that D k = U k ∩ h(C k ). Consider the exact sequence of the pair (U k , U k \ D k ): → H 1 (U k ) → H 1 (U k , U k \ D k ) → H 0 (U k \ D k ) → H 0 (U k ) → H 0 (U k , U k \ D k ) → 0. Since U k is an open (n + 2)-manifold, H 1 (U k ) ∼ =Ȟ n+1 c (U k ) by the Poincaré duality, whereȞ c denotes theČech cohomology with compact supports. Also [9,VIII,7.14], where L = ∅, K = D k and X = U k ). H 1 (U k , U k \ D k ) ∼ =Ȟ n+1 c (D k ) (see We know that H 0 (U k , U k \ D k ) = 0 because U k is arc-connected and U k \ D k = ∅. Therefore we can consider the exact sequence →Ȟ n+1 c (U k ) →Ȟ n+1 c (D k ) → H 0 (U k \ D k ) → H 0 (U k ) → 0. Next we show by induction that the mapȞ n+1 c (U k ) →Ȟ n+1 c (D k ) is trivial. If k = 2 then D k is an open (n + 1)-ball. Then H 0 (U k \ D k ) ∼ = Z 2 , by Lemma 3.1. SinceȞ n+1 c (D k ) ∼ = Z and H 0 (U k ) ∼ = Z, we obtain the exact sequencě H n+1 c (U k ) → Z → Z 2 → Z → 0. Hence the mapȞ n+1 c (U k ) →Ȟ n+1 c (D k ) is indeed trivial, as asserted. SinceȞ n+1 c (D 2 ) ∼ = Z, we obtain by induction thatȞ n+1 c (D k ) ∼ =Ȟ n+1 c (D k−1 ) ⊕Ȟ n+1 c (D ′ 2 ) ∼ = Z k−2 ⊕ Z, where D ′ 2 = h(C n (I 1 ∪ I k ) ∩ B(x 0 , δ)). The mapȞ n+1 c (U k ) →Ȟ n+1 c (D k ) ∼ =Ȟ n+1 c (h(D k−1 )) ⊕Ȟ n+1 c (D ′ 2 ) is trivial because both of its coordinates are trivial, by inductive hypothesis. Therefore the sequence 0 →Ȟ n+1 c (D k ) → H 0 (U k \ D k ) → H 0 (U k ) → 0 is exact. So the sequence 0 → Z k−1 → H 0 (U k \ D k ) → Z → 0 is also exact. Hence H 0 (U k \ D k ) ∼ = Z k and U k \ D k0 → H 0 (U k \ X k ) → H 0 (U k ) → 0 is exact, therefore H 0 (U k \ X k ) ∼ = Z. Proof of Theorem 1.1 We shall need two more lemmata: Lemma 4.1. Consider the Kuratowski curve K 1 and let n ∈ N. Then C n (K 1 ) is not embeddable in R n+2 . Proof. Suppose to the contrary, that there exists an embedding h : C n (K 1 ) → R n+2 . Consider K 1 ⊂ R 3 and denote (see Figure 2) If X = i I i , then σ n−1 * X = i (σ n−1 * I i ) is a union of (n + 1)-disks. Let x 0 ∈ Int σ n−1 and choose ε > 0 so that (see Figure 3) I 1 = [c, a] ∪ [C 1 = h(σ n−1 * (I 1 ∪ I 3 )) locally separates B(h(x 0 ), ε) into B 1 , A 1 at h(x 0 ), C 2 = h(σ n−1 * (I 1 ∪ I 2 )) locally separates B(h(x 0 ), ε) into B 2 , A 2 at h(x 0 ), C 3 = h(σ n−1 * (I 2 ∪ I 3 )) locally separates B(h(x 0 ), ε) into B 3 , A 3 at h(x 0 ). x 0 I 1 I 2 I 3 Figure 3. Local separation at h(x 0 ) By Lemma 3.2 we have that C = h(C n (I 1 ∪ I 2 ∪ I 3 )) = h(σ n−1 * 3 i=1 I i ) = h( 3 i=1 σ n−1 * I i ) locally separates B(h(x 0 ) , ε) into three components. We will show that we can adopt the notation for these three components to be B 1 , A 2 and A 3 . We use abstract linear combinations for describing our joins, e.g. σ n−1 * K 1 = {xt + y(1 − t); x ∈ σ n−1 , y ∈ K 1 , t ∈ [0, 1]}. For σ n−1 ⊂ σ n−1 * K 1 , we have that h(σ n−1 ) is a subset of C 1 , but that h| σ n−1 * I 2 maps all linear combinations with t = 1, but sufficiently close to 1, to a subset that is connected but disjoint from C 1 . Hence this subset can only be contained either in A 1 or in B 1 . We may assume that it is in A 1 . Since the entire neighbourhood of σ n−1 in σ n−1 * I 2 is mapped by h into A 1 , we have h(σ n−1 * I 2 ) ∩ B 1 = ∅, provided ε > 0 is small enough. Then B 1 is not divided by C, so it is one of the three components. Analogously, by considering C 2 (resp. C 3 ) we can make sure that A 2 and A 3 are the other two components and that h(σ n−1 * I 3 )∩A 2 = ∅ and h(σ n−1 * I 1 )∩A 3 = ∅. Since C ∪B 1 ∪A 2 ∪A 3 and C ∪A 1 ∪B 1 are both disjoint decompositions of a neighbourhood of h(x 0 ), the set h(σ n−1 * I 2 )∪C 1 separates the component A 1 into components A 2 and A 3 . Note that x 0 * K 1 = {x 0 t + x(1 − t); x ∈ K 1 , t ∈ [0, 1]} ⊂ C n (K 1 ). Choose t 0 near 1 so that h({x 0 t + x(1 − t); x ∈ K 1 , t ≥ t 0 }) ⊂ B(h(x 0 ), ε). Let p ′ = h(x 0 t 0 + p(1 − t 0 )) ∈ A 1 . The arc H = h({x 0 t 0 + x(1 − t 0 ); x ∈ (p, q)}) is contained in B(h(x 0 ), ε) \ h(C) . Therefore points p ′ and q ′ = h(x 0 t 0 + q(1 − t 0 )) are in the same component. Hence q ′ ∈ A 2 or q ′ ∈ A 3 . So the arc I = h({x 0 t 0 + x(1 − t 0 ); x ∈ (a, q] ∪ [q, d)}) is contained either in A 2 or in A 3 . But this yields a contradiction since a ′ = h(x 0 t 0 + a(1 − t 0 )) / ∈ A 3 (so I ⊂ A 3 ) and d ′ = h(x 0 t 0 + a(1 − t 0 )) / ∈ A 2 (so I ⊂ A 2 ) . The proof of the next lemma can be obtained by changing the proof of [14,Lemma 4] in the same way as we did it for the proof of Lemma 2.3 using the proof of [14,Lemma 3]. Lemma 4.2. Consider the Kuratowski curve K 2 and let n ∈ N. Then C n (K 2 ) is not embeddable in R n+2 . Proof of Theorem 1.1. By Claytor's theorem (see [6], [7]), it suffices to show that C n (K i ) is not embeddable into R n+2 for any i ∈ {1, 2, 3, 4}. Now, Cauty [4] proved that K i × I n is not embeddable into R n+2 for any i ∈ {3, 4}. Therefore also C n (K i ) is not embeddable into R n+2 for any i ∈ {3, 4}. Hence we only have to consider the cases i = 1 and i = 2. The proof is now completed by application of Lemmata 4.1 and 4.2. Epilogue Repovš, Skopenkov andŠčepin [12] proved that if X × I PL embeds into R n+1 , where X is either an acyclic polyhedron and dim X ≤ 2n 3 −1 or a homologically (2 dim X −n−1)-connected manifold and dim X ≤ 2n 3 − 1 or a collapsible polyhedron, then X PL embeds into R n . Question 5.1. What can one say about embeddability of X into Euclidean spaces if one considers C(X) or C n (X) or Σ(X) or Σ n (X) instead of X × I for X in [12]? It follows by [12] that if X is a contractible polyhedron such that X × I embeds into R n+1 then X embeds into R n . So if X is contractible and C(X) ⊂ R n+1 then X embeds into R n . Note that there exists a polyhedron P n such that P n is not embeddable into R n but C 2 (P n ) is embeddable in R n+2 . Namely, Cannon [3] proved that if H n is a homology n-sphere then its double suspension Σ 2 (H n )is the (n + 2)-sphere (see [8] [11] for a far reaching generalization of this result). So if P n = H n \ B n where B n is an n-ball then the double cone C 2 (P n ) embeds in R n+2 . The polyhedron P n is acyclic but not contractible. Question 5.2. Does there exist a contractible n-dimensional polyhedron X n such that C k (X n ) embeds into R n+k , but X n does not embed into R n ? In [14,Theorem 2] contractible continua X n were constructed, such that X n is not embeddable in R n , C(X n ) is embeddable in R n+1 , and X n is not a polyhedron. By [12], if X is an npolyhedron then X × I embeds into R 2n+1 . If X is an n-polyhedron then C(X) need not embed into R 2n+1 . For example, the Kuratowski curves K 1 and K 2 are 1-polyhedra but the cones C(K 1 ) and C(K 2 ) do not embed into R 3 . Question 5.3. Suppose that X is a compact contractible n-dimensional polyhedron. Does the cone C(X) embed into R 2n+1 ? Does the same hold if X is only acyclic? has k components. The point h(x 0 ) belongs to the closure of each of them. Indeed, if X k is D k with a small open neighbourhood of h(x 0 ) removed thenȞ n+1 c (X k ) ∼ = 0 and the sequence Figure 2 . 2a, b], I 2 = [c, p] ∪ [p, b], and I 3 = [c, d] ∪ [d, Kuratowski curve K 1 Acknowledgements[5]for the hint communicated to the second author, and the referee for comments and suggestions. Über stetige Abbildungen der euklidischen Räume. K Borsuk, Fund. Math. 21K. Borsuk,Über stetige Abbildungen der euklidischen Räume, Fund. Math. 21 (1933), 236-246. . C E Burgess, Math. Rev. 26144ReviewC. E. Burgess, Review of [6], Math. Rev. 26 (1963), #749, p. 144 Shrinking cell-like decompositions of manifolds. Codimension three. J W Cannon, Ann. of Math. 1102J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (2) 110:1 (1979), 83-112. Sur le plongement de X ×I n−2 dans une n-variété. R Cauty, Proc. Amer. Math. Soc. 94R. Cauty, Sur le plongement de X ×I n−2 dans une n-variété, Proc. Amer. Math. Soc. 94 (1985), 516-522. . R Cauty, Personal communicationR. Cauty, Personal communication (1993). Topological immersions of Peanian continua in a spherical surface. S Claytor, Ann. of Math. 2S. Claytor, Topological immersions of Peanian continua in a spherical surface, Ann. of Math. (2) 35 (1934), 809-835. Peanian continua not embeddable in a spherical surface. S Claytor, Ann. of Math. 2S. Claytor, Peanian continua not embeddable in a spherical surface, Ann. of Math. (2) 38 (1937), 631-646. R J Daverman, Decompositions of Manifolds. New YorkAcademic PressR. J. Daverman, Decompositions of Manifolds, Academic Press, New York, 1986. A Dold, Lectures on Algebraic Topology. BerlinSpringer-VerlagA. Dold, Lectures on Algebraic Topology, Springer-Verlag, Berlin, 1995. R Engelking, Dimension Theory. North-Holland, AmsterdamR. Engelking, Dimension Theory, North-Holland, Amsterdam, 1978. R D Edwards, arXiv:math.GT/0610573v1Suspensions of homology spheres. preprintR. D. Edwards, Suspensions of homology spheres, preprint arXiv:math.GT/0610573v1 (2006). On embeddability of X × I into Euclidean space. D Repovš, A B Skopenkov, E V Ščepin, Houston J. Math. 21D. Repovš, A. B. Skopenkov and E. V.Ščepin, On embeddability of X × I into Euclidean space, Houston J. Math. 21 (1995), 199-204. On topological factors of 3-dimensional locally connected continuum embeddable in E 3. W Rosicki, Fund. Math. 99W. Rosicki, On topological factors of 3-dimensional locally connected continuum embeddable in E 3 , Fund. Math. 99 (1978), 141-154. On embeddability of cones in Euclidean spaces. W Rosicki, Colloq. Math. 64W. Rosicki, On embeddability of cones in Euclidean spaces, Colloq. Math. 64 (1993), 141-147. Some imbedding and non-imbedding theorems for N-manifolds. B Stubblefield, Trans. Amer. Math. Soc. 103B. Stubblefield, Some imbedding and non-imbedding theorems for N-manifolds, Trans. Amer. Math. Soc. 103 (1962), 403-420.
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[ "S P Jones \nMax-Planck-Institute for Physics\nFöhringer Ring 680805MünchenGermany\n", "A D Martin \nInstitute for Particle Physics Phenomenology\nDurham University\nDH1 3LEDurham\n\nU.K. c Petersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia\n", "M G Ryskin \nInstitute for Particle Physics Phenomenology\nDurham University\nDH1 3LEDurham\n\nU.K. c Petersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia\n", "T Teubner \nDepartment of Mathematical Sciences\nUniversity of Liverpool\nL69 3BXLiverpoolU.K\n" ]
[ "Max-Planck-Institute for Physics\nFöhringer Ring 680805MünchenGermany", "Institute for Particle Physics Phenomenology\nDurham University\nDH1 3LEDurham", "U.K. c Petersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia", "Institute for Particle Physics Phenomenology\nDurham University\nDH1 3LEDurham", "U.K. c Petersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia", "Department of Mathematical Sciences\nUniversity of Liverpool\nL69 3BXLiverpoolU.K" ]
[]
The perturbative QCD expansion for J/ψ photoproduction appears to be unstable: the NLO correction is large (and of opposite sign) to the LO contribution. Moreover, the predictions are very sensitive to the choice of factorization and renormalization scales. Here we show that perturbative stability is greatly improved by imposing a 'Q 0 cut' on the NLO coefficient functions; a cut which is required to avoid double counting. Q 0 is the input scale used in the parton DGLAP evolution. This result opens the possibility of high precision exclusive J/ψ data in the forward direction at the LHC being able to determine the low x gluon distribution at low scales.
10.1140/epjc/s10052-016-4493-y
[ "https://arxiv.org/pdf/1610.02272v2.pdf" ]
14,617,734
1610.02272
8da0794f84f20586514d6314d0f88ab4063f487f
August 8, 2018 7 Nov 2016 S P Jones Max-Planck-Institute for Physics Föhringer Ring 680805MünchenGermany A D Martin Institute for Particle Physics Phenomenology Durham University DH1 3LEDurham U.K. c Petersburg Nuclear Physics Institute NRC Kurchatov Institute 188300Gatchina, St. PetersburgRussia M G Ryskin Institute for Particle Physics Phenomenology Durham University DH1 3LEDurham U.K. c Petersburg Nuclear Physics Institute NRC Kurchatov Institute 188300Gatchina, St. PetersburgRussia T Teubner Department of Mathematical Sciences University of Liverpool L69 3BXLiverpoolU.K August 8, 2018 7 Nov 2016The exclusive J/ψ process at the LHC tamed to probe the low x gluon The perturbative QCD expansion for J/ψ photoproduction appears to be unstable: the NLO correction is large (and of opposite sign) to the LO contribution. Moreover, the predictions are very sensitive to the choice of factorization and renormalization scales. Here we show that perturbative stability is greatly improved by imposing a 'Q 0 cut' on the NLO coefficient functions; a cut which is required to avoid double counting. Q 0 is the input scale used in the parton DGLAP evolution. This result opens the possibility of high precision exclusive J/ψ data in the forward direction at the LHC being able to determine the low x gluon distribution at low scales. W +                                p p (x + ξ)P + (x − ξ)P + J/ψ γ W −              J/ψ γ Figure 1 : dσ(pp → p + J/ψ + p)/dy driven by the subprocess γp → J/ψ + p at two different γp centre-of-mass energies, W ± . Introduction It would be valuable to be able to constrain the gluon parton distribution function (PDF) at low x using J/ψ photoproduction data measured at HERA and at the LHC, via exclusive pp → p + J/ψ + p events, especially events in the forward region measured by the LHCb collaboration. Indeed, for LHCb kinematics at 13 TeV we can reach down to x 3 × 10 −6 . Exclusive J/ψ production is driven by the subprocess γ * p → J/ψ +p, see Fig. 1. Unfortunately, it turns out that the NLO corrections calculated in the conventional MS collinear approach are found to be very large and to depend strongly on the choice of factorization and renormalization scales [1,2,3]. Indeed, for an 'optimum' choice of scales it is found that the NLO correction has the opposite sign to the LO contribution and even changes the sign of the whole amplitude, see the continuous curves in Fig. 2. Thus one may doubt the convergence of the whole perturbation series. Optimum scale What do we mean by the 'optimum' scale? It was shown in Ref. [3] that it is possible to find a scale (namely µ F = m c ) which resums all the double logarithmic corrections enhanced by large values of ln(1/ξ) into the gluon and quark PDFs, where ξ is the skewedness parameter of the Generalised Parton Distributions (GPDs) describing the proton-gluon (and proton-quark) vertices. That is, it is possible to take the (α S ln(1/ξ)ln(µ 2 F )) term from the NLO gluon (and quark) coefficient functions and to move it to the LO GPDs. This allows a resummation of all the double log (α S ln(1/ξ)ln(µ 2 F )) n terms in the LO contribution by choosing the factorization scale to be µ F = m c . The details are given in Ref. [3], see also Ref. [5]. The result is that the γp → J/ψ + p amplitudes are schematically of the form A(µ f ) = C LO ⊗ GPD(µ F ) + C NLO rem (µ F ) ⊗ GPD(µ f ),(1)µ 2 f = 2m 2 c , m 2 c , Q 2 0 respectively where m 2 c ≡ M 2 ψ /4 = 2.4 GeV 2 . where the GPD can be related to the conventional PDF via the Shuvaev transform for ξ < |x| 1 [6]. With the choice µ F = m c there is a smaller remaining term in the NLO coefficent funcions, and so the residual dependence on the scale µ f is reduced. Unfortunately, even after this, the NLO corrections, and their variations with scale, although reduced, are still unacceptably large, as shown in Fig. 2. The dashed and dot-dashed curves correspond to NLO predictions for two different values of the residual scale µ f : namely µ 2 f = 4.8 and 1.7 GeV 2 respectively, while the continuous curves correspond to the 'optimum' scale choice µ 2 F = µ 2 R = m 2 c = M 2 ψ /4 = 2.4 GeV 2 . 1 The choice µ R = µ F is justified in subsection 3.1. Double counting So for the QCD prediction to be useful we should search for some other sizeable physical contribution to the NLO correction. Here we consider a power correction which may further reduce the NLO correction and, moreover, may reduce the sensitivity to the choice of scale. The correction is O(Q 2 0 /M 2 ψ ) where Q 0 denotes the input scale in the parton evolution. It turns out to be important for the relatively light charm quark, m c M ψ /2. Let us explain the origin of this 'Q 0 correction'. We begin with the collinear factorization approach at LO. Figure 3: (a) LO contribution to γp → V + p. (b) NLO quark contribution. For these graphs all permutations of the parton lines and couplings of the gluon lines to the heavy-quark pair are to be understood. Here P ≡ (p + p )/2 and l is the loop momentum. γ C LO g V (x + ξ)P + (x − ξ)P + F g p p γ C NLO q V (x + ξ)P + (x − ξ)P + F q p p l Here, we never consider parton distributions at low virtualities, that is for Q 2 < Q 2 0 . We start the PDF evolution from some phenomenological PDF input at Q 2 = Q 2 0 . In other words, the contribution from |l 2 | < Q 2 0 of Fig. 3(b) (which can be considered as the LO diagram, Fig. 3(a), supplemented by one step of DGLAP evolution from quark to gluon, P gq ) is already included in the input gluon GPD at Q 0 . That is, to avoid double counting, we must exclude from the NLO diagram the contribution coming from virtualities less than Q 2 0 . At large scales, Q 2 Q 2 0 this double-counting correction will give small power suppressed terms of O(Q 2 0 /Q 2 ), since there is no infrared divergence in the corresponding integrals. On the other hand, with Q 0 ∼ 1 GeV and µ F = m c (∼ M ψ /2) a correction of O(Q 2 0 /m 2 c ) may be crucial. In the present paper we re-calculate the NLO contribution for J/ψ photoproduction excluding the contribution coming from the low virtuality domain (< Q 2 0 ). We find that for J/ψ this procedure substantially reduces the resulting NLO contribution and, moreover, reduces the scale dependence of the predictions. It indicates the convergence of the perturbative series. An outline of the procedure is given in [9], where also the NLO description of exclusive J/ψ production in the k T factorization and collinear factorization schemes are compared. 2 Avoiding double counting of the low Q 2 contribution The NLO quark contribution We start with the NLO quark contribution to the γp → J/ψ + p process. The corresponding Feynman diagrams are that of Fig. 3(b) together with the diagram where both gluons couple to the same heavy quark line. Here we will use the non-relativistic approximation for the J/ψ wave function. Since the momentum fractions (x + ξ) and (x − ξ) carried by the left and right quarks are different we have to use the skewed (generalized) parton distribution (GPD), F q (x, ξ, Q 2 ). The skewedness parameter ξ = M 2 ψ /(2W 2 − M 2 ψ ), where W is the γp energy. We see that the upper part of diagram Fig. 3(b) is the same as the diagram for the LO gluon Fig. 3(a) contribution. For the LO contribution the integral over the gluon virtuality |l 2 | starts from the input scale Q 2 0 , while all the contributions from low virtualities |l 2 | < Q 2 0 are collected in the input gluon GPD, F g (x, ξ, Q 2 0 ). Note that this input distribution already includes that part of the quark contribution of Fig. 3(b) coming from |l 2 | < Q 2 0 . Thus to avoid double counting when computing the NLO quark coefficient function, C NLO q , of Fig. 3(b) we have to include the theta function Θ(|l 2 | > Q 2 0 ) in the integration over l 2 . Depending on the ratio Q 2 0 /m 2 c = 4Q 2 0 /M 2 ψ this can be a significant correction. The corresponding integral has no infrared or ultraviolet divergence and can be calculated in D = 4 dimensions. Actually, the calculation is performed in the physical scheme (with D = 4). On the other hand, parton distributions are usually presented in the MS factorization scheme where dimensional regularization is used. The problem is that when we calculate the coefficient function in D = 4 + 2 we have finite contributions of / origin. Formally these / terms come from unphysically large distances ∝ O(1/ ). In fact, these / terms are compensated by a corresponding re-definition of the PDFs. In order to retain the / terms and to use the MS scheme we do not calculate diagram 3(b) in D = 4 dimensions for |l 2 | > Q 2 0 , but instead calculate the part corresponding to small |l 2 | < Q 2 0 . We consider this part as the correction which should be subtracted from the known NLO MS coefficient function [1,10]. Recall that after the subtraction of the contribution generated by the last step of the LO evolution, P LO ⊗ C LO , there is no infrared divergence and the subtracted part of C NLO coming from |l 2 | < Q 2 0 does not contain / terms. The NLO gluon contribution The NLO 'Q 0 corrections' for the gluon coefficient function are more complicated. Besides the ladder-type diagrams analogous to Fig. 3(b), but with the light quark line replaced by a gluon line, there are other diagrams which have a structure similar to vertex corrections, see [1,10]. However the 'dangerous' contribution is again from the ladder-type diagrams, where to avoid double counting we have to exclude the |l 2 | < Q 2 0 domain whose contribution is already included in the LO term using the input gluon GPD, F g (x, ξ, Q 2 0 ). Qualitatively this is exactly the same calculation as that for the NLO quark. The only difference is that the lower line in the diagrams of Fig. 5 is now replaced by a gluon line and the lower part of the diagram is now given by the product of two three-gluon vertices averaged over the incoming gluon transverse polarizations. The notation is identical to that for the quark contribution. Both the quark-and the gluon-induced contributions are determined as described in the Appendix. They involve the calculation of the diagrams of Fig. 5 (given in the Appendix), and the analogous diagrams for the gluon-induced contribution. 3 Results Fig. 4 shows the LO and NLO contributions to the imaginary part of the J/ψ photoproduction amplitude when the Q 0 cut in the NLO contribution is taken into account. It should be compared to Fig. 2 which had exactly the same scale choices, but without the Q 0 cut imposed. The improvement in going from Fig. 2 to Fig. 4 is dramatic. First, the NLO contribution is now much smaller than the LO contribution. Second, the scale variation is much smaller. The continuous curves in Figs. 2 and 4 show the LO and NLO comparison for the choice of scales µ F = µ R = m c ≡ M ψ /2, which we had previously argued was optimal [3]. The stability achieved by imposing the Q 0 cut means that J/ψ photoproduction (γp → J/ψ p) data and LHC exclusive J/ψ (pp → p + J/ψ + p) data can now be included in the global parton analyses. Im A / W 2 [GeV -2 ] W [GeV] A g (0) A g (0) +A g (1) +A q (1) -0.5 -0. The choice of scales Let us discuss the above scale choices in more detail. By choosing the 'optimal' factorization scale µ F = m c we resum all the higher-order double-logarithmic corrections (α s ln(1/ξ) ln µ 2 F ) n (enhanced at high energies by the large value of ln(1/ξ)) into the gluon generalized parton distribution (gluon GPD) [3]. The renormalization scale is taken to be µ R = µ F . The arguments are as follows. First, this corresponds to the BLM prescription [11]; such a choice eliminates from the NLO terms the contribution proportional to β 0 (i.e. the term β 0 ln(µ 2 R /µ 2 F ) in eq. (3.95) of [1]). Second, following the discussion in [12] for the analogous QED case, we note that the new quark loop insertion into the gluon propagator appears twice in the calculation. The part with scales µ < µ F is generated by the virtual component (∝ δ (1 − z)) of the LO splitting during DGLAP evolution, while the part with scales µ > µ R accounts for the running α s behaviour obtained after the regularization of the ultraviolet divergence. In order not to miss some contribution and/or to avoid double counting we take the renormalization scale equal to the factorization scale, µ R = µ F . Discussion of the results Note that in the present paper we have calculated the imaginary part of the γp → J/ψ p amplitude. The real part of the amplitude can be restored via dispersion relations assuming positive signature, as in eq. (5) of Ref. [13]. Recall that we obtain the necessary GPDs from the CTEQ6.6 parton set [4] using the Shuvaev transform [6]. We use a relatively old parton set [4] in which the low x gluons are forced to be positive so as to make a meaningful comparison with our earlier work. The goal of this paper is not to make a quantitative description of the data, but to demonstrate that we can achieve stability of the perturbative QCD description of relatively low scale J/ψ production by imposing the Q 0 cut. We have shown this is a power correction -a correction which is needed to avoid double counting. This will allow future high precision exclusive J/ψ production data obtained at the LHC to be incorporated in global parton analyses. The general procedure to include the HERA γp → J/ψ p data and, in particular, the LHCb data for exclusive J/ψ production, pp → p + J/ψ + p, in a global analysis follows that used to produce Fig. 4 of Ref. [13]. These processes are driven by the gluon PDF and the LHCb data probe the gluon at very low values of x. However, in Ref. [13] we approximated the NLO corrections to the coefficient functions by accounting for the explicit l ⊥ integration in the last step of the interaction. Moreover, we just fitted the J/ψ data and used a parametric form for the gluon which approximated its x and Q 2 dependence. So the analysis of Ref. [13] was quite simplified, although very informative; see, for example, Fig. 5 of [13] which compared the resulting gluon PDF with those of different global analyses 2 . The present paper, on the other hand, retains collinear factorization and calculates the complete NLO contribution. We may expect the high γp energy, W , data points in the updated version of Fig. 4 of Ref. [13] to require a larger gluon distribution in the region from x < ∼ 10 −3 down to 10 −5 , at low scales, than coming from extrapolations of the NLO gluon PDFs from global fits to data not including the J/ψ data. An indication in favour of a larger gluon PDF in this domain comes also from the recent LHCb data on open charm (and beauty) [14]. Finally, it is useful to compare our approach with that of [15], where it was demonstrated that the re-summation of the BFKL-induced (α S ln(1/ξ)) n terms in the coefficient functions additionally reduces the factorization scale dependence. Recall that our choice of Figure 5: Two diagrams (a,b) computed for the NLO quark coefficient function. Note that p and p refer to the incoming and outgoing quark lines. In the corresponding diagrams computed for the NLO gluon coefficient function the light quark line is replaced by a gluon. The other two diagrams of the different coupling of the two t-channel gluons to the heavy quarks are implicitly included. µ F = M ψ /2 l ↑ p ↑ ↓ l ↓ p n → h 2 → h 1 → q l ↑ p ↑ ↓ l ↓ p n → h 2 → h 1 → q resums only the double logarithmic, (α S ln(1/ξ) ln µ F ) n contributions 3 . The remaining part, which does not contain ln µ F , should be considered, in the collinear factorization approach, as higher-order, NNLO, N 3 LO, ... corrections. Of course, it would be good to account for these corrections as well. However, to properly calculate these corrections one has to exclude the low (< Q 2 0 ) virtuality contribution. Otherwise we will face the problem of double counting again. The present paper shows these (power) corrections (necessary to avoid double counting) are crucial to achieve perturbative stability. Appendix Here we describe the calculation of the piece that we subtract from the full result. Only the imaginary part of the ladder-type cut diagrams shown in Fig. 5 and the corresponding diagrams where the light-quark line is replaced by gluons is computed. All momenta appearing in the calculation may be decomposed in terms of light-like momenta p, n and a transverse four-momentum l ⊥ , l µ = βp µ + αn µ + l µ ⊥ , h µ 1 = h µ 2 = β h p µ + α h n µ ,(2) where l is the loop momentum and h 1 , h 2 are the momenta of the outgoing heavy quark and heavy anti-quark, respectively. Here p can be chosen as the momentum of the incoming light parton and n the momentum of the incoming on-shell photon. With this convention we have p · p = n · n = 0, p · n =ŝ/2, p · l ⊥ = n · l ⊥ = 0, whereŝ is the photon-parton centre-of-mass energy squared. The four momenta of the incoming and the outgoing light partons are proportional. We may write p µ and p µ = Xp µ with X = x − ξ x + ξ =ŝ − M 2 ψ s = y 1 + y , where y = x − ξ 2ξ =ŝ M 2 ψ − 1.(4) To leading order in the heavy quark relative velocity, the S-wave spin-triplet component of J/ψ can be computed using the projection [16,17,18] v α (h 2 )ū β (h 1 ) → N J/ψ ( / h 2 − m c )/ * J/ψ ( / K + M ψ )( / h 1 + m c ) αβ .(5) Hereū, v are the spinors of the outgoing heavy quark and anti-quark which form the J/ψ. The indices α and β label their spin. N J/ψ is an overall factor which contains the non-perturbative NRQCD matrix element describing the J/ψ formation. The vector J/ψ describes the polarisation of the J/ψ with momentum K = h 1 + h 2 and mass M ψ = 2m c . The projections onto the quark and gluon GPDs are given by [19,20,21,22], u α (p)ū β (p ) → N q/ p αβ , µ 1 * 2 ν → N g g µν ⊥ = N g g µν − 2 s p µ n ν − 2 s n µ p ν ,(6)β = 4(1 + l 2 /ŝ)m 2 c /ŝ − 2l 2 /ŝ, α = l 2 /ŝ,(7)l 2 = l 2 ⊥ /(1 − β), l 2 = l 2 (1 − 4m 2 c /ŝ).(8) Additionally, we obtain q 2 = −m 2 c for diagram 5(a) and q 2 = l 2 ⊥ − ŝ 2 − l 2 β − 2m 2 ĉ s = 3m 2 c − βŝ(9) for diagram 5(b). In our calculation we split each diagram of Fig. 5 into two parts. An "upper" part which contains a trace over the heavy quark fermion line and a "lower" part which in the quark channel contains a trace over the light quark line and in the gluon case consists of two triple gluon vertices contracted with g µν ⊥ . First we discuss the "upper" part which is different for the diagrams (a) and (b) of Fig. 5 but identical for the quark and gluon channels. Where it appears, we replace the contraction of l ⊥ with the polarisation vectors using (l ⊥ · * J/ψ )(l ⊥ · γ ) = ( γ · * J/ψ )l 2 ⊥ /2(10) which follows from tensor decomposing the l ⊥ integral after the integration over the l azimuthal angle. We can simplify the calculation by noting that the sum of the "upper" parts of diagrams (a) and (b) obey the gauge condition T(h.loop) µν l µ = T(h.loop) µν l ν = 0,(11) where T(h.loop) µν = 1 (−2m 2 c ) Tr(h.loop) µν a + 1 (2m 2 c − βŝ) Tr(h.loop) µν b .(12) Here T(h.loop) is the upper part of the amplitude, which besides the trace over the quark loop, includes the heavy quark propagator 1/(q 2 − m 2 c ). Using the gauge condition the only contractions of the "upper" part that appear in the sum of diagrams are Tr(h.loop) µν a g µν = N J/ψ 4m c ( γ · * J/ψ )(6m 2 c −ŝβ),(13)Tr(h.loop) µν a p µ p ν = N J/ψ 4m c ( γ · * J/ψ )ŝ 2 (1/2 + α)/2,(14) Tr(h.loop) µν a p µ l ⊥ν = N J/ψ 4m c ( γ · * J/ψ )l 2 ⊥ŝ /2, Tr(h.loop) µν a l ⊥µ p ν = N J/ψ 4m c ( γ · * J/ψ )l 2 ⊥ŝ /2, Tr(h.loop) µν b g µν = N J/ψ 4m c ( γ · * J/ψ )2(ŝαβ − m 2 c (2α + 1)),(16)Tr(h.loop) µν b p µ p ν = −N J/ψ 4m c ( γ · * J/ψ )ŝ 2 /4,(17)Tr(h.loop) µν b p µ l ⊥ν = −N J/ψ 4m c ( γ · * J/ψ )l 2 ⊥ (ŝ − l 2 )/2,(18)Tr(h.loop) µν b l ⊥µ p ν = N J/ψ 4m c ( γ · * J/ψ )l 2 ⊥ (ŝ + l 2 )/2.(19) The contractions involving p µ n ν , n µ p ν , n µ n ν , n µ l ν ⊥ , l µ ⊥ n ν , l µ ⊥ l ν ⊥ appear in the computation of individual diagrams but cancel for the sum of diagrams. Quark-induced NLO correction For an unpolarized light quark the trace over the "lower" light quark line gives A q µν = 4N q [p µ (p − l) ν + (p − l) µ p ν + g µν (p · l)] ,(21) where the normalization factor N q = C F F q (x, ξ, µ F )(22) includes the colour factor C F and the quark GPD, F q . This light quark part should be contracted with the trace, Tr(h.loop) µν , given by the heavy quark (upper) loop. Due to the gauge condition (11) we have that (p − l) µ acts as p µ , while (p − l) ν acts as p ν = Xp ν giving M q a = 4N q (−2m 2 c ) l 2 l 2 Tr(h.loop) µν a g µν αŝ 2 + Tr(h.loop) µν a p µ p ν (1 + X) + M q (23) = 4N q N J/ψ (2m c )( γ · * J/ψ ) (−2m 2 c ) l 2 l 2 (6m 2 c −ŝβ)αŝ +ŝ 2 (1/2 + α)(1 + X) + M q ,(24) for diagram (a) and M q b = 4N q (2m 2 c − βŝ) l 2 l 2 Tr(h.loop) µν b g µν αŝ 2 + Tr(h.loop) µν b p µ p ν (1 + X) − M q (25) = 4N q N J/ψ m c ( γ · * J/ψ ) (2m 2 c − βŝ) l 2 l 2 [4(ŝαβ − m 2 c (2α + 1))αŝ − (1 + X)ŝ 2 ] − M q ,(26) for diagram (b). The term M q accounts for terms which cancel between the two diagrams. The denominators come from the uncut propagators: 1/l 2 for the left, and 1/l 2 for the right gluon and 1/(q 2 − m 2 c ) for the uncut heavy quark propagator. The result is to be integrated over the gluon transverse momentum (dl 2 ⊥ ) while the longitudinal components are fixed by the quark on-mass-shell conditions. It is easy to perform this integral numerically accounting for the condition which was introduced in Section 2 in order to avoid double counting. Recall, however, that we are not going to calculate the whole NLO contribution, but just the correction to the known MS coefficient function. So, in order to compute the correction, in the integration over the l ⊥ we only consider the region of |l 2 | < Q 2 0 . Actually, we integrate over dl 2 directly; the factor (1 − β) coming from the relation l 2 = l 2 ⊥ /(1 − β) is exactly cancelled by the residue from the light quark on-mass-shell pole. So we obtain the correction to the quark-induced part of the γp → J/ψ + p amplitude ∆ImM q = α 2 s 2π 1 ξ dx (F q (x, ξ, m c ) − F q (−x, ξ, m c )) Q 2 0 0 (M q a + M q b ) 2πm 4 ĉ s 2 dl 2(27) where the 'hard matrix elements' M q a,b are given by (24) and (26). The factor 1/ŝ 2 comes from the delta functions needed to put the lower light quark and the heavy quark coupled to the right gluon in Fig. 5 on-mass-shell. The factor m 4 c accounts for the normalization N J/ψ , defined to be consistent with the normalization of eqs. (3.93) and (3.95) of [1] for which the correction was calculated; actually the last factor (...) is the correction to f q of (3.93) of [1]. 4 For the gluon correction ∆M g there is an additional factorŝ/2m 2 c = 1/ξ due to the definition of the gluon GPD, F g ; see the extra factor of ξ in eq. (3.94) of [1], see also [10]. Note that we have explicitly calculated the NLO diagrams (a) and (b) of Fig. 5 which contain both LO 5 and NLO contributions. To identify the NLO part we therefore have to subtract the contribution generated by the LO evolution equation, which is of the form of the convolution P LO ⊗ C LO , before we integrate over l 2 ⊥ . This subtraction completely cancels the logarithmic infrared divergence dl 2 /l 2 . Note that the subtraction must be done only in the region of |l 2 | < µ 2 F since at the factorization scale µ F the DGLAP evolution stops. 6 Also note that in the LO approximation the convolution P LO ⊗ C LO is larger than the value of the matrix element given by explicit calculation of the diagrams shown in Fig. 5. Thus the final result has the sign opposite to that for the LO amplitude. In this way we obtain the quark NLO coefficient function. Since we are looking for the power correction needed to avoid double counting of the low |l 2 | < Q 2 0 contribution 7 , we actually have to integrate the matrix element M q over |l 2 | < Q 2 0 only (as explained above) and to subtract the result from the known NLO coefficient function given in the MS scheme. In the notation of Ref. [1] this should be considered as the new form of Imf q (y) of their eq. (3.93), after allowing for the changes made by our introduction of the 'Q 0 cut'. Gluon NLO correction In the gluon case the tensor A g µν corresponding to the lower part of Fig. 5 diagrams (with the lower quark line replaced by a gluon line) was calculated explicitly. It can be written in the form A g µν = N g (ag µν + b 11 p µ p ν + b 22 h µ h ν + b 12 p µ h ν + b 21 h µ p ν + c 1 p µ l ⊥ν + d 1 l ⊥µ p ν + c 2 h µ l ⊥ν + d 2 l ⊥µ h ν ) , Here the normalization factor is 8 N g = C A 8 F g (x, ξ, µ F ) (x + ξ − i )(x − ξ + i ) . (30) 5 The integration of the pure logarithmic form dl 2 /l 2 up to µ F actually reproduces the LO contribution already included in Fig. 3(a). On the other hand some non-logarithmic corrections originating from higher powers of l 2 , together with the whole contribution above µ F , are NLO α s corrections which are not enhanced by the large collinear (l 2 ) logarithms. 6 This is the origin of the ln(4m 2 /µ 2 F ) factor in the first term of f q (y) of eq. (3.93) of [1]. Since now we integrate over the |l 2 | < Q 2 0 < µ 2 F the correction does not depend on µ F . 7 This contribution is already included in the input value GPD(Q 0 ). 8 Here the denominator (x + ξ − i )(x − ξ + i ) arises from the particular definition of the gluon GPD. Note that X is defined in (4) and β is given by (7). Recall that we are looking for the imaginary part of the amplitude (i.e. s-channel discontinuity). This expression should be convoluted with the "upper" part of the diagram. The result for the sum of diagrams can again be simplified using the gauge conditions (11). That is vector h µ = (p − l) µ acts as p µ , while h ν acts as p ν = Xp ν . As before, the result is multiplied by the terms 1/l 2 and 1/l 2 from the t-channel gluon propagators and by the term 1/(q 2 − m 2 c ) from the corresponding heavy quark propagator. Then we have to subtract the part generated by the LO evolution equation which is given by the convolution P LO ⊗ C LO . Finally we integrate over l 2 ⊥ , accounting for the condition |l 2 | < Q 2 0 (the longitudinal components are fixed by the heavy quark and gluon (p − l) 2 = 0 on mass-shell conditions). In this way we obtain the power correction which should be subtracted from the known NLO gluon coefficient function Imf g (y) given by eq. (3.95) of [1] (see also [10]), which we then use to obtain the Q 0 subtracted NLO gluon contribution. Figure 4 : 4The predictions for the LO and NLO contributions to the imaginary part of the J/ψ photoproduction amplitude calculated exactly as inFig. 2except that now the Q 0 cut is imposed. respectively. Here u,ū are the spinors of the light quarks connected to the quark GPD and 1 , * 2 are the polarisation vectors of gluons connected to the gluon GPD. N q , N g are overall factors containing the quark and gluon GPDs.The on-shell conditions h 2 1 − m 2 c = 0 and h 2 2 − m 2 c = 0 for outgoing heavy quarks and the cut-constraints, (p − l) 2 = 0 and (n − h 2 + l) 2 − m 2 c = 0 for Fig. 5 diagram (a), (p − l) 2 = 0 and (h 1 − l − n) 2 − m 2 c = 0 for Fig. 5 diagram (b), allow us to choose α h = 1/2, β h = 2m 2 /ŝ and fix α, β in terms of l 2 ⊥ ,ŝ, m c . Specifically, where h µ = p µ − l µ anda = l 2 (1 + X + 4(1 − β)), b 11 = X(4β − 2) − 4(1 − β) ,(29)b 22 = 2, b 12 = 2X + 4, b 21 = 2 + 4X, c 1 = 3 − 2X, d 1 = 3X − 2, c 2 = 3, d 2 = 3 . Figure 2: The dotted and continuous curves are the LO and NLO predictions, respectively, of ImA/W 2 for the γp → J/ψ + p amplitude, A, as a function of the γp centre-of-mass energy W , obtained using CTEQ6.6 partons[4] (with input Q 0 = 1.3 GeV) for the optimal scale choice µ F = µ R = m c . The top three curves correspond to the NLO prediction for various values of the residual factorization scale µ f , namely:Im A / W 2 [GeV -2 ] W [GeV] A g (0) A g (0) +A g (1) +A q (1) -0.5 -0.25 0 0.25 0.5 0.75 100 200 300 400 500 600 700 800 900 1000 Recall that the choice m c = M ψ /2 effectively accounts for the relativistic corrections to the J/ψ wave function, see[7,8]. Recall, however, that strictly speaking the global analyses use the MS collinear factorization scheme whereas k T factorization uses the physicsl scheme, see, for example,[9]. This result for the optimal scale (see Section 1.1) is confirmed by the formula after eq. (8) in[15]. Note that, in[15], L(L − ln 16) + ln 2 4 = (L − ln 4) 2 = ln 2 (M 2 ψ /4µ 2 F ), since L ≡ ln(M 2 ψ /µ 2 F ). The overall normalization has been checked against[1] and correctly reproduces the leading log term ∝ ln(4m 2 c /µ 2 F ). AcknowledgementsWe thank Ronan McNulty for interesting discussions and for encouraging us to make these predictions. MGR thanks the IPPP at the University of Durham for hospitality. MGR is supported by the RSCF grant 14-02-00281. SPJ is supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement pitn-ga2012316704 (HiggsTools), and TT is supported by STFC under the consolidated grant ST/L000431/1. . D Yu, A Ivanov, L Schäfer, G Szymanowski, Krasnikov, arXiv:hep-ph/0401131v3Eur. Phys. J. 34297Erratum ibid. C75 (2015) 2, 75, ErratumD.Yu. Ivanov, A. Schäfer, L. Szymanowski and G. Krasnikov, Eur. Phys. J. C34 (2004) 3, 297, Erratum ibid. C75 (2015) 2, 75, Erratum arXiv:hep-ph/0401131v3. . 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Nockles, M.G. Ryskin and T. Teubner, Phys. Lett. B682 (2008) 252. . S P Jones, A D Martin, M G Ryskin, T Teubner, arXiv:1609.09738hep-phS.P. Jones, A.D. Martin, M.G. Ryskin and T. Teubner, arXiv:1609.09738 [hep-ph]. A Study of Exclusive Processes to NLO and Small-x PDFs from LHC Data. S P Jones, University of LiverpoolPhD thesisunpublishedS.P. Jones, A Study of Exclusive Processes to NLO and Small-x PDFs from LHC Data, PhD thesis, University of Liverpool, September 2014 (unpublished). . S J Brodsky, G P Lepage, P B Mackenzie, Phys. Rev. D. 28228S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, Phys. Rev. D 28 (1983) 228. . L A Harland-Lang, M G Ryskin, V A Khoze, Phys. Lett. 76120L.A. Harland-Lang, M.G. Ryskin and V.A. Khoze, Phys. Lett. B761 (2016) 20. . S P Jones, A D Martin, M G Ryskin, T Teubner, arXiv:1307.7099JHEP. 1185hep-phS.P. Jones, A.D. Martin, M.G. Ryskin and T. Teubner, JHEP 11 (2013) 085, arXiv:1307.7099 [hep-ph]. . 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[ "Excited Baryons in Large N c QCD Revisited: The Resonance Picture Versus Single-Quark Excitations", "Excited Baryons in Large N c QCD Revisited: The Resonance Picture Versus Single-Quark Excitations" ]
[ "Doe / Er ", "Thomas D Cohen ", "Richard F Lebed ", "\nDepartment of Physics\nDepartment of Physics and Astronomy\nUniversity of Maryland\n20742-4111College ParkMD\n", "\nArizona State University\n85287-1504TempeAZ\n" ]
[ "Department of Physics\nDepartment of Physics and Astronomy\nUniversity of Maryland\n20742-4111College ParkMD", "Arizona State University\n85287-1504TempeAZ" ]
[]
We analyze excited baryon properties via a 1/Nc expansion from two perspectives: as resonances in meson-nucleon scattering, and as single-quark excitations in the context of a simple quark model. For both types of analysis one can derive novel patterns of degeneracy that emerge as Nc → ∞, and that are shown to be compatible with one another. This helps justify the single-quark excitation picture and may give some insight into its successes. We also find that in the large Nc limit one of the S11 baryons does not couple to the π-N channel but couples to the η-N channel. This is empirically observed in the N (1535), which couples very weakly to the π-N channel and quite strongly to the η-N channel. The comparatively strong coupling of the N (1650) to the π-N channel and weak coupling to η-N channel is also predicted. In the context of the simple quark model picture we reproduce expressions for mixing angles that are accurate up to O(1/Nc) corrections and are in good agreement with mixing angles extracted phenomenologically.
10.1103/physrevd.67.096008
[ "https://export.arxiv.org/pdf/hep-ph/0301219v1.pdf" ]
119,329,749
hep-ph/0301219
45217aa08741246d0bf6c92fc881b4a5c34636e9
Excited Baryons in Large N c QCD Revisited: The Resonance Picture Versus Single-Quark Excitations 24 Jan 2003 Doe / Er Thomas D Cohen Richard F Lebed Department of Physics Department of Physics and Astronomy University of Maryland 20742-4111College ParkMD Arizona State University 85287-1504TempeAZ Excited Baryons in Large N c QCD Revisited: The Resonance Picture Versus Single-Quark Excitations 24 Jan 2003(Dated: January, 2003)numbers: 1115Pg1239-x1375Gx1420Gk * Electronic address: cohen@physicsumdedu † Electronic address: RichardLebed@asuedu 2 We analyze excited baryon properties via a 1/Nc expansion from two perspectives: as resonances in meson-nucleon scattering, and as single-quark excitations in the context of a simple quark model. For both types of analysis one can derive novel patterns of degeneracy that emerge as Nc → ∞, and that are shown to be compatible with one another. This helps justify the single-quark excitation picture and may give some insight into its successes. We also find that in the large Nc limit one of the S11 baryons does not couple to the π-N channel but couples to the η-N channel. This is empirically observed in the N (1535), which couples very weakly to the π-N channel and quite strongly to the η-N channel. The comparatively strong coupling of the N (1650) to the π-N channel and weak coupling to η-N channel is also predicted. In the context of the simple quark model picture we reproduce expressions for mixing angles that are accurate up to O(1/Nc) corrections and are in good agreement with mixing angles extracted phenomenologically. I. INTRODUCTION The intellectual history of the quark model is a study in irony. In the 1960's the quark model provided a concrete dynamical model that incorporated SU(3) flavor in a natural way, and thus organized the known hadrons according to an intelligible scheme. As such, it provided an essential role in the evolution of ideas that ultimately led to the formulation of QCD in the early 1970's as the underlying theory of strong-interaction dynamics. However, once QCD was discovered, the status of the quark model became quite problematic, since it was not known how this model was connected to QCD, as there was no known way to derive the quark model from QCD as some type of approximate treatment of the full theory. This irony would be of interest only to historians of science, if not for the fact that fully three decades since the QCD Lagrangian was first written down the quark model remains the principal tool used by the community for describing hadronic resonances. The reason is simple: Direct ab initio calculation of excited states from QCD remains an exceptionally difficult problem (although new lattice techniques are showing some promise [1]). This paper focuses on the relationship of the quark model treatment of excited baryons to QCD, using techniques based on the 1/N c expansion [2,3]. We study excited baryons because the contracted SU(2N f ) spin-flavor symmetry that emerges for baryons in large N c [4,5] provides a powerful tool for comparing results known to be generally true with those arising in the quark model. This analysis is useful for two reasons: i) It helps to justify the quark model, which is of utility since the quark model is the basis of considerable intuition about the excited states; and ii) it gives direct insight into certain aspects of the phenomenology of excited states. The usefulness of the 1/N c expansion for analyzing the properties of baryons in the ground-state band (i.e., the N 's and ∆'s) has been demonstrated for numerous observables including masses, electromagnetic moments, and axial couplings [5,6,7,8,9,10,11,12], with significant recent interest in properties of the ∆ and the N -∆ transition [13,14,15,16,17]. The spin-flavor symmetry gives significant insight into the validity of the quark model. On one hand it is derived by imposing consistency between the N c scaling of the π-N coupling (N 1/2 c ) and that of π-N scattering (N 0 c ) [4,5]. On the other hand, the simplest way to compute the consequences of the symmetry is to treat the baryon as though it were a collection of N c quark fields in identical S-wave single-particle orbitals and then keep track of the color, spin, and flavor [8,10,11], with the quarks combined into a color singlet and into a particular representation of combined spin-flavor symmetry. The N c counting can then be implemented by introducing operators that break the symmetry and are accompanied by a characteristic power of 1/N c . This organizes operators contributing to a particular observable according to a well defined power-counting scheme. The key point is that for these ground-state band baryons, large N c QCD has the same spin-flavor symmetry as the simple quark model and has the same pattern of symmetry breaking. Thus, at least for those ground-state baryon properties that are insensitive to the dynamical details and are essentially fixed by the symmetries, large N c QCD provides a justification of the quark model. In a similar way we wish to investigate the extent to which large N c QCD treats excited baryons in an analogous manner. That is, we wish to understand what large N c rules can be obtained from large N c consistency rules at the purely hadronic level, and compare these results to what is obtained for a quark picture, to see what aspects of the quark picture can be justified. A central issue in using the 1/N c expansion to study quantities in our N c = 3 universe is whether to include quantities that vanish for N c = 3. For example, should the quantity 1−(3/N c ) be evaluated by immediately taking the N c → ∞ limit and retained as unity, or should one note that eventually one imposes N c = 3 in the 1/N c expansion and treat it as vanishing? This question was discussed in the case of baryon charge radii and quadrupole moments in Ref. [15]. In the present context we believe it is useful to work in the N c → ∞ world directly, because we are attempting to make qualitative comparisons between two different pictures whose large N c behaviors are known. We caution, however, that it is by no means yet entirely clear how to handle 1/N c corrections to the two pictures in a consistent way. This issue is particularly thorny because there exist states at large N c that do not exist at large N c , and the role of these large N c artifacts must be isolated before one attempts to draw phenomenological conclusions. Before starting, we must note a key distinction between excited baryons and their ground-state band cousins. The ground-state baryons are stable in the large N c limit (Of course, in the real world the ∆ decays due to the anomalous lightness of the π, thanks to approximate chiral symmetry), while excited baryons are all resonances. Now, from standard counting based on Witten's original arguments [3], it is clear that the characteristic N c scaling for the excitation energy of an excited baryon is N 0 c , while the scaling of the three-point coupling between the excited baryon, a ground-state band baryon, and a meson is also N 0 c . Thus, the resonance width also scales as N 0 c . In this respect, baryons in large N c QCD are fundamentally different from mesons; in the meson case we know that widths scale as 1/N c , so that well-defined narrow meson states exist at large N c . Indeed, from the perspective of large N c counting alone, one must be agnostic about the very existence of baryon resonances that are narrow enough to isolate. Here we simply note that the empirical evidence indicates identifiable resonances. From the purely hadronic perspective the fact that excited baryons are associated with resonances simply suggests that the appropriate first step is to describe scattering processes, such as meson-nucleon scattering, in channels for which such resonances may reveal themselves. The role of large N c QCD is then simply to relate scattering in various channels [up to O(1/N c ) corrections], in the sense that various linear combinations of channels are equal [18,19,20,21]. We note that, while these relations were initially derived in the context of chiral soliton models, they are in fact model independent. An outline of a derivation of these relations directly from large N c consistency relations along the lines of Ref. [8] appears in the Appendix. From these relations one can deduce patterns of degeneracy among resonances in various channels that are valid up to corrections of O(1/N c ). We note that, although the linear relations in scattering amplitudes have been known for a long time, the patterns of mass degeneracies among the excited baryons reported here are brand new, their existence first mentioned in our recent Letter [22]. The fact that the excited baryons are resonances has always been an awkward fact for quark models. If one defines a quark model for a baryon as a description in which there are N c constituent quarks interacting through potentials and with no mechanism for pair creation, then as a matter of principle there is no way that such a model can ever describe a physical state that is a meson-nucleon resonant scattering state. Implicitly, what is done is to assume that the quark model describes a state that is relatively weakly coupled via some quark pair-creation mechanism to a larger Hilbert space that includes the asymptotic two-hadron state. If such a coupling is weak enough, one expects the position of the resonance to be close to the bound-state energy of the uncoupled system. Thus, any quark model treatment that fits parameters so that the energies computed in the model match resonance masses implicitly makes a weak-coupling assumption. This is worth stressing in the present context, if only to remind ourselves that from a large N c perspective there is no reason to assume such a coupling is weak (Note that this is not the case for mesons, where every type of pair creation is suppressed by 1/N c ). Again, we assume here that the coupling is weak for reasons not connected to N c , and proceed. Let us look in a bit more detail at how the quark model for baryons is realized. The system can be solved as a true three-body problem, with the potential consisting of two-body or three-body interactions between the quarks that depend only on their relative coordinates. Such a treatment has the virtue of being consistent with the underlying spirit of the model and has the technical advantage that the center-of-mass coordinate automatically separates from the relative coordinates, allowing for a description of internal excitations that is not contaminated by any spurious center-of-mass motion. However, full three-body calculations are technically difficult. Moreover, the wave functions are complicated, and thus it is hard to obtain much intuition from them. Accordingly, many simple calculations are based on a single-particle potential-type model, where this potential is thought of as arising from the interactions of the other quarks. In the simplest version of this model (for N c = 3), the ground-state baryons have all three quarks in the lowest S-wave orbital (yielding a 56-plet in spin-flavor for N f = 3) while the first excited group of baryons has one quark in a P wave and fills a mixed-symmetry 70-plet spin-flavor state [23]. Both types of models can be called quark models, but for our purposes it is useful to distinguish between them. Accordingly, we refer to the second variant as the quark-shell model, since it has similarities with the shell model of nuclear physics. A few additional comments about the quark-shell model are in order. First, much of the intuition many people have about excited baryons and much of the language used to describe them are based on the quark-shell model rather than more sophisticated treatments. The reason is that the simplicity of the model allows one to form a comprehensible picture of the state. Second, one may create more sophisticated versions of the quark-shell model that include admixtures of different single-particle descriptions, in order to include some of the correlations; such admixtures are called configuration mixing. Of course, if all possible configuration mixing is allowed and if the interactions being used are the full quark-quark potentials of the underlying quark model, then the quark-shell model is equivalent to the full three-body quark model, representing just a convenient basis in which to work rather than a distinct model (The situation is completely analogous to that of the case in the nuclear many-body problem [24]). Here, when we refer to the quark-shell model, we mean models in which configuration mixing is neglected or taken to be small. It is also worth noting that the simple quark-shell model (with little or no configuration mixing) does a good job of describing the spectrum of the lowest-lying observed N *'s. In the present context, we note that to date all quark-based treatments that describe excited baryons in the large N c limit of QCD were quark-shell models that neglect configuration mixing [25,26,27,28,29,30,31,32]. In such a picture, the first excited states are either radial excitations of the symmetric ground-state multiplet (such as the Roper), or orbital excitations (with quantum number ℓ) of a single quark with respect to the other N c −1 quarks remaining in a spin-flavor symmetrized "core." Again using SU(6) terminology, these states fill representations analogous to the three-color 56 and 70, respectively [25,26,27,28,29,30,31,32]. In this paper we compare the physical content of the two pictures-excited baryons as resonances in meson-nucleon scattering, and as single-quark excitations in a quark-shell model-to test whether the two are consistent with each other in a large N c world. We consider the lowest positive-and negative-parity nonstrange excited baryon resonances. We find that generically the two pictures are compatible. That is, both pictures predict patterns of mass degeneracy at leading order in the 1/N c expansion and these patterns are identical. We note that this compatibility is nontrivial and may help justify the use of the quark-shell model and explain its qualitative success. We should comment briefly on the role of model dependence in what follows. The relations that follow from our treatment of meson-baryon scattering are truly model independent and are direct results of large N c QCD. There is a subtlety in results obtained for excited baryons using the quark model language: For the ground-state band, studies of the N c dependence of operator matrix elements in a quark model picture [8] completely reproduce results of the large N c consistency conditions. Something similar can be seen here-the multiplet structure we obtain using quark model language below is identical to that obtained by Pirjol and Yan [26] using large N c consistency relations. In this sense, their results are model independent. However, both the analysis here and that of Ref. [26] are based on treatments of matrix elements of operators between excited baryon states. This is strictly only well defined for stable states. However, generically, as noted above the excited baryons are not stable in the large N c limit; they have widths of O(N 0 c ). Thus, the relations derived in quark model language are known to be valid and model independent only for stable excited states. One could imagine a world in which the quark masses and the pion were so heavy that the N (1535) was stable. In such a world the quark model results and the large N c consistency relations would agree for these stable states and would be truly model independent as N c → ∞. Unfortunately, with realistic quark masses there is no reason to think there are any stable baryon resonances at large N c . This raises the question of whether any of results are in fact valid in the real world for the unstable baryons. Of course, it is not implausible that some results derived in a model-independent way assuming the states are stable may nevertheless be valid for the resonances. In essence, the question of whether that is true is at the heart of this paper. As we shall see, the multiplet structure of excited states seen in this quark model-type language, derived assuming that states are stable, are in fact seen in full large N c QCD derived without this assumption. This is one of the principal results of this paper. We also use the large N c results to explore directly aspects of the phenomenology of the negative-parity baryons, and find two rather interesting phenomenological results. The first concerns the N 1/2 negative-parity states: The large N c analysis predicts the existence of a N 1/2 negative-parity state whose coupling to the π-N channel is weak (vanishing at large N c ), but couples strongly to the η-N channel. In fact, N (1535) has precisely this character. We similarly predict the other N 1/2 negative-parity state to couple weakly to the η-N channel while coupling strongly to the π-N channel, which is seen in the N (1650). The second result concerns the mixing angles between the various excited nucleon states in the quark-shell model context. From large N c emerges an analytic result predicting the value of these mixings, and we find that these predicted values are in good agreement with phenomenological extractions. In Sec. II we discuss the resonance picture. The key point is the existence of model-independent linear relations between meson-nucleon scattering in various channels that become exact in the large N c limit of QCD. These relations imply degeneracy patterns for excited baryons. In Sec. III we discuss the quark-shell model picture for the lowest-lying negative-parity baryons. This discussion is based on the methods of Refs. [28,29]. However, we discover an important analytic result not elucidated in these works, namely, that at leading order in the 1/N c expansion the mixing angles between various states are fixed and that various states are degenerate up to this order. Finally, in Sec. IV we discuss the implication of these results both in justifying the quark-shell model and directly in terms of phenomenology. II. MESON-NUCLEON SCATTERING PICTURE It has long been known, primarily through the work of Hayashi, Eckart, Holzwarth, and Walliser [18], and of Mattis and collaborators [19,20,21,33], that the S matrices of various channels in meson-nucleon scattering (or more generally scattering of mesons off ground-state band baryons) are linearly related in the large N c limit. For the present purpose it is sufficient to consider the case of π or η mesons scattering off a ground-state band baryon. In this case the S matrices are given by S π LL ′ RR ′ IJ = K (−1) R ′ −R (2R + 1)(2R ′ + 1)(2K + 1) K I J R ′ L ′ 1 K I J R L 1 s π KL ′ L , (2.1) S η LRJ = K δ KL δ(LRJ) s η K . (2.2) For π scattering, the incoming baryon spin (which equals its isospin) is denoted as R, that of the final baryon is denoted R ′ , the incident (final) π is in a partial wave of orbital angular momentum L (L ′ ), and I and J represent the (conserved) total isospin and angular momentum, respectively, of the initial and final states (and hence represent isospin and angular momentum of the state in the s channel). S π LL ′ RR ′ IJ is the (isospin-and angular momentumreduced) S matrix for this channel in the sense of the Wigner-Eckart theorem, the factors in braces are 6j coefficients, and s π KL ′ L are universal amplitudes that are independent of I, J, R, and R ′ . For η-meson scattering, since I η = 0 many of the quantum numbers are more tightly constrained. The isospin (= spin) R of the baryon is unchanged and moreover equals the total isospin I of the state. The orbital angular momentum L of the η remains unchanged in the process due to large N c constraints, and J denotes the total angular momentum of the state, which is constrained by the triangle rule δ(LRJ). S η LRJ is the reduced scattering amplitude, and s η K are universal amplitudes independent of J. The reason that various scattering amplitudes are linearly related is clear from the structure of Eqs. (2.1) and (2.2): There are more amplitudes S π LL ′ RR ′ IJ than there are s π KL ′ L amplitudes, and thus there are linear constraints between them that hold to leading order in the 1/N c expansion; similarly, there are more S η LRJ amplitudes than s η K amplitudes. Equations (2.1) and (2.2) are the starting point for our analysis of meson-nucleon resonances. These equations were first derived in the context of the chiral soliton model [18,19,20,21,33]. In this picture, the quantum number K has a simple interpretation-the soliton at the classical or mean-field level breaks both the rotational and isospin symmetries but preserves the length K of K ≡ I + J. Thus, the Hamiltonian describing the intrinsic dynamics of the soliton (that not associated with collective zero modes) commutes with the "grand spin" K, and excitations can be labeled by K. It is important to stress, however, that Eqs. (2.1) and (2.2) are exact results in large N c QCD and are independent of any model assumptions. A derivation directly from the large N c consistency rules [8] exploiting the famous I t = J t rule [20] can be found in the Appendix. The key point for our purposes is that, in order for a resonance to occur in one channel S π LL ′ RR ′ IJ in π-N scattering, there must be a resonance in at least one of the contributing amplitudes s π KL ′ L . However, since the same s π KL ′ L contributes to amplitudes in more than one channel, all of them resonate at the same energy, and this implies degeneracies in the excited baryon spectrum. An analogous argument holds for η-N scattering. To make the preceding argument concrete, one needs a method to extract the resonance position from the S matrix for scattering. Here we adopt the usual theoretical prescription: One analytically continues the scattering amplitude in the complex plane to unphysical but on-shell kinematics and defines the complex resonance position to be the position of a pole in the scattering amplitude. One can then simply relate the real and imaginary parts to the mass and width of the resonance. Using this definition of the resonance position, the argument given above is quite clean. Analytically continuing Eqs. (2.1) and (2.2) to the complex plane, it is apparent that if there is a pole in the complex plane on the left-hand side for π-N scattering, then at this pole position the right-hand side must also diverge, implying a pole in one of the s π KL ′ L amplitudes on the right-hand side. One can then turn this around and argue, as above, that since s π KL ′ L contributes to multiple channels, all of them must resonate at the same point (unless other selection rules, e.g., parity, forbid them to mix); similarly for η-N scattering. We note in passing that, although this theoretical definition of the resonance position is valid, there is a practical difficulty in extracting the precise resonance positions from scattering since one cannot directly probe unphysical kinematics, and thus there is always some model dependence in any extraction of the resonance position from data. Consequences of Eqs. (2.1) and (2.2) for negative-parity partial waves can be seen in the right column of Table I, which lists the linear combinations of the s π KL ′ L or s η K "K-amplitudes" contributing to a particular partial wave of fixed I and J in the s channel. In using this table to deduce patterns of degeneracy of the baryon states, it is important to clarify whether any degeneracy might be expected in the s KL ′ L and s η K amplitudes themselves. On one hand it is clear that various K sectors ought to be dynamically distinct, since they are distinct in the large N c limit. This is particularly clear in the context of the chiral soliton models, where the various K sectors are completely separate. More generally, there is no reason to suspect degeneracy between different K sectors, and it would be unnatural to impose any such degeneracies. On the other hand, it is quite plausible that there may be degeneracies in the poles of amplitudes with the same K but different values of L or L ′ : The orbital angular momentum of the π is not an immutable quantity in the same sense as I or J. Thus, if there is a resonance of fixed I and J but accessible by various L (e.g., by scattering off a ∆ rather than N ), one would expect resonances in channels of different L to be degenerate. Similarly, scattering partial waves involving different mesons that nevertheless contain amplitudes in the same K channel (e.g., s π 222 and s η 2 ) can produce degenerate poles. Examples of degenerate negative-parity multiplets at large N c that one can infer from the meson-baryon scattering relations include: N 1/2 , ∆ 3/2 , · · · (s η 0 ), (2.3) N 1/2 , ∆ 1/2 , N 3/2 , ∆ 3/2 , ∆ 5/2 , · · · (s π 100 , s π 122 ), (2.4) ∆ 1/2 , N 3/2 , ∆ 3/2 , N 5/2 , ∆ 5/2 , ∆ 7/2 , · · · (s π 222 , s η 2 ), (2.5) ∆ 3/2 , N 5/2 , ∆ 5/2 , ∆ 7/2 , · · · (s π 322 ), (2.6) where the states are listed on the left and the contributing amplitudes on the right. The ellipses indicate that in the large N c world the multiplets are infinite dimensional, and we have simply listed the low-spin and -isospin members of the multiplet. The preceding multiplets are one of the principle results of this work. A few comments about them are in order. First note that at large N c the poles of the scattering for these various members of the multiplet occur at the same point in the complex plane. Thus, the states have both the same mass and the same width as N c → ∞. Next, let us look at the multiplet in Eq. (2.4). Note that we have included the ∆ 5/2 for the s 100 channel; this may be surprising in light of the fact that Table I has no partial wave for ∆ 5/2 with a contribution from s 100 . It is because the table is restricted to partial waves in scattering off the N and ∆, while the s 100 contribution to ∆ 5/2 is seen in .2). Note that these states are those appropriate to a large Nc world (As discussed in the text, some do not occur for Nc = 3). We only list states with quantum numbers consistent with a single quark excited to ℓ = 1 and with total isospin of 3/2 or less; and we only list partial waves sufficient to accommodate all the given resonances (hence in particular L = L ′ ≤ 2). State Quark Model Mass Partial Wave, K-Amplitudes scattering off a ground-state band baryon with I = J = 5/2, which though absent in our world, exists at large N c . Note also that the degeneracy pattern seen in Eq. (2.4) occurs for both the s 100 contributions and the s 122 . There are two logical possibilities for this to occur: Either there are two distinct multiplets with the same degeneracy patterns, or the two multiplets are in fact the same. The second possibility is clearly more economical, and we believe it to be correct. Similarly, the multiplet pattern seen in Eq. (2.5) occurs for both π and η scattering, strongly suggesting that the same physical states occur in both. These observations tie in neatly with our previous discussion. We should note that the degeneracy patterns shown above are newly derived in the context of meson-baryon scattering. This is somewhat surprising, since nearly 20 years ago Mattis and Karliner [33] computed the excited baryon spectrum in the Skyrme model directly from pion-baryon scattering using Eq. (2.1). In that work the only degeneracy found was between the N 1/2 and the ∆ 1/2 . How can we understand these degeneracy rules in light of the fact that explicit calculations based on the same fundamental formula missed them? The answer lies in the algorithm used in Ref. [33] to extract resonance positions. The technique first computed the scattering amplitudes and then used motion in the Argand plots to fix the resonance position. This is essentially the technique used by experimentalists, and is highly appropriate when used in comparison with experimental extractions. However, by restricting attention to physical kinematics, one cannot directly access the pole position. N 1/2 m0, III. NEW RESULTS IN THE QUARK-SHELL MODEL PICTURE Although the multiplet structure in Eqs. (2.3)-(2.6) is newly derived here in the context of meson-baryon scattering, the existence of such towers has been known for some time in the context of quark model-type treatments in the work or Pirjol and Yan [26]. Although the formalism used in Ref. [26] is apparently model independent, as discussed in the Introduction it makes use of a strong dynamical assumption characteristic of the quark model, namely that the excited states are stable. This assumption is generally not compatible with large N c QCD. The derivation of these degenerate multiplets in [26] is quite beautiful conceptually, essentially applying the reasoning of Refs. [4,5]. However, it is computationally somewhat involved. Here we rederive these results using a more explicit quark-shell model language (but an analogous dynamical assumption of stable baryons) using the methods of [28,29]. As noted in the Introduction, our approach here is to work directly in the N c → ∞ world. This observation has a direct bearing on the enumeration of states in the quark-shell model. States in the first negative-parity multiplet (ℓ = 1) have an (N c −1)-quark core that is completely symmetric under spin × flavor, and thus have the quantum numbers S c = I c in the nonstrange case. Multiple states with the same fixed values of I and J are distinguished by the total spin S carried by the quarks, and this is denoted [28,29] by primes (no primes for S = 1/2, one for S = 3/2, etc.). It is an elementary exercise in combining angular momenta and isospin to show that, for N c ≥ 5, the states with I ≤ 3/2 are N 1/2 , N ′ 1/2 , N 3/2 , N ′ 3/2 , N ′ 5/2 , ∆ 1/2 , ∆ ′ 1/2 , ∆ 3/2 , ∆ ′ 3/2 , ∆ ′′ 3/2 , ∆ ′ 5/2 , ∆ ′′ 5/2 , and ∆ ′′ 7/2 . For N c = 3, the states ∆ ′ 1/2 , ∆ ′ 3/2 , ∆ ′′ 3/2 , ∆ ′ 5/2 , ∆ ′′ 5/2 , and ∆ ′′ 7/2 do not occur, but must be included in a full 1/N c analysis until the final step of setting N c = 3. Working up to O(N 0 c ) in the quark-shell model picture, one finds 3 operators contributing to the Hamiltonian, denoted in Refs. [28,29] as H = c 1 1 1 + c 2 ℓs + c 3 ℓ (2) gG c /N c . To remind the reader (Refer to Refs. [28,29] if the following notation is not familiar), lowercase indicates operators acting upon the excited quark, and subscript c indicates those acting upon the core. G ia denotes the combined spin-flavor operator ∝ q † σ i τ a q, and ℓ (2) is the ∆ℓ = 2 tensor operator. This is to be contrasted with Refs. [26], which effectively include only one spin×flavor-breaking operator at this order. The Hamiltonian up to O(N 0 c ) for the mixed N 1/2 states reads H N 1/2 = N 1/2 N ′ 1/2 M N 1/2 N 1/2 N ′ 1/2 , (3.1) as may be obtained from Eqs. (A6)-(A8) or Table II of Ref. [29], again including only contributions up to O(N 0 c ). The mass matrix M N 1/2 is diagonalized by the unitary matrix U N 1/2 : U N 1/2 M N 1/2 U † N 1/2 = diag M (1) N 1/2 , M(2)N 1/2 , where M N 1/2 = c 1 N c − 2 3 c 2 − 1 3 √ 2 c 2 − 5 24 √ 2 c 3 − 1 3 √ 2 c 2 − 5 24 √ 2 c 3 c 1 N c − 5 6 c 2 − 5 48 c 3 , (3.2) U N 1/2 = cos θ N1 sin θ N1 − sin θ N1 cos θ N1 ; (3.3) N 1/2 and N ′ 1/2 refer to unmixed negative-parity spin-1/2 nucleon states in the initial quark-shell model basis. Anticipating a remarkable result, let us define 3 particular combinations of the parameters: m 0 ≡ c 1 N c − c 2 + 5 24 c 3 , m 1 ≡ c 1 N c − 1 2 c 2 − 5 24 c 3 , m 2 ≡ c 1 N c + 1 2 c 2 − 1 24 c 3 . (3.4) One finds that M (1) N 1/2 = m 0 , M(2) N 1/2 = m 1 , and tan θ N1 = √ 2. Note first that, had the numerical coefficients in Eq. (3.2) been arbitrary, the eigenvalues would in general contain square roots of terms quadratic in c 2 and c 3 , and the mixing angle would have been a complicated function of their ratio. The simplicity of the actual results indicates something deep is happening. Indeed, this simple mixing angle result at large N c was earlier noticed in the work of Pirjol and Yan [26]. As will be seen shortly, results for states with other quantum numbers are equally simple. Note that the simplicity of the present result-analytic expressions for both the masses and the mixing anglewas not noted in Refs. [28,29]; the reason is simply that previous work always included the O(1/N c ) terms in the Hamiltonian and then diagonalized numerically. The simple result given above, however, only holds to O(N 0 c ), and the inclusion of the 1/N c correction terms in the Hamiltonian obscured the simple leading result. Using an analogous notation for the other states, , (3.9) from which M (1) ∆ 5/2 = m 0 , M(2) ∆ 5/2 = m 1 , and tan θ ∆5 = 3/7. Finally, ∆ 7/2 is unmixed and has eigenvalue M 7/2 = m 2 . The pattern, masses being equal to one of three eigenvalues and mixing angles having simple expressions, obviously extends into the ∆ sector. The compilation of mass eigenvalues into states of various quantum numbers is given in Table I. Clearly, the fact that all of the masses described by the model are given by either m 0 , m 1 , or m 2 to leading order in the 1/N c expansion implies that at large N c the various states fall into degenerate multiplets. These multiplets are given by (3.12) where the states are listed on the left and the masses on the right. After a draft of the present paper was completed we became aware of similar work by Pirjol and Schat [34], who obtained similar results and computed 1/N c corrections. N 1/2 , ∆ 3/2 , · · · (m 0 ), (3.10) N 1/2 , ∆ 1/2 , N 3/2 , ∆ 3/2 , ∆ 5/2 , · · · (m 1 ), (3.11) ∆ 1/2 , N 3/2 , ∆ 3/2 , N 5/2 , ∆ 5/2 , ∆ 7/2 , · · · (m 2 ), IV. DISCUSSION A. On the Compatibility of the Resonance and Quark-Based Pictures The results of the previous two sections are quite striking. In both the resonance pole picture and the quark-shell model picture, one finds that the excited baryons are organized into multiplets of states that are degenerate modulo splittings arising at next-to-leading order, O(1/N c ). Let us compare the multiplet structures predicted by the two pictures as given in Eqs. (2.3)-(2.6) and Eqs. (3.10)- (3.12). It is clear that the multiplet structure in Eq. (2.3) is identical to that of Eq. (3.10); that in Eq. (2.4) is identical to that of Eq. (3.11); and that in Eq. (2.5) is identical to that of Eq. (3.12). The two pictures are compatible. The interpretation is quite simple: the multiplet states with mass m 0 in the quark-shell model are those states for which the resonance occurs in the K = 0 scattering channel and analogously for m 1 states (K = 1) and m 2 states (K = 2). This result is highly significant in justifying the quark model. Before turning to the question of just how strong this justification is, we note that the resonance picture can have poles for K = 3, as seen in Eq. (2.6), for which we have not reported any analogous states in the quark-shell model. This in no way spoils the compatibility of the two pictures. Rather, it merely reflects the fact that the quark-shell model studies were limited to single-quark excitations in the lowest ℓ = 1 orbital. Had we considered higher excitations in a quark-shell model, such as two-quark excitations or excitations in the ℓ = 3 orbital, presumably we would have seen a degenerate multiplet consistent with Eq. (2.6). This prediction, that such a multiplet structure will be found higher in the spectrum of a large N c quark-shell model, is a stringent test of our interpretation. Studies of higher excited states in the quark-shell model are presently under way [35]. While studies of higher negative-parity states remain to be completed, we have also computed the degeneracy patterns for the lowest-lying excited positive-parity states. These states include N (1440) (P 11 ) and ∆(1600) (P 33 ). One finds, using Eq. (2.1), the relations P πN N 11 = (s π 011 + 2s π 111 )/3, P πN N 13 = (s π 111 + 5s π 211 )/6, P πN N 31 = (s π 111 + 5s π 211 )/6, P πN N 33 = (2s π 011 + 5s π 111 + 5s π 211 )/12. Since no poles are observed in the lowest multiplet for P 13 or P 31 , consistency between the resonance and quark-shell model pictures is achieved simply by placing a pole in s π 011 but not in s π 111 or s π 211 for this multiplet. We note that this result holds more generally, being valid for any state that in the quark-shell model lies in a spin-flavor symmetric multiplet. The present interpretation-that quark shell model states correspond to a well-defined K quantum number (up to 1/N c corrections)-is based on meson-baryon scattering, with scattering restricted to π-baryon and η-baryon channels. Of course, if the interpretation is correct, one should also find consistency between the multiplet structure of the quark-shell model and resonances deduced from mixed scattering, in which the initial meson is a π and final meson is an η. Although not presented here, it is straightforward to verify that this is true. Let us now turn to the question of just how strongly the present result justifies the quark-shell model picture. We start by observing that, whatever justification there is for the quark-shell model at N c = 3 should become increasingly reliable as N c becomes large. The basic point is simply that there are more quarks available that combine to generate the effective single-body potential seen by the last quark. Thus, if there is justification for the model at N c = 3, it is likely stronger for large N c . Conversely, a clear failure of the model at large N c would suggest that the picture is unlikely to be valid for N c = 3. Thus, the fact that the quark-shell picture produces the same qualitative spectrum in terms of the multiplet structure as is seen in large N c QCD from scattering is a real test, in the sense that the failure to do so would have cast serious doubts on the model. Of course, the fact that the multiplet structure in the quark-shell model agrees with that of large N c QCD does not justify all aspects of the model. In particular, it does not justify the dynamical details of the model in general, and certainly does not justify an approach that neglects open channels for decay. Rather, as for the ground-state band, it justifies those aspects of the model that essentially follow from the contracted SU(2N f ) symmetry. Finally, we note that compatibility of the excitation spectrum of the quark-shell model with large N c QCD is highly nontrivial. At first blush, one might think the result is trivial. After all, Witten's initial derivation [3] of large N c rules for baryons was done using heavy quarks, which essentially defines a quark model in the first sense described in the Introduction. Moreover, the counting used by Witten was essentially combinatoric and applied independently of the dynamical details. Thus, one expects that any relations that apply for large N c QCD should hold in generic large N c quark models. However, as noted in the Introduction, we are not studying a full quark model; rather we are using a quark-shell model. Of course, one can again appeal to Witten's original argument and argue that the Hartree approximation emerging at large N c is a single-particle picture in exactly the same manner as the quark-shell model. However, the preceding argument is specious. It is certainly true that the Hartree approximation becomes valid at large N c for the ground state. Whether it is valid for excited states is a bit more subtle. It is well known [36] in many-body theory that mean-field theories such as the Hartree approach respect the underlying symmetries of the theory or spontaneously break them. However, the use of a mean-field potential for excited states, for example as in the Tamm-Dancoff approximation, involves ad hoc truncations that generically violate the symmetries. In contrast, symmetry-conserving approximations to treat excited states, such as linear response theory or random phase approximation, typically go beyond the mean-field potential and take into account ground-state correlations. Since the results of our studies are essentially group theoretic-and thus entirely dependent on the treatment of the symmetries-it is by no means a priori obvious that the use of the mean-field potential for treatments of the excited states is adequate. Thus, the success of the quark-shell model in replicating the multiplet structure is quite significant. Again we note that it is important that the quark-shell model is justified (and not just the full treatment of the N c -body quark model), as much intuition has been gleaned from the quark-shell model over the years. B. Phenomenological Consequences The results of Secs. II and III may be used to gain phenomenological insight into the low-lying excited baryons. A certain amount of care must be exercised when doing this, however, since the physical world of N c = 3 cannot be regarded as an approximately large N c world for all purposes. In particular, consider the most striking formal results of this work-the existence of nearly degenerate multiplets of states associated with a fixed K. Unfortunately, it will be very difficult to extract this structure directly in the baryon spectrum for the low-lying odd-parity states in the real world. The key difficulty concerns the scales of the problem: Note that the physical states in question vary in mass from 1520 MeV (N 3/2 ) to 1700 (∆ 3/2 ), as listed by the Particle Data Group [37]. Thus, all of the observed states lie within a 200 MeV window. However, ∆-N mass splitting, a 1/N c effect, is ∼ 290 MeV. Thus, the actual splittings are not large enough to resolve, given the characteristic size of the 1/N c effects. Studies of constraints on these next-to-leading 1/N c effects are underway [35]. Fortunately, there are phenomenological predictions of the preceding analysis that may well be meaningful in the physical world. Consider Eqs. (2.3) and (3.10). With the interpretation above, we would say that the quark-shell states with mass m 0 in the large N c world correspond to states accessible in scattering experiments characterized by modes with K = 0. Now, it so happens that π-N scattering does not couple to negative-parity K = 0 modes. This can be seen directly from the structure of the 6j coefficients in Eq. (2.1), which implies that K ≥ |L − 1|. Clearly, K = 0 can only happen for L = 1, but L = 1 makes even-parity states. Thus, in Eq. (2.3) we see that the K = 0 multiplet is accessible via η-N scattering but not via π-N . Of course, this result that the K = 0 negative-parity states couple to the η-N channel but not to the π-N only holds to leading in order in 1/N c , so the actual prediction is that the coupling to the π-N channel is weak for these states. In Eq. (2.3) we list two such states in the multiplet, N 1/2 and ∆ 3/2 . Of these, only the N 1/2 can be clearly discerned at N c = 3. Recall that at large N c there are three ∆ 3/2 negative-parity states in the quark-shell model with a single ℓ = 1 quark excitation, but for N c = 3 there is only one. Thus, one cannot associate the physical negative-parity ∆ 3/2 state with a given K. In contrast, there are two negative-parity N 1/2 states, both at large N c and for N c = 3. Thus we predict that, to the extent the 1/N c expansion is useful, one of these two states couples weakly to pions. Similarly, the other state has K = 1, and by an analogous argument, it should be clear that this state couples strongly to the π-N channel but weakly to the η-N channel. How well are these prediction borne out in nature? According to the Particle Data Group [37], the N (1535) has a decay fraction to the π-N channel (35-55%) that is virtually equal to its decay fraction to the η-N channel (30-55%). Now, this is striking since the phase space factor for the decay is nominally ∼ 2.6 times larger for the π-N channel. Moreover, the nominal phase space obtained by assuming that the decaying particle has its Breit-Wigner mass presumably understates the relative advantage the pion has in phase space: Since the η-N threshold is only 50 MeV from the nominal mass of the N (1535), if one averages over the width of the resonance in estimating the effective phase space, one substantially reduces the phase space for the η-N channel but not for the π-N channel. Thus, the N (1535) clearly is much more strongly coupled to the η-N channel than to π-N . Next, consider the N (1650); according to the Particle Data Group [37], N (1650) has a decay fraction to the π-N channel (55-90%) that is much larger than the decay fraction to the η-N channel (3-10%). The dominance of the π-N channel is far larger than what one estimates purely from the phase space, since the π-N channel has a phase space factor ∼ 1.6 times that of the η-N channel. Thus, the qualitative large N c predictions for the dominant decay modes of the N 1/2 states are apparent in the data. The success in describing the decay modes of the N 1/2 states tell us that the leading large N c result does a good job in describing the mixing between these two states. One can also ask about the mixing in the context of the quarkshell model, where it is parameterized by mixing angles. Thus, another check on the phenomenological usefulness of the leading-order large N c treatment is its ability to predict these mixing angles in a manner that is qualitatively consistent with traditional quark models. Again, we restrict our attention to those sectors for which large N c and N c = 3 have the same number of quark model states-namely the N 1/2 and N 3/2 states. From Sec. III we see that the mixing angles are given by θ N1 = tan −1 ( √ 2) ≈ 0.96 and θ N3 = tan −1 (1/ √ 5) ≈ 2.72 (angles in radians). In comparison, fits using decay data give θ N1 = 0.56, θ N3 = 2.96 using SU(6) [38]. The fact that N 1 mixing in large N c is not too far from the phenomenological one is, of course, not surprising since the latter was fit to decays; as we have seen, this is qualitatively consistent with the large N c result for the N 1/2 states. It is encouraging, however, to note that the phenomenological fits for the N 3/2 also work well. We note also that similar results were obtained directly from N c = 3 quark model calculations [39], as well as large N c quark-shell model approaches in which 1/N c corrections are included within individual operator matrix elements [25,28,29,31,32]. It will be interesting to see if similar predictions can be obtained in the strange sector [35]. As mentioned in Sec. II, the relations between various channels in pion-baryon scattering in Eq. (2.1) were first derived in the context of a chiral soliton model [18,19,20,21,33]. However, the result is exact in the large N c limit of QCD. In this Appendix we show how this result can be derived directly from large N c consistency rules with essentially no model-dependent assumptions. Before doing this, we note that it has long been recognized these results are in fact model independent in the large N c limit. As noted long ago by Adkins and Nappi [40], the Skyrme model (treated via semi-classical projection, as appropriate for the large N c limit of QCD) has the striking feature that many relations are completely independent of the details of the Skyrme model Lagrangian-i.e., they are independent of the values of the parameters in the Lagrangian, the number and type of terms included in the Lagrangian, or even the number and types of degrees of freedom. For example, the π-N coupling is precisely 2/3 of the π-N -∆ coupling in the leading-order treatment of any chiral soliton model. It was a reasonable conjecture that such relations are fully model independent and are exact results of QCD in the large N c limit. This conjecture was subsequently shown to be correct for all such relations tested. The method used the large N c consistency rules discovered by Gervais and Sakita in the 1980's [4] and then rediscovered and greatly extended by Dashen and Manohar [5] in the 1990's. The only assumptions used with this method are: i) Baryonic quantities in large N c QCD scale according to the generic large N c rules of Witten [3], or more slowly (if there are cancellations); ii) there exists a hadronic description that reproduces the large N c QCD results; iii) the π-N coupling scaling is generic (without cancellations), scaling as N 1/2 c ; and iv) nature is realized in the most symmetric representation of the contracted SU(4) group that emerges from the previous assumptions. The key to the method is to compare the scaling of π-N scattering (which scales as N 0 c ) with the π-N coupling constant (which scales as N 1/2 c ), suggesting that the sum of the Born term in π-N scattering plus the cross graph scales as N 1 c . This mismatch in scattering implies the need for cancellations, and these in turn are only possible if the baryons form nearly degenerate bands of states with I = J (in the two-flavor case) and lie in irreducible representations of a contracted SU(2N f ). The consequences of this symmetry for various matrix elements are worked out in detail in a series of papers by Dashen, Jenkins, and Manohar [7]. One consequence is that all of the relations derived in the Skyrme model, but that are insensitive to Skyrme model details, are in fact results of large N c QCD. Now we note that that Eqs. (2.1) and (2.2) are, in fact, completely independent of the details of the Skyrme model, and thus one expects that it should be possible to derive it using the methods of Ref. [7]. We focus first on the case of π scattering, hence the explicit I π = 1 below. The first step is to express the amplitudes in terms of t-channel rather than s-channel exchange. This can be done simply via a standard relation between 6j coefficients [41]: K I J R ′ L ′ 1 K I J R L 1 = J (−1) I+J+L+L ′ +R+R ′ +K+J (2J + 1) 1 R ′ I R 1 J R ′ J L ′ L J R 1 L ′ K L 1 J .(A1) Inserting this identity into the first of Eqs. (2.1) yields S LL ′ RR ′ IJ = J 1 R ′ I R 1 J R ′ J L ′ L J R s t J LL ′ ,(A2) with s t J LL ′ ≡ K (−1) J+I+L+L ′ +K+J +1 (2J + 1)(2K + 1) (2R + 1)(2R ′ + 1) 1 L ′ K L 1 J s π KLL ′ .(A3) From the first 6j coefficient in Eq. (A2), it is apparent that J is the isospin exchanged in the t-channel, while the second 6j coefficient implies that J is angular momentum exchanged in the t channel. The superscript t in the function s t J LL ′ indicates that this function is given in terms of the angular momentum and isospin in the t-channel. The fact that t channel-exchanged isospin and the t channel-exchanged angular momentum are both equal to J , of course, implies that they are equal to each other. Thus Eq. (A2) encodes the celebrated I t = J t rule of Mattis and Mukerjee [20]. Now, the preceding argument shows that Eq. (2.1) implies the I t = J t rule for pion-baryon scattering. For our purpose, the important point is that the converse is also true: The I t = J t rule implies Eq. (2.1), since the right-hand side of Eq. (A2) is the most general form for a scattering amplitude consistent with I t = J t . Thus, if one can establish the I t = J t rule directly from the large N c consistency rules, then one has established Eq. (2.1) directly from large N c QCD. However, as shown in Ref. [42] it is straightforward to establish the I t = J t rule using the techniques of Ref. [7]. An analogous argument works for the case of η-baryon scattering. TABLE I : INegative-parity mass eigenvalues in the quark-shell model picture, corresponding partial waves, and their expansions in terms of K-amplitudes. The association of masses with states is from the large Nc quark-shell model relations in Sec. III. The superscripts πN N , πN ∆, π∆∆, ηN N , and η∆∆ refer to the scattered meson and the initial and final baryons, respectively. The partial-wave amplitudes are derived from Eqs. (2.1) and(2 with M(1)The mixing angles are parameterized by . F X See For Example, S J Lee, T Dong, I Draper, K F Horvath, N Liu, J B Mathur, Zhang, hep-lat/0208070See for example F. X. Lee , S. J. Dong, T. Draper, I. Horvath, K.F. Liu, N. Mathur, J. B. Zhang,hep-lat/0208070. . G Hooft, Nucl. Phys. 72461G. 't Hooft, Nucl. Phys. B72, 461 (1974). . E Witten, Nucl. Phys. 16057E. Witten Nucl. Phys. B160, 57 (1979). . J.-L Gervais, B Sakita, Phys. Rev. Lett. 5287J.-L. Gervais and B. Sakita Phys. Rev. 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Karliner, Phys. Rev. D 31, 2833 (1985). . D Pirjol, C Schat, hep-ph/0301187D. Pirjol and C. Schat, hep-ph/0301187. . T D Cohen, R F Lebed, in preparationT.D. Cohen and R.F. Lebed, in preparation. See for example Quantum Theory of Finite Systems. J.-P. Blaizot and G. RipkaMIT PressCambridgeSee for example Quantum Theory of Finite Systems, J.-P. Blaizot and G. Ripka, (MIT Press, Cambridge, 1986). . Phys. Rev. D. 6610001Particle Data GroupParticle Data Group, Phys. Rev. D 66, 010001 (2002). . A J G Hey, P J Litchfield, R J Cashmore, Nucl. Phys. 95516A.J.G. Hey, P.J. Litchfield, and R.J. Cashmore, Nucl. Phys. B95, 516 (1975). . N Isgur, G Karl, Phys. Lett. 72109N. Isgur and G. Karl, Phys. Lett. 72B, 109 (1977). . G S Adkins, C R Nappi, Nucl. Phys. 249507G.S. Adkins and C.R. Nappi, Nucl. Phys. B249, 507 (1985). A R Edmonds, Angular Momentum in Quantum Mechanics. Princeton, NJPrinceton Univ. PressA.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. 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[]
[ "On Unification of RR Couplings", "On Unification of RR Couplings" ]
[ "Emil T Akhmedov ", "Anton A Gerasimov ", "Samson L Shatashvili ", "\nDepartment of Physics and Astronomy\nInstitute of Theoretical and Experimental Physics\nUniversity of British Columbia\nB. Cheremushkinskaya 25V6T 1Z1, 117259Vancouver, MoscowBritish ColumbiaCanada, Russia\n", "\nDepartment of Physics\nYale University\nP. O. Box 20812006520-8120New HavenCT\n" ]
[ "Department of Physics and Astronomy\nInstitute of Theoretical and Experimental Physics\nUniversity of British Columbia\nB. Cheremushkinskaya 25V6T 1Z1, 117259Vancouver, MoscowBritish ColumbiaCanada, Russia", "Department of Physics\nYale University\nP. O. Box 20812006520-8120New HavenCT" ]
[]
We consider the couplings of RR fields with open string sector for Dp-Dp backgrounds of various p. The proposed approach, based on the approximation of the open string algebra by the algebra of differential operators, provides the unified description of these couplings and their interrelations.
10.1088/1126-6708/2001/07/040
[ "https://export.arxiv.org/pdf/hep-th/0105228v3.pdf" ]
1,520,014
hep-th/0105228
016614d94accc0947030d8097f0885740d1be3a6
On Unification of RR Couplings arXiv:hep-th/0105228v3 30 May 2001 Emil T Akhmedov Anton A Gerasimov Samson L Shatashvili Department of Physics and Astronomy Institute of Theoretical and Experimental Physics University of British Columbia B. Cheremushkinskaya 25V6T 1Z1, 117259Vancouver, MoscowBritish ColumbiaCanada, Russia Department of Physics Yale University P. O. Box 20812006520-8120New HavenCT On Unification of RR Couplings arXiv:hep-th/0105228v3 30 May 2001 We consider the couplings of RR fields with open string sector for Dp-Dp backgrounds of various p. The proposed approach, based on the approximation of the open string algebra by the algebra of differential operators, provides the unified description of these couplings and their interrelations. Introduction The process of brane annihilation [1] leads to the unified description of the backgrounds with various D-brane configurations [2]. However, despite the recent progress in understanding the tachyon low energy action and the related qualitative phenomena [3,4,5,6], the picture of the dynamics of the D-brane formation/annihilation is still not very clear. One of the most important points seems to be the interrelation of the open and closed string sectors in this process, which is one of the main mysteries of string theory. The satisfactory understanding of this relation most likely will provide the clue regarding the symmetries of string theory [6]. In this paper we reconsider the couplings of the RR gauge fields of the closed string sector with the fields of the open string sector. These couplings [7] are especially simple due to their anomalous origin [8] and thus provide a suitable framework for the discussion of the various open string backgrounds. These couplings were extensively studied (see e. g. [9,10,11,12,13,14,15,16,17,18,19]). Our ultimate goal will be the unified description of these couplings for various D − D backgrounds 3 . The picture that emerges could be considered as an explicit realization of some proposals from [20] (some related remarks have been made in [16]). In an attempt to provide the unified description of the RR gauge field couplings for various background brane configurations it is natural to consider the off-shell interpolation of these backgrounds. The connection with K-theory uncovered in [21,2] leads to the description of the RR-gauge field couplings in terms of the superconnection [22]. Thus a natural framework for the universal description of RR couplings would be in terms of a superconnection in some universal (infinite) dimensional bundle E over space-time M which provides interpolation between various brane/anti-brane configurations. This infinite-dimensional bundle E should be naturally connected with the geometry of M and thus it is not very surprising to find the "tautological" bundle with infinite dimensional fiber isomorphic to space of functions on the base manifolds. This bundle has a rich structure connected with the action of the differential operators in the fiber. This is in perfect agreement with the appearance of the differential operators in the explicit description of the elements of the K-homolgy groups. The appearance of the differential operators is also natural from string field theory point of view. Configuration space of the open string theory is roughly given by the maps of the interval into space-time. If we approximate the strings by straight lines the configuration space reduces to space of the pairs of the points (ends of the intervals) in space-time. Therefore the functionals on configuration space become the functions of two points and could be interpreted as the kernels of the integral operators. The expansion around diagonal (image of the interval is a point) leads naturally to the differential operators. In this note we discuss very briefly the approximation of the open string algebra by the differential operators. The more systematic exposition will is given in [23]. Here we describe explicit interpolation of the various RR gauge field couplings using the formalism of superconnections. For simplicity we always consider the case of the flat space-time. The generalizations to curved manifolds are rather obvious. In the process of describing our proposal we will focus on the local properties. The organization of this paper is as follows. In section 2 we consider the interpolation between the RR gauge field couplings with open strings for various brane/anti-brane backgrounds in terms of superconnections acting in appropriate finite dimensional (Z 2 -graded) bundles. The physical interpretation of the construction is through the appearance of D-branes of higher codimension in the process of annihilation of D − D-branes with the corresponding nontrivial low-dimensional RR charges. In order to get an arbitrary brane configuration one could use only D-branes of the highest possible dimension. For simplicity we will illustrate these phenomena looking at type IIB theory and D9 − D9 system [2]. It should be noted that the configurations with non equal number of D9 and D9 branes is anomalous in string theory. But in our approach to RR-couplings all D branes are more or less on equal footing and we will not discriminate the case of D9 branes in the following presentation. As an application we derive explicit formula for the topological density of the "fat" D5brane inside the D9-brane, which coincides with the well known formula for the densities of the instanton charge (see e.g. [24]). The main deficiency of the approach of this section is the necessity to consider special configurations of the D9 − D9-branes depending on the desired ultimate configuration of the lower dimensional brane/anti-brane system we would like to get. The lesson learned from these examples suggests the idea to search for the universal description by taking the number of D9 − D9 branes infinite [20] (see [13] for earlier discussions) 4 . Our main proposal is presented in Section 3. Here we describe the concrete procedure of taking the limit by considering the superconnection in the appropriate geometrically defined infinite dimensional bundle. We show how the explicit formulas from the section 2 appear in this formalism. At the end we briefly discuss the interpretations of the proposed universal description in terms of string theory. The issues related to the appearance of non-abelian Chan-Paton factors in the proposed universal description and its connection with background independent open string field theory [27], [28] will be published separately [23]. RR-couplings from the finite number of D9−D9-branes In this section we consider the RR gauge field couplings with open strings for the backgrounds with the finite number D9 − D9-branes and the low dimensional brane backgrounds arising in the process of the annihilation. The general coupling of the D-brane fields with the RR fields may be characterized as follows. Any configuration of D-branes in type IIB theory defines some element of the K 0 -group of space-time [21,2] which may be represented as a formal difference of the vector bundles up to some equivalence. This was interpreted in [2] as gauge bundles on D9 and D9 branes filling whole space-time. There is a canonical map of the K 0 -group to the cohomology of the manifold defined with the help of the Chern character. In order to get explicit representation of this map in terms of the (closed) differential forms one should supply the difference of the bundles with an appropriate analog of the connection -superconnection [22]. The superconnection corresponding to a given configuration of D9 − D9-branes may be constructed in terms of the open string field. The coupling of the RR gauge fields with open string sector of the D9 − D9branes filling whole space-time has the following form: S RR = µ 9 M (10) [C RR ∧ Ch(A)] top (2.1) Here µ 9 is the D9-and D9-brane charge, M (10) is ten-dimensional flat target space, Ch(A) is the Chern character of the superconnection A constructed from the open string modes, C RR is defined as a sum of the RR gauge fields of the given parity (even in IIB theory): C RR = k C (2k) , C (2k) = C µ 1 ...µ 2k dx µ 1 ∧ · · · ∧ dx µ 2k (2.2) and the subscript top means that we should integrate the top-dimensional differential form over M (10) . We use the normalization of RR gauge fields consistent with the standard definition of the superconnection used below. The gauge bundles on D9 and D9 branes may be combined in a Z 2 graded bundle with the superconnection defined in terms of the gauge field A µ on the D, the gauge field A µ on the D and the complex tachyon field T corresponding to the lowest energy excitation of the strings stretched between D and D by the following: A = dx µ ∇ + µ T T dx µ ∇ − µ , ∇ + µ = ∂ µ + i A µ , and ∇ − µ = ∂ µ + i A µ . (2.3) The Chern character of the superconnection is defined by the standard formula: Ch(A) = Str e − A 2 2π . (2.4) Here we use the supertrace on the matrices acting on Z 2 -vector spaces which is defined as follows. Let V = V + ⊕ V − be a Z 2 -vector space and O ∈ End(V ): O ++ O +− O −+ O −− . (2.5) Then the supertrace of O is defined as: Str V O = Tr V τ O = tr V + O ++ − tr V − O −− , (2.6) where τ is an operator defining Z 2 -structure. In the following subsections we consider various families of the background open string fields and show how the RR gauge field couplings for the lower dimensional D-branes appear in some limits. D-instantons from D9 − D9-branes We start with the simple case of k D-instantons inside ten-dimensional flat space, which appear as the result of the D9−D9 annihilation. In order to construct k D-instantons from the D9−D9 system one has to consider 16 k D9-branes and 16 k D9-branes [1,2,29], where 16 Chan-Paton indexes from both sides are embedded into the spinor bundles of the target space with opposite chirality. The corresponding superconnection may be described as follows [2]. Consider a Z 2 -graded vector bundle V = V + ⊕ V − , where V ± = S ± ⊗ E k and S ± are the spinor bundles of the definite chirality and E k is a k -dimensional vector bundle. The superconnection is defined as in (2.3) with: ∇ + : V + → V + ∇ − : V − → V − T : V + → V − T : V − → V + (2.7) We take a trivialization of the vector bundle E k , i.e. ∇ ± = d. To begin we describe the simplest situation of coincident D-instantons which are left after the D9 − D9 annihilation. The corresponding superconnection may be constructed as the limit t → 0 of the superconnection (2.3) with: ∇ = d and T = 1 √ t x µ σ µ ⊗ 1 k×k , (2.8) where σ µ are ten-dimensional γ-matrices in the Majorana-Weyl representation and 1 k×k is the identity matrix acting in the space E k (k is the number of the instantons). Description of Ch(A) in terms of such a family of superconnections is a standard procedure and appeared in mathematical literature long ago [22]. In the language of string theory it may be interpreted as off-shell interpolation (one shall compare this tachyon profile T with the one from [2]). The family (2.8) corresponds to the smearing of the D-instanton charge over a region of the size ∝ √ t (see eq. (2.13) below). The square of the superconnection in question has the form: A 2 t = 1 t |x| 2 1 √ t dx µ σ µ 1 √ t dx µ σ µ 1 t |x| 2 ⊗ 1 [k×k] = 1 √ t dx µ γ µ + 1 t |x| 2 ⊗ 1 [k×k] . (2.9) Taking into account that in the spinor notations supertrace of the arbitrary operator O may be represented in terms of the trace over spinor representation, Sp, as StrO = tr k Sp γ 11 O, (2.10) we have: Ch(A t ) = k Sp γ 11 e − A 2 t 2π . (2.11) Here k = tr 1 [k×k] . Thus, using the identities for d dimensional space: Spγ d+1 γ µ 1 . . . γ µ d = (2i) d 2 ǫ µ 1 ...µ d (2.12) and lim t→0 1 (πt) d/2 exp − |y| 2 t = δ (d) (y),(2.13) we obtain: Ch(A) = lim t→0 Ch(A t ) = k δ (10) (x) vol(M 10 ). (2.14) After the substitution into (2.1) this gives S RR = k µ −1 C (0) (x = 0), i.e. the correct coupling of k D-instantons with RR gauge fields. Consider a more general situation given by the deformation of the tachyon profile (2.8) as follows: T = 1 √ t x µ γ µ ⊗ 1 k×k − Φ µ γ µ ,(2.15) with Φ µ -constant k × k hermitian matrix, µ = 0, ..., 9, which capture the fluctuations of the relative positions of k D-instantons. These matrices naturally appear in the low energy descriptions of the D-instantons in string theory [30]. For the square of the superconnection we have: A 2 t = 1 √ t dx µ γ µ ⊗ 1 k×k + 1 2t [Φ µ , Φ ν ]γ µ γ ν + 1 t |x µ ⊗ 1 k×k − Φ µ | 2 . (2.16) Let us define the symmetric trace "SymTr" following [31,32] as the symmetrisation of the usual trace. Thus we have: Tr e A+B+C = Tr n (A + B + C) n n! = SymTr n (A + B + C) n n! = SymTr e A+B+C (2.17) Taking into account (2.16) and (2.17) we obtain: S RR = µ −1 M (10) C RR (x) SymTr e −[i Φ , i Φ ]/2π δ (10) (x ⊗ 1 µ − Φ µ ) = = µ −1 M (10) C RR (x) SymTr e −[i Φ , i Φ ]/2π e −iΦµ∂µ δ (10) (x) = = µ −1 SymTr e −[i Φ , i Φ ]/2π C RR (Φ) (2.18) where i Φ is defined in [32] -the indexes of Φ's in the commutator in the exponent are contracted with those of C (2k) . We also use the following definition of the δ-function: δ (d) (x µ ⊗ 1 − Φ µ ) = lim t→0 1 (πt) d/2 exp{− 1 t d µ=1 |x µ ⊗ 1 − Φ µ | 2 }. (2.19) Note that the expression (2.18) coincides with S RR obtained in [32]. One could consider more general deformations of T : T ∝ x µ σ µ ⊗ 1 + T µ 1 ···µ k (k) (x) σ µ 1 ···µ k + · · · ,(2.20) which apparently correspond to massive open string excitations. Dp-branes from the D9 − D9 It is not hard to generalize the above construction to the case of k Dp-branes with p < 9. Let us separate indexes µ = 0, ..., 9 into m = 0, ..., p and i = p + 1, ..., 9 and consider the following background values of the tachyon and gauge field: ∇ = d and T = 1 √ t x i σ i ⊗ 1 k×k . (2.21) Here now σ i are (9 − p)-dimensional σ-matrices. Now, in order to construct k Dp-branes from the D9 − D9 system, one has to consider [2] 2 (9−p)/2 × k D9-branes and the same number of D9-branes. Similarly to the case of the instantons the superconnection (2.21) leads to the coupling: S RR = kµ p M (p+1) C (p+1) (x m ), which is the source corresponding to k Dp-branes; M (p+1) here is the world-volume of the Dp-branes. Let us consider the fluctuations around (2.21) of the form: ∇ = ∂ m + i A m (x m ) dx m + ∂ i dx i and T = 1 √ t x i σ i ⊗ 1 k×k − Φ i (x m ) σ i . (2.22) Here A and Φ are the gauge field and scalar field living on the Dp-brane world volume 5 . The explicit expression for the square of the superconnection is: A 2 t = [∇ m , ∇ n ] dx m dx n + 1 √ t dx i γ i ⊗ 1 k×k + 1 √ t [∇ m , Φ i ]γ i dx m + + 1 2t [Φ i , Φ j ]γ i γ j + 1 t |x i ⊗ 1 k×k − Φ i | 2 . (2.23) After substitution of this expression into (2.1) and taking t → 0, one finds: S RR = µ p M (p+1) SymTr C RR (x m , Φ i ) e − 1 2π (F(2)(xm)+[∇(1)(xm), i Φ ]+[i Φ , i Φ ]) top , (2.24) with F (2) = [∇ m , ∇ n ] dx m ∧ dx n and ∇ (1) = ∇ m dx m . This expression also coincides with S RR considered in [32]. General configuration of D-branes of various codimensions The general case is now rather obvious. For illustration we will construct a very simple configuration of low dimensional D-branes. Consider D-branes oriented along coordinate hyperplanes in d = 10 dimensional flat space. One describes such a configuration with k p Dp-branes and k p Dp-branes for all p by considering the collection of D9 − D9 which represents Z 2 -graded bundle: V = V + ⊕ V − = ⊕ p (E kp ⊕ E kp ) ⊗ S(9 − p). (2.25) Here E kp and E kp are vector bundles of dimension k p and k p and S(9 −p) are the spinor bundles in the directions transverse to the Dp-branes (S(0) is a trivial Z 2 -even bundle). The sum in (2.25) is taken over the odd p's (we are considering type IIB string theory). Z 2 -grading is given by the operator τ acting on spinor bundles S(9 − p) as τ = (−1) p/4 γ 10−p where γ 10−p is a product of all gamma-matrices acting in S(9 − p) (analog of γ 5 in (9 − p) dimensions). It acts on the bundles E kp and E kp as follows: τ E kp = E kp τ E kp = −E kp (2.26) Then the obvious generalization of the superconnection (2.22) is: A t, t = ∇ + p 1 √ t p 10 i=p+2 (x i γ i ⊗ 1 kp×kp − Φ i γ i ) + (2.27) + p 1 t p 10 i=p+2 (x i γ i ⊗ 1 kp× kp − Φ i γ i ). (2.28) Here the matrices 1 kp×kp (1 kp× kp ) acts as unit matrices on E kp (E kp ) and by zero otherwise. It gives rise to the density of the Chern character which in the limit t p , t p → 0 describes the coupling of the collection of the Dp-branes with the RR fields. The general superconnection (2.28) interpolates between various configurations of D-branes capturing all their low-energy fluctuations. Note an interesting novel feature of the general case. The general deformations of this superconnection includes operators transforming in the spinor representations of the transversal Lorentz groups. These tachyonic excitations correspond to the lowest energy modes of the strings stretched between the Dp-and Dp-branes with the mixed Neumann-Dirichlet boundary conditions. They are responsible for the "thickening" of the lower dimensional D-branes inside bigger D-branes. In the next subsection we proceed with the discussion of some example of this situation. D5-branes inside D9 -branes and the instanton charge density Consider the special case of the k D5-branes inside N D9-branes as the result of the D9 − D9 annihilation. The theory with N > 0 is anomalous. However, this is not important for our further considerations and we proceed with this case in order to make the presentation most transparent. Otherwise one could consider D3 − D7-brane system using the machinery of the previous subsection. The Z 2 -graded vector bundle in this case has the form: V = V − ⊕ V + = (W (9) N ⊕ E (5) k ⊗ S(4) − ) ⊕ (E(5) k ⊗ S(4) + ), (2.29) where S ± are definite chiral, Weyl fermion bundles in four dimensional space transverse to the D5-branes. We take a trivialization of the E k bundles ∇ = d. In order to make a comparison with on-shell description in terms of the instanton moduli space we restrict the superconnection under consideration by the condition that it is invariant with respect to the natural action of Sp(1)-group on S + . Then its moduli space is parameterized by ADHM data (see e.g. [33]): T = 1 √ t ∆ ∆ : S − (4) ⊗ E (5) k ⊕ W (9) N → S + (4) ⊗ E (5) k . (2.30) Here ∆ is the standard ADHM matrix [33]: ∆ = [x i σ i ⊗ 1 k×k + B i (x m ) σ i , h(x m )],(2.31) where i = 6, ..., 9 are directions transversal to the D5-brane and x i σ i naturally acts as the operator x : S − → S + , similarly constant (independent of x i ) operator B acts as B : S − ⊗E k → S + ⊗ E k and the constant operator h acts as h : W N → S + ⊗ E k . In the language of string theory B i 's are scalars localized on the D5-brane world-volume, which appear as the massless excitations of the strings with both their ends on the D5-brane. They parameterize transversal fluctuations of the D5-brane. In these terms h appears as the massless excitations of strings stretched between D5-and D9-branes. The condition on the matrix ∆ in ADHM construction is equivalent to invariance with respect to the action of Sp(1)-group on S + and thus we cover the whole ADHM moduli space. The derivation of this tachyon soliton from the condition of SUSY restoration after the annihilation was given in [34]. The Chern character of the superconnection A t at the limit t → 0 reduces to the standard expression for the Chern character of the instanton solutions. Consider for the illustration the simple case of one instanton in the Sp(1) gauge theory: k = 1, dim W = 2. It is useful to identify W ∼ = S + . Choose the configuration of the instanton with the center at x = 0 and of the radius h. Then the corresponding superconnection is: A t = d ⊗ 1 4×4 + 1 √ t x i γ i ⊕ 0 W + h p w + h p * w . (2.32) Here B = 0 and p w : W → S + identifies W and S + . Note that 1−γ 5 2 p w = p w and 1+γ 5 2 p w = 0. The square of the superconnection is: A 2 t = 1 √ t dx i γ i ⊕ 0 W + 1 t |x| 2 1 S + h 2 1 S + + h 2 1 W + h x i σ i p W + h p * W x i σ i ,(2.33) where we use the notations: 1 S is the projector on the spinor subspace, 1 S + = 1+γ 5 2 is the projector on the S + subspace and 1 W is the projector on W . Thus, in matrix form, such A superconnection looks as: A 2 t =   1 t (|x| 2 + h 2 ) 1 √ t dx i σ i 0 1 √ t dx i σ i 1 t |x| 2 1 t h x i σ i 0 1 t x i σ i h 1 t h 2   . (2.34) One can use the identity 6 : e A+B = P exp and obvious commutation relations between γ i , 1 S , 1 S + , 1 W and p w , in order to expand the expression exp(−A 2 t ) in the components having the differential forms of the definite degree and prove that: Ch(A) = lim t→0 Sp tr (γ 5 ⊕ 1 W ) e − A 2 t 2π ∝ dim(W ) + 1 8π h 4 (|x| 2 + h 2 ) 4 vol(R 4 ). ( 2.36) This is the correct expression for the topological charge density of one Sp(1) YM instanton [24]. The more conceptual derivation of this formula and its generalizations to the multiple instantons with higher range gauge groups using the results of [35] will be published elsewhere [36]. 3 Universal description and algebra of differential operators The construction described in the previous section has an obvious drawback. Auxiliary (super)vector bundles that appear in the description of D-branes do not have clear geometrical meaning. In particular for each D-brane configuration one needs to consider the specific collection of vector bundles. On the other hand it is obvious that different vector bundles with the same topological invariants lead to the equivalent description of the D-branes. In order to get a universal description of D-brane backgrounds it is natural to look at the bundles of the infinite rank [20]. We will use a concrete realization of this limiting bundle as the bundle where fiber is identified with the space of functions (more exactly sections of the finite-dimensional bundles) on an auxiliary space. As an auxiliary space it appears natural to take a copy of the space where string theory is defined. The idea to use this realization of the universal bundle comes from simple qualitative arguments concerning the configuration space of the open strings that were discussed in the introduction. Note that the natural algebra acting on the fiber of this bundle is the algebra of differential operators on auxiliary space. This construction may be considered as an application of the general approach of [23] to the description of the RR gauge field couplings. In the next subsection we show how to derive the corresponding RR field coupling for the D9-brane filling the whole space-time. Then we proceed with the derivation of the RR gauge field coupling for an arbitrary case. This reproduces the expressions considered in Section 2. Description of D9 branes We start the description of the universal construction with simple physical motivations. Consider 16 k D9-branes and 16 k D9-branes in the flat space -time M = R 10 and take k → ∞. In this way we construct k → ∞ D(D)-instantons. We could use these D(D)-instantons to construct various Dp-branes. The configuration of k non-coincident D-instantons may be parameterized by k points of space-time. More correctly it is the space of 10 mutually commuting k × k−matrices up to conjugation. It is easy to see that as the number k of D-instantons tends to infinity the k-dimensional vector space on which these matrices act becomes more and more like Hilbert space of the square integrable functions on space-time M where string theory is defined. In these terms the matrices themselves may be considered as differential operators (more generally integral operators) acting on functions on M. The traces of matrices in this limit may be reduced to the integrals over the space M. For any operator function F ( q, p) of the coordinates q µ and momentum p µ one has: Tr H : F ( q, p) := dp ∧ dq q| : F ( q, p) : |p p|q = dp ∧ dq F (q, p), (3.37) where : F : is the normal ordering of the operators such that the momentum operators act on the right and the position operators act on the left. Note that to have a well defined trace the operators should have special property (to be of the trace class). A particular class of such operators is given by the expressions F ( p, q) ∝ δ (10) ( p)f ( q). It's trace is naturally represented by the integral over the lagrangian submanifold M ∈ T * M rather than over T * M and thus the operators of this kind and its smooth deformations are natural candidates for the limits of the appropriate matrix variables. This type of operator has a simple qualitative interpretation in terms of string backgrounds. Let us symbolically denote open string by the matrix K x,y (or more exactly integral operators K with the kernel K(x, y) acting on the space of functions on M) where possible end points of the strings are enumerated by the indexes "x" and "y". The trace invariants naturally correspond to the open strings with identified ends and thus may be considered as closed strings. Now let us consider the trace of the operator K with additional insertion of the operator δ( p) ( p is momentum operator) which projects on the states with zero eigenvalue of p. It is easy to see that: δ( p) = |p = 0 p = 0| = ( x |x )( x ′ x ′ |) (3.38) (note that the identity operator is 1 = x |x x|) and we have: Tr H δ( p) K ∼ x,x ′ K(x, x ′ ) (3.39) We see that in the presence of the operator we should sum over the positions of the "ends" of the string independently. Thus the insertion of the operator δ( p) may be interpreted as the creation of the D-brane filling the whole space-time. We have replaced the consideration of the infinite number of D-branes by the consideration of the infinite-dimensional vector bundles over the space-time M with the fiber -the space of functions on the copy M of M. The Lie algebra of the differential operators acts naturally in the fibers of the bundle. Note that one may consider infinite dimensional bundles with the fiber given by the sections of a finite dimensional (super)-bundle. We encounter examples with the space of sections of spin bundles as a fiber. Now we give an explicit construction of the superconnection in this bundle which leads to the desired description of RR-couplings. Let us start with few remarks on the notations. In order to distinguish the space-time M from the auxiliary space M we will use the coordinates (x µ ) on M ≡ M x and the coordinates (y µ ) on M ≡ M y . Thus the matrices realize the quantization of the space of functions on T * M y and the corresponding Hilbert space may be constructed in terms of the sections of the spinor bundle over M y . Having in mind more general cases treated below we consider Clifford algebra for the total space M x ⊗ T * M y : {γ µ , γ ν } = δ µν {Γ µ , Γ ν } = δ µν { Γ µ , Γ ν } = δ µν {Γ µ , γ ν } = { Γ µ , Γ ν } = { Γ µ , γ ν } = 0. (3.40) The gamma matrices are defined with respect to the explicit coordinates as follows: (x µ , y ν , p ρ ) ↔ (γ µ , Γ ν , Γ ρ ) First we give a simple example of the superconnection describing the D9-brane filling the whole space-time. Taking into account the heuristic description in terms of the infinite number D-instantons considered at the beginning of this subsection and the results of the first part of the paper we could propose the following superconnection: A s,t = d + 1 √ t γ µ (x µ − y µ ) + 1 √ s Γ µ ∂ ∂y µ (3.41) This superconnection takes values in the space of differential operators acting in the auxiliary space of sections of spinor bundle on M y with standard Z 2 -structure defined by the chirality. The first term is a direct analog of the tachyonic profile (2.15). Note also that the last term is a Dirac operator on this auxiliary space. The superconnection (3.41) corresponds to the tachyon field of the following form: T x| y, p = 1 √ t σ µ (x µ − y µ ) + 1 √ s Σ µ p µ (3.42) where p = ∂/∂y. The first term is rather obvious and one of the reasons for introducing the last term is to fulfill the condition of being trace class. The square of (3.41) is given by the expression: A 2 s,t = 1 t |x − y| 2 + 1 s | p µ | 2 + 1 √ st γ µ Γ µ + 1 √ t dx µ γ µ (3.43) and the Chern character is given by: Ch(A) = lim t→0 lim s→0 Ch(A s,t ) (3.44) Note that here "Str" is taken over the representation of the full Clifford algebra (3.40) and includes Tr H as well. By the trivial considerations it may be reduced to the following trace over the space of y-dependent functions: Ch(A) = Tr H δ (10) ( p µ ) δ (10) (x − y) = 1 (3.45) Thus we have reproduced the correct coupling with the top-degree RR-form for the D9-brane filling whole space-time: S RR = µ 9 M (10) C (10) (x). One could include a non-trivial connection over y-variables. Let us show that this is equivalent to the inclusion of the same connection on M x . Consider the following superconnection: A s,t = d + 1 √ t γ µ (x µ − y µ ) + 1 √ s Γ µ ∂ µ + i A µ (y) (3.46) Simple calculation gives: Ch(A) = Tr H δ (10) (∂ µ + i A µ ) δ (10) (x − y) exp − 1 2π F (2) [A(y)] = = exp − 1 2π F (2) [A(x)] (3.47) At the last step we use the following identity: let us represent δ-function as: δ (10) ∂ µ + i A µ (y) = dq e i q µ ∂µ + i Aµ(y) (3.48) and use: e i q µ ∂µ + i Aµ(y) = e i q µ ∂µ e − 1 0 Aν (y+t q) q ν dt . (3.49) this follows from (2.35). Furthermore, for an arbitrary function F (p, y), we have Tr H dq e i q µ ∂µ F (q, y) = dp dy dq y|e i q µ ∂µ |p p|F (q, y)|y = = dp dq dy e i q µ pµ F (q, y) = dy F (0, y). S RR = µ 9 M (10) [C RR ∧ e − 1 2π F (2) [A] ] top ,(3.51) which is what we expect for D9-branes. In conclusion, we could equivalently turn on the gauge fields in x-space and in y-space. Towards Universal Description of Dp-brane RR-couplings In this subsection we give a general construction of the RR gauge field couplings for arbitrary D-brane background. Let us start with a fixed infinite-dimensional bundle over the space-time with the fiber identified with the space of sections of the spinor bundle over the base space times the twodimensional vector space. We start with the construction of a superconnection corresponding to n + D7 branes and n − of D7 parallel branes. Let (y a = y 9 , y 8 ) be coordinates in the orthogonal plane and W (y 9 , y 8 ) is a function defined by the condition that its only critical points are the positions of D7 and D7 branes in the plane (y 9 , y 8 ) (the sign of the Hessian ∂ 2 W defines the sign of the RR charge). Consider now the superconnection: A s,t = d + 1 √ t γ µ (x µ − y µ ) + 1 √ s 1 Γ a ∂ a W (y) + 1 √ s 2 Γ µ p µ (3.52) In this formula the additional two-dimensional vector space is interpreted as the representation of the Clifford algebra of the cotangent space to the transverse space (the notations are in agreement with (3.40). Note that this representation is very close to the expression that enter the Supersymmetric Quantum Mechanic interpretation of Morse theory [37]. 7 In fact, the calculation of the Chern character for this connection in the limit t, s i → 0 gives the expression with the insertion of the projector in the y-integral: det(∂ a ∂ b W ) δ (2) (∂ a W (y)) (3.53) 7 One could say that we have N = 1 SUSY Quantum mechanic in the directions orthogonal to D-brane and N = 1 2 SUSY Quantum mechanic in the directions parallel to D-brane along with δ (10) (x − y) and δ (10) ( p). Trivial calculations reduce the full expression to the one given in the first part of the paper and gives rise to the Chern character form representing n + D7-branes n − D7-branes (n ± is number of critical points of W (y) with the positive and negative indexes). Note that at the intermediate step after taking the limit s 1 → 0 we get an infinite dimensional bundle with the fiber naturally identified with the space of sections of the super bundle of spinors on the auxiliary space of two dimensions lower multiplied by the finite dimensional super-bundle E = E + ⊕E − with dim E ± = n ± . Let us represent the ten-dimensional space-time R 10 in the factorized form R 10 = R 2 ⊗ R 8−p−1 ⊗ R p+1 . One could identify the bundle E with the bundle of the representations of Clifford algebra of the cotangent space to R 8−p−1 (similar to identification of the gauge bundle and tangent bundle in string compactifications). Now we could again use the same localization procedure. Suppose we would like to get the superconnection leading to the RR-coupling with Dp parallel branes orthogonal to R 2 ⊗ R 8−p−1 . Let W (p) (y) be a function on R 8−p−1 whose critical points define the positions of Dp-branes. Then the following superconnection leads to the RR coupling with Dp branes: A s,t = d + 1 √ t γ µ (x µ − y µ ) + (Γ a (∂ a ( 1 √ s 1 W (y 9 , y 8 ) + 1 √ s p W (p) (y 7 , · · · , y p+2 )))) + (3.54) + 1 √ s 2 Γ µ p µ Here the summation is over a = 9, 8, · · · (p + 2). Iteration of this procedure leads to the expressions for RR gauge field couplings discussed at the first part of the paper. The set functions W (p) emerged in this iterative procedure is rather special and connected with the special geometry of D brane configurations. Consideration of arbitrary functions leads to the expression for RR gauge field coupling with an arbitrary configuration of D − D branes. Conclusions In this note we have proposed a universal description of the couplings of RR gauge fields with open strings in arbitrary backgrounds of the Dp − Dp for p < 9 in terms of the superconnection on some naturally defined infinite-dimensional bundle. This description is given in terms of the differential operators in the auxiliary space (which is a kind of a double of genuine space-time where string theory is defined) depending on the point of space-time. From the mathematical point of view this construction is very close to Beilinson construction of the holomorphic bundles on P n [38]. In conclusion let us give some comments on the interpretation of this construction within string theory. As was already discussed in the introduction, open strings (with unspecified boundary conditions) lead to the differential operators as the crude approximation of the open string algebra -only end points of the open string are taken into account. The real picture is obviously more difficult and the first correction is that we should represent an open string state not as an interval but as a "slice of pizza" part of a closed string world-sheet [39,40]. It is defined in the first approximation by three points -two of them are "ends" of the open string and the third one which is at the "apex" of the slice is responsible for the interaction with closed string vertexes. The description we encounter in the last section is very close to this qualitative picture. Curiously enough the very rough approximation of the open string algebra leads to a satisfactory description of the couplings of the open strings to RR gauge fields. Let us note at the end that the considerations of this paper, though inspired by string field theory, were rather formal. This allows us to present the logic behind the construction in the most explicit way. The detailed comparison/derivation of these results in terms of the first quantized string theory will be given elsewhere. It is also interesting to construct the natural "geometric" action for the algebra of differential operators in the lines of [41]. −t B Ae t B e B (2.35) Permanent Address: Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow, Russia. 2 On leave of absence from St. Petersburg Branch of Steklov Mathematical Institute, Fontanka, St. Petersburg, Russia. We were informed by G. Moore that some related studies have been undertaken by him in collaboration with Duiliu-Emanuel Diaconescu. Another motivation for this comes from the search of closed string in the field theory of open strings[25]-N D1-branes thought as multi-soliton on space-filling D9 brane, or as D9 compactified on T 8 , in large N limit and IR describes type II closed string via Matrix String construction[26]. 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[ "A Counter-Forensic Method for CNN-Based Camera Model Identification", "A Counter-Forensic Method for CNN-Based Camera Model Identification" ]
[ "David Güera \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Yu Wang \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Luca Bondi \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Politecnico Di \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "ItalyMilano Milan \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Paolo Bestagini \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Politecnico Di \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "ItalyMilano Milan \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Stefano Tubaro \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Politecnico Di \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "ItalyMilano Milan \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n", "Edward J Delp \nPurdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana\n" ]
[ "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana", "Purdue University West Lafayette\nPurdue University West Lafayette\nPurdue University West Lafayette\nIndiana, Indiana, Indiana" ]
[]
An increasing number of digital images are being shared and accessed through websites, media, and social applications. Many of these images have been modified and are not authentic. Recent advances in the use of deep convolutional neural networks (CNNs) have facilitated the task of analyzing the veracity and authenticity of largely distributed image datasets. We examine in this paper the problem of identifying the camera model or type that was used to take an image and that can be spoofed. Due to the linear nature of CNNs and the high-dimensionality of images, neural networks are vulnerable to attacks with adversarial examples. These examples are imperceptibly different from correctly classified images but are misclassified with high confidence by CNNs. In this paper, we describe a counter-forensic method capable of subtly altering images to change their estimated camera model when they are analyzed by any CNNbased camera model detector. Our method can use both the Fast Gradient Sign Method (FGSM) or the Jacobian-based Saliency Map Attack (JSMA) to craft these adversarial images and does not require direct access to the CNN. Our results show that even advanced deep learning architectures trained to analyze images and obtain camera model information are still vulnerable to our proposed method.
10.1109/cvprw.2017.230
[ "https://arxiv.org/pdf/1805.02131v1.pdf" ]
19,159,569
1805.02131
151114b926ca8b83d93d270e3d1015a2f7f64a7a
A Counter-Forensic Method for CNN-Based Camera Model Identification David Güera Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Yu Wang Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Luca Bondi Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Politecnico Di Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana ItalyMilano Milan Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Paolo Bestagini Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Politecnico Di Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana ItalyMilano Milan Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Stefano Tubaro Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Politecnico Di Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana ItalyMilano Milan Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana Edward J Delp Purdue University West Lafayette Purdue University West Lafayette Purdue University West Lafayette Indiana, Indiana, Indiana A Counter-Forensic Method for CNN-Based Camera Model Identification An increasing number of digital images are being shared and accessed through websites, media, and social applications. Many of these images have been modified and are not authentic. Recent advances in the use of deep convolutional neural networks (CNNs) have facilitated the task of analyzing the veracity and authenticity of largely distributed image datasets. We examine in this paper the problem of identifying the camera model or type that was used to take an image and that can be spoofed. Due to the linear nature of CNNs and the high-dimensionality of images, neural networks are vulnerable to attacks with adversarial examples. These examples are imperceptibly different from correctly classified images but are misclassified with high confidence by CNNs. In this paper, we describe a counter-forensic method capable of subtly altering images to change their estimated camera model when they are analyzed by any CNNbased camera model detector. Our method can use both the Fast Gradient Sign Method (FGSM) or the Jacobian-based Saliency Map Attack (JSMA) to craft these adversarial images and does not require direct access to the CNN. Our results show that even advanced deep learning architectures trained to analyze images and obtain camera model information are still vulnerable to our proposed method. Introduction The recent increase in the number of digital images that are being uploaded and shared online has given rise to unique privacy and forensic challenges [1]. Among those challenges, verifying the integrity and authenticity of these widely circulated pictures is one of the most critical and complex tasks [2,3]. In the last few years, the digital media forensic community has explored several techniques to evaluate the truthfulness of digital images and media [4,5]. Due to its mul-tiple applicable scenarios, research efforts have focused on camera model identification [6,7,8,9,10]. Determining the camera model used to take a picture can be very important in criminal investigations such as copyright infringement cases or where it is required to identify the authors of pedo-pornographic material. Camera model identification can also be considered an important preliminary step to reduce the set of camera instances when we try to detect a unique camera instance rather than just the make and model [8]. In addition, being able to identify the camera model by inspecting small image regions is a viable method to uncover manipulation operations that could have been done to the image (e.g. splicing) [11]. Current camera model identification detectors make use of the fact that each camera model completes a distinctive set of tasks on each image when the device acquires the image. Examples of these tasks include the use of different JPEG compression schemes, application of proprietary methods for CFA demosaicing, and "defects" in the optical image path. Due to these characteristic operations, a singular "footprint" is embedded in each picture. This information can be utilized to identify the camera model, and perhaps the exact camera, that has been used to capture an image or record a video sequence. Due to the inherent and growing complexity of the image acquisition pipeline of modern image capturing devices, it is a difficult challenge to adequately model the set of operations that a camera has to execute to capture an image. Successful attempts that use hand-crafted features to model the traces left by some of these operations can be found in [7,12,13,14,15,10,16,17]. The use of deep learning techniques for image and video classification tasks [18,19,20] has shown that it is also possible to learn characteristic features that model a problem space directly from the data itself. This offers a viable path to leverage the growing amount of available image data. These modern approaches are data-driven in that they learn directly from the data rather than imposing a predetermined analytical model. The data-driven model has recently proved valuable for forensics applications [21,22,23,24]. Initial exploratory solutions targeting camera model identification [25,26,27] show that it is possible to use CNNs to learn discriminant features directly from the observed known images, rather than having to use hand-crafted features. The use of CNNs also makes it possible to capture characteristic traces left by non-linear and hard to model operations present in the acquisition pipeline. With the introduction of CNNs as detectors for camera model identification, a new vector for counter-forensic attacks is presented for a malevolent skilled individual. The idea of counter-forensics was first introduced in [28], where the authors presented the concept of fighting against image forensics with a practical application, namely a method for resampling an image without introducing pixel correlations. An up-to-date survey of the last counter-forensics advances can be found in [29]. Before exploring the vulnerabilities of CNN-based camera model detectors, it is important to note that detectors that rely on hand-crafted features are not immune to similar counter-forensics attacks. As explained in [30], digital camera fingerprints are vulnerable to forging. In particular, if an attacker obtains access to images from a given camera, they can estimate its fingerprint and "paste" it into an arbitrary image to make it look as if the image came from the camera with the stolen fingerprint. An early attempt to investigate such counter-forensic methods appeared in [31]. As presented in [32], several machine learning models, including state-of-the-art convolutional neural networks, are vulnerable to adversarial attacks. This means that these machine learning models misclassify images that are only slightly different from correctly classified images. In many cases, an ample collection of models with different architectures trained on different subsets of the training data misclassify the same adversarial example [33]. Although there are techniques such as adversarial training [32] or defensive distillation [34] that can slightly reduce the incidence of adversarial examples in CNN-based detectors, defending against adversarial examples is still an on-going challenge in the deep learning community. Adversarial attacks are hard to defend against because they require machine learning models that produce correct outputs for every possible input. The imposition of linear behavior when presented with inputs similar to the training data, though desirable, is precisely the main weakness of CNNs [33]. Due to the massive amount of possible inputs that a CNN can be presented with, it is remarkably simple to find adversarial examples that look unmodified to us but are misclassified by the network. Designing a truly adaptive de-fense against adversarial images remains an open problem. In this paper, we propose a counter-forensic method to subtly change an image to induce an error in its estimated camera model when analyzed by a CNN-based camera model detector. We leverage the recent developments to rapidly generate adversarial images. We test our counterforensic method, using two well established adversarial image crafting techniques [33,35], against an advanced deep learning architecture [36] carefully trained on a reference camera model dataset. Our results show that even modern and properly trained CNNs are susceptible to simple adversarial attacks. Note that our method only requires access to the predictions of the CNN-based camera model identification detector and does not need access to the weights of the CNN. CNN-Based Camera Model Identification In this section, we provide a brief overview of convolutional neural networks sufficient to understand the rest of this paper and show how they can be used as camera model detectors. For a more detailed description, please refer to one of the several available tutorials in the literature [37,38]. Convolutional neural networks are a special type of neural networks, biologically inspired by the human visual cortex system, that consist of a very high number of interconnected nodes, or neurons. The architecture of a CNN is designed to take advantage of the 2D structure of an input image. This is achieved with local connections and tied weights followed by some forms of pooling which results in translation invariant features. The nodes of the network are organized in multiple stacked layers, each performing a simple operation on the input. The set of operations in a CNN typically comprises convolution, intensity normalization, non-linear activation and thresholding, and local pooling. By minimizing a cost function at the output of the last layer, the weights of the network are tuned so that they are able to capture patterns in the input data and extract distinctive features. In a CNN, the features are learned using backpropagation [39] coupled with an optimization method such as gradient descent [40] and the use of large annotated training datasets. The shallower layers of the networks usually learn low-level visual features such as edges, simple shapes and color contrast, whereas deeper layers combine these features to identify complex visual patterns. Finally, fullyconnected layers coupled with a softmax layer are commonly used to generate an output class label that minimizes the cost function. For example, in the context of image classification, the last layer is composed of N nodes, where N is the number of classes, that define a probability distribution over the N visual category. The value of a given node p i , i = 1, . . . , N belonging to the last layer represents the probability of the input image to belong to the visual class c i . To train a CNN model for a specific image classification task we need to define the hyperparameters of the CNN, which range from the sequence of operations to be performed, to the number of layers or the number and shape of the filters in convolutional layers. We must also define a proper cost function to be minimized during the training process. Finally, a dataset of training and test images, annotated with labels according to the specific task (e.g. camera models in our work) needs to be prepared. Figure 1 shows an example of a CNN-based pipeline for camera model identification similar to the one presented in [26]. To train the CNN architecture, we use a given set of training and validation labeled image patches coming from N known camera models. For each color image I, associated to a specific camera model L, K non-overlapping patches P k , k ∈ [1, K], of size 32 × 32 pixels are randomly extracted. Each patch P k inherits the same label L of the source image. As trained CNN model M, we select the one that provides the smallest loss on validation patches. When a new image I is under analysis, the camera model used to acquire it is estimated as follows. A set of K patches is obtained from image I as described above. Each patch P k is processed by CNN model M in order to assign a label L k to each patch. The predicted modelL for image I is obtained through majority voting onL k , k ∈ [1, K]. Figure 2 shows the block diagram of our proposed counter-forensic method. Our method consists of an adversarial image generator module that can be added to a CNNbased camera model evaluation pipeline. In Figure 2, we assume a similar structure to the previously presented pipeline in Section 2. Our adversarial image generator module takes as input the set of K patches that have been extracted from the image I that is being analyzed. When presented with new image patches, our module can work in two different modes. Proposed Method In the first operation mode, the adversarial image generator module does an untargeted image manipulation, that is, it does not try to perturb the image patches to produce a specific misclassification class. Instead, we use the derivative of the loss function of the CNN with respect to the input image patches to add a perturbation to the images. The derivative is computed using backpropagation with the labelsL k , k ∈ [1, K] that are given by the CNN detector when it first processes the unmodified image patches. This procedure is known as the fast gradient sign method (FGSM) [33]. In the second operation mode, the adversarial image generator module does a targeted image manipulation. In this case, we try to perturb the image patches to produce a specific misclassification class L , different from the true real label L that is associated with the analyzed image I and its associated P k patches. In this mode of operation, we exploit the forward derivative of a CNN to find an adversarial perturbation that will force the network to misclassify the image patch into the target class by computing the adversarial saliency map. Starting with an unmodified image patch, we perturb each feature by a constant offset . This process is repeated iteratively until the target misclassification is achieved. This procedure is known as the Jacobian-based saliency map attack (JSMA) [35]. We present a detailed overview of both FGSM and JSMA techniques as follows. Fast Gradient Sign Method In [33], the fast gradient sign method was introduced for generating adversarial examples using the derivative of the loss function of the CNN with respect to the input feature vector. Given an input feature vector (e.g. an image), FGSM perturbs each feature in the direction of the gradient by magnitude , where is a parameter that determines the perturbation size. For a network with loss J(Θ, x, y), where Θ represents the CNN predictions for an input x and y is the correct label of x, the adversarial example is generated as x * = x + sign(∇ x J(Θ, x, y)) With small , it is possible to generate adversarial images that are consistently misclassified by CNNs trained using the MNIST and CIFAR-10 image classification datasets with a high success rate [33]. Jacobian-Based Saliency Map Attack In [35], an iterative method for targeted misclassification was proposed. By exploiting the forward derivative of a CNN, it is possible to find an adversarial perturbation that will force the network to misclassify into a specific target class. For an input x and a convolutional neural network C, the output for class j is denoted C j (x). To achieve an output of target class t, C t (x) must be increased while the probabilities C j (x) of all other classes j = t decrease, until t = arg max j C j (x). This is accomplished by exploiting the adversarial saliency map, which is defined as S(x, t)[i] = 0, if ∂Ct(x) ∂xi < 0 or j =t ∂Cj (x) ∂xi > 0 ( ∂Ct(x) ∂xi )| j =t ∂Cj (x) ∂xi |, otherwise for an input feature i. Because we work with images in this paper, in our case each input feature i corresponds to a pixel i in the image input x. Starting with a normal sample x, we locate the pair of pixels {i, j} that maximize S(x, t)[i] + S(x, t)[j] , and perturb each pixel by a constant offset . This process is repeated iteratively until the target misclassification is achieved. This method can effectively produce MNIST dataset examples that are correctly classified by human subjects but misclassified into a specific target class by a CNN with a high confidence. Implementation Details To implement our counter-forensic method, we have used the software library cleverhans [41]. The library provides standardized reference implementations of adversarial image generation techniques and adversarial training. The library can be used to develop more robust CNN architectures and to provide standardized benchmarks of CNNs performance in an adversarial setting. As noted in [41], benchmarks constructed without a standardized implementation of adversarial image generation techniques are not comparable to each other, because a good result may indicate a robust CNN or it may merely indicate a weak implementation of the adversarial image generation procedure. Experimental Results In this section, we evaluate our proposed method and compare the results of the two techniques for generating the adversarial images. First, we create a reference dataset specially designed to exploit the traces left by the operations of the acquisition pipeline of different image capturing devices. Then, we train an advanced deep learning architecture to have a baseline to compare the accuracy results in the presence of adversarial images. Finally, we generate several adversarial image examples to demonstrate the performance of our proposed method. Experimental Setup As part of DARPA's MediFor Program, PAR Government Systems collected an initial dataset of 1611 images acquired by 10 different camera models, ranging from DSLRs to phone cameras, with a mixture of indoor and outdoor flatfield scenes. We focus on a flat-field image dataset because flat-field images are more difficult to modify without inserting visual distortions due to the absence of texture content. Throughout the rest of the paper, we refer to this dataset as PRNU-PAR. Using the PRNU-PAR dataset, we create a patch dataset, composed by image patches of 32 × 32 pixels randomly extracted from the original images. Specifically, 500 patches are uniformly sampled from each original image in the PRNU-PAR dataset, which results in a patch dataset that contains 805,500 patches in total. The training, validation and test sets are created following a 70/20/10 split, while we ensure that the patches in each dataset split only contain patches from different images. Table 1 shows the statistics of the patch dataset. As can be seen, due to the difference in the number of images per camera model class in the PRNU-PAR dataset, our dataset of image patches has an unequal number of patches for each of the camera models. Figure 3 shows a representative example of the images that are present in the PRNU-PAR dataset next to one of their randomly extracted patches. In this case, both camera models PAR-A075 and PAR-A106 have been used to capture images of a cloudy sky. Other camera models such AS-One or ES-D5100 have taken images of a white screen. All the image scenes that are captured in the PRNU-PAR dataset are mostly flat and bright. As it has been shown in the literature [7], these largely uniform images are ideal candidates to be used for the ex-traction of the "fingerprint" (e.g. the characteristic PRNU noise of the camera model) left in the image by the camera. CNN Architecture In order to do a fair evaluation of our counter-forensic method, we use a CNN-based camera model detector that has been trained to achieve state-of-the-art accuracy results in the patch dataset. CNN architecture designs have tended to explore deeper models. Networks which can be hundreds of layers deep are now commonplace in the literature. This design trend has been motivated by the fact that for many applications such as image classification tasks, an increase in the depth of the CNN architecture translates into higher accuracy performance if sufficient amounts of training data are available. A first approach to design a CNN architecture may be to simply stack convolutional or fully-connected layers together. This naive strategy works initially, but gains in accuracy performance quickly diminish the deeper this kind of architecture becomes. This phenomenon is due to the way in which conventional CNNs are trained through backpropogation. During the training phase of a CNN, gradient information must be propagated backwards through the network. This gradient information slightly diminishes as it passes through each layer of the neural network. For a CNN with a reduced number of layers, this is not a problem. For an architecture with a large number of layers, the gradient signal essentially becomes noise by the time it reaches the first layer of the network again. The problem is to design a CNN in which the gradient information can be easily distributed to all the layers without degradation. ResNets and DenseNets are modern CNN architectures that try to address this problem. A Residual Network [42], or ResNet is a deep CNN which tackles the problem of the vanishing gradient using a straightforward approach. It adds a direct connection at each layer of the CNN. In previous CNN models, the gradient always has to go through the activations of the layers, which modify the gradient information due to the nonlinear activation functions that are commonly used. With this direct connection, the gradient could theoretically skip over all the intermediate layers and be propagated through the network without being disturbed. A Dense Network [36], or DenseNet generalizes the idea of a direct connection between layers. Instead of only adding a connection from the previous layer to the next, it connects every layer to every other layer. For each layer, the feature maps of all preceding layers are treated as separate inputs whereas its own feature maps are passed on as inputs to all subsequent layers. The increased number of connections ensures that there is always a direct route for the information backwards through the network. The connectivity pattern of DenseNets yields state-of-the-art accuracies on the CIFAR10 image classification dataset, which is composed by images of 32 × 32 pixels in size. Motivated by the accuracy performance of DenseNet in the CIFAR10 dataset and the fact that we also work with image patches of 32 × 32 pixels, we select a DenseNet model with 40 layers as our CNN camera model detector. To prevent the network from growing too wide and to improve the parameter efficiency, we limit the growth rate of the network, this is, the maximum number of input feature-maps that each layer can produce, to k = 12. To train the CNN, we use the Adam optimizer with a learning rate of 0.0001 and a batch size of 512 images. After 5 training epochs, we reach a plateau in the accuracy in our validation set. Table 2 shows the single patch accuracy results for our training, validation and test splits of the patch dataset. Dataset Split Train Validation Test Adversarial Image Generation In order to evaluate the performance of our counterforensic method, we test the DenseNet model trained on the patch dataset using untargeted attacks with FGSM and targeted attacks with JSMA. To properly evaluate our method, we only perturb images from the test split which were correctly classified by our CNN in their original states. To be clear, what we refer as the average confidence score in this paper is the average value of the probability that is associated with the candidate camera model label for each of the image patches in the test split. The probability for each candidate camera model label corresponds with the highest probability value assigned by the softmax layer of our trained DenseNet model. For untargeted attacks with FGSM, we report in Table 3 the error rate and the average confidence score on the test split of the patch dataset for different values of which have been shown to generate high misclassified adversarial images while not producing appreciable visual changes. We find that using = 0.005 offers the best compromise between error rate and visual changes in the image, causing the trained DenseNet model detector to have a error rate of 93.1% with an average confidence of 95.3% on the patch test split. It should be noted that as we increase the value of , the manipulations become more visually apparent. Figure 4 shows an example of the adversarial images that our proposed method can generate when we use FGSM. The modifications done to the images by FGSM are performed on 32-bit floating point values, which are used for the input of the DenseNet model. The gradient computed for Figure 4 uses 8-bit signed integers. To publish the sign of the gradient image in the paper, we have done a custom conversion from 8-bit signed integers to 8-bit unsigned integers. To increase the range of each color channel, we represent the −1s values as 0 and the 1s as 255. For the possible 0's, we have treated them as positive values (they are represented by 255). For targeted attacks with JSMA, we report in Table 4 the error rate and the average confidence score for each possible camera model target class. Figure 5 shows an example of the images that JSMA allows us to generate when we perform a targeted attack. In this case, an image patch captured by camera ES-D5100 that is correctly classified when is analyzed by our trained DenseNet model is manipulated to be misclassified as an image patch that had been generated by camera model PAR-1233. It is important to appreciate that although JSMA allows us to generate image patches that get misclassified into a specific camera model with high error rates and confidence scores, the modifications that it applies to the images can usually be spotted through visual inspection. This effect is due to the fact that JSMA crafts the adversarial images by flipping pixels to their minimum or maximum values. Because our patch dataset is composed of image patches with mostly flat scene content, the effect can be clearly observed, for example, in the upper corners of the manipulated image patch in Figure 5. Conclusions This paper described a counter-forensic method to subtly alter images to change their estimated camera model when they are analyzed by a CNN-based camera model detector. We tested our method on a reference dataset with images from multiple cameras that show highly similar indoor and outdoor scenes. The results demonstrate that we can generate imperceptibly altered adversarial images that are misclassified with high confidence by the CNN. In the future, we will extend our method to apply it to video sequences and we will explore viable adversarial example detection methods and defense techniques to increase the robustness of CNN-based camera model detectors. Acknowledgments This material is based on research sponsored by the Defense Advanced Research Projects Agency (DARPA) and the Air Force Research Laboratory (AFRL) under agreement number FA8750-16-2-0173. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA, AFRL or the U.S. Government. Figure 1 . 1Example of a pipeline for camera model identification. The patches extracted from each training image I (bottom) inherit the same label L of the image. These patches are used in the CNN training process. For each patch P k from the image I under analysis (top), a candidate labelL k is produced by a trained CNN model M. The predicted labelL for analyzed image I is obtained by majority voting. Figure 2 . 2Block diagram of our proposed method. Figure 3 . 3Example of images from the training set of the patch dataset. (Top) Image from camera model PAR-A075 and one of the randomly selected patches associated with it. (Bottom) Image from camera model PAR-A106 and one of the randomly selected patches associated with it. Figure 4 . 4An example of untargeted fast adversarial image generation using FGSM applied to our trained DenseNet model on the patch dataset. By adding an imperceptibly small vector whose elements are equal to the sign of the elements of the gradient of the cost function with respect to the input, we can change DenseNet's classification of the image patch. Figure 5 . 5An example of targeted adversarial image generation using JSMA applied to our trained DenseNet model on the patch dataset. (Left) Original image patch correctly classified as ES-D5100. (Right) Altered image patch with target camera model PAR-1233 Table 3. Error rate and confidence score values of our trained DenseNet model after an untargeted attack with FGSM to the test split with different values of .value Error rate (%) Confidence Score (%) 0.001 91.4 97.7 0.002 91.7 97.2 0.003 92.2 96.7 0.004 92.7 95.8 0.005 93.1 95.3 0.006 94.1 95.1 0.007 94.5 94.2 0.008 95.3 93.6 0.009 95.9 93.0 0.01 96.2 92.3 Target Camera Model Error rate (%) Confidence Score (%) AS-One 99.5 87.7 ES-D5100 99.3 88.6 MK-Powershot 99.3 88.4 MK-s860 99.7 88.5 PAR-1233 99.7 87.9 PAR-1476 99.4 88.1 PAR-1477 99.5 88.2 PAR-A015 99.6 88.4 PAR-A075 99.3 87.8 PAR-A106 99.2 87.9 Table 4. Error rates and confidence scores of our trained DenseNet model for each possible target camera model after applying a targeted attack with JSMA to the test split. Privacy concerns for photo sharing in online social networks. K Liang, J K Liu, R Lu, D S Wong, IEEE Internet Computing. 192K. Liang, J. K. Liu, R. Lu, and D. S. 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[]
[ "The structure of pentaquarks P + c in the chiral quark model", "The structure of pentaquarks P + c in the chiral quark model" ]
[ "Gang Yang \nDepartment of Physics\nJiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems\nNanjing Normal University\n210023NanjingP. R. China\n", "Jialun Ping \nDepartment of Physics\nJiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems\nNanjing Normal University\n210023NanjingP. R. China\n" ]
[ "Department of Physics\nJiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems\nNanjing Normal University\n210023NanjingP. R. China", "Department of Physics\nJiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems\nNanjing Normal University\n210023NanjingP. R. China" ]
[]
The recent experimental results of LHCb collaboration suggested the existence of pentaquark states with charmonium. To understand the structure of the states, a dynamical calculation of 5-quark systems with quantum numbers IJ P = 1 2 ( 1 2 ) ± , 1 2 ( 3 2 ) ± and 1 2 ( 5 2 ) ± is performed in the framework of chiral quark model with the help of gaussian expansion method. The results show that the negative parity states can be bound states while all of the positive parity states are the scattering states. The Pc(4380) state is suggested to be the bound state of Σ * c D. Although the energy of ΣcD * is very close to the mass of Pc(4450), the inconsistent parity prevents the assignment. The calculated distances between quarks confirm the molecular nature of the states. Other five-quark bound states of the combination of ΣcD and Σ * c D * are also found in the region about 4.3GeV and 4.5GeV.
10.1103/physrevd.95.014010
[ "https://arxiv.org/pdf/1511.09053v2.pdf" ]
119,229,495
1511.09053
55bad1d6d72c122bd721e32523de52c5c4ad7843
The structure of pentaquarks P + c in the chiral quark model 6 Dec 2015 Gang Yang Department of Physics Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems Nanjing Normal University 210023NanjingP. R. China Jialun Ping Department of Physics Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems Nanjing Normal University 210023NanjingP. R. China The structure of pentaquarks P + c in the chiral quark model 6 Dec 2015numbers: 1375Cs1239Pn1239Jh The recent experimental results of LHCb collaboration suggested the existence of pentaquark states with charmonium. To understand the structure of the states, a dynamical calculation of 5-quark systems with quantum numbers IJ P = 1 2 ( 1 2 ) ± , 1 2 ( 3 2 ) ± and 1 2 ( 5 2 ) ± is performed in the framework of chiral quark model with the help of gaussian expansion method. The results show that the negative parity states can be bound states while all of the positive parity states are the scattering states. The Pc(4380) state is suggested to be the bound state of Σ * c D. Although the energy of ΣcD * is very close to the mass of Pc(4450), the inconsistent parity prevents the assignment. The calculated distances between quarks confirm the molecular nature of the states. Other five-quark bound states of the combination of ΣcD and Σ * c D * are also found in the region about 4.3GeV and 4.5GeV. The recent experimental results of LHCb collaboration suggested the existence of pentaquark states with charmonium. To understand the structure of the states, a dynamical calculation of 5-quark systems with quantum numbers IJ P = 1 2 ( 1 2 ) ± , 1 2 ( 3 2 ) ± and 1 2 ( 5 2 ) ± is performed in the framework of chiral quark model with the help of gaussian expansion method. The results show that the negative parity states can be bound states while all of the positive parity states are the scattering states. The Pc(4380) state is suggested to be the bound state of Σ * c D. Although the energy of ΣcD * is very close to the mass of Pc(4450), the inconsistent parity prevents the assignment. The calculated distances between quarks confirm the molecular nature of the states. Other five-quark bound states of the combination of ΣcD and Σ * c D * are also found in the region about 4.3GeV and 4.5GeV. I. INTRODUCTION Since the report of Θ + (1540) by several groups [1][2][3] about ten years ago, it brought lots of arguments during that time. Although the observation of pentaquark state Θ + (1540) was not confirmed by the further experiments [4] (LEPS Collaboration still insisted on the existence of pentaquark Θ + (1540) [5]), the study of pentaquark inspires new structures for hadrons, beyond the conventional quark configuration (qqq or qq). Five-quark components in baryons was also studied which showed that the qqqqq in ground state is more favorable than qqq with L = 1 for 1/2 − baryons [6]. Recently, the interesting in pentaquark is revived, because LHCb experiment reported the observation of two pentaquark states, denoted as P + c (4380) and P + c (4450), in the decay of Λ 0 b , Λ 0 b → J/ψK − p [7]. The masses and widths of these two structures, appeared in the J/ψp invariant mass, are determined to be 4380±8 ± 29 MeV, 205±18 ± 86 MeV, and 4449.8±1.7 ± 2.5 MeV, 39±5 ± 19 MeV. The pentaquark nature of the structures comes from the valence structure, uudcc, of J/ψp. The possible quantum numbers J P of these two states are (3/2 − , 5/2 + ) or (5/2 − , 3/2 + ). In fact, the hiddencharm pentaquark states have been predicted several years ago. In 2010, J. J. Wu et al. predicted several narrow resonances with hidden charm above 4 GeV, N * cc (4265), N * cc (4415), and Λ * cc (4210), in the framework of the coupled-channel unitary approach [8]. Z. C. Yang et al. also studied the possible existence of very loosely bound hidden-charm molecular baryons in the one-boson-exchange model, Σ cD * and Σ cD states are proposed [9]. After the report of LHCb, a lot of theoretical are devoted to explain the nature of the two states. By using the boson exchange model, R. Chen et al. interpreted the two states as the molecular states, Σ c (2455)D * and Σ * c (2520)D * with spin-parity J P = 3/2 − and J P = 5/2 − , respectively [10]. The Bethe-Salpeter equation method was employed to study thē DΣ * andD * Σ c interactions, and the two states P + c (4380) and P + c (4450) are identified as Σ * cD and Σ c (2455)D * molecular states with quantum numbers J P = 3/2 − and J P = 5/2 + , respectively [11]. In QCD sum rule approach, P + c (4380) and P + c (4450) were explained as hidden-charm pentaquark states with quantum numbers J P = 3/2 − and J P = 5/2 + , respectively, by using diquark-diquark-antiquark type interpolating currents [12,13]. By analyzing the reaction Λ 0 b → J/ψK − p with coupled-channel calculation, L. Roca et al. assigned the quantum numbers J P = 3/2 − to the state P + c (4450) and concluded that P c (4450) + state is a molecular state of mostly Σ cD * and Σ * cD * with 3/2 − [14]. In the soliton approach, the hidden-charm state with quantum numbers IJ P = 1 2 3 2 − was shown to exist and is compatible with P + c (4380), but the state with IJ P = 1 2 5 2 + has much higher mass compared with that of P + c (4450) [15]. The small mass splitting between P + c (4380) and P + c (4450) can be understood in the diquark-triquark model by using an effective diquark-triquark Hamiltonian based on spin-orbital interaction [16]. Non-resonance explanations of the structures observed experimentally were also proposed [17,18]. Based on the theory of QCD, it is possible to excite quark-antiquark pairs from vacuum to form hadronic state. For the light quark-antiquark pair excitation, the effect can be absorbed into the parameters in the quark model description. For the heavy quark-antiquark pair excitation, it is too difficult to occur in the light hadron system and its effect cannot be absorbed by the model parameters. So the states P + c reported by LHCb should be genuine pentaquarks. Its study will provide us more information of the underlying fundamental theory of strong interaction, QCD. The most common approach to multiquark system is quark model. After fifty years development and with the accumulation of experimental data on multiquark states, to tackle the problem of multiquark seriously in the framework of quark model is expected. In the present work, the chiral quark model is used to study the pentaquark states with hidden-charm. Different from other approaches, no prior spacial structure of the state is assumed, the structure is determined by the system dynamics. For this purpose, a powerful method of fewbody system, gaussian expansion mthod (GEM) [19] is employed to do the calculation. The GEM has been successfully applied to many few-body systems, light nuclei, hypernuclei, hadron physics and so on [19]. It suits for both of compact multi-quark systems and loosely bound molecular states. The structure of the paper is as follows. In section II the quark model, wavefunctions and calculation method is presented. Section III is devoted to the calculated re-sults and discussions. A brief summary is given in the last section. II. MODEL AND WAVE FUNCTION The chiral quark model has acquired great achievement both in describing the hadron spectra and hadronhadron interaction. Here we apply it to 5-quark system. The details of the model can be found in Ref. [20]. The Hamiltonian for multiquark system takes the form H = n i=1 m i + p 2 i 2m i − T CM + n j>i=1 [V CON (r ij ) + V OGE (r ij ) + V χ (r ij ) + V σ (r ij )] ,(1)V CON (r ij ) = λ c i · λ c j −a c (1 − e −µcrij ) + ∆ , V OGE (r ij ) = 1 4 α s λ c i · λ c j 1 r ij − 1 6m i m j σ i · σ j e −rij /r0(µ) r ij r 2 0 (µ) , r 0 (µ) =r 0 /µ, α s = α 0 ln( µ 2 +µ 2 0 Λ 2 0 ) . V σ (r ij ) = − g 2 ch 4π Λ 2 σ Λ 2 σ − m 2 σ m σ Y (m σ r ij ) − Λ σ m σ Y (Λ σ r ij ) ,(2)V χ (r ij ) = v π (r ij ) 3 a=1 (λ a i · λ a j ) + v K (r ij ) 7 a=4 (λ a i · λ a j ) + v η (r ij )[cos θ P (λ 8 i · λ 8 j ) − sinθ P ], v χ (r ij ) = g 2 ch 4π m 2 χ 12m i m j Λ 2 χ Λ 2 χ − m 2 χ m χ Y (m χ r ij ) − Λ 3 χ m 3 χ Y (Λ χ r ij ) (σ i · σ j ). χ = π, K, η(3) Here T CM is the center of mass kinetic energy, µ is the reduced mass of two interacting quark pair. Because we are interested in the ground state of multiquark system, only the central part of the interaction is given above. All the symbols take their usual meanings. The model parameters of the model are taken from Ref. [20] and are listed in Table I. The wavefunctions for the system are constructed in the following way. First the 5-quark system is separated into two clusters, one with 3 quarks and another with quark-antiquark. The wavefunctions for these subclusters can be easily written down. Then two clusters are coupled and anti-symmetrized (if necessary) to form the total wavefunction of 5-quark system. Clearly, there are other ways to construct the wavefunctions of the system. However, it makes no difference by choosing any one configuration that if we use enough bases for system during the calculation. For the 5-quark system with quark content uudcc, there are types of separation, one is (udc)(cu)+(uuc)(cd) and the other is (uud)(cc). Due to the large mass difference between c quark and u, d and s quarks, the flavor SU(4) symmetry is strongly broken. Here we just construct the flavor wavefunctions of system based on the flavor SU(2) symmetry. The flavor wavefunctions for the sub-clusters constructed are shown below. |B 11 = uuc, |B 10 = 1 √ 2 (ud + du)c, |B 1−1 = ddc, |B 00 = 1 √ 2 (ud − du)c, |B 1 1 2 , 1 2 = 1 √ 6 (2uud − udu − duu),(4)|B 2 1 2 , 1 2 = 1 √ 2 (ud − du)u, |M 1 2 , 1 2 =cu, |M 1 2 ,− 1 2 =cd, |M 00 =cc. The flavor wavefunctions for 5-quark system with isospin I = 1/2 are obtained by the following couplings, |χ f 1 = 2 3 |B 11 |M 1 2 ,− 1 2 − 1 3 |B 10 |M 1 2 , 1 2 , |χ f 2 = |B 00 |M 1 2 , 1 2 ,(5)|χ f 3 = |B 1 1 2 , 1 2 |M 00 , |χ f 4 = |B 2 1 2 , 1 2 |M 00 ,g 2 ch /(4π) 0.54 θP ( • ) -15 ac (MeV) 430 Confinement µc (fm −1 ) 0.70 ∆ (MeV) 181.10 αs 0.777 α0 2.118 Λ0 (fm −1 ) 0.113 OGE µ0 (MeV) 36.976 r0 (MeV fm) 28.170 rg (MeV fm) 34.500 In a similar way, the spin wavefunctions for 5-quark system can be constructed, |χ σ1 1 2 , 1 2 (5) = 1 6 |χ σ 3 2 ,− 1 2 (3) |χ σ 11 − 1 3 |χ σ 3 2 , 1 2 (3) |χ σ 10 + 1 2 |χ σ 3 2 , 3 2 (3) |χ σ 1−1 |χ σ2 1 2 , 1 2 (5) = 1 3 |χ σ1= 1 √ 6 (2ααβ − αβα − βαα), |χ σ2 1 2 , 1 2 (3) = 1 √ 2 (αβα − βαα), |χ σ1 1 2 ,− 1 2 (3) = 1 √ 6 (αββ − αββ − 2ββα), |χ σ2 1 2 ,− 1 2 (3) = 1 √ 2 (αββ − βαβ), |χ σ 11 = αα, |χ σ 10 = 1 √ 2 (αβ + βα), |χ σ 1−1 = ββ, |χ σ 00 = 1 √ 2 (αβ − βα).(3) For the color wavefunction, only the color singlet channel, two clusters are all colorless, is used here. The reason for this simplification comes from that the color singlet channel states are complete when all the excitation of other degrees of freedom are included for the multiquark systems, the energies for the excited states are rather high and have small effect on the group states. Then the color wavefunction of the system is χ c = 1 √ 6 (rgb−rbg+gbr−grb+brg−bgr) 1 √ 3 (rr+ḡg+bb). (7) As for the orbital wavefunctions, we do not separate the motions of particles in the system into internal and relative ones and freeze the internal motion, the structure of the system is assumed priorly, as the most work did. In the present work, the orbital wavefunctions for each motion of the system are determined by the dynamics of the system, so does the structure. Another reason for not including the hidden-color channels in the calculation is that the direct extension of interactions between quark pairs from colorless states to colorful states are questionable, it will lead to too much bound states [21]. The orbital wavefunctions for this purpose is obtained as follows, ψ LML = [[φ n1l1 (ρ)φ n2l2 (λ)] l φ n3l3 (r)] l ′ φ n4l4 (R) LML(8) where the Jacobi coordinates are defined as, ρ = x 1 − x 2 , λ = x 3 − ( m 1 x 1 + m 2 x 2 m 1 + m 2 ),(9)r = x 4 − x 5 , R = ( m 1 x 1 + m 2 x 2 + m 3 x 3 m 1 + m 2 + m 3 ) − ( m 4 x 4 + m 5 x 5 m 4 + m 5 ). To find the orbital wavefunctions, the Gaussian expansion method (GEM) is employed, i.e., each φ is expanded by gaussians with various sizes [19] φ nlm (r) = nmax n=1 c n N nl r l e −(r/rn) 2 Y lm (r), where N nl is normalization constants N nl = 2 l+2 (2ν n ) l+ 3 2 √ π(2l + 1) 1 2 .(11) The size parameters r n are taken as the geometric progression numbers ν n = 1/r 2 n r n = r min a n−1 . c n is the variational parameters, which is determined by the dynamics of the system. Finally, the complete channel wave function for the 5-quark system is written as Ψ JM,i,j,n = A [χ σi S (5)ψ L ] JMJ χ f j χ c (13) where the A is the antisymmetry operator of the system, it can be written as A = 1 − (15) − (25)(14) for (udc)(cu) case and A = 1 − (13) − (23)(15) for (uud)(cc) case. The eigen-energy of the system is obtained by solving the following eigen-equation HΨ JM = EΨ JM ,(16) by using variational principle. The eigen functions Ψ JM are the linear combination of the above channel wavefunctions. When the angular momenta are not all zero, the calculation of the matrix elements of Hamiltonian is rather complicated. Here a new useful method named the infinitesimally-shifted Gaussian (ISG) are used [19]. In this method, the orbital wavefunctions are written as φ nlm (r) = N nl lim ε→0 1 (νε) l kmax k=1 C lm,k e −νn(r−εD lm,k ) 2 (17) the coefficients C lm,k and shift-direction vector D lm,k are dimensionless numbers. By absorbing the spherical harmonic function into the shifted gaussians, the calculation becomes easy with no tedious angular-momentum algebra required. III. RESULTS AND DISCUSSIONS In the present calculation, we are interested in the lowlying states of uudcc pentaquark system, so the total orbital angular momentum L is limited to be 0 and 1. For L = 0, all of l 1 , l 2 , l 3 , l 4 are 0 and for L = 1, only one of l 1 , l 2 , l 3 , l 4 can be 1. In this way, the total angular momentum J can take values 1/2, 3/2 and 5/2. The possible channels under the consideration are listed in Table II. The single channel and channel coupling calculations are performed in this work. The results are shown in Tables III. The tables gives the eigen-energies of the states (column 3), along with the theoretical (column 4) and experimental thresholds (column 6), the binding energies (column 5) and the corrected energies of the states (column 7), which are obtained by taking the sum of experimental thresholds and the binding energies. Table IV gives the spacial configurations of the states. In the following we analyse the results in detail. For the parity negative states, the results are shown in Table III. (a) J P = 1 2 − : For N η c , N J/ψ, Λ c D and Λ c D * states, the single-channel calculation shows that no bound state can be formed. For Σ c D, Σ c D * and Σ * c D states, bound states with −3 ∼ −4 MeV biding energies appear. The channel-coupling is rather weak, it does not push the state N η c or Λ c D down enough to form a bound state, and it also does not push the state Σ c D up above the threshold. So the present calculation shows that there is a resonance Σ c D with resonance energy 4315 MeV for J P = 1 2 − . The result is in agreement with that of Ref. [9]. Although the energy of Σ c D * is very close to that of P c (4450), it is difficult to make the assignment because of the different parities. Bound state is obtained as before. Although the binding energy is small, −3 MeV, the decay width of the state may be small, 10 ∼ 20 MeV, due to the small decay widths of its constituents, Σ * c (Γ Σ * c →Λcπ ∼ 15 MeV) and D * (Γ D * →Dπ ∼ 1 MeV). So it is a good candidate of the χ σi 1/2 χ f j i = 2, 3, j = 1 4518 4521 −3 4462(ΣcD * ) 4459 χ σi 1/2 χ f j i = 1, j = 1 4563 4566 −3 4527(Σ * c D * ) 4524 mixed 4397 3 2 − χ σi 3/2 χ f j i = 3, 4, j = 3, 4 3841 3841 0 4036(N J/ψ) 4036 χ σi 3/2 χ f j i = 3, 4, j = 2 4115 4115 0 4293(ΛcD * ) 4293 χ σi 3/2 χ f j i = 3, 4, j = 1 4518 4520 −2 4462(ΣcD * ) 4460 χ σi 3/2 χ f j i = 2, j = 1 4444 4447 −3 4385(Σ * c D) 4382 χ σi 3/2 χ f j i = 1, j = 1 4564 4566 −2 4527(Σ * c D * ) 4525 mixed 4442 5 2 − χ σi 5/2 χ f j i = 1, j = 1 4563 4566 −3 4527(Σ * c D * ) 4524 heavy pentaquark with high spin. For the parity positive states, one of the orbital angular momenta is 1, Generally it is difficult to form a bound state in this case. Our calculation shows that all the states under investigation are not bounded. So if the state P + c (4450) is identified as a pentaquark state with positive parity, the chiral quark model may be needed to be modified for multi-quark system. The non-resonance explanation of the narrow structure at 4.45 GeV was also proposed, Guo et al. showed that the structure was compatible with the kinematical effects of the rescattering from χ c1 p to J/ψp [18]. To find the structure of the resonances obtained in the present work, the distances between any two quarks are calculated. The results are shown in Table IV. From the table, we can see that the distances among quarks 1, 2 and 3 is around 0.8 fm, and the distances between quark 4 and antiquark are 0.6-0.7 fm, while the distances between quarks 1, 2, 3 and 4, 5 are 1.5-2.4 fm. Clearly, quark 1,2,3 form a cluster, Σ( * ) and quark 4,5 form another cluster, D( * ), then two clusters combine to a pentaquark state. To describe P + c (4380) as molecular state of Σ * c D is reasonable. Fig. 1 shows the correlation functions of two clusters for J P = 3 2 − . Typical behavior of the wavefunction for bound state is obtained. IV. SUMMARY In the framework of chiral quark model, the 5-quark systems with quark contents uddcc are investigated by means of Gaussian expansion method. The calculation shows that there are several resonances for IJ P = with configuration Σ * c D is very close to that of state P + c (4380)), a pentaquark announced by LHCb collaboration. The distances between quark pairs suggest a molecular structure for these resonances. A sound interpretation of P + c (4380) is the molecule of Σ * c D with IJ P = − is also close to that of P + c (4450), another pentaquark reported by LHCb collaboration. Nevertheless, the opposite parity of the state to the P + c (4380) may prevent this assignment. Meanwhile all the positive parity states are all unbound in our calculation. In the present calculation, the spacial structure of a − χ σi 5/2 χ f j i = 1, j = 1 (Σ * c D * ) 0.9 0.8 1.9 1.8 2.4 2.0 0.7 5-quark system is not assumed in advance. Although the two colorless sub-clusters are used, the internal structures of the sub-clusters are not fixed. Generally the state in this approach will have smaller energy than it in other approaches, because of the larger space. As a preliminary work, the spin-orbit and tensor interactions are not included in the calculation. For parity negative states, their effects are expected to be zero or small. For parity positive state, it will play a minor role. To understand the nature of P + c (4450) in quark model approach, the calculation with including the spin-orbit interaction is needed, which in progress in our group. PACS numbers: 13.75.Cs, 12.39.Pn, 12.39.Jh −− are obtained. N J/ψ, Λ c D * states are unbound and all Σ c D's are bound states. The channel-coupling pushes down the state Σ * c D a little. So a resonance, Σ * c D, is shown up. After the correction, the resonance energy is 4382 MeV, which is very close the mass of P + c (4380), which was claimed by LHCb collaboration[1]. However, the large decay width of P + c (4380) cannot be explained in the present calculation. The decay width of Σ * c D state to N J/ψ, Λ c D * are estimated to several MeVs due to the weak channel-coupling. Because of the missing of the spin-orbit interaction in the present calculation, the energies of N J/ψ, Σ c D * with J P = 3 2 − are the same as that of Σ c D * , N J/ψ with J P : Only one channel: Σ * c D * , remains in this case if all orbital angular momenta are set to zero. FIG. 1 : 1The correlation functions of ΣcD * , Σ * c D and Σ * c D * with IJ P − . However, the large decay width of P + c (4380)), 205±18 ± 86 MeV, is out of reach of the present picture. The mass of molecule state Σ c D * with IJ P TABLE I : IQuark model parameters. The masses of mesons take their experimental mass, i.e., mπ = 0.70 fm −1 , mσ = 3.42 fm −1 , mη = 2.77 fm −1 , mK = 2.51 fm −1 . Λσ (fm −1 ) 4.20Goldstone bosons Λη = ΛK (fm −1 ) 5.20Quark masses mu=m d (M eV ) 313 mc (M eV ) 1752 Λπ = TABLE II : IIThe channels under the consideration.J P |LML; SMS TABLE III : IIIEigen-energies of the udccu system with parity negative (unit: MeV).J P Channel Eigen Energy E th (Theo.) Binding Energy E th (Exp.) Corrected Energy 1 2 − χ σi 1/2 χ f j i = 4, 5, j = 3, 4 3745 3745 0 3919(N ηc) 3919 χ σi 1/2 χ f j i = 2, 3, j = 3, 4 3841 3841 0 4036(N J/ψ) 4036 χ σi 1/2 χ f j i = 4, 5, j = 2 3996 3996 0 4151(ΛcD) 4151 χ σi 1/2 χ f j i = 2, 3, j = 2 4115 4115 0 4293(ΛcD * ) 4293 χ σi 1/2 χ f j i = 4, 5, j = 1 4398 4402 −4 4320(ΣcD) 4316 TABLE IV : IVDistances between any two quarks (unit: fm).J P Channel r12 r13 r14 r15 r34 r35 r45 1 2 − χ σi 1/2 χ f j i = 4, 5, j = 1 (ΣcD) 0.8 0.7 1.5 1.6 1.9 1.8 0.6 χ σi 1/2 χ f j i = 2, 3, j = 1 (ΣcD * ) 0.8 0.7 1.7 1.6 2.1 1.8 0.7 χ σi 1/2 χ f j i = 1, j = 1 (Σ * c D * ) 0.9 0.8 1.6 1.6 2.0 1.7 0.7 3 2 − χ σi 3/2 χ f j i = 3, 4, j = 1 (ΣcD * ) 0.8 0.7 1.9 1.7 2.3 2.0 0.7 χ σi 3/2 χ f j i = 2, j = 1 (ΣcD * ) 0.9 0.8 1.5 1.8 2.1 2.0 0.6 χ σi 3/2 χ f j i = 1, j = 1 (Σ * c D * ) 0.9 0.8 2.1 2.1 2.7 2.3 0.7 5 2 AcknowledgmentsThe work is supported partly by the National Natural Science Foundation of China under Grant Nos. 11175088, 11535005 and 11205091. . T Nakano, LEPS CollaborationPhys. Rev. Lett. 9112002T. Nakano et al. (LEPS Collaboration), Phys. Rev. Lett. 91, 012002 (2003). . V V Barmin, DIANA CollaborationPhys. At. Nucl. 661715Yad. Fiz.V. V. Barmin et al. (DIANA Collaboration), Yad. Fiz. 66, 1763 (2003) [Phys. At. Nucl. 66, 1715 (2003)]. . S Stepanyan, CLAS CollaborationPhys. Rev. Lett. 91252001S. Stepanyan et al. (CLAS Collaboration), Phys. Rev. Lett. 91, 252001 (2003). . M Battaglieri, CLAS CollaborationPhys. Rev. Lett. 9642001and references thereinM. Battaglieri et al. (CLAS Collaboration), Phys. Rev. Lett. 96, 042001 (2006) and references therein. . T Nakano, LEPS CollaborationPhys. Rev. C. 7925210T. Nakano et al. (LEPS Collaboration), Phys. Rev. C 79, 025210 (2009). Chin. Phys. C (High Ener. Phys. Nucl. Phys.), bf. B S Zou, 331113B. S. Zou, Chin. Phys. C (High Ener. Phys. Nucl. Phys.), bf 33, 1113 (2009). . R Aaij, LHCb CollaborationPhys. Rev. Lett. 11572001R. Aaij et al. (LHCb Collaboration), Phys. Rev. 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R L Zhu, C F Qiao, arXiv:1510.08693R. L. Zhu, C. F. Qiao, arXiv:1510.08693. . X H Liu, Q Wang, Q Zhao, arXiv:1507.05359X. H. Liu, Q. Wang and Q. Zhao, arXiv:1507.05359. . F K Guo, U G Meißner, W Wang, Z Yang, Phys. Rev. D. 9271502F. K. Guo, U. G. Meißner, W. Wang and Z. Yang, Phys. Rev. D 92 071502 (2015). . E Hiyama, Y Kino, M Kamimura, Prog. Part. Nucl. Phys. 51223E. Hiyama, Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys. 51, 223 (2003). . J Vijande, F Fernandez, A Valcarce, J. Phys. G. 31481J. Vijande, F. Fernandez and A. Valcarce, J. Phys. G 31, 481 (2005). . Y C Yang, J L Ping, C R Deng, H S Zong, J. Phys. G. 31105001Y. C. Yang, J. L. Ping, C. R. Deng and H. S. Zong, J. Phys. G 31 105001 (2005).
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[ "Sample Complexity Using Infinite Multiview Models", "Sample Complexity Using Infinite Multiview Models", "Sample Complexity Using Infinite Multiview Models", "Sample Complexity Using Infinite Multiview Models" ]
[ "Robert A Vandermeulen \nMachine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n\n", "Vandermeulen@tu-Berlin De \nMachine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n\n", "Robert A Vandermeulen \nMachine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n\n", "Vandermeulen@tu-Berlin De \nMachine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n\n" ]
[ "Machine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n", "Machine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n", "Machine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n", "Machine Learning Group Berlin Institute for the Foundations of Learning and Data\nTechnische Universität Berlin\n" ]
[]
Recent works have demonstrated that the convergence rate of a nonparametric density estimator can be greatly improved by using a low-rank estimator when the target density is a convex combination of separable probability densities with Lipschitz continuous marginals, i.e. a multiview model. However, this assumption is very restrictive and it is not clear to what degree these findings can be extended to general pdfs. This work answers this question by introducing a new way of characterizing a pdf's complexity, the non-negative Lipschitz spectrum (NL-spectrum), which, unlike smoothness properties, can be used to characterize virtually any pdf. Finite sample bounds are presented that are dependent on the target density's NL-spectrum. From this dimension-independent rates of convergence are derived that characterize when an NL-spectrum allows for a fast rate of convergence.1. As a technical convenience it will always be assumed that w1 > 0. 1dλ = ε/2.InductionStep: Suppose the lemma holds for some d ∈ N. Let ε > 0 andNote that f × f ′ ∈ F d+1 and ww ′ f × f ′ ≤ I. The following completes the proof
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[ "https://export.arxiv.org/pdf/2302.04292v1.pdf" ]
256,697,223
2302.04292
c87004769a54307fe259046e9ddcd759c108d531
Sample Complexity Using Infinite Multiview Models 8 Feb 2023 Robert A Vandermeulen Machine Learning Group Berlin Institute for the Foundations of Learning and Data Technische Universität Berlin Vandermeulen@tu-Berlin De Machine Learning Group Berlin Institute for the Foundations of Learning and Data Technische Universität Berlin Sample Complexity Using Infinite Multiview Models 8 Feb 2023Nonparametric Density EstimationLow-Rank ModelDensity EstimationTensor FactorizationSample Complexity Recent works have demonstrated that the convergence rate of a nonparametric density estimator can be greatly improved by using a low-rank estimator when the target density is a convex combination of separable probability densities with Lipschitz continuous marginals, i.e. a multiview model. However, this assumption is very restrictive and it is not clear to what degree these findings can be extended to general pdfs. This work answers this question by introducing a new way of characterizing a pdf's complexity, the non-negative Lipschitz spectrum (NL-spectrum), which, unlike smoothness properties, can be used to characterize virtually any pdf. Finite sample bounds are presented that are dependent on the target density's NL-spectrum. From this dimension-independent rates of convergence are derived that characterize when an NL-spectrum allows for a fast rate of convergence.1. As a technical convenience it will always be assumed that w1 > 0. 1dλ = ε/2.InductionStep: Suppose the lemma holds for some d ∈ N. Let ε > 0 andNote that f × f ′ ∈ F d+1 and ww ′ f × f ′ ≤ I. The following completes the proof Introduction Nonparametric density estimation is a statistical task whose mathematical properties are very wellstudied. The universal consistency of popular nonparametric density estimators, like the histogram and kernel density estimator, has been known for some time. In addition, there exist finite sample bounds and rates of convergence for nonparametric density estimators when the target density is known to come from a smooth class of densities (Silverman, 1978;Györfi et al., 1985;Devroye and Lugosi, 2001;Tsybakov, 2008;Vandermeulen and Scott, 2013;Dasgupta and Kpotufe, 2014;Jiang, 2017). It is well-known that nonparametric density estimation suffers strongly from the curse of dimensionality. In theoretic works this typically manifests as a dimensionality exponent somewhere in rates or bounds (see Theorem 1 in Jiang (2017), for example). Recently it has been proven that combining smoothness assumptions with a low-rank/multiview assumption can drastically improve the rate of convergence of a nonparametric density estimator, obviating the curse of dimensionality. In particular, it has been proven that there exist universally consistent nonparametric density estimators that converge at rateÕ n (1/ 3 √ n) whenever the target density satisfies a multiview model assumption (Vandermeulen and Ledent, 2021), p (x 1 , . . . , x d ) = k i=1 w i d j=1 p i,j (x j ) ,(1) where p i,j are Lipschitz continuous probability density functions (pdfs) and w lies in the probability simplex. Remarkably this rate is independent of dimension, d, the number of components, k, or the Lipschitz constants of the component marginal densities. On the same class of densities it was also shown that the standard histogram estimator converges at rate ω (1/ d √ n) regardless of choice of rate on bin width (Vandermeulen and Ledent, 2021), which is a clear instance of the curse of dimensionality. While the result above demonstrates the potential benefits of incorporating multiview structure into density estimation, the assumption (1) is quite strong. This work extends the analysis of Vandermeulen and Ledent (2021) so that it is applicable to virtually any pdf. To do this, a new characterization of density complexity, the non-negative Lipschitz spectrum (NL-spectrum) is introduced. The NL-spectrum is an infinite sum (k = ∞) extension of (1), that characterizes how fast w i decays and the Lipschitz constants grow. The NL-spectrum is then shown to be applicable to an extremely general class of pdfs: every (Lebesgue) almost everywhere (a.e.) continuous pdf has an NL-spectrum. This enables the extension of the analysis in Vandermeulen and Ledent (2021) to a much larger class of pdfs. A finite sample bound depending on NL-spectra is then derived for the low-rank histogram estimators introduced in Vandermeulen and Ledent (2021). Finally this bound is used to show rates of convergence of estimators based on NL-spectra rates of growth and decay. In particular, in the infinite component version of (1), if it is known that ∞ i=k+1 w i ∈ O k (k −α ) and the Lipschitz constants of p k,j , L k satisfy, L k ∈ O k k β then there exists a universally consistent estimator that converges at rateÕ n n −α/(3α+β+1) . Related Work The first work to investigate nonparametric density estimators for multiview models was Song et al. (2014), which proposed a method for factorizing a kernel density estimate to recover a multiview model. While Song et al. (2014) contained theoretical guarantees for the rate of recovery of the multiview components, it did not demonstrate that the multiview assumption could be leveraged to improve estimator convergence. Instead it was proposed an approach to nonparametric mixture modeling. Other approaches to nonparametric mixture modeling with strong theoretical guarantees include assuming the data is grouped with respect to mixture components (Vandermeulen and Scott, 2015;Vandermeulen and Scott, 2019;Ritchie et al., 2020;Vandermeulen and Saitenmacher, 2022) or that the are mixture components satisfy concentration assumptions (Dan et al., 2018;Aragam et al., 2020;Aragam and Yang, 2021;Aragam and Tai, 2022). Like Song et al. (2014), these approaches did not produce rates of convergence that beat typical nonparametric rates. Other investigations into low-rank density estimation have focused on identifiability and recoverability (Allman et al., 2009;Kargas and Sidiropoulos, 2019), while other works have proposed low-rank methods that are empirically shown to improve nonparametric estimation without rate guarantees (Song and Dai, 2013;Novikov et al., 2021;Amiridi et al., 2022). Low-rank approaches to matrix estimation have also been studied extensively. Non-negative matrix or tensor factorization is a task similar to multiview density estimation since it can be used to recover a low-rank probability matrix/tensor (Lee and Seung, 1999;Donoho and Stodden, 2004;Kim and Choi, 2007;Arora et al., 2012). Low-rank matrix methods have been studied extensively in the field of compressed sensing which has produced improved matrix estimators with strong theoretical guarantees using the restricted isometry property (Recht et al., 2010) or restricted strong convexity Wainwright, 2011, 2012). Although a bit different than the methods presented so far, enforcing low non-negative rank for coupling measures in Wasserstein distance estimation has also been investigated. This has been shown to yield improved statistical estimation with computational benefits (Scetbon et al., 2021;Scetbon and Cuturi, 2022). While distance estimation is fairly different from density estimation, this method is noteworthy since it optimizes over a class of low-rank probability measures and has strong theoretical analysis demonstrating improved estimator convergence. The works Vandermeulen (2020); Vandermeulen and Ledent (2021) are the first to show, via strong theoretic guarantees, that a multiview assumption can be used to improve the rate of convergence of nonparametric density estimators. Those works also show that a non-negative Tucker factorization can also be used to this effect and found that Tucker factorization produced better estimators in practice. Similarly to the proofs in those works, the results in this paper are not adaptations of techniques developed for compressed sensing or non-negative matrix factorization. Results This section presents and discusses the main results of this paper. Proofs of all results can be found in Section 3. Before introducing the results, some notation and terminology needs to be introduced. Other notation will be introduced intermittently through this work and a table summarizing notation in this work can be found in Appendix A. For a pair of sets, A and B, A × B denotes the Cartesian product. For a pair of real-valued functions, f : A → R and g : B → R, their product is defined as f × g : A × B → R: (a, b) → f (a)g(b). Note that if A = B in this case then f × g is a function on A × A, not a function on A; f · g : x → f (x)g(x) . N denotes all integers greater than 0. For a set A, ½ A denotes the indicator function on A. For n ∈ N, [n] = {1, 2, . . . , n}. For sets and functions, the product operator and power operator · ×n will always mean the products, ×, defined above. The term almost everywhere (a.e.) will always refer to the Lebesuge measure. A pdf is an a.e non-negative function with Lebesuge integral equal to one. Equalities (or inequalities) of functions will always mean equality (or inequality) almost everywhere. The Non-negative Lipschitz Spectrum In density estimation the smoothness of a pdf is often used as a measure of its complexity or how difficult the density is to estimate. The following is a characterization of a density's complexity in manner similar to the spectrum of a linear operator and is the focus of the rest of this work. Definition 1 A pdf p has a non-negative Lipschitz spectrum, (w, L), with w (w i ) ∞ i=1 and L (L i ) ∞ i=1 , if p = ∞ i=1 w i d j=1 p i,j ,(2)where (w i ) ∞ i=1 is a non-negative sequence with ∞ i=1 w i = 1 and (p i,j ) (i,j)∈N×[d] are Lipschitz continuous pdfs with the Lipschitz constant of p i,j equal to L i for all i, j. 1 A non-negative Lipschitz spectrum will be called smooth if the marginals p i,j in (2) are all smooth. Intuitively, an NL-spectrum where w decays slowly and the L i 's are large indicates a more complex pdf. Later results will describe this precisely. The "smooth" descriptor of an NL-spectrum will not play any role for any of the results in the rest of this work. It is simply included because it is an additional regularity property that was simple to include in the proofs and may perhaps be useful in future works. It's worth noting that the NL-spectrum of a pdf is not unique. This lack of uniqueness is not only due to trivial modifications of the spectrum, e.g. reordering or repeated summands, but may also occur since, unlike the singular value decomposition of a matrix, minimal non-negative factorizations and factorizations of tensors are not necessarily unique up to scaling and reordering. There are many works investigating the uniqueness or lack of uniqueness of (non-negative) matrix/tensor/measure factorizations (Kruskal, 1977;Sidiropoulos and Bro, 2000;Donoho and Stodden, 2004;Comon et al., 2008;Allman et al., 2009;Vandermeulen and Scott, 2015;Tahmasebi et al., 2018;Vandermeulen and Scott, 2019;Vandermeulen and Saitenmacher, 2022). This lack of uniqueness might be considered a potential disadvantage of the NL-spectrum compared smoothness-based characterizations of complexity such as Hölder, Sobolev, and Nikol'ski smoothness that are characterized by one or two scalar values rather than a pair of infinite series (Tsybakov, 2008). The following theorem shows that NL-spectra describe a very rich class of pdfs. Theorem 2 If a pdf is a.e. continuous then it has a smooth non-negative Lipschitz spectrum. This set of pdfs arguably contains all pdfs that are of practical interest; it is difficult to imagine a real-life situation where one would be interested in estimating a pdf that is essentially discontinuous on a set of positive measure. In comparison the Hölder, Sobolev, and Nikol'ski smoothness classes are not applicable to pdfs that contain a single discontinuity. Decompositions or approximations reminiscent of the NL-spectrum exist elsewhere in analysis and probability theory. For example, a multivariate Riemann sum has a form similar to (1), k i=1 w i ½ (a i,1 ,b i,1 )×···×(a i,d ,b i,d) = k i=1 w i ½ (a i,1 ,b i,1 ) × · · · × ½ (ai,d,bi,d) = k i=1 w i d j=1 ½ (a i,j ,b i,j ) . A similar decomposition has been mentioned in works on stochastic processes. The following quote is from Kendall and Montana (2002): ...one can show existence for state-space a smooth manifold when the kernel has a continuous density p(x, y), and indeed then one can show small sets of order 1 abound, in the sense that they can be used to produce a representation p( x, y) = ∞ i=1 f i (x)g i (y) , where the f i (x) are non-negative continuous functions supported on small sets, and the g i (y) are probability density functions. Theorem 2 stands apart from previous results because the NL-spectrum decomposition is exact, not an approximation, it is a mixture of pdfs, the component marginals have strong regularity properties, and it describes a very rich and practically useful class of pdfs. Nonparametric Estimator Results The following theorems show the existence of nonparametric density estimators whose performance depends on the NL-spectrum of the target density. Let D d be the set of all pdfs supported on the d-dimensional unit cube, [0, 1] ×d . The following theorem is derived using a low-rank histogram estimator introduced in Vandermeulen and Ledent (2021). Theorem 3 Let d, b, k, n ∈ N and 0 < δ ≤ 1. There exists an estimator V n ∈ D d such that, for any density p ∈ D d , with NL-spectrum (w, L), the following holds P   p − V n 1 > √ 3d 2b k i=1 w i L i + 6 ∞ i=k+1 w i + 7 2bdk log(4bdkn) n + 7 log( 3 δ ) 2n   < δ, where V n is a function of X 1 , . . . , X n iid ∼ p. While the estimators in this work are technically implementable, they are computationally intractable. It will be helpful to introduce a slightly different way of characterizing NL-spectra. Definition 4 An NL-class, denoted by V W,L , with W (W i ) ∞ i=1 a non-negative, non-increasing sequence that converges to 0, andL L i ∞ i=1 a non-negative, non-decreasing sequence, is the set of all pdfs with an NL-spectrum (w, L) satisfying L k ≤L k and ∞ i=k+1 w i ≤ W k for all k. There exist estimators with with following behavior on NL-classes. Theorem 5 Let d, b, k, n ∈ N and 0 < δ ≤ 1. There exists an estimator V n ∈ D d such that, for any density p ∈ D d V (W, L), the following holds, P   p − V n 1 > √ 3d 2b L k + 6W k + 7 2bdk log(4bdkn) n + 7 log( 3 δ ) 2n   < δ, where V n is a function of X 1 , . . . , X n iid ∼ p. One can apply this to get estimators that are adapted to NL-classes with polynomial rates. Proposition 6 Let α, β > 0 and d ∈ N. There exists a universally consistent sequence of estimators, (V n ) ∞ n=1 , on D d , such that, for a fixed sampling density, p ∈ D d ∩ V(W, L), with W k ∈ O k (k −α ) and L k ∈ O k k β , the following holds, V n − p 1 ∈Õ n n −α/(3α+β+1) . Universal consistency holds on D d ; if the sampling density p ∈ D d is fixed then p − V n 1 p → 0. Here the decay in w works against growth in L with regards to convergence rate. Leaving β fixed and letting α → ∞ this estimator approaches a rate ofÕ n (1/ 3 √ n) which matches the rate of convergence in Vandermeulen and Ledent (2021) for finite rank densities. This approximately matches the optimal rate of convergence for one-dimensional histograms in Györfi et al. (1985), "for smooth densities, the average L 1 error for the histogram estimate must vary at least as n −1/3 ." Proofs This section is broken up into two subsections. The fist subsection builds towards and then proves Theorem 2 and the second subsection proves estimator bounds and rates. Before proving the results of this paper, more notation must be introduced. Let F be the set of smooth, Lipschitz continuous pdfs on R. Let F d {f 1 × · · · × f d | f i ∈ F }, note that these are also pdfs. Recall that the conical hull of a set S in a real-valued vector space is cone (S) { n i=1 w i s i | n ∈ N, w i ≥ 0, s i ∈ S}. The cone operator will only be applied to F d . Though standard notation, the reader is reminded that, for a set S, S N is the set of all infinite sequences of elements of S: S N {(s i ) ∞ i=1 | s i ∈ S}. Finally λ denotes the Lebesgue measure where dimension will always be clear from context. Towards a Proof of Theorem 2 The following lemma states that under an indicator function on a multivariate interval, d i=1 ½ (a i ,b i ) , one can fit a positively scaled element of F d that approximates it arbitrarily well in L 1 distance. Lemma 7 Let ε > 0 and I = d i=1 ½ (a i ,b i ) . There exists f ∈ F d and w ≥ 0 such that wf ≤ I and I − wf dλ ≤ ε. Proof of Lemma 7 The proof will proceed by induction on dimension, d. Base Case, d = 1: If ε ≥ b 1 − a 1 one can simply select any f ∈ F and let w = 0, giving wf = 0 ≤ I and I − wf dλ = b 1 − a 1 − 0 ≤ ε, so the lemma holds when ε ≥ b 1 − a 1 . The remainder of this case will proceed with ε < b 1 − a 1 and the "1" subscript dropped. The base case will be proven using smooth bump functions; see Section 13.1 in Tu (2010) for a technical treatment of bump functions. This proof only necessitates a very simple set of bump functions: for any c 1 < c 2 < c 3 < c 4 ∈ R there exists a smooth function, ρ, with ρ equal to zero on (c 1 , c 4 ) C , ρ equal to 1 on (c 2 , c 3 ), and ρ in [0, 1] elsewhere. Using this, letf be a smooth function like ρ with c 1 = a, c 2 = a + ε/4, c 3 = b − ε/4, c 4 = b. Let f =f / f dλ and w = f dλ sõ f = wf and f is a pdf with wf ≤ I. Because f is smooth its first derivative exists everywhere and is continuous. Since df is continuous on the compact set [a, b], df bounded on [a, b]. Because f is zero on [a, b] C , df is identically zero on that set. From this it follows that df is bounded and therefore f is Lipschitz continuous, in addition to smooth, and thus f ∈ F . The following inequality then finishes the base case d = 1, I − wf dλ = (a+ε/4,b−ε/4) ½ (a,b) − wf dλ + (a,b) C ½ (a,b) − wf dλ + (a,a+ε/4)∪(b−ε/4,b) ½ (a,b) − wf dλ = (a,a+ε/4)∪(b−ε/4,b) ½ (a,b) − wf dλ ≤ (a,a+ε/4)∪(b−ε/4,b) The following lemma shows that, for any function in a class of sufficiently regular non-negative functions, one can find an element in the conical hull of F d that fits under the function and approximates that function arbitrarily well in L 1 distance. Lemma 8 Let p : R d → R be non-negative, compactly supported, bounded, and a.e. continuous and let ε > 0. There exists k ∈ N, w 1 , . . . , w k ≥ 0, and f 1 , . . . , f k ∈ F d such that p − k i=1 w i f i dλ ≤ ε and p ≥ k i=1 w i f i . Proof of Lemma 8 Because p is compactly supported, bounded, and a.e. continuous it is Riemann integrable as a consequence of the Riemann-Lebesgue Theorem and therefore Darboux integrable 2 . Thus, using a lower Darboux sum, there exists a "step function, " φ = k i=1 w ′ i d j=1 ½ (a i,j ,b i,j ) , with w ′ i ≥ 0, such that φ ≤ p and p − φdλ ≤ ε/2. Let I i = d j=1 ½ (a i,j ,b i,j ) for brevity. From Lemma 7 there exists f 1 , . . . , f k ∈ F d and w 1 , . . . , w k ≥ 0 such that w i f i ≤ w ′ i I i and w ′ i I i − w i f i dλ ≤ ε/(2n). The following now completes the proof, p − k i=1 w i f i dλ = p − k i=1 w ′ i I i + k i=1 w ′ i I i − k i=1 w i f i dλ = p − k i=1 w ′ i I i dλ + k i=1 w ′ i I i − w i f i dλ ≤ ε/2 + n ε 2n = ε. Lemma 9 Let p : R d → R be non-negative, compactly supported, bounded, and a.e. continuous. There exists a non-negative sequence (w i ) ∞ i=1 and (f i ) ∞ i=1 ∈ F d N such that p = ∞ i=1 w i f i . Proof of Lemma 9 From Lemma 8 there exists q 1 ∈ cone F d such that p ≥ q 1 and p − q 1 dλ ≤ 1/2. Note that p − q 1 is non-negative, compactly supported, bounded, and a.e continuous. Proceeding by induction there exists a sequence ( q i ) ∞ i=1 ∈ cone F d N satisfying p − k i=1 q i dλ ≤ 1/2 k and p ≥ k i=1 q i , for all k. Since, p − k i=1 q i dλ = p − k i=1 q i dλ = p − k i=1 q i 1 , the summation is a Cauchy sequence in L 1 and ∞ i=1 q i = p. Since q i ∈ cone F d one can write q i = k i j=1 w i,j f i,j , with w i,j ≥ 0, f i,j ∈ F d . From this, p = ∞ i=1 k i j=1 w i,j f i,j . Observe that, p 1 = ∞ i=1 k i j=1 w i,j f i,j 1 < ∞, and since w i,j f i,j are non-negative functions it follows that ∞ > ∞ i=1 k i j=1 w i,j f i,j 1 = ∞ i=1 k i j=1 w i,j f i,j 1 = ∞ i=1 k i j=1 w i,j f i,j 1 .(3) 2. The facts used here are common in single variable analysis texts. For a complete treatment of the multivariate versions of these facts see Sections 2 and 3 in Chapter IV in Edwards (1994). Note that the terminology in that work is somewhat nonstandard. So the set of all w i,j f i,j is absolutely summable in L 1 . Thus, by relabeling, there exist w i ≥ 0 and f i ∈ F d such that ∞ i=1 w i f i = p. Lemma 9 can now be generalized to prove Theorem 2. Proof of Theorem 2 Let p be an a.e. continuous pdf. One can clearly decompose p into the absolutely summable summation p = ∞ i=1 p i where p i are all non-negative, a.e. continuous, compactly supported functions. For example, one can simply break p into components supported on d i=1 [a i , a i + 1] | a 1 , . . . , a d ∈ Z which is a countable set. One summand in this summation will be considered, so fix some p i . It will now be shown that p i = ∞ j=0 q i,j where q i,j are bounded, a.e. continuous, compactly supported, non-negative functions. For this define q i,j = max (min (p i − j, 1) , 0). Clearly 1 ≥ q i,j ≥ 0 for all j so q i,j are bounded and non-negative. For arbitrary x, if p i (x) = 0 then q i,j (x) = max (min (p i (x) − j, 1) , 0) = max (min (0 − j, 1) , 0) = max (0 − j, 0) = 0, so p i (x) = 0 ⇒ q i,j (x) = 0, and thus supp (q i,j ) ⊆ supp (p i ) and since the support of p i is compact the support of q i,j is compact. The q i,j are a.e continuous by virtue of the fact that max, min, and subtraction preserve pointwise continuity. It will now be shown that ∞ j=0 q i,j = p i . Consider some arbitrary x ∈ R d . There exists some ℓ ∈ N ∪ {0} such that p i (x) ∈ [ℓ, ℓ + 1]. It follows that ∞ j=0 q i,j (x) = ℓ−1 j=0 max (min (p i (x) − j, 1) , 0) + max (min (p i (x) − ℓ, 1) , 0) · · · + ∞ j=ℓ+1 max (min (p i (x) − j, 1) , 0) = ℓ−1 j=0 max (1, 0) + max (p i (x) − ℓ, 0) + ∞ j=ℓ+1 max (p i (x) − j, 0) =ℓ + p i (x) − ℓ + ∞ j=ℓ+1 0 = p i (x). Since the q i,j are all non-negative, using the same argument as (3) in the proof of Lemma 9, it follows that the sum ∞ i=1 ∞ j=0 q i,j is L 1 absolutely summable. From Lemma 9 it follows that for all q i,j = ∞ ℓ=1 w i,j,ℓ f i,j,ℓ with w i,j,ℓ ≥ 0 and f i,j,ℓ ∈ F d and p. Substituting these in yields p = ∞ i=1 ∞ j=0 ∞ ℓ=1 w i,j,ℓ f i,j,ℓ = p, which again is absolutely summable. Because this summation is over a countable set, one can relabel p = ∞ i=1 w i f i . Because f i ∈ F d , f i = d j=1 f i,j with f i,j being Lipschitz continuous, a sequence can be constructed, (L i ) ∞ i=1 , such that f i,j are all L i - Lipschitz continuous. Finally note 1 = ∞ i=1 w i f i = ∞ i=1 w i . It follows that p has NL-spectrum (w, L). Proofs of Estimator Bounds and Rates The estimator results build upon results in Vandermeulen and Ledent (2021) that analyzed histogramstyle estimators of densities on d-dimensional unit cubes. Before presenting these results a bit more notation is required. In Vandermeulen and Ledent (2021) the authors define a notion of a low-rank histogram. For this context a "histogram" refers to a pdf that is constant on a partition containing equally spaced cubes. In particular they define H 1,b to be the set of all histograms on the unit interval with b evenly spaced bins i.e. H 1,b b i=1 w i ½ [(i−1)/b,i/b] | w i ≥ 0, b i=1 w i ½ [(i−1)/b,i/b] dλ = 1 . A low-rank histogram on d-dimensional space, with b bins per dimension, is then defined as H k d,b    k i=1 w i d j=1 h i,j | 0 ≤ w i , k i=1 w i = 1, h i,j ∈ H 1,b    . Let Lip L be the set of L-Lipschitz continuous from R to R. The following proposition was proven in Vandermeulen and Ledent (2021) and gives a bound on how well one can select estimators from H k d,b . Proposition 10 (Proposition 2.1 from Vandermeulen and Ledent (2021)) Let d, b, k, n ∈ N and 0 < δ ≤ 1. There exists an estimator V n ∈ H k d,b such that sup p∈D d P   p − V n 1 > 3 min q∈H k d,b p − q 1 + 7 2bdk log(4bdkn) n + 7 log( 3 δ ) 2n   < δ, (4) where V n is a function of X 1 , . . . , X n iid ∼ p. To begin proving Theorem 3, the bias term, min q∈H k d,b p − q 1 , is analyzed via NL-spectrum. The following is a bound on rank-one bias that simplifies the analysis in Vandermeulen and Ledent (2021) (c.f. Theorem 2.5 in that paper). Lemma 11 Let f 1 , . . . , f d ∈ Lip L ∩D 1 . Then min h∈H 1 d,b d i=1 f i − h 1 ≤ dL √ 12b . Note that a minimizer exists due to the compactness of the set H 1 d,b (see Appendix A.1 in Vandermeulen and Ledent (2021)). Proof of Lemma 11 To begin, min h∈H 1 d,b d i=1 f i − h 1 = min h 1 ,...,h d ∈H 1,b d i=1 f i − d j=1 h j 1 ≤ min h 1 ,...,h d ∈H 1,b d i=1 f i − h i 1 (5) = min h 1 ,...,h d ∈H 1,b d i=1 ½ [0,1] · (f i − h i ) 1 ≤ min h 1 ,...,h d ∈H 1,b d i=1 ½ [0,1] 2 f i − h i 2 (6) ≤ min h 1 ,...,h d ∈H 1,b d i=1 L √ 12b = dL √ 12b ,(7) where (5) is a consequence of Lemma 3.3.7 in Reiss (1989) (see Appendix B) and (6) follows from Hölder's inequality. To see (7) note that Lemma C.2 and C.4 in Vandermeulen and Ledent (2021) state that, for f i ∈ D 1 ∩ Lip L , f i − Proj span(H 1,b) f i 2 ≤ L √ 12b and f i − Proj span(H 1,b) f i 2 = min h∈H 1,b f i − h 2 , with the projection being in L 2 distance. This finishes the proof. In Vandermeulen and Ledent (2021) the authors investigated pdfs whose domains were [0, 1] ×d , not R d . This is a rather subtle point, but this means that the density p(x) = 2x½ [0,1] (x) would be a valid 2-Lipschitz continuous density in that paper, but it is not in this work. The results from that work are applicable to densities in D d ∩ Lip L . The results in this work could be adjusted to be slightly more general on the domain [0, 1] ×d , but this was omitted for simplicity. The next lemma characterizes the bias in terms of NL-spectrum. Lemma 12 Let p ∈ D d have NL-spectrum (w, L), then min H k d,b p − h 1 ≤ d √ 12b k i=1 w i L i + 2 ∞ i=k+1 w i . Proof of Lemma 12 Because p has NL-spectrum (w, L), p = ∞ i=1 w i d j=1 p i,j , with p i,j ∈ D d ∩ Lip L i and w i ≥ 0 for all i. Letp i = arg min h∈H 1 d,b d j=1 p i,j − h 1 andw i = w i / k j=1 w j (p i exists since Lemma 11 is a minimum and not an infimum recall that w 1 > 0 in Definition 1 sõ w i all exist). So k i=1w ipi ∈ H k d,b . Now the following bound can be shown, ∞ i=1 w i p i − k j=1w jpj 1 ≤ k i=1 w i p i − k j=1w jpj 1 + ∞ i=k+1 w i p i 1 = k i=1 w i p i − w ipi + w ipi −w ipi 1 + ∞ i=k+1 w i ≤ k i=1 w i p i − w ipi 1 + k i=1 w ipi −w ipi 1 + ∞ i=k+1 w i .(8) Using Lemma 11, the following bounds the left term of (8) k i=1 w i p i − w ipi 1 ≤ k i=1 w i p i −p i 1 ≤ k i=1 w i dL i √ 12b = d √ 12b k i=1 w i L i . The following bounds the center term of (8), finishing the proof, k i=1 w ipi −w ipi 1 ≤ k i=1 |w i −w i | p i 1 = k i=1 w i − w i /   k j=1 w j   (9) = k i=1 w i 1 −   k j=1 w j   −1 = k i=1 w i − 1 = ∞ i=k+1 w i . Proof of Theorem 3 From Proposition 10, for all d, b, k, n ∈ N and 0 < δ ≤ 1. There exists an estimator V n ∈ H k d,b such that sup p∈D d P   p − V n 1 > 3 min q∈H k d,b p − q 1 + 7 2bdk log(4bdkn) n + 7 log( 3 δ ) 2n   < δ, where V n is a function of X 1 , . . . , X n iid ∼p. For this V n it thus follows that P   p − V n 1 > 3 min q∈H k d,b p − q 1 + 7 2bdk log(4bdkn) n + 7 log( 3 δ ) 2n   < δ.(10)From Lemma 12, min H k d,b p − h 1 ≤ d √ 12b k i=1 w i L i + 2 ∞ i=k+1 w i . So 3 min q∈H k d,b p − q 1 ≤ 3 d √ 12b k i=1 w i L i + 2 ∞ i=k+1 w i ≤ √ 3d 2b k i=1 w i L i + 6 ∞ i=k+1 w i . The theorem follows from substituting this back into (10). Lemma 13 Let p ∈ D d ∩ V(W,L), then min H k d,b p − h 1 ≤ d √ 12b L k + 2W k . Proof of Lemma 13 Because p ∈ V(W,L), p can be written as, p = ∞ i=1 w i d j=1 p i,j , with, w i ≥ 0, ∞ i=k+1 w i ≤ W k for all k, and p i,j ∈ D d ∩ Lip L i , with L i ≤L k for all i ≤ k. Using this NL-spectrum the proof proceeds exactly as the proof of Lemma 12 to (9), yielding min h∈H k d,b p − h 1 ≤ k i=1 w i dL i √ 12b + k i=1 w i − w i /   k j=1 w j   + ∞ i=k+1 w i . SinceL is non-decreasing and w i ≥ 0 with k i=1 w i ≤ 1 it follows that k i=1 w i dL i √ 12b ≤ k i=1 w i dL i √ 12b ≤ k i=1 w i dL k √ 12b ≤ dL k √ 12b . Using the same argument as in the proof of Lemma 12 it follows that k i=1 w i − w i /   k j=1 w j   + ∞ i=k+1 w i = 2 ∞ i=k+1 w i ≤ 2W k . Proof of Theorem 5 Proceed as in the proof of Theorem 3, with Lemma 13 for Lemma 12. Proof of Proposition 6 Let V n be the estimator from Theorem 5 with k = n 1/(3α+β+1) and b = n (α+β)/(3α+β+1) . Note that (4) also holds for V n . Let p ∈ D d ∩V(W, L) with W k ∈ O (k −α ) and L k ∈ O k β . From Theorem 5 there is the following bound P   p − V n 1 > √ 3d 2b L k + 6W k + 7 2bdk log(4bdkn) n + 7 log( 3 δ ) 2n   < δ. It will be shown that the four terms right of the inequality go to zero at rateÕ n −α/(3α+β+1) . First observe that k ∈ Θ n n 1/(3α+β+1) and b ∈ Θ n n (α+β)/(3α+β+1) . For the first two terms, for n sufficiently large, √ 3d 2b L k + 6W k ≤ C k β n (α+β)/(3α+β+1) + Ck −α (C chosen sufficiently large) ≤ C ′ n β/(3α+β+1) n (α+β)/(3α+β+1) + C ′ n −α/(3α+β+1) (substitute in n; C ′ suf. large) = C ′ n −α/(3α+β+1) + C ′ n −α/(3α+β+1) . For sufficiently large C The third term can bounded as 7 2bdk log(4bdkn) n ≤ C bk n log(4bdkn). For the first term, for sufficiently large n and for C chosen sufficiently large bk n ≤ C n (α+β)/(3α+β+1) n 1/(3α+β+1) n = C n (α+β)/(3α+β+1) n 1/(3α+β+1) n (3α+β+1)/(3α+β+1) = n −2α/(3α+β+1) = n −α/(3α+β+1) . Since 4bdkn ∈ O n (poly(n)), it follows that log(4bdkn) ∈ O n log(n) . Combining the previous two rates yields 7 2bdk log(4bdkn) n ∈ O n n −α/(3α+β+1) log(n) ⊂Õ n n −α/(3α+β+1) . Finally 7 log( 3 δ ) 2n ∈ O n (1/ √ n). Since α, β > 0 it follows that α/(3α + β + 1) < 1/3 and 7 log( 3 δ ) 2n ∈ O n n −1/2 ⊂ O n n −α/(3α+β+1) , which finishes the rate portion of this proof. For universal consistency (4) will be used. It has already been shown that the right two summands in (4) go to zero. The following lemma is also from Vandermeulen and Ledent (2021) and demonstrates that the bias term goes to zero for all densities in D d , finishing the proof. Lemma 14 (Lemma 2.1 from Vandermeulen and Ledent (2021)) Let p ∈ D d . If k → ∞ and b → ∞ then min q∈H k d,b p − q 1 → 0. © R.A. Vandermeulen. AcknowledgmentsThis work was supported by the Federal Ministry of Education and Research (BMBF) for the Berlin Institute for the Foundations of Learning and Data (BIFOLD) (01IS18037A).Appendix A. Notation{1, 2, . . . , n} D d set of all pdfs whose support is contained in [0, 1] ×d F the set of all smooth, Lipschitz continuous pdfs on R.Appendix B. 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[]
[ "Direction-dependent Dynamics of Colloidal Particle Pairs and the Stokes-Einstein Relation in Quasi-Two-Dimensional Fluids", "Direction-dependent Dynamics of Colloidal Particle Pairs and the Stokes-Einstein Relation in Quasi-Two-Dimensional Fluids" ]
[ "Noman Hanif Barbhuiya \nDiscipline of Physics\nIndian Institute of Technology Gandhinagar\n382055Palaj, GandhinagarGujaratIndia\n", "A G Yodh [email protected] \nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA\n", "Chandan K Mishra \nDiscipline of Physics\nIndian Institute of Technology Gandhinagar\n382055Palaj, GandhinagarGujaratIndia\n" ]
[ "Discipline of Physics\nIndian Institute of Technology Gandhinagar\n382055Palaj, GandhinagarGujaratIndia", "Department of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA", "Discipline of Physics\nIndian Institute of Technology Gandhinagar\n382055Palaj, GandhinagarGujaratIndia" ]
[]
Hydrodynamic interactions are important for diverse fluids especially those with low Reynold's number such as microbial and particleladen suspensions, and proteins diffusing in membranes. Unfortunately, while far-field (asymptotic) hydrodynamic interactions are fully understood in two-and three-dimensions, near-field interactions are not, and thus our understanding of motions in dense fluid suspensions is still lacking. In this contribution, we experimentally explore the hydrodynamic correlations between particles in quasi-two-dimensional colloidal fluids in the near-field. Surprisingly, the measured displacement and relaxation of particle pairs in the body frame exhibit direction-dependent dynamics that can be connected quantitatively to the measured near-field hydrodynamic interactions. These findings, in turn, suggest a mechanism for how and when hydrodynamics can lead to a breakdown of the ubiquitous Stokes-Einstein relation (SER). We observe this breakdown, and interestingly, we show that the direction-dependent breakdown of the SER is ameliorated along directions where hydrodynamic correlations are smallest. In total, the work uncovers significant ramifications of near-field hydrodynamics on transport and dynamic restructuring of fluids in two-dimensions.The investigation of hydrodynamics in fluids at low Reynold numbers has a venerable history and continues to yield surprises[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Generally, particle transport in such fluids is influenced by hydrodynamic interactions which span near-and far-field length scales[4][5][6], and which depend strongly on spatial confinement (dimension) and fluid boundary conditions[17,18]. In three dimensions (3D), monopole-like hydrodynamic interactions give rise to drag forces on particles in particle-pairs of the same sign in both longitudinal and transverse directions[10,17]. By contrast, in two-dimensions (2D), the asymptotic far-field hydrodynamic solutions exhibit a dipolar flow profile with longitudinal drag and transverse anti-drag coupling between particles in particle-pairs[4,6]. In addition, as the particle packing fraction increases, nearfield drag correlations exhibit oscillatory modulations with respect to particle separation that are in-phase with structural signatures such as the particle pair correlation function[19]. However, the nature of transverse anti-drag coupling in the near-field deviates from the far-field dipolar flow profile and remains largely unexplored; for example, phase differences between transverse and longitudinal correlations could exist and, if so, could have consequences in dense suspensions and confined geometries. Our study of rigidly confined colloidal suspensions in 2D sheds light on these issues, revealing direction-dependent transport of particles in colloid-pairs caused both by the contrast in strength between longitudinal drag and transverse anti-drag, and by the phase difference between the particle-separation-dependent oscillations of longitudinal drag and transverse anti-drag. Moreover, the impact of this anisotropy on the Stokes-Einstein relation is elucidated.
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[ "https://export.arxiv.org/pdf/2212.13481v2.pdf" ]
256,697,658
2212.13481
85e5f169a841bb7d525c78fdba0647bb3e9dd9ee
Direction-dependent Dynamics of Colloidal Particle Pairs and the Stokes-Einstein Relation in Quasi-Two-Dimensional Fluids 9 Feb 2023 Noman Hanif Barbhuiya Discipline of Physics Indian Institute of Technology Gandhinagar 382055Palaj, GandhinagarGujaratIndia A G Yodh [email protected] Department of Physics and Astronomy University of Pennsylvania 19104PhiladelphiaPennsylvaniaUSA Chandan K Mishra Discipline of Physics Indian Institute of Technology Gandhinagar 382055Palaj, GandhinagarGujaratIndia Direction-dependent Dynamics of Colloidal Particle Pairs and the Stokes-Einstein Relation in Quasi-Two-Dimensional Fluids 9 Feb 2023 Hydrodynamic interactions are important for diverse fluids especially those with low Reynold's number such as microbial and particleladen suspensions, and proteins diffusing in membranes. Unfortunately, while far-field (asymptotic) hydrodynamic interactions are fully understood in two-and three-dimensions, near-field interactions are not, and thus our understanding of motions in dense fluid suspensions is still lacking. In this contribution, we experimentally explore the hydrodynamic correlations between particles in quasi-two-dimensional colloidal fluids in the near-field. Surprisingly, the measured displacement and relaxation of particle pairs in the body frame exhibit direction-dependent dynamics that can be connected quantitatively to the measured near-field hydrodynamic interactions. These findings, in turn, suggest a mechanism for how and when hydrodynamics can lead to a breakdown of the ubiquitous Stokes-Einstein relation (SER). We observe this breakdown, and interestingly, we show that the direction-dependent breakdown of the SER is ameliorated along directions where hydrodynamic correlations are smallest. In total, the work uncovers significant ramifications of near-field hydrodynamics on transport and dynamic restructuring of fluids in two-dimensions.The investigation of hydrodynamics in fluids at low Reynold numbers has a venerable history and continues to yield surprises[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Generally, particle transport in such fluids is influenced by hydrodynamic interactions which span near-and far-field length scales[4][5][6], and which depend strongly on spatial confinement (dimension) and fluid boundary conditions[17,18]. In three dimensions (3D), monopole-like hydrodynamic interactions give rise to drag forces on particles in particle-pairs of the same sign in both longitudinal and transverse directions[10,17]. By contrast, in two-dimensions (2D), the asymptotic far-field hydrodynamic solutions exhibit a dipolar flow profile with longitudinal drag and transverse anti-drag coupling between particles in particle-pairs[4,6]. In addition, as the particle packing fraction increases, nearfield drag correlations exhibit oscillatory modulations with respect to particle separation that are in-phase with structural signatures such as the particle pair correlation function[19]. However, the nature of transverse anti-drag coupling in the near-field deviates from the far-field dipolar flow profile and remains largely unexplored; for example, phase differences between transverse and longitudinal correlations could exist and, if so, could have consequences in dense suspensions and confined geometries. Our study of rigidly confined colloidal suspensions in 2D sheds light on these issues, revealing direction-dependent transport of particles in colloid-pairs caused both by the contrast in strength between longitudinal drag and transverse anti-drag, and by the phase difference between the particle-separation-dependent oscillations of longitudinal drag and transverse anti-drag. Moreover, the impact of this anisotropy on the Stokes-Einstein relation is elucidated. We employ optical video microscopy to probe the hydrodynamic interactions of colloidal suspensions in quasi-two-dimensional (quasi-2D) samples. The experiments were performed with suspensions of micron-size polystyrene latex beads (with diameter, σ) in their liquid phase as a function of packing area-fraction, φ. Cursory examination of the basic displacement correlation data ( Fig. 1) reveals central observations of the experiment. In Fig. 1a, we show the single particle displacement distribution after lag time, t, for a pair of particles {i, j} located at positions (in 2D) {r ′ , r + r ′ }, respectively; the particles are spatially separated by, r, at initial time, t 0 . The single particle displacement distributions for each particle are P (∆r i (r ′ , t)|r j = r ′ + r) and P (∆r j (r ′ + r, t)|r i = r ′ ). Here, displacement distributions are obtained from all t 0 and for all pairs of particles separated by distance r at t 0 . As expected, since particle velocity distributions (and positions) are isotropic, these distributions are measured to be spatially symmetric about their initial position at t 0 regardless of r and packing fraction, φ. The nature and influence of the hydrodynamic interactions between particles in a colloid-pair are revealed in other panels of Fig. 1 (and in other figures) starting with measurements of the conditional probability distribution for particle displacements: P (∆r j (r ′ + r, t)|∆r L,T i (r ′ , t)). Here L and T denote the longitudinal and transverse axes of the particle pair in the body frame, oriented respectively, along and perpendicular to the line joining the particles separated by r at t 0 . P (∆r j (r ′ + r, t)|∆r L,T i (r ′ , t)) represents the probability that the j th particle will experience displacement ∆r j (t), when the i th particle, separated by r, is displaced by ∆r i (t) along the longitudinal (L) or transverse (T ) direction in the body frame. Visualizing hydrodynamic modes and spatiotemporal evolution of pairs. a, Colormap of P (∆r i (r ′ , t)|r j = r ′ + r) and P (∆r j (r ′ + r, t)|r i = r ′ ) for particles in pairs separated by r = 2.0σ. Conditional P (∆r(t)) measured for the particle on the right of the pair when the particle on the left displaces by ≥ 1.0 σ b, along L and r = 2.0 σ c, along T and r = 2.5 σ and d, along T and r = 1.1 σ, depicted by yellow arrows. Solid red and white circles represent the mean positions of the particles at t 0 and (t 0 + t), respectively. The displacement color map for a−d are normalized by the maximum displacements in each case. e, Polar colormap, r(r, θ), of hydrodynamic flow profile when the colloid at origin (solid red circle) moves towards the right (open red circle). The dashed radial circles represent r = {2, 3, 4}σ. Representative red arrows, with their head and length represent the direction and strength (shown also in the background), respectively, of the field. The measurements for a−e were performed at φ = 0.15 and t = 0.5 s. f, H L (solid symbols) and H T (open symbols) versus r for φ = 0.15 (green circles) and φ = 0.61 (red squares). The black solid and dashed lines show ±A/r 2 dependencies, respectively, where A is a constant. The inset shows (H T ) 3 at φ = 0.61. g, Z rel versus r at φ = 0.61 for t = 0.5 s (black squares), t = 1.5 s (red circles) and t = 3.0 s (blue triangles). The inset shows typical schematics depicting configurations corresponding to the peak position in Z rel . h, "Most-probable" schematic construction of spatiotemporal evolution of a pair of particles due to near-field hydrodynamics. Note, the relative magnitudes and directions of the yellow arrows correspond to expectation for each configuration. along T (Fig. 1c & d), and the distribution of the j th particle displacement is shown. These conditional probability distributions provide exemplar exhibits of the well-known hydrodynamic dipolar interaction modes [4,6], showing codiffusion (drag) of particles in a pair along L (Fig. 1b & SI Video S1) and anti-symmetric diffusion (anti-drag) of particles in a pair along T (Fig. 1c & SI Video S2). Notice, in Fig. 1d, the linear superposition of near-field drag and anti-drag hydrodynamic fields produces circumferential motion of one particle around the other, thereby leading to a "mass void filling" motion of one particle created by the motion of its partner ( Fig. S1). At higher density, φ = 0.61, local structural features emerge in the near-field (SI Fig. S2). Specifically, oscillatory spatial modulation of the amplitude appears in both H L and H T , and the hydrodynamic correlation functions deviate from the dipolar form. Nevertheless, in the far-field (r > 8σ) even at large packing fraction, the profiles decay as 1/r 2 ( Figure 1f). As reported in previous studies, our measurements find that the spatial modulation of H L (in dense suspensions) is in-phase with oscillation of the colloidal fluid structural pair correlation function (SI Fig. S2) [4,19]. Surprisingly, we find that the anti-drag spatial modulations associated with H T exhibit a spatial phase-shift (phase difference/lag) of around 0.25σ with respect to H L ; this behaviour is most easily seen in the inset to Fig. 1f which plots the cube of H T . This effect is also revealed by the function Z rel (r, t) ≡ r(t+t0) r(t0) r ′ ,t0 The insights offered by H L and H T (and Z rel ) suggest a "most-probable" spatiotemporal evolution of particles in a colloid-pair as a function of the particle separation, r(t). This evolution is schematically shown in Figure 1h. Initially, the particles in the pair are separated by a small distance, r ∼ 1.0σ; they then diffuse and separate to r ∼ 1.25σ. When r ∼ 1.25σ, the pairs are in their most stable state, i.e., they reside in first minima of Z rel (Fig. 1g), and longitudinal drag is dominant. When the separation between particles increases further to r ∼ 1.5σ, then transverse rotation of the particles begins and leads to further radial separation. When Z rel reaches a maximum at r ∼ 1.75σ, H T is comparatively stronger and the pair configuration destabilizes. Over time, as r increases, anti-drag weakens, and drag becomes dominant again at r ∼ 2.0σ. The cycle will then repeat, but the hydrodynamic interactions become attenuated at larger r ( Fig. 1f & g). The emergent spatiotemporal mobility landscape, revealed by our experiments in the near-field, leads to particular local viscosity and diffusivity associated with the motions of particles in the colloid-pair. The experiments thus offer an opportunity to explore the validity of basic physics rules such as the Stokes-Einstein relation (SER) in quasi-2D. Recall, the SER relates the particle diffusion coefficient, D, to the viscosity, η, of the suspending fluid: D = kB T 6πη(σ/2) , where k B T is the thermal energy. In practice, the structural relaxation time, τ α , is often used as a proxy for η [20][21][22]. Simulations of D and τ α in 3D liquids demonstrate D ∝ τ −ξ α , with expected SER exponent ξ = 1 [22]. However, recent computer simulations and experiments in 2D fluids have observed ξ > 1 [22][23][24]. The origin of this unusual behaviour, which apparently violates the SER, is unresolved. One interesting suggestion alludes to the presence of long-wavelength Mermin-Wagner fluctuations in 2D liquids [25]. These correlations due to Mermin-Wagner fluctuations can be removed by considering the relative motion of particles with respect to their cages [26,27], which, after implementation, recovers ξ ∼ 1 [25], and thereby suggests that Mermin-Wagner fluctuations cause the anomalous SER exponent. However, strictly speaking, this approach to filter out correlated motions necessarily assumes that D and τ α are isotropic, i.e., the approach assumes that near-field dynamics have zero angular dependence. Since we have measured longitudinal and transverse hydrodynamic correlations and particle displacements in the near-and far-field, our experiments offer means to revisit the SER in 2D and to directly investigate the influence of spatial phase differences between longitudinal and transverse hydrodynamic modes. Specifically, we study the separation and angular dependence associated with colloid-pair dynamics based on measurements of single particle diffusion, D(r, θ), and relaxation, τ α (r, θ) (Methods). Here, r is the particle separation distance in a pair at t 0 , and θ is the angle between probing direction and the longitudinal axis in the body frame, L (Methods). At low φ (φ = 0.15), D(r, θ) is measured to be isotropic (Fig. 2a & c), but τ α (r, θ) is not; for r < 2.0σ, τ α (r, θ) is found to be anisotropic ( Fig. 2b & d). This θ−dependence is readily understood. Since drag leads to co-diffusion of particles in colloid-pairs along the longitudinal direction, when r ∼ 1.25σ, H T is substantially smaller than H L , and particles in the colloid-pairs will take longer to relax along L than along T : τ L α (r ∼ 1.25σ) > τ T α (r ∼ 1.25σ) (Fig. 2b & d). The oscillatory structural features in the near-field become more pronounced when the particle packing area-fraction is increased. D(r, θ) = D(r) is still measured to be angularly isotropic and exhibits oscillations as a function of r that are in-phase with H L (Fig. 2e & g), but τ α (r, θ) is measured to be oscillatory with r and strongly anisotropic ( To explore influence of anisotropy in the hydrodynamic correlations on the validity of the SER, we measured D(r, θ) and τ α (r, θ) for all packing fractions, φ. For simplicity, our discussion will focus on data taken along the longitudinal (L, θ = 0 • ) and transverse (T, θ = 90 • ) directions as a function of φ for fixed r. Specifically, for each r, we measure the power-law relationship between D L,T and τ L,T α using all φ. Exemplar plots and extracted exponents ξ L,T are shown in the top panel of Figure 3. The resultant variations of D and τ α with r, and the anisotropy in τ α along L and T are reflected in the SER exponents, ξ L and ξ T , respectively (Fig. 3). Notably, the SER exponents associated with the spatial directions L and T differ from unity and differ from each other. Moreover, the spatial phase lag of ∼ 0.25σ observed for H L and H T is also apparent in ξ L and ξ T . By contrast, if instead we derive ξ from measurements of D and τ α along two randomly chosen orthogonal directions in the lab frame (different from L and T in the body frame), then we find that the ξ are essentially in-phase and identical within experimental certainty (SI Fig. S3). Together, these observations suggest that the unusual trends of ξ(r, θ) that are apparent in the body frame are due to the distinct motional modes associated with near-field hydrodynamic correlations that arise in 2D colloidal fluids. The consequences of the near-and far-field hydrodynamic correlations persist in our measurements of the traditional single particle diffusivity, D self , and the traditional fluid structural relaxation, τ self α . The φ−dependent D self and τ self α yield ξ self = 1.16 ± 0.03 (Inset to Fig. 3). Notice, the spatial modulations of the body-frame ξ decay with r and converge to ξ self in the far-field, r > 8σ (ξ L,T → 1.18 ± 0.03). Evidently, the hydrodynamic interactions in quasi-2D that lead to direction-dependent dynamics and SER violation in the body frame, also lead to violation of the SER in the lab frame. Finally, we consider whether it might be possible to recover the ξ ∼ 1 SER exponent for quasi-2D colloidal fluids, perhaps along special directions. To this end, we propose a simple approach again based on the colloid-pairs and their correlated interactions and displacements. Generally, validity of the SER (ξ = 1) is expected for purely random processes. Since hydrodynamic correlations are smallest along the direction perpendicular to the centre-ofmass displacement (CM ⊥ ) of the colloid-pair (inset to Fig. 4a), one might expect that extraction of D CM ⊥ and τ CM ⊥ α along this direction could yield ξ CM ⊥ ∼ 1. The data in Figure 4a corroborates these hypotheses. In the farfield, r > 8σ, where the spatial modulations in H L and H T are diminished (Fig. 1f), we find that ξ CM ⊥ decays and saturates to 1.01 ± 0.02 (Fig. 4a). In the near field, r < 8σ, ξ CM ⊥ oscillates around unity. Interestingly, in this regime (r < 8σ), ξ CM ⊥ → 1 at specific r that correspond to the extrema of dH dr (or extrema of Z rel ) wherein the net hydrodynamic correlations are weakest in direction orthogonal to the centre-of-mass displacements (Fig. 4b) and in other studies [23,25]. Looking forward, these insights about near-field hydrodynamics of spherical particles could prove even more interesting for anisotropic particles, both passive and active, and may lead to novel ideas for affecting self-assembly and structural relaxation [12,[28][29][30]. Broadly, we expect that these near-field hydrodynamics could impact phenomena in dense, spatially constrained systems such as arise in cluster aggregation [31], translocation of proteins [15,16], nucleation and growth kinetics of crystals [7,9], active systems [11,12], and clogging and jamming of channels [32,33]. Methods Experimental details. We used polystyrene microspheres of diameter 2σ = 1.04 µm with polydispersity of < 5% suspended in water. The particles were loaded into a wedge-shaped cell and allowed to sediment under gravity into the thin quasi-two-dimensional (quasi-2D) region of the cell. Once a desired packing area-fraction, φ, was achieved, the cell was equilibrated for at least six hours before performing video microscopy. Data for all φs were taken from the same region of the cell. The images, at each φ, were captured at 10 frames per second (fps) for 20 minutes. The trajectories of the particles were obtained using standard tracking algorithms [34]. The dynamic spatial resolution was found to be 20 nm. All subsequent analyses were performed using in-house developed codes. Hydrodynamic flow profile. The hydrodynamic field shown in 1e was determined as follows [5]. Briefly, we first computed the displacements of each particle, ∆r i (t = 0.5s). Next consider the pair of particles, {i, j}. ∆r i (t = 0.5s), ∆r j (t = 0.5s), and the unit vector (r) pointing from i to j subtend angles α i , α j , and β ij with respect to the positive x−axis (ranging between 0 and 2π). For each reference particle i we rotate the 2D coordinate system through an angle α i so that ∆r i (t = 0.5s) is aligned in the positive x (horizontal) direction. The other vectors then also rotate through α i . We define the θ as the polar angle between the positive x−axis (now aligned with ∆r i (t = 0.5s)) and r (SI Fig. S4). The polar plot (r, θ) in 1e is derived from the ensemble and initial time, t 0 , average of the displacements. Dynamics measurement. Single particle diffusivity, D self , in the lab frame were measured from the mean squared displacements, ∆r(t) 2 (SI Fig. S5). ∆r(t) 2 = 1 N N k=1 (∆r k (t)) 2 . Here, N is the total number of particles, ∆r k (t) is the displacement of k th particle during the lag time, t, and the averaging, , were performed over t 0 . The diffusion of single particles, D(r, θ), with respect to the colloid-pair body-frame were measured from dynamical quantities, ∆r 2 (t; r ′ , r + r ′ , θ), obtained using displacement of either of the particle {i, j} in a pair alonĝ R(θ)r(t 0 , r ′ , r + r ′ )); herer is the unit vector along the line joining the particles in pair located at {r ′ , r + r ′ } and at initial time t 0 , andR(θ) is the rotation matrix. D is obtained from the linear regime of ∆r(t) 2 plot. Note, even D(r) at low φ, φ ≤ 0.35, are anisotropic when extracted from duration timescales (t < 20 s) where hydrodynamic interactions are significant (SI Fig. S6). At higher φ (φ ≥ 0.58), the dynamics become mildly sub-diffusive on short timescales, precluding extraction of D(r, θ), and hence, we cannot comment on whether D(r) continues to be anisotropic at these densities at short timescales. The structural relaxation time, τ self α , in the lab frame were measured from self-intermediate scattering functions, F s (q, t) [23] (SI Fig. S5). F s (q, t) = 1 N N k=1 e iq.∆r k (t) , where symbols have usual meanings as explained above. For all the analyses presented in this study (including pair dynamics), the magnitude of probe wave-vector, q = 2π/a, where a is the position of the first peak in the pair correlation function, g(r), at φ = 0.61. The direction of q is chosen to be along x−axis. The structural relaxation time, τ α (r, θ), associated with particle motion with respect to the colloid-pair body-frame were measured using F s (q θ , t; r ′ , r + r ′ ), by using ∆r(t; r ′ , r + r ′ , θ) of either of the particle {i, j} in a pair alongR(θ)r(t 0 , r ′ , r + r ′ )). q θ is alongR(θ)r. The time for which the decay of F s (q, t) drops to 1/e is read-off as structural relaxation time, τ α , i.e., Figure 1b-d show example data associated with the conditional distribution of particle displacements. Here, the i th particle is displaced by ≥ σ during a time lag of t = 0.5 s, either along L(Fig. 1b)or Fig. 1d & 1dSI Video S3). To the best of our knowledge, the full character of this type of mass void filling motion (Fig. 1d) has not been directly observed in experiment. The motional effects caused by near-field drag and anti-drag (and their linear combination) are apparent in the polar plot of the hydrodynamic field profile, which we derive using the ensemble-averaged displacement correlations of particles in a pair with separation vector, r(r, θ), at t 0 and at low φ = 0.15 (Methods, Fig. 1e). Figure 1 1also shows the measured longitudinal and transverse displacement correlation functions, H L and H T , respectively, versus r. The longitudinal (transverse) hydrodynamic correlation, H L(T ) (r, t) = ∆r L(T ) i (r ′ , t) · ∆r L(T ) j (r ′ + r, t) /D self [4, 6]. Here, ∆r L i (∆r T i ) is the displacement in lag time, t, of the i th −particle in the {i, j}−pair along the L (T ) direction; the averaging, , is performed over all initial times, t 0 , and all possible unique pairs {i, j}. Normalization of H L,T by the φ−dependent single-particle diffusivity, D self , facilitates comparison of H L,T across different φ (Methods). Figure 1f shows H L and H T versus r for two different packing fractions. At low φ (φ = 0.15), H L and H T exhibit expected dipolar decay profiles in the far-field, i.e., they decay as 1/r 2 . A distinguishing feature of quasi-2D fluid confinement is the positive and negative value of the correlation function amplitude of H L and H T , respectively; when we remove one cell wall and thereby increase sample dimensionality to 3D, the amplitudes of both H L and H T become positive (SI − 1 ( 1Fig. 1g). Z rel (r, t) represents the fractional change in separation of particles in the colloid-pair during lag time t. Z rel clearly captures the anti-drag influence on pair-rotation and colloid-pair separation. At the highest φ (φ = 0.61), Z rel shows oscillatory decaying modulations that are in-phase with the modulations of H T in the near-field(Fig. 1g). Fig. 2f & h) due to contrasting magnitude of H L and H T , and the spatial ∼ 0.25σ phase-lag of H T with respect to H L . This behaviour is readily apparent in Figure 2d & h which shows the different variations of τ L α and τ T α . Fig. 2 2Elucidating the influence of hydrodynamics on transport quantifiers. Polar colormaps versus r with color-scale on left for D(r, θ) at a, φ = 0.15 and e, φ = 0.61, and τα(r, θ) at b, φ = 0.15 and f, φ = 0.61. The dashed radial circles are at r = {2, 3, 4, ...}σ. Plots along L and T directions corresponding to θ = 0 • and θ = 90 • , respectively, for D(r) at c, φ = 0.15 and g, φ = 0.61, and τα(r) at d, φ = 0.15 and h, φ = 0.61. The error bars in D are from fittings. Fig. 3 3Influence of hydrodynamics on Stokes-Einstein relation in near-field. SE exponents ξ L and ξ T versus r. The inset plots the particle self-diffusivity (derived from measurements in the lab frame), D self , versus relaxation, τ self α ; the solid line shows D self ∝ τ −1.16±0.03 α . Black dashed and dotted lines at ξ = −1.00 and ξ = −1.18 depict the ideally expected and measured asymptotic values of ξ, respectively. Top panel shows representative D L and D T versus τ L α and τ T α , respectively, for different r as shown in the figures. The solid lines depict linear fits to determine ξ. Standard error from power-law fittings between D L,T and τ L,T α are used in ξ L,T versus r plots; systematic errors, obtained by extraction of D L,T from different time-windows, are found to be larger than standard error and are used when quoting the value of ξ L,T in the main text. Fig. 4 4Recovery of the expected Stokes-Einstein exponent. a, ξ CM ⊥ versus r. ξ CM ⊥ → −1.01±0.02 for r > 8σ; the ideal expected value of ξ is shown by black dashed line. Standard error from power-law fittings between D CM ⊥ and τ CM ⊥ α are used in ξ CM ⊥ versus r plot; systematic errors, obtained by extraction of D CM ⊥ from different time-windows, are found to be larger than standard error and are used when quoting the value of ξ CM ⊥ in the main text. Inset: schematic to visualize the direction in which the displacements of particles in the pair are least correlated, i.e., direction perpendicular to the centre-of-mass displacement direction. b, dH dr versus r at φ = 0.61; H = H L + H T . Inset: comparison of ξ L with ξ CM ⊥ at discrete values of r corresponding to the extrema of dH dr . . In the inset to Figure 4b, ξ CM ⊥ at extrema of dH dr are compared to the corresponding value of ξ L in the near-field. Evidently, thermal forces are the dominant fluctuations experienced by particles in the direction orthogonal to the centre-of-mass displacements, and thus the SER is recovered. To conclude, we have measured and studied the near-and far-field longitudinal and transverse hydrodynamic modes in quasi-2D colloidal fluids. The findings highlight the importance of the contrasting magnitudes and phaseshift between these modes. 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[ "TALE DEGREE MAP AND 0-CYCLES", "TALE DEGREE MAP AND 0-CYCLES" ]
[ "Iván Rosas-Soto " ]
[]
[]
By using the triangulated category ofétale motives over a field k, for a smooth projective variety X over k, we define the group CHé t 0 (X) as anétale analogue of 0-cycles. We study the properties of CHé t 0 (X), giving a description about the birational invariance of such group. We define and present theétale degree map by using Gysin morphisms inétale motivic cohomology and theétale index as an analogue to the classical case. We give examples of smooth projective varieties over a field k without zero cycles of degree one but withétale zero cycles of degree one, however, this property is not always true as we present examples where theétale degree map is not surjective.
null
[ "https://export.arxiv.org/pdf/2305.06444v1.pdf" ]
258,615,453
2305.06444
0f9b063df99bee82fd0525c0473f2e19c9b8234b
TALE DEGREE MAP AND 0-CYCLES 10 May 2023´E Iván Rosas-Soto TALE DEGREE MAP AND 0-CYCLES 10 May 2023´EarXiv:2305.06444v1 [math.AG] By using the triangulated category ofétale motives over a field k, for a smooth projective variety X over k, we define the group CHé t 0 (X) as anétale analogue of 0-cycles. We study the properties of CHé t 0 (X), giving a description about the birational invariance of such group. We define and present theétale degree map by using Gysin morphisms inétale motivic cohomology and theétale index as an analogue to the classical case. We give examples of smooth projective varieties over a field k without zero cycles of degree one but withétale zero cycles of degree one, however, this property is not always true as we present examples where theétale degree map is not surjective. Let X be a smooth projective variety over a field k. We define the zero cycles of X, denoted by Z 0 (X), as the free the abelian group generated by sums x n x x with x a closed point of X and n x ∈ Z equals to zero for every but finitely many x. The degree map is defined by deg : Z 0 (X) → Z x n x x → x n x [k(x) : k]. This map is compatible with the quotient by rational equivalence, so we can define it over CH 0 (X). By definition, it coincides with the push-forward along the structural map p : X → Spec(k) as p * : CH 0 (X) → CH 0 (Spec(k)) = Z. We define the index of a variety X over k as follows I(X) := gcd {[k(x) : k] | x ∈ X} . If the field is algebraically closed then there exists a k-rational point and the map is surjective. However if the base field is not algebraically closed the existence of a k-rational point, or even of a zero cycle of degree 1, is not guaranteed. Let us remark that the existence of a k-rational point implies the existence of a zero cycle of degree 1, but the converse does not always hold. As it is shown in [CM04] for d = 2, 3, 4 there exist del Pezzo surfaces of degree d over a field of cohomological dimension 1 which do not have a zero cycle of degree 1. Or as is presented in [Col05,Theorem 5.1] a hypersurface whose index I(X) = p, for a prime p ≥ 5. Date: May 2023. The study of zero cycles of a smooth projective variety X has played an important part inside algebraic geometry. For instance, if C/k is a smooth projective curve which has a k-rational point then the Chow motive h(C) admits an integral Chow-Künneth decomposition, for this see [MNP13,Chapter 2]. In general if X is a smooth projective variety which has a zero cycle of degree 1 then the integral Chow motive decomposes as h(X) = h 0 (X) ⊕ h + (X) ⊕ h 2d (X) with h 0 (X) ≃ L and h 2d (X) ≃ L d where L is the Lefschetz motive. Another important fact about zero cycles concerns birational invariance, i.e. if f : X → Y is a birational map between smooth varieties over a field k then CH 0 (X) ≃ CH 0 (Y ). We can say even more, if f : X → Y is stably birational, meaning that there exists r, s ∈ N such that X × k P r k → Y × k P s k is birational, then CH 0 (X) ≃ CH 0 (Y ). This gives another tool to study rationality problems and leading to the notions of CH 0 -universally triviality, which induces a decomposition of the diagonal in the Bloch-Srinivas case as in [ For a smooth projective variety X of dimension d over a field k we define the group CH L 0 (X) as follows: CH L 0 (X) := CH d L (X) = H 2d L (X, Z(d)). In the present article we focus on the study of some properties of the group CH L 0 (X) (or CHé t 0 (X) after inverting the characteristic of k), by looking at it with the purpose of obtaining a refinement of classical facts such as birational invariance property and the existence of Lichtenbaum zero cycles of degree 1 which will induce a decomposition of the diagonal. For a smooth projective variety X, we defineétale degree map, denoted by degé t , as the pushforward of X → k using the category DMé t (k), and theétale index of X as an analogue of I(X), as follows: Ié t (X) := gcd degé t (CHé t 0 (X)) ∩ Z . The main results of this article concern the existence of smooth and projective varieties X over a field of cohomological dimension ≤ 1 whose index I(X) > 1 but Ié t (X) = 1, as the following theorems show: Theorem 1 (Theorem 5.4). There exists a smooth projective surface S over a field k, with char(k) = 0 of cohomological dimension ≤ 1, without zero cycles of degree one but Ié t (X) = 1. Theorem 2 (Theorem 5.5). For each prime p ≥ 5 there exists a field k such that char(k) = 0 with cd(k) = 1 and a smooth projective hypersurface X ⊂ P p k with Ié t (X) = 1 but index I(X) = p. To find this kind of varieties, we use Proposition 5.3 which characterizes some smooth varieties X over a field k of cohomological dimension ≤ 1, the ones such that Alb(Xk) tors = 0, whoseétale degree map is surjective. The proof relies in the fact that the condition Alb(Xk) tors = 0 impose that CH L 0 (Xk) hom is uniquely divisible, thus, with trivial Galois cohomology in positive degrees. After that we remark that the varieties presented in [ These results give us the first refinement for the existence of hé t (X) = h 0 et (X) ⊕ h + et (X) ⊕ h 2d et (X) in the category of integralétale motives but not in the category of integral Chow motives. Despite this new refinement of the index of a smooth projective variety, we give an example of how the property Ié t (X) = 1 is not always achieved. For Severi-Brauer varieties X we show that Ié t (X) is greater or equal to the order of the class [X] ∈ Br(k) as follows: After that, we prove that this bound also holds for the product of Severi-Brauer varieties. In order to prove that, we give the following generalization of [GS06,Theorem 5.4.10]: Lemma (Lemma 5.14). Let X be a Severi-Brauer variety of dimension d over a field k. For the product X ×n := n−times X × . . . × X we then obtain an exact sequence 0 → Pic(X ×n ) → Pic(P d k × . . . × P d k ) G k ≃ Z ⊕ . . . ⊕ Z s − → Br(k) → Br(X ×n ) where s sends (a 1 , . . . , a n ) → n i=1 a i [X] ∈ Br(k). With this lemma, we can state and prove the following result for a product of Severi-Brauer varieties: Theorem 4 (Theorem 5.15). Let k be a field and let X be a Severi-Brauer variety over k of dimension d. Then Ié t (X ×n ) ≥ Ié t (X) ≥ ord([X]). With this goal in mind, we start recalling some properties of Lichtenbaum anétale motivic cohomology, presenting in Lemma 2.4 vanishing results depending on theétale cohomological dimension of the concerning variety, and then by stating in Proposition 2.5 the Hochschild-Serre spectral sequence for Lichtenbaum cohomology, i.e. for a finiteétale morphism f : Y → X with Galois group G and for every degree n ∈ Z we have a spectral sequence E p,q 2 (n) = H p (G, H q L (Y, Z(n))) =⇒ H p+q L (X, Z(n)) . Along with this, in Lemma 2.8 we give a proof of classical formulas, such as the ones for projective bundle, blow-up with smooth center and varieties which admit a cellular decomposition foŕ etale motivic and Lichtenbaum cohomology. These formulas and the vanishing lemma for Lichtenbaum cohomology play a main role in the study of birationality properties of the zero cycles of Lichtenbaum cohomology, giving counter-examples where the stability and birationality fail for CHé t 0 (X). After that, we define theétale degree map, denoted as degé t , as the push-forward of the structural morphism p : X → k, in other words degé t : CHé t 0 (X) → Z[1/p] with p the exponential characteristic of the base field k. One of the main tools that we use for obtaining the results is that degé t factors through the term E 0,2d ∞ (d) given by the Hochschild-Serre spectral sequence. This is pretty important to give a description about the nature of CH L 0 (X) since E 0,2d This article is organized as follows: in section 2, we present the preliminaries. In the beginning we present the definition of profinite cohomology and a particular case of it, as Galois cohomology. We start by definingétale motivic and Lichtenbaum cohomology as an analogue of motivic cohomology and state some facts that we use throughout the present article. Then we construct the Hochschild-Serre spectral sequence for Lichtenbaum cohomology. After that we discuss and recall the formulas of a projective vector bundle, blow-up with smooth center and varieties X which admit cellular decomposition, for motivic,étale motivic and Lichtenbaum cohomology. ∞ (d) ֒→ E 0,2d 2 (d) = CH d (Xk)[1/p] G k , Section 3 is devoted to the discussion of birational invariance of CH L 0 (X) presenting cases where it is known to be a birational invariant and showing that this is not always the case for some non-algebraically closed fields. In sections 4 and 5, we study the group of zero cycles forétale Chow groups. In section 4, we give the definition of theétale degree map and relate the filtration on CHé t 0 (X) induced by the Hochschild-Serre spectral sequence and its factorization through the E 0,2d ∞ (d) term i.e. a subgroup of the fixed points of the Galois action on CH 0 (Xk). Section 5 deals with the examples of varieties X over a field k with Ié t (X) = 1 but I(X) > 1. For that, we present Proposition 5.3 which is a general statement which characterizes some of the varieties X whoseétale degree equals to 1. After that we apply Proposition 5.3 to the examples given in [CM04] and [Col05,Theorem 5.1] to obtain Theorem 5.4 and 5.5. Then we move to varieties X which do not have Ié t (X) equals to one. We give the bound for suchétale degree of Severi-Brauer variety as in Theorem 5.10 and then we continue with the product of a Severi-Brauer varieties. In order to achieve this we present the generalization Lemma 5.14 and then in Theorem 5.15 we obtain the bound for Ié t (X n ). Conventions Let k be a field, we denote as k s andk the separable and algebraic closure of k respectively. For a prime number ℓ, we denote the ℓ−cohomological dimension of k as cd ℓ (k), and we set the cohomological dimension of k to be cd(k) := sup ℓ {cd ℓ (k)}. Let G be an abelian group, ℓ a prime number and r ≥ 1, then we denote G[ℓ r ] := {g ∈ G | ℓ r · g = 0}, G{ℓ} := r G[ℓ r ], G tors denotes the torsion sub-group of G. Continuing with the same hypothesis for G, for an integer p, we set G[1/p] := G ⊗ Z Z[1/p]. The prefix "L-" indicates the respective version of some result, conjecture, group, etc... in the Lichtenbaum setting. If now G is a profinite group, i.e. can be written as G = lim ← − G i with G i finite groups, and A is a G-module we will consider its cohomology group H j (G, A) as the continuous cohomology group of G with coefficients in A defined as H j (G, A) := lim − → H j (G i , A Hi ) with H i running over the open normal subgroups of G such that G/H i ≃ G i . Sm k will denote the category of smooth schemes over k and Xé t denotes the smallétale site of X. Preliminaries We start considering a profinite group G = lim ← − G i , we present a useful fact about continuous cohomology of profinite groups with coefficients in a uniquely divisible module, which is used several times throughout the present article: Let us recall the definition of Galois cohomology. Let k be a field, fix a separable closure denoted by k s and denote by G k its Galois group. Our main interest is the study of the cohomology of the group G k . For a finite Galois extension K/k we denote as Gal(K/k) the Galois group of K and recall that G k ≃ lim ← − Gal(K/k) where K runs through the finite Galois extensions of k, thus, a profinite group. The importance of this fact throughout the paper is reflected in the relationship between Galois cohomology and Lichtenbaum cohomology groups through a Hoschschild-Serre spectral sequence. Now let us give a quick review ofétale motivic cohomology and their most used properties during the present article. In this subsection we use the category ofétale motives, since we do not mention much more details about the construction and/or functorial behaviour of the category, for further details about these properties we refer the reader to [Ayo14] and [CD16]. Let k be a perfect field and let R be a commutative ring. We denote the category of effective motivicétale sheaves with coefficients in R over the field k as DM ef et (k, R) and if we invert the Lefschetz motive, we then obtain the category of motivicétale sheaves with coeffcients in R denoted by DMé t (k, R). For simplicity, we denote DM where Mé t (X) = ρ * M (X) with ρ is the canonical map associated to the change of topology ρ : (Sm k )é t → (Sm k ) Nis which induces an adjunction ρ * := Lρ * : DM(k) ⇄ DMé t (k) : Rρ * =: ρ * . In particular we define theétale Chow groups of codimension n as theétale motivic cohomology in bi-degree (2n, n) with coefficients in Z, i.e. CH ń et (X) : = H 2n M,ét (X, Z(n)) = Hom DMé t (k) (Mé t (X), Z(n)[2n]). Remark 2.2. (1) Let k be a field, let ℓ be a prime number different from the characteristic of k and r ∈ N. By the rigidity theorem for torsion motives, see [CD16,Theorem 4.5.2], we have an isomorphism H m M,ét (X, Z/ℓ r (n)) ≃ H ḿ et (X, µ ⊗n ℓ r ). (2) Notice that Mé t (X) (and also M (X)) can also be defined when X is singular, but for simplicity during this work we consider X smooth. We consider a second notion of theétale version of Chow groups, which is the well known Lichtenbaum cohomology groups, groups defined by the hypercohomology of theétale sheafification of Bloch's complex sheaf. These groups are characterized by Rosenschon and Srinivas in [RS16] usingétale hypercoverings. In this context, we consider Sm k as the category of smooth separated k−schemes over a field k. We denote z n (X, •) the cycle complex of abelian groups defined by Bloch z n (X, •) : · · · → z n (X, i) → · · · → z n (X, 1) → z n (X, 0) → 0 where the differentials are given by the alternating sum of the pull-backs of the face maps and whose homology groups define the higher Chow groups CH n (X, m) = H m (z n (X, •)). Let us recall that z n (X, i) and the complex z n (X, •) is covariant functorial for proper maps and contravariant functorial for flat morphisms between smooth k-schemes, see [Blo10, Proposition 1.3], therefore for a topology t ∈ {flat,ét, Nis, Zar} we have a complex of t-presheaves z n (−, •) : U → z n (U, •). In particular the presheaf z n (−, i) : U → z n (U, i) is a sheaf for t ∈ {flat,ét, Nis, Zar}, see [Gei04, Lemma 3.1], and then z n (−, •) is a complex of sheaves for the smallétale, Nisnevich and Zariski sites of X. We set the complex of t-sheaves R X (n) t = (z n (−, •) t ⊗ R) [−2n] where R is an abelian group and for our purposes we just consider t = Zar orét and then we compute the hypercohomology groups H m t (X, R X (n) t ). For example, setting t = Zar and R = Z the hypercohomology of the complex allows us to recover the higher Chow groups CH n (X, 2n−m) ≃ H m Zar (X, Z(n)). We denote the motivic and Lichtenbaum cohomology groups with coefficients in R as H m M (X, R(n)) = H m Zar (X, R(n)), H m L (X, R(n)) = H ḿ et (X, R(n) ) and in particular we set CH n L (X) := H 2n L (X, Z(n)). Let ρ : Xé t → X Zar be the canonical morphism of sites, then the associated adjunction formula Z X (n) → Rρ * ρ * Z X (n) = Rρ * Z X (n)é t induces comparison morphisms H m M (X, Z(n)) κ m,n − −− → H m L (X, Z(n)) for all bi-degrees (m, n) ∈ Z 2 . We can say more about the comparison map: due to [Voe11, Theorem 6.18], the map κ m,n : H m M (X, Z(n)) → H m L (X, Z(n)) is an isomorphism for m ≤ n + 1 and a monomorphism for m ≤ n + 2. If R is torsion then we can compute the Lichtenbaum cohomology as anétale cohomology. To be more precise for a prime number ℓ, r ∈ N ≥ 1 and R = Z/ℓ r then we have the following quasi-isomorphisms (Z/ℓ r ) X (n)é t ∼ − → µ ⊗n ℓ r if char(k) = ℓ ν r (n)[−n] if char(k) = ℓ where ν r (n) is the logarithmic de Rham-Witt sheaf. After passing to direct limit we have also quasi-isomorphisms (Q ℓ /Z ℓ ) X (n)é t ∼ − → lim − →r µ ⊗n ℓ r if char(k) = ℓ lim − →r ν r (n)[−n] if char(k) = ℓ and finally set (Q/Z) X (n)é t = (Q ℓ /Z ℓ ) X (n)é t ∼ − → Q/Z(n)é t . In the case when k =k then for a smooth projective variety X and n ≥ dim(X) by the Suslin rigidity theorem, the morphism Z X (n) → Rρ * Z X (n)é t is a quasi-isomorphism. For this, see [VSF00, Section 6, Theo. 4.2] and [Gei18, Section 2], and for a proof we refer to [Ros22, Lemma 2.2.2]. Another important reminder concerns the vanishing of higher Chow groups. Following [MVW06, Theorem 3.6] for every smooth scheme and any abelian group R, we have H m M (X, R(n)) = 0 when m > n + dim(X). Also we have a second vanishing theorem for motivic cohomology, presented in [MVW06,Theorem 19.2], for X and R under the same assumptions as before, we have that H m M (X, R(n)) = 0 when m > 2n. Remark 2.3. Let k =k. Since the map Z k (n) → Rρ * Z k (n)é t is a quasi-isomorphism for all n ≥ 0 we obtain that H m L (Spec(k), Z(n)) ≃ H m M (Spec(k), Z(n)) for all (m, n) ∈ Z × N. In particular H m L (Spec(k), Z(n)) = 0 for m > n ≥ 0. By pursuing a similar vanishing theorem for Lichtenbaum cohomology is that we obtain the following results about the vanishing of the cohomology groups: Lemma 2.4. Let k be a field and let X be in SmProj k . Consider a bi-degree (m, n) ∈ Z 2 we then have the following: (1) if m > n and m > cd(k) + 1 we have that H m L (Spec(k), Z(n)) = 0. (2) More generally if m > n + cd(X) then H m L (X, Z(n)) = 0. Proof. Let k be a field of characteristic exponent p and let (m, n) ∈ Z 2 . For (1) and following [Voe11, Theorem 6.18] we obtain that if m ≤ n + 1 then H m M (k, Z[1/p](n)) ≃ H m M,ét (k, Z(n)) and in particular H n+1 M,ét (k, Z(n)) = 0. Now consider the exact triangle Z(n)é t → Q(n)é t → Q/Z(n)é t +1 − − → which induces a long exact sequence . . . → H m L (k, Z(n)) → H m L (k, Q(n)) → H m L (k, Q/Z(n)) → H m+1 L (k, Z(n)) → . . . Considering the previous remark concerning the vanishing of higher Chow group and Lichtenbaum cohomology we obtain an isomorphism H m L (k, Z(n)) ≃ H m−1 et (k, Q/Z(n)) where n ∈ N and m > n, with the later isomorphism we conclude that H m L (k, Z(n)) = 0 if m > n and m > cd(k) + 1. For the more general case presented in (2) consider X be SmProj k and consider the motivic complex Z(n). The complex vanishes for degrees greater than n. Let us consider the canonical map ρ : Xé t → X Zar , the functor that is induced by the change as ρ * : D(AbShv Zar (Sm k )) ⇆ D(AbShvé t (Sm k )) : Rρ * . Recall that H m L (X, Z(n)) is the hypercohomology of the complex ofétale sheaves Z X (n)é t . Since the functor ρ * is exact, theétale cohomology sheaves of Z X (n)é t vanish in cohomological degree > n. Thus, we conclude that H m L (X, Z(n)) = 0 for m > n + cd(X). Let us denote the Suslin-Voevodsky motivic complex of Nisnevich sheaves in Sm k as Z SV (n). Since Z X (n)é t ∼ − → Z SV (n) Xé t is a quasi-isomorphism then we have a comparison map ρ m,n : H m L (X, Z(n)) → H m M,ét (X, Z(n)) which is induced by the quasi-isomorphism Z X (n)é t ∼ − → Z SV (n) Xé t and Z SV (n)é t → L A 1 (Z SV (n)é t ) where L A 1 is the A 1 −localization functor ofétale motivic complexes. According to [CD16, Theorem 7.1.2] the morphism ρ m,n becomes an isomorphism after inverting the characteristic exponent of k. If p equals the field characteristic, therefore by using Z[1/p] X (n)é t we can recover the functorial properties ofétale motivic cohomology for Lichtenbaum cohomology. The later isomorphism after inverting the exponential characteristic of the field gives us an important tool for the study of theétale motivic cohomology, which is the relationship between Galois cohomology and Lichtenbaum cohomology groups via the Hochschild-Serre spectral sequence for Lichtenbaum cohomology: E p,q 2 (n) = H p (G, H q L (Y, Z(n))) =⇒ H p+q L (X, Z(n)) . Proof. Let p : Y → X be a Galois covering with X be a smooth projective k−variety and G the Galois group associated to the covering. Let C X := Ch(Shvé t (X)) be the category of cochain complexes of abelianétale sheaves. Consider the composite functor Shvé t (X) → Z[G]-mod → Ab F → F (Y ) → F (Y ) G which is Γ(X, −) by [Mil80, Proposition II.1.4] , therefore for C • ∈ C X we have a spectral sequence, [Wei94, Section 5.7] associated with such functor E p,q 2 = H p (G, H q et (Y, C • )) =⇒ H p+q et (X, C • ). Our main interest is the case when we consider C • as theétale sheafification of Bloch complex Z X (n)é t for some n, so from now on we consider C • = Z X (n)é t . To argue about the convergence of the spectral sequence we use the arguments given in [Kah12, Section 2]. Due to the fact that H ḿ et (X, Z X (n)é t ) ≃ H m Zar (X, Rρ * Z X (n)é t ) with ρ : Xé t → X Zar and consider the exact triangle Rρ * Z X (n) → Rρ * Q X (n) → Rρ * Q/Z X (n) +1 − − →. Since Q X (n) ≃ Rρ * Q X (n)é t then the hypercohomology of the second and third terms are convergent and so are their respective hypercohomology spectral sequences. Remark 2.6. Let k be a field and k s be a separable closure. Since cohomology commutes with inverse limits, and considering that the absolute Galois group of k is defined as the inverse limit over the finite separable field extensions G k = lim ← −k⊂K⊂k s Gal(K/k), then again for [K : k] < ∞ we have a convergent spectral sequence H p (Gal(K/k), H q L (X K , Z(n))) =⇒ H p+q L (X, Z(n)). Mixing the compatibility of hypercohomology with inverse limits we obtain a spectral sequence for the absolute Galois group H p (G k , H q L (X k s , Z(n))) =⇒ H p+q L (X, Z(n)). In the followings, we recall a few facts about the structure of Lichtenbaum cohomology group of smooth projective varieties over an algebraically closed field. For further details about the structure and properties about Lichtenbaum cohomology we refer the reader to [Kah12,Proposition 4.17], [Gei17, Theorem 1.1] and [RS16, Theorem 3.1]. Consider X ∈ SmProj k with k =k of characteristic exponent p and consider a bi-degree (m, n) ∈ Z 2 . If m = 2n then according to [RS16,Theorem 3 .1] H m L (X, Z(n)) ⊗ Q ℓ /Z ℓ = 0 for all ℓ = p. Denoting (Q/Z) ′ = ℓ =p Q ℓ /Z ℓ we have that H m L (X, Z[1/p](n)) ⊗ (Q/Z) ′ = 0 and then 0 → H m L (X, Z(n)) tors → H m L (X, Z(n)) → H m L (X, Z(n)) ⊗ Q → 0. In fact this short exact sequence splits, so for m = 2n, H m L (X, Z(n)) is the direct sum of a uniquely divisible group and a torsion group. For the case when m = 2n + 1 we have an isomorphism H m L (X, Z(n)){ℓ} ≃ H m−1 et (X, Q ℓ /Z ℓ (n) ) again considering ℓ = p. Since for any n we have an exact triangle Z X (n)é t → Q X (n)é t → (Q/Z) X (n)é t +1 − − → and for m < 0 the group H ḿ et (X, Q/Z(n)) vanishes, then we conclude that for such m we have isomorphisms H m L (X, Z(n)) ≃ H m L (X, Q(n)) i.e. the Lichtenbaum cohomology groups with integral coefficients are Q-vector spaces, thus uniquely divisible groups. Now let us come-back to the Hochschild-Serre spectral sequence for Lichtenbaum cohomology. Assume that X is a smooth projective geometrically integral k−variety of dimension d with k a perfect field of characteristic exponentp, and letk be an algebraic closure of k with Galois group G k and denote Xk = X × Spec(k) Spec(k). For such X consider the Hochschild-Serre spectral sequence E p,q 2 (n) := H p (G k , H q L (Xk, Z[1/p](n))) =⇒ H p+q L (X, Z[1/p](n) ) with the previous recall, we can give information about the vanishing of some E p,q 2 (n)−terms: • E p,q 2 (n) = 0 for p < 0 because we work with the cohomology of a profinite group. • E p,q 2 (n) = 0 for p > 0 and q < 0 by the uniquely divisibility of H q L (Xk, Z[1/p](n)). • E p,q 2 (n) = 0 for p > cd(k) and q = 2n. Indeed, as q = 2n then H q L (Xk, Z[1/p](n)) ≃ H q L (Xk, Q(n)) ⊕ H q L (Xk, Z[1/p](n)) tors , since H q L (Xk, Q(n)) is uniquely divisible, so for a pair (p, q) satisfying the above restrictions, we have that H p (G k , H q L (Xk, Z[1/p](n))) ≃ H p (G k , H q L (Xk, Z[1/p](n)) tors ). Now, if p > cd(k), the group H p (G k , H q L (Xk, Z[1/p](n)) tors ) vanishes. Example 2.7. For instance if we assume that cd(k) ≤ 2 and q < 2n, then we have the following isomorphisms E 0,q ∞ (n) = ker d 2 : E 0,q 2 (n) → E 2,q−1 2 (n) = ker d 2 : H q (Xk, Z(n)) G k → H 2 (G k , H q−1 L (Xk, Z(n))) E 1,q ∞ (n) ≃ E 1,q 2 (n) E 2,q ∞ (n) ≃ E 2,q 2 (n)/im E 0,q+1 2 (n) → E 2,q 2 (n) = H 2 (G k , H q L (Xk, Z(n)))/im H q+1 L (Xk, Z(n)) G k → H 2 (G k , H q L (Xk, Z(n))) . We conclude this section by mentioning some well-known results about the structure ofétale motivic and Lichtenbaum cohomology groups of projective bundles, smooth blow-ups and varieties with cellular decomposition: Lemma 2.8. Let k be a field of characteristic p ≥ 0 and let X be a smooth projective scheme over k. Let ε ∈ {L, {M,ét}} and consider a bi-degree (m, n) ∈ Z 2 , then there exists the following characterizations: (1) If r ≥ 0 and let P r X be the projective space of dimension r over X, then the canonical map P r X → X induces an isomorphism: H m ε (P r X , Z(n)) ≃ r i=0 H m−2i ε (X, Z(n − i)). (2) Let Z be a smooth projective sub-scheme of X of codimension c ≥ 2. Denote the blow-up of X along Z as Bl Z (X), then H m ε (Bl Z (X), Z(n)) ≃ H m ε (X, Z(n)) ⊕ c−1 i=1 H m−2i ε (Z, Z(n − i)). (3) Assume that k = k s and that there exists a map f : X → S which is a flat of relative dimension r over a smooth base S. Assume as well that X has a filtration X = X p ⊃ X p−1 ⊃ . . . ⊃ X 0 ⊃ X −1 = ∅ where X i is smooth and projective for all i and U i : = X i − X i−1 ≃ A r−di S then we obtain the following formula: H m L (X, Z[1/p](n)) ≃ p i=0 H m−2di L (S, Z[1/p](n − d i )). Proof. The statements (1) and (2) M (X)(i)[2i] ≃ − → M (P r X ) and M (Bl Z (X)) ≃ M (X) ⊕ c−1 i=1 M (Z)(i)[2i] . When ε = L both formulas (1) holds because for R = Q we recover the formulas for rational coefficients whereas for finite coefficients we invoke [ Meanwhile for ε = {M,ét} this holds because of the previous isomorphisms when R = Z and the fact that the functor ρ * : DM(k) → DMé t (k) is exact. For (3) we have to invert the characteristic of k. We will proceed as in [Köc91,Appendix], by induction and use the localization long exact sequence. By the notation we have that dim(X j ) = dim(S) + n − d j and for k < j we have codim(X k , X j ) = d k − d j , denote the last quantity as c k,j . Notice that d j is the codimension of X j in X. Let us start saying that due to the homotopy invariance of higher Chow groups and [Mil80, VI, Corollary 4.20], the map π * 0 : H m L (S, Z[1/p](n)) → H m L (X 0 , Z[1/p](n)) is an isomorphism for all bi-degree. Denote π j : U j → S. Consider the following commutative diagram with exact rows: 0 j−1 i=0 H m−2cj,i,n−cj,i L (S) j i=0 H m−2cj,i,n−cj,i L (S) H m,n L (S) 0 . . . H m−2cj−1,j ,n−cj−1,j L (X j−1 ) H m,n L (X j ) H m,n L (U j ) . . . ≃ (πj ) * i * j * where H m,n L (Y ) := H m L (Y, Z[1/p](n)). By inductive hypothesis the right vertical arrow is an isomorphism, while the left one is because of the invariance of theétale motivic cohomology, therefore the map j * is surjective and i * is injective. Thus we obtain the desired formula. But in both cases we need to work away from the characteristic of k. Example 2.10. By Lemma 2.8, the Lichtenbaum cohomology groups of the projective space over a field k are the following CH m L (P l k ) ≃ m j=0 CH j L (Spec(k)) . For i ≥ 2 we have that CH i L (Spec(k)) ≃ H 2i Zar (Spec(k), τ ≥i+2 Rπ * Z(i)é t ). Meanwhile, by the vanishing of motivic cohomology, we have that CH j (Spec(k)) ≃ 0 for j ≥ 1, thus we obtain that CH i L (Spec(k)) ≃ H 2i−1 et (Spec(k), Q/Z(i)). Birational invariance Let us recall some definitions of birational geometry. Let X, Y smooth k-varieties, we say that a rational map f : X → Y is birational if there exists open subsets U ⊂ X and V ⊂ Y such that f : U → V is an isomorphism. We say that X is stably birational to Y if there exist r, s ∈ N such that X × P r k → Y × P s k is a birational morphism. The importance of CH 0 (X) relies in its birational invariance, for which we refer to [Ful98, Example 16.1.11]. Now assume that X → Y is stably birational, then there exist r, s ∈ N such that X × P r k → Y × P s k is a birational and since CH 0 is a birational invariant we obtain an isomorphism CH 0 (X × P r k ) ≃ − → CH 0 (Y × P s k ), but by the projective bundle formula for Chow groups and the vanishing properties we obtain that CH 0 (X × P r k ) ≃ CH 0 (X) and CH 0 (Y × P s k ) ≃ CH 0 (Y ) so CH 0 (X) ≃ CH 0 (Y ). So CH 0 is also a stable birational invariant. Remark 3.1. The reference [Ful98, Example 16.1.11] is given for algebraically closed fields, but the same argument works for fields in general. The first question that arises is whether or not CH L 0 (X) (or CHé t 0 (X)) is a birational invariant or a stably birational invariant. Let X be a smooth projective variety over a field k, because of the comparison map CH 0 (X) → CH L 0 (X) we can say a few words about the invariance depending on the field and the dimension of X: if k =k then CH d (X) ≃ CH d L (X), thus we can use the stable birational invariance of zero cycles in the classical setting cited above, for the category SmProj k . If the field is not algebraically closed, then we lost many of the properties. For example, consider k a number field which can be embedded into R and d ≥ 2, by invoking Lemma 2.8 and the vanishing properties of Lemma 2.4, we see immediately that CH 0 L (Spec(k)) = CH d L (P d k ) ≃ d i=0 CH i L (Spec(k)). Thus CH L 0 is not a stable birational invariant. If now we just focus in the study about the birational invariance of CH L 0 (X), we then have the following result: Proposition 3.2. Let k be an arbitrary field and let X be a smooth projective scheme of dimension d over k. Then CH L 0 is a birational invariant if d ∈ {0, 1, 2}. Proof. The case d = 0 is trivial. If d = 1, we use the isomorphism CH 1 (X) ≃ CH 1 L (X) and the birational invariance of zero cycles in the classical case. For d = 2 we have a short exact sequence 0 → CH 2 (X) → CH 2 L (X) → H 3 nr (X, Q/Z(2)) → 0. The group CH 2 (X) is a birational invariant for surfaces and the unramified cohomology groups H 3 nr (X, Q/Z(2)) is birational invariant for any dimension, this is a consequence of the Gersten's conjecture, see [CV12, Théorème 2.8], therefore CH 2 L (X) is a birational invariant. If we go to higher dimension the argument using the comparison map fails. To illustrate that consider the following: Let X be a smooth projective variety of dimension three over a field k. There is a long exact sequence → H 5 Zar (X, τ ≥5 Rπ * Z(3)é t ) → CH 3 (X) → CH 3 L (X) → H 6 Zar (X, τ ≥5 Rπ * Z(3)é t ) → 0. We have that H 5 Zar (X, τ ≥5 Rπ * Z(3)é t ) ≃ H 4 nr (X, Q/Z(3)) is a birational invariant. Therefore CH 3 L ( X) is a birational invariant if and only if H 6 Zar (X, τ ≥5 Rπ * Z(3)é t ) is a birational invariant. We obtain a short exact sequence, 0 → H 1 Zar (X, H 4 et (Q/Z(3))) → H 6 Zar (X, τ ≥5 Rπ * Z(3)é t ) → E 0,6 ∞ → 0 where E 0,6 ∞ = ker H 5 nr (X, Q/Z(3)) → H 2 Zar (X, H 4 et (Q/Z(3)) ) . In fact, one can find the first counter-example in dimension 3. Recall that by the formulas given in Lemma 2.8 we have the following: let X be a smooth projective variety and let Z ⊂ X a smooth sub-variety of codimension c. Then for the blow-upX Z of X along Z, Lichtenbaum cohomology decomposes as follows CH d L (X Z ) ≃ CH d L (X) ⊕ c−1 j=1 CH d−j L (Z) Notice that d − j > d − c = dim(Z), therefore the groups CH d−j L (Z) are just torsion isomorphic to H 2(d−j) Zar (X, τ ≥d−j+2 Rπ * Z(d − j)é t ). The next example shows how to exploit this fact to get the announced counter-example. Example 3.3. Consider X a smooth threefold with a rational point over K, with K an algebraic number field which is not totally imaginary, and let Z = Spec(K). LetX Z be the blow-up with center Z, then we have that CH 3 L (X Z ) = CH 3 L (X) ⊕ CH 2 L (Spec(K)). Since CH 2 L (Spec(K)) ≃ H 3 et (Spec(K), Q/Z(2)) then we can conclude that CH 3 L (X Z ) = CH 3 L (X). In general we have the proposition: Proposition 3.4. Let k be a field and assume that there exists n ≥ 2 such that the group H 2n−1 et (Spec(k), µ ⊗n ℓ r ) = 0 for some prime number ℓ and r ∈ N, then CH L 0 is not a birational invariant for SmProj k . Proof. Let us consider the field k such that H 2n−1 et (Spec(k), µ ⊗n ℓ r ) = 0 for some prime number ℓ, r ∈ N and n ≥ 2. Consider X a smooth projective variety over k of dimension d ≥ n + 1 such that X has a k-rational point. LetX be the blow-up of X along a point Z = Spec(k) → X. Invoking Lemma 2.8 we obtain CH d L (X) ≃ CH d L (X) ⊕ d−1 j=1 CH d−j L (Z) As CH i L (Z) ≃ H 2i−1 et (Z, Q/Z(i)) for i ≥ 2, the hypothesis implies that CH n L (Z) = 0 and thus CH d L (X) = CH d L (X). Remark 3.5. We need to remark, that the hypothesis of last proposition impose the restriction that cohomological dimension of k should be ≥ 3. Thus the previous argument does not give a counter-example for fields with cohomological dimension ≤ 2. 4.Étale degree map Let p : X → Spec(k) be the structural morphism associated to a smooth and projective k-scheme of dimension d. Recall that the degree map is defined as deg := p * : CH 0 (X) → CH 0 (Spec(k)) = Z. We can reformulate this definition due to the existence of Gysin morphisms in DM(k) as is described in [Dég12] and [Dég08]. With this formalism we obtain the pull-back of the morphism p defined as (1) Let k be a field of characteristic exponent p. Due to functoriality properties we have the following commutative diagram Hom DM(k) (M (Y ), Z(d)[2d]) Hom DM(k) (Z(d)[2d], Z(d)[2d]) Hom DMét(k) (Mé t (Y ), Z(d)[2d]) Hom DMét(k) (Z(d)[2d], Z(d)[2d])CH d (X K ) CH d (X) Z Z CH d et (X K ) CH d et (X) f * deg deg [K:k]· f * degé t degé t with [K : k] the degree of the extension. (3) It is possible to define theétale degree map for Lichtenbaum cohomology over a field k =k. This follows because for X a smooth and proper projective variety of dimension d there is a quasi-isomorphism Z X (n) Zar → Rπ * Z X (n)é t for n ≥ d. In general we have to invert the characteristic exponent of k and use the isomorphism between Lichtenbaum andétale Chow groups. Let f : X → Y be a projective morphism of smooth varieties of relative dimension c. Again by the existence of Gygin morphisms in DMé t (k), we obtain push-forwards forétale motivic cohomology f * : H m+2c M,ét (X, Z(n + c)) → H m M,ét (Y, Z(n)) . Combining the existence of push-forward maps forétale motivic cohomology and the functoriality of the Hochschild-Serre spectral sequence we obtain the following diagram In fact we will prove that the subgroup in question is given by the E 0,2d ∞ -term of the Hochschild-Serre spectral sequence associated to X. To see this, consider the structural morphism f : X → k, then we have an induced morphism of E 2 -terms E p,q 2 := H p (G k , H q L (Xk, Z[1/p](d))) → H p (G k , H q−2d L (Spec(k), Z[1/p](0))) but as q − 2d ≤ 0 we have that H q−2dX L (Spec(k), Z[1/p](0)) ≃ 0 for q = 2d Z[1/p] for q = 2d that gives us H p L (k, Z[1/p](0)) ≃ H p (G k , H 0 (k, Z[1/p](0))) and hence we conclude that degé t : CH d et (X) → Z[1/p] factors as CH d et (X) E 0,2d ∞ Z[1/p] degé t deg where deg is the composite map E 0,2d ∞ ֒→ E 0,2d 2 = CH d (Xk)[1/p] G k ֒→ CH d (Xk)[1/p] deg − − → Z[1/p]. Lichtenbaum zero cycles 5.1. Varieties where Ié t (X) = 1. This subsection aims to construct examples where theétale degree map is surjective but its classical counterpart is not. In order to achieve that, we start by giving a lemma about the divisibility of the zero cycles of degree 0 of a variety over an algebraically closed field: Lemma 5.1. Let X be a complete scheme over an algebraically closed field k of characteristic p ≥ 0. Define A 0 (X) = ker {deg : CH 0 (X) → Z}, then A 0 (X) is a divisible group. If X is a smooth quasi-projective scheme and H 2d−1 et (X, Q ℓ /Z ℓ (d)) = 0 for ℓ = p then A 0 (X) ·ℓ r − − → A 0 (X) is an isomorphism for all r ∈ N. Proof. The first statement is classic, see [Ful98, Example 1.6.6], the argument goes as follows: as A 0 (X) is generated by the image of the maps of the form: f * : A 0 (C) → A 0 (X) [P ] − [Q] → f * ([P ] − [Q]) where f : C → X a smooth projective curve with P , Q points in C. Since A 0 (C) ≃ J(C) and the Jacobian of a smooth projective curve is divisible over an algebraically closed field k, we obtain the desired result. We prove the second assertion. Notice that by the assumption that k is an algebraically closed field, one gets that CH d (X) ≃ CH d L (X) and that CH d L (X){ℓ} ≃ H 2d−1 et (X, Q ℓ /Z ℓ (d)). Therefore CH 0 (X){ℓ} = CH 2d (X){ℓ} ≃ H 2d−1 et (X, Q ℓ /Z ℓ (d)) = 0 and CH 0 (X){ℓ} ≃ A 0 (X){ℓ}, so one deduces that under the assumption, A 0 (X) is ℓ r -divisible for any r > 0. Remark 5.2. Notice that with the previous statement, if H 2d−1 et (X, Q ℓ /Z ℓ (d)) = 0 for all ℓ different from the characteristic of k, we conclude that A 0 (X) is uniquely ℓ r -divisible. For X a smooth and projective variety over a field k of exponential characteristic equal to p, we set A 0,ét (X) := ker degé t : CH d et (X) → Z[1/p] . Notice that if k is algebraically closed then we have an isomorphism A 0,ét (X) ≃ A 0 (X)[1/p]. Proposition 5.3. Let X be a geometrically integral smooth projective variety of dimension d ≥ 2 over a perfect field k with cd(k) ≤ 1 and p the exponential characteristic of k. Letk be the algebraic closure of k and assume that H 2d−1 et (Xk, Q ℓ /Z ℓ (d)) = 0 for every prime ℓ = char(k), then degé t : CH d et (X) → Z[1/p] is surjective. Proof. First assume that char(k) = 0, then CH n L (X) ≃ CH ń et (X) for all n ∈ N. Using the notation given in Proposition 2.5, if cd(k) ≤ 1 then E 2,q 2 (n) = 0 for 1 < q < 2n so by the characterizations of the infinity terms given in Example 2.7, we obtain a short exact sequence 0 → H 1 (G, H 2n−1 L (Xk, Z(n))) → CH n L (X) → CH n L (Xk) G k → 0. For n = d we have that CH d L (X) → CH d L (Xk) G k is always surjective, now consider the short exact sequence 0 → A 0 (Xk) → CH d L (Xk) degé t − −− → Z → 0 where A 0 (Xk) := ker degé t : CH d et (Xk) → Z , i.e. the numerically trivial zero cycles of Xk, which induces a long exact sequence 0 → A 0 (Xk) G k → CH d L (Xk) G k deg − − → Z → H 1 (G k , A 0 (Xk)) → . . . Where the factor Z is obtained by using the fact that CH 0 (Spec(k)) G k ≃ CH 0 (Spec(k)). By [RS16, Proposition 3.1(a)] we have that CH d L (Xk){ℓ} ≃ H 2d−1 et (Xk, Q ℓ /Z ℓ (d)) so A 0 (Xk) tors ≃ CH d L (Xk) tors = 0 and then the group A 0 (Xk) is uniquely divisible, so we conclude that H 1 (G, A 0 (Xk)) = 0. Consequently the map degé t : CH d L (X) → CH d L (Xk) G k → Z is surjective. Now assume that char(k) = p > 0, in this case it is necessary to invert the exponential characteristic of the field. For an abelian group A we put A[1/p] := A ⊗ Z Z[1/p]. Setting q = 2d, we have that H q L (Xk, Z(d)) is a extension of a divisible groups D by a torsion groups T . Using the convention for tensor product, we notice that 0 → D → H q L (Xk, Z(d))[1/p] → T [1/p] → 0 where the last map kills the p-primary part of the torsion group T . Also the spectral sequence holds for the complex ofétale sheaves Z[1/p](n)é t , for the convergence we use the same arguments with the exact triangle Z[1/p] X (d)é t → Q X (d)é t → ℓ =char(k) Q ℓ /Z ℓ (d) +1 − − → therefore we have a similar short exact sequence 0 → H 1 (G k , H 2n−1 L (Xk, Z[1/p](n))) → CH n L (X)[1/p] → CH n L (Xk)[1/p] G k → 0 and also 0 → A 0 (Xk)[1/p] → CH d L (Xk)[1/p] deg L − −− → Z[1/p] → 0 therefore we can conclude. Theorem 5.4. There exist a smooth projective surface S over a field k, with char(k) = 0 of cohomological dimension ≤ 1, without zero cycles of degree one but Ié t (X) = 1. Proof. By [CM04, Théorème 1.1] and [CM04, Théorème 1.2] there exist del Pezzo surfaces of degree 2, 3 and 4 over a field k of characteristic zero and cd(k) = 1 without zero cycles of degree 1. Let S be one of such surfaces of degree d ∈ {2, 3, 4}. Since S is a del Pezzo surface, thus for all field extension K/k the variety S K is a del Pezzo surface of degree d as well, so in particular for K =k. As Sk is del Pezzo, we have that H 1 (Sk, O Sk ) = H 2 (Sk, O Sk ) = 0 therefore Alb(Sk) = 0. Since we are working over an algebraically closed field, CH 2 (Sk) ≃ CH 2 L (Sk) and then by Roitman's theorem CH 2 L (Sk) tors = N 2 (Sk) tors = 0 so the group N 2 (Sk) is uniquely divisible and consequently by Proposition 5.3 the map CH 2 L (S) → CH 2 L (Sk) G k → Z is surjective while CH 2 (S) → Z is not a surjective map. Theorem 5.5. For each prime p ≥ 5 there exist a field F such that char(k) = 0 with cd(F ) = 1 and a smooth projective hypersurface X ⊂ P p F with Ié t (X) = 1 but index I(X) = p. Proof. Let us consider n ≥ 2, a field k such that cd(k) ≤ 1 and a hypersurface X ⊂ P n+1 k that is geometrically integral. Consider the hypersurface Xk ⊂ P Remark 5.6. Assume that k is a field with cd(k) ≤ 1. Consider S a smooth geometrically integral k-surface with H 1 (S, O S ) = 0 , therefore Alb(S) = 0 so again by Roitman theorem CH 2 L (Sk) is torsion free and then uniquely divisible, so H 1 (G, N d (Sk)) = 0 and then CH 2 L (S) → Z is surjective. In general if N d (Xk) is a divisible group then CH d L (X) → Z is surjective. 5.2.Étale degree of Severi-Brauer varieties. In the following, we will see non-trivial examples where theétale degree map is not surjective. For that we will study the Lichtenbaum cohomology groups of Severi-Brauer varieties by giving an explicit characterization of the zero cycles of Lichtenbaum groups of Severi-Brauer varieties. Definition 5.7. A variety X over a field k is called a Severi-Brauer variety of dimension n if and only if Xk ≃ P n k . If X is a Severi-Brauer variety of dimension n and there exists an algebraic extension k ⊂ k ′ ⊂k such that X k ′ ≃ P n k ′ we say that X splits over k ′ . If for a field k, such that, the Brauer group Br(k) = 0, there exists a unique Severi-Brauer variety modulo isomorphisms to P n k . Some cases of such fields are the following: • a field k with cd(k) ≤ 1. In this category we can find fields such as algebraically and separable closed fields, finite fields, extensions of transcendence degree 1 of an algebraically closed field. • if k is a field extension of Q containing all the roots of unity, see [Ser68,§7] and [Ser02, II. §3, Proposition 9]. Lemma 5.8. Let X be a Severi-Brauer variety of dimension d over k which splits over a field k ′ . Then for all 0 ≤ n ≤ d the group CH n (X k ′ ) ≃ CH n (P d k ′ ) is a trivial Gal(k ′ /k)-module. Proof. First consider d = 1, then CH 1 (P d k ′ ) ≃ Pic(P d k ′ ) ≃ Z. Following the argument given in [GS06,Proposition 5.4.4] the action of Gal(k ′ /k) over Pic(P d k ′ ) is trivial as the only non-trivial action would permute 1 with −1. The vector bundles of P d k ′ with Chern class 1 can not permute with the one in the class −1 due to the existence of global sections for the first case. For the general cases when n = 1 we consider the isomorphisms CH 1 (P d k ′ ) ≃ CH n (P d k ′ ) given by the intersection with hyperplanes. Remark 5.9. (1) We can similarly deduce that for all m, n ∈ N the group Pic(P m k × P n k ) ≃ Z[α] ⊕ Z[β], where α and β are the generators of Pic(P m k ) and Pic(P n k ) respectively, is a trivial G k -module. (2) Let k be a perfect field of characteristic exponent equals top and let X be a Severi-Brauer variety of dimension d over k. The fact Xk ≃ P d k simplifies several computations for the Hochschild-Serre spectral sequence given in Proposition 2.5. For instance if m = 2n + 1, then for ℓ =p we can characterize the ℓ-primary torsion groups as follows H m L (Xk, Z(n)){ℓ} ≃ H m−1 et (P d k , Q ℓ /Z ℓ (n)) ≃ Q ℓ /Z ℓ if m is odd 0 otherwise. Therefore for m < 2n and even the group H m L (Xk, Z(n)) is uniquely divisible, thus some of the E 2 (n)-terms associated to the Hochschild-Serre spectral sequence of H p+q L (X, Z[1/p](n)) can be characterized in the following way E p,q 2 (n) =      H q (Pk, Z(n)) G k if p = 0, H p (G k , H q−1 et (Pk, (Q/Z) ′ (n))) if q is odd and p > 0, 0 if q is even and p > 0. Now let us set n = 1 and X a Severi-Brauer variety over k of dimension d. If we use the Hoschschild-Serre spectral sequence given in 2.5 and Lemma 5.8, we recover a classical result of Lichtenbaum, see [GS06,Theorem 5.4.10], concerning the Picard group of X and Brauer groups there is an exact sequence 0 → Pic(X) → Pic(P d k ) G k δ − → Br(k) → Br(k(X)),(1) where the map δ sends 1 to the class of X in Br(k). For an arbitrary integer n, if we apply the projective bundle formula we obtain the following H m L (P d k , Z(n)) ≃ d i=0 H m−2i L (Spec(k), Z(n − i)). As we change the base field to its algebraic closure we have that H m L (P Proof. Let X be a Severi-Brauer variety of dimension d, and consider the Hochschild-Serre spectral sequence for Lichtenbaum cohomology in two cases: when n = 1 and n = d. For n = 1 we recover (1), where some of the terms of the exact sequence come from E 0,2 2 (1) = Pic(P d k ) G k and E 2,1 2 (1) ≃ Br(k). For the case when n = d, and using the computations from the previous discussion, we obtain the following terms: E 0,2d 2 (d) = CH d (P d k ) G k and E 2,2d−1 2 (d) ≃ Br(k). Notice that the isomorphisms H 2n L (P d k , Z(n)) ≃ d i=0 H 2(n−i) L (k, Z(n − i)) for n = 1 and d induced by the map P d k → Spec(k), give us a commutative diagram where the vertical arrows are given by the intersection with the hyperplane section P d k Pic(P d k ) G k Br(k) CH d (P d k ) G k Br(k) δ ≃ ≃ d 0,2d 2 (d) Since the vertical arrows are isomorphisms, then E 0,2d 3 (d) = ker(d 0,2d 2 (d)) ≃ ker(δ) ≃ Pic(X). Now by Proposition 4.3, the map degé t factors through E 0,2d ∞ (d) which is a subgroup of E 0,2d 3 (d). The assumption about the cohomological dimension of k gives us that E 0,2d ∞ (d) ≃ E 0,2d 3 (d). For further details about this computation see the next example and proposition. Remark 5.11. Notice the following: Consider 1 ≤ n ≤ d and consider the Hochschild-Serre spectral sequence associated to CH n L (X). From the projective bundle formula we have that H 2n−1 L (P d k ) ≃ K M 1 (k) , thus E 0,2n 2 (n) ≃ Br(k) and consequently by the commutative diagram Pic(P d k ) G k Br(k) CH n (P d k ) G k Br(k) δ ≃ ≃ d 0,2n 2 (n) the term E 0,2n ∞ (n) is isomorphic to a subgroup of Pic(X). Example 5.12. Let X be a Severi-Brauer variety of dimension d = 2 over a perfect field k with Galois group G k . Using the previous characterizations through the projective bundle formula, we then describe the E 2 -terms associated to X in the following way: E p,0 2 = H p (G k , H 0 M (Spec(k), Z(2))), E p,1 2 = H p (G k , H 1 M (Spec(k), Z(2))), E p,2 2 = H p (G k , K M 2 (k)), E p,3 2 = H p (G k , K M 1 (k)), E p,4 2 = H p (G k , CH 2 L (P 2 k ) ) and E p,q 2 = 0 for q ≥ 5. By Remark 5.9 (2), we have that E p,0 2 = E p,2 2 = 0 for p > 0, also E 1,3 2 = 0 by Hilbert 90 theorem and E 2,3 2 = Br(k), obtaining with this the following terms: by trivial reasons E 1,3 ∞ = E 2,2 ∞ = E 4,0 ∞ = 0 and: E 3,1 ∞ = H 3 (G k , H 1 M (k, Z(2)))/im K M 1 (k) G k → H 3 (G k , H 1 M (k, Z(2))) The only remaining piece of the filtration of CH = ker E 0,4 4 → H 4 (G k , H 1 M (Spec(k), Z(2))) . Therefore CH 2 L (X) fits into a short exact sequence given by the filtration induced by the Hochschild-Serre spectral sequence 0 → E 3,1 ∞ → CH 2 L (X) → E 0,4 ∞ → 0. If we want to generalize such result for higher dimension, we need to impose a condition on the cohomological dimension of k: Proposition 5.13. Let X be a Severi-Brauer variety of dimension d over a perfect field k of cohomological dimension cd(k) ≤ 4. Then the group CH L 0 (X) fits in the following exact sequence 0 → E 3,2d−1 ∞ → CH d L (X) → E 0,2d ∞ → 0. with E 0,2d ∞ = ker CH d et (P d k ) G k → Br(k) and in particular Ié t (X) = ord([X] ). Proof. We follow the arguments given in example 5.12. Consider such k and X, then by hypothesis Xk ≃ P d k . By the projective bundle formula for Lichtenbaum cohomology we have that H m (P d k , Z(d)) ≃ d i=0 H m−2i L (Spec(k), Z(d − i)). Notice that by divisibility arguments we have that E p,2k 2 = 0 for 0 ≤ k ≤ d and p > 0. Under the assumption about the cohomological dimension of k we have that E p,q 2 = 0 for p > 4 and q < 2n, this results that E 0,2d ∞ ≃ E 0,2d 3 = ker CH d (P d k ) G k → H 2 (G k ,k * ) and the other E p,q 2 -terms with p + q = 2d that could not vanish are E 1,2d−1 2 and E 3,2d−3 2 , but H 2d−1 L (P d k , Z(d)) ≃ K M 1 (k) therefore E 1,2d−1 2 = 0. On the other hand, the remaining element of the filtration, which is E 3,2d−3 ∞ = E 3,2d−3 4 , is defined as E 3,2d−3 4 = E 3,2d−3 3 /im E 0,2d−1 3 → E 3,2d−3 3 = H 3 (G k , H 2d−3 M (P d k , Z(d)))/im K M 1 (k) G k → H 3 (G k , H 2d−3 M (P d k , Z(d)) ) . Using the recursive formula H m L (P n k , Z(n)) ≃ H m L (k, Z(n)) ⊕ H m−2 L (P n−1 k , Z(n − 1)). we can obtain easily H 2d−3 M (P d k , Z(d)) ≃      0 if d = 1 H 1 M (k, Z(2)) if d = 2 H 1 M (k, Z(2)) ⊕ K M 3 (k) if d ≥ 3. Again as in example 5.12, the group CH d et (X) fits into the following short exact sequence 0 → E 3,2d−3 ∞ → CH d et (X) → E 0,2d ∞ → 0. We should emphasize that as we mentioned in Proposition 4.3, theétale degree map factors through E 0,2d ∞ , having the following commutative diagram 0 E 3,2d−3 ∞ CH d et (X) E 0,2d ∞ 0 Z degé t deg where deg : E 0,2d ∞ → Z is the composition of the following maps: E 0,2d ∞ ֒→ CH d (P d k ) G k ≃ − → CH d (P d k ) deg − − → Z. As we may expect, theétale index of a product of Severi-Brauer is again bounded by the order of the Brauer class in Br(k). For the sequel we denote X ×n := n-times X × . . . × X Lemma 5.14. Let X be a Severi-Brauer variety of dimension d over a field k. Then there exists an exact sequence 0 → Pic(X × X) → Pic(P d k × P d k ) G k ≃ Z ⊕ Z s − → Br(k) → Br(X × X) where s sends (a, b) → (a + b) [X] ∈ Br(k) and [X] is the Brauer class associated to X. In general for a product X ×n we then obtain an exact sequence 0 → Pic(X ×n ) → Pic(P d k × . . . × P d k ) G k ≃ Z ⊕ . . . ⊕ Z s − → Br(k) → Br(X ×n ) where s sends (a 1 , . . . , a n ) → n i=1 a i [X] ∈ Br(k). Proof. Let Y a smooth projective variety over k. Considering the Hochschild-Serre spectral sequence E p,q 2 = H p (G k , H q L (Yk, Z(1))) =⇒ H p+q L (Y, Z(1)) we obtain the following exact sequence 0 → E 2 ∞ → E 0,2 2 → E 2,1 2 → E 3 ∞ . If Y = X ×n then Yk ≃ P d k × . . . × P d k and consequently Pic(P d k × . . . × P d k ) ≃ Z ⊕ . . . ⊕ Z and by remark 5.9 we obtain an isomorphism Pic(P d k × . . . × P d k ) G k ≃ Z ⊕ . . . ⊕ Z giving us the exact sequences of the statement. Now let us see the easiest case for Y = X × X. Consider the arrows X X × X X ∆ pr 1 pr 2 where ∆ : X → X × X is the diagonal embedding and pr i : X × X → X is the projection in the i-th component. Notice that the composition gives the identity on X. Notice that the morphism pr i : X × X → X induces a morphism pr * i : H m L (X, Z(n)) → H m L (X × X, Z(n)) and pr * i : H m L (P d k , Z(n)) → H m L (P d k × P d k , Z(n)) for all bi-degree (m, n). By functoriality properties of the Hochschild-Serre spectral sequence we have a diagram 0 Pic(X) Z Br(k) Br(X) 0 Pic(X × X) Z ⊕ Z Br(k) Br(X × X) ff s where the vertical arrows are induced by pr * i . The composition pr i • ∆ is the identity on X, thus id * = ∆ * • pr * i therefore we obtain that the maps f : Z → Z ⊕ Z andf : Br(k) → Br(k) are injective and then, the elements of the form (a, 0) and (0, b) are sent through s to a [X] and b [X] ∈ Br(k) respectively. For the general case we take the arrows where∆ is the n-diagonal morphism and pr i is the projection in the i-th component, then we conclude as in the case of X × X. Theorem 5.15. Let k be a field and let X be a Severi-Brauer variety over k of dimension d. Then Ié t (X ×n ) ≥ Ié t (X) ≥ ord([X]). Proof. Fix an integer n ≥ 1. By hypothesis we have that Xk ≃ P d k thus (X ×n )k ≃ P d k × . . . × P d k . Considering the Hochschild-Serre spectral sequence for the Lichtenbaum cohomology of X ×n E p,q 2 = H p (G k , H q L ((P d k ) ×n , Z(nd))) =⇒ H p+q L (X ×n , Z(nd)). Notice that by the projective bundle formula for Lichtenbaum cohomology we have By noticing that if 2nd − 1 − 2 d j=1 a j > nd − n j=1 a j then H 2nd−1−2 n j=1 aj L (Spec(k), Z(nd − n j=1 a j )) = 0, this give us a vanishing condition for nd − 1 > n j=1 a j . As 0 ≤ a j ≤ d for all j, then the only n-tuples (a 1 , . . . , a n ) which do not satisfy such condition are ((P d k ) ×n ), Z(nd)). By the functoriality of the Hochschild-Serre spectral sequence we obtain a commutative diagram as follows: Pic((P d k ) ×n ) G k Br(k) CH nd ((P d k ) ×n ) G k n i=1 Br(k), s g where the vertical arrows are induced by δ. According to Lemma 5.14 the arrow s sends Z⊕. . .⊕Z ∋ (α 1 , . . . , α n ) → d i=1 α i [X] ∈ Br(k). Notice that the arrow x i → x d 1 · · · x d−1 i · · · x d n induces an isomorphism CH 1 ((P d k ) ×n ) ≃ CH nd−1 ((P d k ) ×n ) and that CH nd−1 ((P d k ) ×n ) ⊗ H 1 ((P d k ) ×n , Z(1)) ≃ H 2nd−1 L ((P d k ) ×n , Z(nd)) given by the map (α 1 , . . . , α n )⊗β → β(α 1 , . . . , α n ) which is the cup product. Therefore the arrow g maps CH nd ((P d k ) ×n ) G k ∋ a → (a[X], . . . , a[X]) ∈ Br(k) giving us that ker(g) = ord([X])Z. Since E 0,2nd ∞ ֒→ E 0,2nd 3 = ker(g) and degé t factors through E 0,2nd ∞ we conclude the proof. The natural question that arises is when this bound is reached, this is the case for the product C × C when C is a smooth, geometrically connected curve of genus 0 over a field k such that Ck ≃ P 1 k as the following proposition shows: Proposition 5.16. Let k be a perfect field of characteristic p ≥ 0 with Galois group G k , and let C be a smooth, geometrically connected curve of genus 0 over the field k such that Ck ≃ P 1 k , then Ié t (C × C) = ord([C]). in other words, it is a subgroup of the zero cycles in Xk which are Galois invariant. Lemma 2 . 1 . 21Let G be a profinite commutative group and let A be a G-module which is uniquely divisible. Then H n (G, A) = 0 for all n ≥ 1. Proof. Let G be a profinite group and let H be an open normal subgroup of G. By definition we have that H j (G, A) = lim − → H H j (G/H, A H ), as G/H is a finite group, invoking [Wei94, Proposition 6.1.10] we have that the result holds for Hmodules where multiplication is an isomorphism , in particular uniquely divisible modules, therefore H j (G/H, A H ) = 0 for all H and all j > 0. The result follows from the definition of continuous cohomology. k, Z). One defines theétale motivic cohomology group of bi-degree (m, n) with coefficients in a commutative ring R as H m M,ét (X, R(n)) := Hom DMé t (k,R) (Mé t (X), R(n)[m]). Remark 2. 9 . 9When k is not separably closed, by using the homotopy invariance property in the category DMé t (k) we can extend Lemma 2.8.(3) for the higherétale Chow groups H m M,ét (X, Z(n)) ≃ p i=0 H m−2di M,ét (S, Z(n − d i )). p * : M (Spec(k))(d)[2d] = Z(d)[2d] → M (X) in the category DM eff Nis (k). Applying the contravariant functor Hom DM eff Nis (k) (−, Z(d)[2d]) we re-obtain the previous definition. From this, we can extend the existence of Gysin morphisms for DMé t (k), giving us anétale analogue of the degree map forétale Chow groups:Definition 4.1. Let X be a smooth and projective scheme of dimension d over k, where k is a field of exponential characteristic equals to p. Then we define theétale degree map degé t : CH d et (X) → CH 0 et (Spec(k)) ≃ Z[1/p] as degé t := p * where p is the structure morphism p : X → Spec(k). We define theétale index of X as the greatest common divisor of the subgroup degé t (CH d et (X)) ∩ Z, denoted by Ié t (X). CH 0 τ (Spec(k)) with τ ∈ {Nis,ét}, there are isomorphismsHom DM(k) (Z(d)[2d], Z(d)[2d]) = H 0,0 M (Spec(k)) ≃ Z and Hom DMét(k) (Z(d)[2d], Z(d)[2d]) = H 0,0 M,ét (Spec(k)) ≃ Z[1/p] (2)By the previous point, if char(k) = 0, K/k is a finite Galois extension and X → k a smooth projective k-scheme then the morphism f : X K → X is a finiteétale morphism. Seeing that f is proper, so there exists an induced map f * : CH d et (X K ) → CH d et (X) which fits in the following commutative diagram the exponential characteristic of k andf : Xk → Yk. For the particular case of theétale degree map we have the following: Proposition 4.3. Let X be a smooth and projective k-scheme of dimension d with char(k) =p ≥ 0. Then the map degé t : CH d et (X) → Z[1/p] factors through a subgroup of CH d (Xk)[1/p] G k . Proof. ( Xk, Q ℓ /Z ℓ (n)) = 0 so by Proposition 5.3 the morphism degé t : CH ń et (X) → Z is surjective. Now if we fix a prime number p ≥ 5 then by [Col05, Theorem 1.1] there exist a field F with cd(F ) = 1 and a smooth projective hypersurface X ⊂ P p F with index equals to p. d k , Z(d)) ≃ H m M (P d k , Z(d)) for all m ∈ Z and in particular H m−2i L (Spec(k), Z(d − i)) = 0 if m − d > i. For instance if m = 2d − 1 then H 2d−1 L (P d k , Z(d)) ≃ K M 1 (k) or for m = 2d− 2 we have H 2d−2 L (P d k , Z(d)) ≃ K M 2(k) and hence for a Severi-Brauer variety and applying Proposition 2.5, we obtain that E 1,2d−1 2 (d) = H 1 (G k ,k * ) = 0, by Hilbert 90, and E 2,2d−1 2 (d) = H 2 (G k ,k * ) = Br(k). ǫ i = (d, . . . , d, i-th pos. d −1 , d, . . . , d) for all i, and (d, . . . , d). For such cases, if a j = d for all j then H 2nd−1−2nd L (Spec(k), Z(nd − nd)) = H −1 L (Spec(k), Z(0)) = 0, and if (a 1 , . . . , a n ) = ǫ i to ker CH nd ((P d k ) ×n ) x i is the pull-back of the generator class of Pic(P d k ) through the map pr i : X ×n → X. The intersection product with δ defines morphisms Pic((P d k ) ×n ) ∪δ − − → CH nd ((P d k ) ×n ) and H 1 L ((P d k ) ×n ), CM04, Théorème 1.1], [CM04, Théorème 1.2] and [Col05, Theorem 5.1] fulfill the hypothesis of Proposition 5.3. Theorem 3 (Theorem 5.10). Let X be a Severi-Brauer variety of dimension d over a field k.Then the image of degé t : CH d et (X) → Z is isomorphic to a subgroup of Pic(X), and in particular Ié t (X) ≥ ord([X]) where [X] is the Brauer class of X in Br(k). Moreover, if cd(k) ≤ 4 then this subgroup is isomorphic to Pic(X) i.e. Ié t (X) = ord([X]). Proposition 2.5.[Mil80, Th. III.2.20],[RS18, Let p : Y → X be a finite Galois covering of X with Galois group G, then there exists a convergent Hochschild-Serre spectral sequence with abutment the Lichtenbaum cohomology group Mil80, VI, Lemma 10.2] for coefficients away from the characteristic and [Gro85, I, Théorème 2.1.11] for the logarithmic Hodge-Witt complex. The formula (2) holds again because it holds for R = Q and for finite coefficients by the proper base change [Mil80, VI, Corollary 2.3] and [Gro85, IV, Corollaire 1.3.6] for the logarithmic Hodge-Witt complex. Theorem 5.10. Let X be a Severi-Brauer variety of dimension d over a field k. Then the imageof degé t : CH d et (X) → Z isisomorphic to a subgroup of Pic(X) and in particular Ié t (X) ≥ ord([X]) where [X] is the Brauer class of X in Br(k). Moreover, if cd(k) ≤ 4 then this subgroup is isomorphic to Pic(X) i.e. Ié t (X) = ord([X]). AcknowledgementsThe author thanks his advisors Frédéric Déglise and Johannes Nagel for their suggestions, useful discussions and the time for reading this article. This work is supported by the EIPHI Graduate School (contract ANR-17-EURE-0002). We thank the French "Investissements d'Avenir" project ISITE-BFC (ANR-15-IDEX-0008) and the French ANR project "HQ-DIAG" (ANR-21-CE40-0015).Proof. By our assumptions we have that Ck ≃ P 1 k then (C × C)k ≃ P 1 k × P 1 k . Considering the Hochschild-Serre spectral sequence for Lichtenbaum cohomology E p,q 2 = H p (G k , H q L (P 1 k × P 1 k , Z(2))) =⇒ H p+q L (C × C, Z(2)). Since H m L (P 1 k × P 1 k , Z(2)) ≃ H m M (P 1 k × P 1 k , Z(2)) for m ≤ 3, using again the projective bundle formula for motivic cohomology we obtain that, Z(2)). As we have mentioned before, H 0 M (Spec(k), Z(2)) and K 2 (k) are uniquely divisible, then for p > 0 we have E p,0 2 = E p,2 2 = 0. Due to the compatibility ofétale cohomology with colimits, and in particular with direct sums, so E p,3In particular, notice that again Hilbert's theorem 90 gives us that E 1,3 2 = 0 and by definition E 2,3 2 ≃ Br(k) ⊕ Br(k). With all this information about the E 2 -terms, we obtain the E ∞ -terms such as E 1,3and then we obtain the characterizationwhere deg : E 0,4 ∞ → Z is the composition of the following maps: (1)) = x + y. Taking the morphisms induced by the intersection product with δ:induced by the cup product. Hence the cup product with the diagonal induces a map Br(k) → Br(k) ⊕ Br(k) defined by a → (a, a) and then we can deduce that CH 2 (P 1 k × P 1 k ) G k → Br(k) ⊕ Br(k) sends the 1 → ([C], [C]). Since E 0,4 ∞ ≃ ord([C])Z we conclude that Ié t (C × C) = ord([C]). Remark 5.17.(1) By Theorem 5.4 and Theorem 5.5 with theétale degree map we can improve the existence of integral projectors of the Künneth decomposition. 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[ "Global fit to b → cτ ν anomalies 2022 mid-autumn", "Global fit to b → cτ ν anomalies 2022 mid-autumn" ]
[ "Syuhei Iguro [email protected] \nInstitute for Astroparticle Physics (IAP)\nKIT, Hermann-von-Helmholtz-Platz 176344Eggenstein-LeopoldshafenGermany (\n", "Teppei Kitahara [email protected] \nInstitute for Advanced Research\nNagoya University\n464-8601NagoyaJapan (\n\nKobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\n464-8602NagoyaJapan (\n\n) KEK Theory Center\nIPNS\n305-0801KEK, TsukubaJapan (\n\nInstitute of Theoretical Physics\n) CAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina (\n", "Ryoutaro Watanabe \nSezione di Pisa\nLargo B. Pontecorvo 356127PisaItaly\n", "\nInstitute for Theoretical Particle Physics (TTP)\nKarlsruhe Institute of Technology (KIT)\nEngesserstraße 776131KarlsruheGermany (\n" ]
[ "Institute for Astroparticle Physics (IAP)\nKIT, Hermann-von-Helmholtz-Platz 176344Eggenstein-LeopoldshafenGermany (", "Institute for Advanced Research\nNagoya University\n464-8601NagoyaJapan (", "Kobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\n464-8602NagoyaJapan (", ") KEK Theory Center\nIPNS\n305-0801KEK, TsukubaJapan (", "Institute of Theoretical Physics\n) CAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina (", "Sezione di Pisa\nLargo B. Pontecorvo 356127PisaItaly", "Institute for Theoretical Particle Physics (TTP)\nKarlsruhe Institute of Technology (KIT)\nEngesserstraße 776131KarlsruheGermany (" ]
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Recently, the LHCb collaboration announced a preliminary result of the test of lepton flavor universality (LFU) in B → D ( * ) semi-leptonic decays: R LHCb2022 D = 0.441 ± 0.089 and R LHCb2022 D * = 0.281 ± 0.030 based on the LHC Run 1 data. This is the first result of R D for the LHCb experiment, and its precision is comparable to the other B-factory data. Interestingly, those data prefer the violation of the LFU again. A new world average of the data from the BaBar, Belle, and LHCb collaborations is R D = 0.358 ± 0.027 and R D * = 0.285 ± 0.013. Including this new data, we update a circumstance of the b → cτ ν measurements and their implications for new physics. Incorporating recent developments for the B → D ( * ) form factors in the Standard Model (SM), we observe a 4.1σ deviation from the SM predictions. Our updates also include; modelindependent new physics (NP) formulae for the related observables; and the global fittings of parameters for leptoquark scenarios as well as single NP operator scenarios. Furthermore, we show future potential to indirectly distinguish different new physics scenarios with the use of the precise measurements of the polarization observables in B → D ( * ) τ ν at the Belle II and the high-p T flavored-tail searches at the LHC. We also discuss an impact on the LFU violation in Υ → l + l − .
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[ "https://export.arxiv.org/pdf/2210.10751v2.pdf" ]
252,992,597
2210.10751
3fb8f2168a3874eb9e30f8ca572e8a6a2b5a39b1
Global fit to b → cτ ν anomalies 2022 mid-autumn Dated: October 18, 2022 23 Jan 2023 Syuhei Iguro [email protected] Institute for Astroparticle Physics (IAP) KIT, Hermann-von-Helmholtz-Platz 176344Eggenstein-LeopoldshafenGermany ( Teppei Kitahara [email protected] Institute for Advanced Research Nagoya University 464-8601NagoyaJapan ( Kobayashi-Maskawa Institute for the Origin of Particles and the Universe Nagoya University 464-8602NagoyaJapan ( ) KEK Theory Center IPNS 305-0801KEK, TsukubaJapan ( Institute of Theoretical Physics ) CAS Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina ( Ryoutaro Watanabe Sezione di Pisa Largo B. Pontecorvo 356127PisaItaly Institute for Theoretical Particle Physics (TTP) Karlsruhe Institute of Technology (KIT) Engesserstraße 776131KarlsruheGermany ( Global fit to b → cτ ν anomalies 2022 mid-autumn Dated: October 18, 2022 23 Jan 2023Beyond Standard ModelB physicsEffective Field Theories Recently, the LHCb collaboration announced a preliminary result of the test of lepton flavor universality (LFU) in B → D ( * ) semi-leptonic decays: R LHCb2022 D = 0.441 ± 0.089 and R LHCb2022 D * = 0.281 ± 0.030 based on the LHC Run 1 data. This is the first result of R D for the LHCb experiment, and its precision is comparable to the other B-factory data. Interestingly, those data prefer the violation of the LFU again. A new world average of the data from the BaBar, Belle, and LHCb collaborations is R D = 0.358 ± 0.027 and R D * = 0.285 ± 0.013. Including this new data, we update a circumstance of the b → cτ ν measurements and their implications for new physics. Incorporating recent developments for the B → D ( * ) form factors in the Standard Model (SM), we observe a 4.1σ deviation from the SM predictions. Our updates also include; modelindependent new physics (NP) formulae for the related observables; and the global fittings of parameters for leptoquark scenarios as well as single NP operator scenarios. Furthermore, we show future potential to indirectly distinguish different new physics scenarios with the use of the precise measurements of the polarization observables in B → D ( * ) τ ν at the Belle II and the high-p T flavored-tail searches at the LHC. We also discuss an impact on the LFU violation in Υ → l + l − . Introduction The semi-tauonic B-meson decays, B → D ( * ) τ ν, have been intriguing processes to measure the lepton flavor universality (LFU): R D ≡ B(B → D τ ν τ ) B(B → D ν ) , R D * ≡ B(B → D * τ ν τ ) B(B → D * ν ) ,(1.1) since it has been reported that the measurements by the BaBar [1,2], Belle [3][4][5][6][7] and LHCb [8][9][10] collaborations indicate deviations from the Standard Model (SM) predictions, where = e, µ for the BaBar/Belle and = µ for the LHCb. See Table 1 for the present summary. A key feature of the deviation is that the measured R D ( * ) are always excesses compared with the SM predictions and thus imply violation of the LFU. Then it has been followed by a ton of theoretical studies to understand its implication from various points of view, e.g., see Ref. [11] and references therein. A confirmation of the LFU violation will provide an evidence of new physics (NP). Summary of the current status: 2022 mid-autumn Three years have passed since the previous experimental report of R D ( * ) measurements from the B factories [7]. In the meantime, the Belle II experiment finally started taking data Experiment R D * R D Correlation BaBar [1,2] 0.332 ± 0.024 ± 0.018 0.440 ± 0.058 ± 0.042 −0.27 Belle [3] 0.293 ± 0.038 ± 0.015 0.375 ± 0.064 ± 0.026 −0.49 Belle [4,5] 0.270 ± 0.035 +0.028 −0.025 --Belle [6,7] 0.283 ± 0.018 ± 0.014 0.307 ± 0.037 ± 0.016 −0.51 LHCb [9,10] 0.280 ± 0.018 ± 0.029 --LHCb [8,18] 0.281 ± 0.018 ± 0.024 0.441 ± 0.060 ± 0.066 −0.43 World average [19] 0.285 ± 0.010 ± 0.008 0.358 ± 0.025 ± 0.012 −0.29 Table 1. Current status of the independent experimental R D ( * ) measurements. The first and second errors are statistical and systematic, respectively. from 2020 [12,13], and the CMS collaboration has developed an innovative data recording method, called "B Parking" since 2019 [14][15][16][17], although their official first results are still being awaited. On the other hand, the LHCb collaboration has shown their results in 2015 and 2017 with the LHCb Run 1 dataset, and thus it was five years passed. Then, now, the LHCb collaboration reported their preliminary result of R D and also R D * with the LHCb Run 1 dataset [18], The τ is reconstructed in τ → µνν and the result supersedes the previous result performed in 2015 [8]. In Table 1, we summarize the current status of the R D ( * ) measurements including the new LHCb result. It is found that the new LHCb result is consistent with the previous world average evaluated in the HFLAV 2021 report [20] within the experimental uncertainty. The combined average of the experimental data gives p(χ 2 ) = 32% with χ 2 /dof = 9.21/8 for the p-value among all data, compared with the previous HFLAV average of 28% with χ 2 /dof = 8.8/7 written in Ref. [20]. The amplified p-value indicates consistency among the data. New world averages of the R D ( * ) measurements are [19] R D = 0.358 ± 0.025 ± 0.012 , R D * = 0.285 ± 0.010 ± 0.008 , (1.3) and R D -R D * correlation of −0.29. Reference R D R D * P D τ −P D * τ F D * L R J/ψ R Λc R Υ(3S) Bernlochner, et al. [22] 0.288(4) 0.249 (3) ------Iguro, Watanabe [23] 0.290(3) 0.248(1) 0.331(4) 0.497 (7) Regarding the combined average, an important analysis is given in Ref. [21]. The authors pointed out that evaluations of the D * * distributions in the SM background involve nontrivial correlations that affect the R D ( * ) measurements. Their sophisticated study shows that the combined R D ( * ) average is slightly sifted, which is beyond the scope of our work. #1 Recent SM predictions for R SM D ( * ) have been obtained in Refs. [20,[23][24][25] as summarized in Table 2. The difference of these SM values is mainly due to development of the B → D ( * ) form factor evaluations both by theoretical studies and experimental fits, whose details can be found in the literature. In our work, we will employ the work of Ref. [23] as explained soon later. A further concern for the SM evaluation is long-distance QED corrections to B → D ( * ) ν, which remains an open question. They depend on the lepton mass as being of O[α ln(m /m B )] and hence it could provide a few percent correction to violation of the LFU in the semileptonic processes [29][30][31][32]. This will be crucial in future when the Belle II experiment reaches such an accuracy. In Fig. 1, we show the latest average of the R D -R D * along with the several recent SM predictions. A general consensus from the figure is that the deviation of the experimental data from the SM expectations still remains. For instance, applying the SM prediction from {HFLAV2021 [20], Ref. [22], Ref. [23], Refs. [24,25] In addition to these deviations in the LFU measurements, τ -and D * -polarization observables in B → D ( * ) τ ν also provide us important and nontrivial information. This is because these observables can potentially help us to pin down the NP structure that causes these deviations [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. We refer to the τ longitudinal-polarization asymmetry in B → D ( * ) τ ν and the fraction of the D * longitudinal mode in B → D * τ ν as P D ( * ) τ and F D * L , respectively. #1 Instead, a comparison among the two previous and new world averages is shown in Fig. 1. See Refs. [42,51,52] for their explicit definitions. In recent years, the first measurements for some of the above polarization observables have been reported by the Belle collaboration. It is summarized as P D * τ = −0.38 ± 0.51 +0. 21 −0.16 [4] and F D * L = 0.60 ± 0.08 ± 0.04 [53]. See also Table 2. Although the experimental uncertainty in F D * L is still large, this result already has important implications for a tensor-operator NP as pointed out in Ref. [42]. Although P D τ is the most striking observable to disentangle the leptoquark (LQ) scenarios that can explain the discrepancy, the experimental study does not exist so far. + Note that the D * longitudinal polarization in the electron mode has also been measured, [53]. This is comparable to the SM prediction of 0.534 ± 0.002 [23]. The F D * L and F D * ,e L measurements have the same level of significance (1.5σ and 1.3σ, respectively). F D * L (B 0 → D * + eν) ≡ F D * ,e L = 0.56 ± 0.02 Preliminaries of our analysis Main points of this paper are that (i) we provide state-of-the-art numerical formulae for the observables relevant to the semi-tauonic B decays and (ii) we revisit to perform global fits to the available R D ( * ) measurements with respect to NP interpretations. It will be given by incorporating following updates and concerns: • The preliminary result from the LHCb collaboration is encoded in our world average as shown in Table 1. • The recent development of the B → D ( * ) transition form factors is taken into account. It is described by the heavy quark effective theory (HQET) taking higher-order corrections up to O(Λ 2 QCD /m 2 c ) as introduced in Refs. [24,54]. We follow the result from the comprehensive theory+experiment fit analysis as obtained in Ref. [23]. #2 • The recent study of Ref. [22] introduced an approximation method to reduce independent parameters involving the O(Λ 2 QCD /m 2 c ) corrections in HQET. Although this affects some of the parameter fits for the form factors, readers can find in Table 2 that our reference values of Ref. [23] are consistent with those of Ref. [22] as for R D ( * ) . Hence we do not take this approximation in our work. • Recently, Fermilab-MILC Collaborations [55] presented the first lattice result of the form factors for B → D * ν at nonzero recoil, with which one obtains R SM D * = 0.2484 ± 0.0013 and again readers can see the consistency with our reference. Since this preliminary result needs to be finalized and to be compared with upcoming lattice results from the other collaborations such as JLQCD and HPQCD, we do not include this update in our analysis. • Another way of constraining the form factor has been discussed in Refs. [56][57][58]. It is free from the parameterization method and obtained by a general property of the unitary bound. They found R SM D = 0.296 ± 0.008 and R SM D * = 0.261 ± 0.020 (R SM D * = 0.275 ± 0.021 by taking the Fermilab-MILC result [55]), slightly larger, but still consistent within the uncertainty. We do not consider this case. • Indirect LHC bounds from the high-p T mono-τ searches with large missing transverse energy [59][60][61][62][63][64][65][66][67][68][69] are concerned. We impose the result of Ref. [67] that directly constrains the NP contributions to the b → cτ ν current and accounts for the NP-scale dependence on the LHC bound, which is not available by the effective-field-theory description. Requiring an additional b-tagged jet also helps to improve the sensitivity [66,68]. We will see how it affects the constraints in the leptoquark scenarios. • Similar sensitivity can be obtained by the τ τ final state [70,71]. It is noted that the about three standard deviation is reported by the CMS collaboration [71], which would imply the existence of leptoquark, while the ATLAS result [70] has not found the similar excess. We need the larger statistics to confirm it, and thus we do not include the constraint to be conservative. #2 To be precise, we employ the "(2/1/0) fit" result, preferred by their fit analysis. See the reference for details. In addition to the above points, we also investigate the following processes that are directly/indirectly related to the b → cτ ν current: • The LFU in B c → J/ψ l ν decays is connected to R D ( * ) . The LHCb collaboration has measured the ratio R J/ψ ≡ B(B c → J/ψ τ ν)/B(B c → J/ψ µ ν) = 0.71±0.17±0.18 [72]. Although the current data includes such a large uncertainty, it would be useful in future to test some NP scenarios for the sake of the R D ( * ) anomalies. We update the numerical formula for R J/ψ in the presence of general NP contributions and put a prediction from our fit study. • The Υ leptonic decays, Υ → l + l − , are potentially connected to R D ( * ) once one specifies NP interactions to the bottom quark and leptons. Although the SM contribution comes from a photon exchange, it is suppressed by the Υ mass squared. The sensitivity to NP is, therefore, not completely negligible, and the LFU of R Υ(nS) ≡ B(Υ(nS) → τ + τ − )/B(Υ(nS) → + − ) can be an important cross check of the R D ( * ) anomalies. Furthermore, the BaBar collaboration has reported a result which slightly violates the LFU: R Υ(3S) = 0.966 ± 0.008 ± 0.014 [73]. We investigate the theoretical correlations in several NP models. This paper is organized as follows. In Sec. 2, we put the numerical formulae for the relevant observables in terms of the effective Hamiltonian. We also summarize the case for single operator analysis. In Sec. 3, based on the generic study with renormalizationgroup running effects, we obtain relations among R D , R D * , and F D * L in the LQ models and discuss their potential to explain the present data. Relations to the τ polarizations are also discussed. In Sec. 4, we also investigate the LFU violation in the Υ decays and show its correlation with b → cτ ν observables. Finally, we conclude in Sec. 5. General formulae for the observables At first, we describe general NP contributions to b → cτ ν in terms of the effective Hamiltonian. The operators relevant to the processes of interest are described as #3 H eff = 2 √ 2G F V cb (1 + C V L )O V L + C V R O V R + C S L O S L + C S R O S R + C T O T ,(2.1) with O V L = (cγ µ P L b)(τ γ µ P L ν τ ) , O V R = (cγ µ P R b)(τ γ µ P L ν τ ) , O S L = (cP L b)(τ P L ν τ ) , O S R = (cP R b)(τ P L ν τ ) , (2.2) O T = (cσ µν P L b)(τ σ µν P L ν τ ) , #3 The different naming scheme of the operators are often used [74][75][76]. Our CV L , CV R , CS L , and CS R correspond to CV 1 , CV 2 , CS 2 , and CS 1 , respectively. where P L = (1 − γ 5 )/2 and P R = (1 + γ 5 )/2. The NP contribution is encoded in the Wilson coefficients (WCs) of C X , normalized by the SM factor of 2 √ 2G F V cb . The SM corresponds to C X = 0 for X = V L,R , S L,R , and T in this description. We assume that the light neutrino is always left-handed, and NP contributions are relevant to only the third-generation neutrino (ν τ ), for simplicity. #4 Note that the leading SU (2) L × U (1) Y invariant operator, to generate the LFU violated type of the O V 2 form, is given in dimension-eight as ( c R γ µ b R )(L 3 γ µ τ A L 3 )(Hτ A H). This implies that C V 2 in a NP model necessarily has an additional suppression compared with the other operators generated from dimension-six operators. See Ref. [85] for a NP model that can generate the C V 2 contributions to R D ( * ) . In the following parts, the observables for B → D ( * ) τ ν, B c → τ ν, and B c → J/ψ τ ν are evaluated with Eq. (2.1) at the scale µ = µ b = 4.18 GeV. The process Υ(nS) → l + l − will be described in detail in Sec. 4. B → D ( * ) τ ν In this work, we follow analytic forms of the differential decay rates for B → D ( * ) τ ν obtained in Refs. [74,86]. Regarding the form factors, we employ the general HQET based description [24], in which the heavy quark expansions [87,88] are taken up to NLO for a = α s /π, b = Λ/(2m b ) and NNLO for c = Λ/(2m c ) by recalling the fact a ∼ b ∼ 2 c . Thanks to HQET property, the form factors for the different Lorenz structures of the NP operators are connected to that for the SM current, which enables us to evaluate the NP contributions to the observables. Two parametrization models have been considered with respect to the z = ( √ w + 1 − √ 2)/( √ w + 1 + √ 2) expansions for the form factors in this description, with which the most general fit analyses of the form-factor parameters and |V cb | have been performed in Ref. [23]. For the present work, we take the (2/1/0) model with a minor update and apply the updated fit result based on Ref. [23]. We have evaluated the ratio observables, R D ( * ) , P D ( * ) τ and F D * L , for the case of the effective Hamiltonian of Eq. (2.1) at the scale µ = µ b . In the end, we find the following #4 See Refs. [77][78][79][80][81][82][83] for models with the right-handed neutrino νR. It is noted that the W is necessarily accompanied by Z and thus the recent di-τ resonance search [70,71] excludes the W R explanation [84]. updated numerical formulae, R D R SM D = |1 + C V L + C V R | 2 + 1.01|C S L + C S R | 2 + 0.84|C T | 2 + 1.49Re[(1 + C V L + C V R )(C * S L + C * S R )] + 1.08Re[(1 + C V L + C V R )C * T ] , (2.3) R D * R SM D * = |1 + C V L | 2 + |C V R | 2 + 0.04|C S L − C S R | 2 + 16.0|C T | 2 − 1.83Re[(1 + C V L )C * V R ] − 0.11Re[(1 + C V L − C V R )(C * S L − C * S R )] − 5.17Re[(1 + C V L )C * T ] + 6.60Re[C V R C * T ] , (2.4) P D τ P D τ, SM = R D R SM D −1 × |1 + C V L + C V R | 2 + 3.04|C S L + C S R | 2 + 0.17|C T | 2 + 4.50Re[(1 + C V L + C V R )(C * S L + C * S R )] − 1.09Re[(1 + C V L + C V R )C * T ] , (2.5) P D * τ P D * τ, SM = R D * R SM D * −1 × |1 + C V 1 | 2 + |C V 2 | 2 − 0.07|C S 1 − C S 2 | 2 − 1.85|C T | 2 − 1.79Re[(1 + C V L )C * V R ] + 0.23Re[(1 + C V L − C V R )(C * S L − C * S R )] − 3.47Re[(1 + C V L )C * T ] + 4.41Re[C V R C * T ] , (2.6) F D * L F D * L, SM = R D * R SM D * −1 × |1 + C V L − C V R | 2 + 0.08|C S L − C S R | 2 + 6.90|C T | 2 − 0.25Re[(1 + C V L − C V R )(C * S L − C * S R )] − 4.30Re[(1 + C V L − C V R )C * T ] , (2.7) which can be compared with those in the literature [42,43,78,82,89]. The SM predictions are obtained as #5 R SM D = 0.290 ± 0.003 , R SM D * = 0.248 ± 0.001 , P D τ, SM = 0.331 ± 0.004 , P D * τ, SM = −0.497 ± 0.007 , F D * L, SM = 0.464 ± 0.003 . (2.8) Furthermore, we have checked uncertainties of the above numerical coefficients in the formulae, based on the fit result from Ref. [23]. The tensor (scalar) terms involve ∼ 4% (10%) uncertainties for the D (D * ) mode, while the others contain less than 1% errors. At present, they are not significant and thus neglected in our following study. #5 We updated the fit analysis with the modification of the formula for unitarity bound [87], pointed out by Ref. [90]. It only affects the last digits of the SM predictions, though. B c → τ ν The significant constraint on the scalar operators O S L,R comes from the B c lifetime measurements (τ Bc ) [91][92][93][94][95]: the branching ratio of B − c → τ ν, which has not yet been observed, is significantly amplified by the NP scalar interactions, and the branching ratio is constrained from measured τ Bc [96]. We obtain an upper bound on the WCs as [96]. The b and c quark mass inputs, which are relevant for scalar contributions, are taken as m b (µ b ) = (4.18 ± 0.03) GeV and m c (µ b ) = (0.92 ± 0.02) GeV. Reference [92] evaluated that the upper bound (UB) on the branching ratio from τ Bc is B(B c → τ ν) UB = 0.3. However, it is pointed out by Ref. [43] and later confirmed by Ref. [95] that there is a sizeable charm-mass dependence on the B c decay rate because the dominant contribution comes from the charmquark decay into strange within the B c meson. A conservative bound is set by Ref. [43] as |1 + C V L − C V R − 4.35 (C S L − C S R )| 2 = B(B c → τ ν) B(B c → τ ν) SM < 27.1 B(B c → τ ν) UB 0.6 , (2.9) with B(B c → τ ν) SM 0.022. Here, |V cb | = (41.0 ± 1.4) × 10 −3 is usedB(B c → τ ν) UB = 0.6. One should note that more aggressive bound B(B c → τ ν) UB = 0.1 has been obtained in Ref. [97] by using LEP data. However, it is pointed out that p T dependence of the fragmentation function, b → B c , has been entirely overlooked, and thus the bound must be overestimated by several factors [43,44,98]. Although the CEPC and FCC-ee experiments are in planning stages, the future Tera-Z machines can directly measure B(B c → τ ν) at O(1%) level [99,100]. Thanks to the conservative bound, the left-handed scalar operator, C S L comes back to the game. For instance a general two-Higgs doublet model is a viable candidate and readers are referred to Refs. [101,102]. B c → J/ψ τ ν We follow the form factor description from the recent lattice result of Ref. [103] for B c → J/ψ τ ν. We also take m b (µ b ) and m c (µ b ) for the scalar and tensor sectors as aforementioned. The formula for R J/ψ is given as R J/ψ R SM J/ψ = |1 + C V L | 2 + |C V R | 2 + 0.04|C S L − C S R | 2 + 14.7|C T | 2 − 1.82Re[(1 + C V L )C * V R ] − 0.10Re[(1 + C V L − C V R )(C * S L − C * S R )] − 5.39Re[(1 + C V L )C * T ] + 6.57Re[C V R C * T ] ,(2.10) where we take R SM J/ψ = 0.258 ± 0.004 [26]. The coefficients potentially have 10-20% uncertainties for C S L,R and C T , while a few percent for C V L,R . Fit analysis In this paper, several NP scenarios are investigated in accordance with the following steps: 1. The measurements of R D , R D * and F D * L are taken in the χ 2 fit, and then the favored regions for the NP parameter space are obtained. 2. We then check whether the above solutions are consistent with the other relevant observables, such as the B c lifetime and the LHC bound. 3. Furthermore, we evaluate NP predictions on P D τ , P D * τ and R J/ψ . 4. If applicable, a combined study with R Υ(3S) is discussed. The χ 2 fit function is defined as χ 2 ≡ i,j (O theory − O exp ) i Cov −1 ij (O theory − O exp ) j , (3.1) where we take into account the R D ( * ) and F D * L measurements for O exp reported by BaBar, LHCb, and Belle collaborations. The covariance is given as Cov ij = ∆O exp i ρ ij ∆O exp j + ∆O theory i δ ij ∆O theory j , where correlation ρ ij is given as in Table 1 while ρ ij = δ ij among the independent measurements. For every observables, we have the theory formulae O theory as shown in Sec. 2, and hence obtain best fit values in terms of the WCs as defined in Eq. (2.1). Given the SM predictions as R SM D = 0.290 ± 0.003, R SM D * = 0.248 ± 0.001, and F D * L, SM = 0.464±0.003, we obtain χ 2 SM = 22.4 (corresponding to 4.0σ) implying a large deviation from the SM. Recall that this chi-square contains the F D * L fit which enlarges the value compared with the R D ( * ) fit shown in Sec. 1. In order to see how NP scenarios improve the fit, we use "Pull" value (defined in, e.g., Refs. [43,104]). For cases of the single WC fits, the Pull is equivalent to Pull ≡ χ 2 SM − χ 2 NP-best (σ) ,(3.2) such that we can discuss quantitative comparisons among the NP scenarios. Regarding the LHC bound to be compared with the above fit result, we refer to the result from Ref. [67], in which the τ + missing searches have been analyzed. Their result is shown in Table 3, where we give the 95% CL upper limit at the µ b scale. #6 It should be emphasized that the LHC bound on the WC has a non-negligible mediator mass dependence, see Ref. [67] for details. This feature is indeed crucial for some NP scenarios as will be seen later. Furthermore it is pointed out that the charge asymmetry of the τ lepton will improve the bound on C X . #6 Note that Table 2 of Ref. [67] shows the LHC bound at µ = ΛLHC. Table 3. The 95% CL upper bounds on the WCs at the µ = µ b scale from the LHC analysis of the τ + missing search [67]. The future prospects with b-tagged jet + τ ν final state assuming 3 ab −1 of accumulated data are given in the parenthesis [68]. The NP mass scale is shown as M LQ = 2 TeV, 4 TeV and Λ EFT > 10 TeV. |C V L | |C V R | |C S L | |C S R | |C T | EFT (> 10 EFT: single operator scenario We begin with the single NP operator scenarios based on the effective field theory (EFT) of Eq. (2.1). Assuming the WC to be real, we immediately obtain the fit results with the Pull values and predictions of P D τ , P D * τ and R J/ψ as shown in Table 4. The allowed regions from the B c lifetime and current LHC bounds are listed as well. For all the NP scenarios, we can see much improvement on the fit compared with the SM. A significant change from the previous conclusion (before the new LHCb result [18] came up [44,105,106]) is that the C S R scenario becomes consistent with the data within 95% CL, i.e., χ 2 best < 8.0 (for three observed data). Furthermore, the C S R (µ b ) 0.2 solution indicates the second best Pull. Unfortunately, it is known that the usual type II two-Higgs doublet model (2HDM) cannot achieve this solution because the sign of C S R must be negative: C S R = −m b m τ tan 2 β/m 2 H ± < 0. It is noted that even in the generic 2HDM, sizable C S R contribution is difficult due to constraints from ∆M s and the LHC search [62,107]. Instead, the C S L scenario is likely to explain the present data. This is the same feature with the previous fit result before the LHCb data is included. The C V L scenario well explains the present data, while C V R gives a lower Pull. The C T solution gives unique predictions on the other observables, which may be able to identify the NP scenario, and it predicts a large shift of F D * L with opposite direction from the present measurement [42,53]. Once we allow complex values of WCs, the complex C V R , C S L , and C T scenarios improve the fits such as C V R 0.02 ± i 0.43 Pull = 4.1 , (3.3) C S L −0.88 ± i 0.88 Pull = 4.3 ,(3.4)C T 0.06 ± i 0.16 Pull = 3.3 ,(3.5) while the complex C V L and C S R scenarios give the same Pulls, compared with those with the real WC scenarios. The complex C T scenario has a similar Pull with the real C T case. The complex C V R result at the above best fit point is, however, not consistent with the LHC bound for the case of EFT, |C V R | < 0.33. Nevertheless, it could be relaxed in some LQ models with the mass of the LQ particle to be M LQ 2 TeV as seen in Table 3. Pull [χ 2 best ] Fitted C X Allowed region of C X Predictions (∆χ 2 ≤ 1) B c → τ ν LHC P D τ −P D * τ R J/ψ As for the complex C S L scenario, it looks that the best fit point in Eq. (3.4) is disfavored by the LHC and B c lifetime constraints. It is noted, however, that the LHC bound is not always proper and depends on the NP model. In the case of the charged Higgs model, for instance, the bound on the s-channel mediator H ± significantly depends on the resonant mass. Experimentally it is not easy to probe the low mass τ ν resonance due to the huge SM W background. Reference [101] points out that the range of m t ≤ m H ± ≤ 400 GeV is still viable for the 1σ explanation, although LHC Run 2 data is already enough to probe this range if the τ ν + b signature is searched [102]. Thus, we leave the LHC bound for the complex C S L scenario below. Once the B c bound of eq. (2.9) is imposed, we find C S L −0.58 ± i 0.88 Pull = 4.2 ,(3.6) for the best Pull within the constraint. It has been pointed out that q 2 distribution in B → D ( * ) τ ν is sensitive to the scalar contribution [86,93]. We do not consider the constraint since the detailed correlation among the bins is not available and the constraints depends on |V cb |. Furthermore, experimental analyses for the q 2 -distribution measurement rely on the theoretical models [2,3]. In any case, the Belle II data will be important to resolve these issues [76]. In Fig. 2, we show predictions on the plane of P D τ -P D * τ from our fit analysis with each complex WC scenario. The allowed regions satisfying ∆χ 2 ≤ 1 (4) are shown in dark (light) orange, brown, and blue for the complex C S L , C S R , and C T scenarios, respectively, where the B c lifetime and LHC bounds based on the EFT framework are also taken into account. The C V L, R scenarios do not deviate P D τ and P D * in the single complex NP operator scenarios. The allowed regions satisfying ∆χ 2 ≤ 1 (4) are shown in dark (light), orange, brown, and blue for the C S L , C S R , and C T scenarios, respectively, whereas the black dot is the case for the C V L,R scenarios. The B c lifetime and LHC bounds are also taken into account. As for the S L scenario, the B c lifetime rules out the region for ∆χ 2 ≤ 1, whereas the LHC bound is not taken as discussed in the main text. black dot in the figure. Also note that each shaded region is based on different Pull values, implying different significance, in Fig. 2. We can see that the correlation in τ polarization observables provide the unique predictions that can identify the NP scenarios. On the other hand, R J/ψ is less helpful to distinguish the different operators. LQ scenarios Finally, we study several LQ scenarios. It is well known that three categories of LQs can address the R D ( * ) anomalies [74], which are referred to as a SU (2) L -singlet vector U µ 1 , a SU (2) L -singlet scalar S 1 , and a SU (2) L -doublet scalar R 2 . The relevant LQ interactions are given in Appendix A. A key feature with respect to the fit is that these LQ scenarios involve three independent couplings relevant for b → cτ ν, which are encoded in terms of the two independent (and complex in general) WCs as U µ 1 : C V L , C S R ,(3.7)S 1 : C V L , C S L = −4C T ,(3.8)R 2 : C V R , C S L = 4C T ,(3.9) at the LQ scale Λ LQ = M LQ . The SU (2) L doublet vector leptoquark V 2 forms C S R [74], equivalent to the single C S R scenario, and hence this LQ has now the viable solution as seen in Sec. 3.1. Flavor and collider phenomenologies of V 2 LQ could be interesting, but we leave it for a future work [108]. The C V L phase in |1 + C V L | 2 can be absorbed [42] in the flavor process. Thus, the absorption of the C V L phase is irrelevant for the fit within the flavor observables and we take C V L in U 1 and S 1 LQs as real without loss of generality. #7 As for C V R in the R 2 LQ, we assume it as pure imaginary from the fact of Eq. (3.3). Therefore, the three LQ scenarios of our interest have three degrees of freedom for the fit and the relevant observables, and then it is expected that fit results could be different from the previous studies. These years, UV completions of the LQ scenarios have been studied in the literature; Refs. [109][110][111][112][113][114][115][116][117][118][119][120][121][122][123][124] for U 1 , Refs. [125][126][127] for S 1 , Refs. [128,129] for R 2 , and see also Refs. [130,131]. In the next subsection, we consider the case if the U 1 LQ is induced by a UV completed theory that gives a specific relation to the LQ couplings, and see how it changes the fit result. Recent re-evaluations on mass differences of the neutral B mesons ∆M d , ∆M s , (improved by HQET sum rule and lattice calculations [132]), would constrain a UV-completed TeVscale LQ model [111,112,124,126,133,134]. In particular, the ratio ∆M d /∆M s provides a striking constraint on the coupling texture of the LQ interactions. Here, we comment that a typical UV completion requires a vector-like lepton (VLL) and it induces LQ-VLL box diagrams that contribute to ∆M d,s . This implies that the constraint of our concern depends on the vector-like fermion mass spectrum, and hence we do not consider ∆M d,s further in our analysis. The LQ mass has been directly constrained as M LQ 1.5 TeV from the LQ pair production searches [135][136][137]. Hence we take M LQ = 2 TeV for our benchmark scale. We recap that the WCs are bounded from the τ + missing search and, as shown in Table 3, the LQ scenarios receive milder constraints than the EFT operators as long as M LQ ≤ 10 TeV. The WCs will be fitted at the µ b scale in our analysis, and then they are related to the WCs defined at the Λ LQ = M LQ scale. The renormalization-group equations (RGEs) (the first matrix below) [138][139][140] and the LQ-charge independent QCD one-loop matching (the second one) [141] give the following relation             C V L (µ b ) C V R (µ b ) C S L (µ b ) C S R (µ b ) C T (µ b )                                                                       C V L (Λ LQ ) C V R (Λ LQ ) C S L (Λ LQ ) C S R (Λ LQ ) C T (Λ LQ )             #7 Now the real CV L fit to the R D ( * ) anomalies gives the minimum |CV L |, and thus is less constrained from the LHC data.                         C V L (Λ LQ ) C V R (Λ LQ ) C S L (Λ LQ ) C S R (Λ LQ ) C T (Λ LQ )             ,(3.10) with Λ LQ = 2 TeV. Using these numbers, we obtain C S L (µ b ) = −8.9 C T (µ b ) and C S L (µ b ) = 8.4 C T (µ b ) for S 1 and R 2 LQs, respectively. With these ingredients, the LQ scenarios in terms of C X (µ b ) up to three degrees of freedom are investigated, where the full variable case is referred to as the general LQ. The results of the best fit points for the general LQ scenarios are then summarized as U 1 LQ : C V L = 0.07 , C S R = 0.06 , Pull = 3.8 , (3.11) S 1 LQ : C V L = 0.06 , C S L = −8.9 C T = 0.06 , Pull = 3.8 , (3.12) R 2 LQ : C V R = ±i0.68 , C S L = 8.4 C T = 0.04 ∓ i0.65 , Pull = 3.8 . (3.13) We observe that these three general LQ scenarios have the same Pull which means equivalently favored by the current data. We also see that C S R (C S L ) is preferred to be real at the best fit point for the U 1 (S 1 ) LQ scenario, whereas C S L for R 2 LQ is given complex. The fit results for S 1 LQ and R 2 LQ with C V L = C V R = 0 are obtained as where the improvements of Pull only come from the benefit of reducing the variables. In turn, we evaluate the LHC bound on the two independent variables, such as (C V L , C S R ), by the following interpretation U 1 LQ : |C V L (µ b )| 2 (0.42) 2 + |C S R (µ b )| 2 (0.77) 2 < 1 , (3.16) S 1 LQ : |C V L (µ b )| 2 (0.42) 2 + |C S L (µ b )| 2 (0.80) 2 < 1 , (3.17) R 2 LQ : |C V R (µ b )| 2 (0.51) 2 + |C S L (µ b )| 2 (0.80) 2 < 1 , (3.18) where the denominators are the current LHC bounds for the single WC scenarios with M LQ = 2 TeV from Table 3. Indeed this is a good approximation since the bound comes from the high-p T region that suppresses the interference term between the V L,R and S L,R operators. It can be seen that the best fit point of Eq. (3.13) for R 2 LQ is not consistent with the LHC bound of Eq. (3.18). Table 5. The fit results of the U 1 , S 1 , and R 2 LQ scenarios for M LQ = 2 TeV. The WCs are given at the µ b scale. The structure is the same as in Table 4. Pull [χ 2 best ] Fitted C X Allowed region of C X Predictions (∆χ 2 ≤ 1) B c → τ ν LHC P D τ −P D * τ R J/ψ In Table 5, we show our fit results and predictions with respect to the LQ scenarios like we did for the EFT cases. It is observed that the general LQ scenarios have less predictive values of the tau polarizations. This can be understood from the fact that the complex scalar WCs give large impacts on the interference terms as can be checked from Eqs. (2.5) and (2.6), which result in the wide ranges of the predictions. Figure 3 visualizes the combined P D τ -P D * τ predictions satisfying ∆χ 2 ≤ 1 (4) and the aforementioned bounds, where the general U 1 , S 1 , and R 2 LQ scenarios are shown in dark (light) green, magenta, and yellow, respectively. The U 1 and R 2 LQ scenarios produce the correlated regions of the P D τ -P D * τ predictions and hence could be distinguished. On the other hand, the S 1 LQ scenario has the less-predictive wide region, which is hard to be identified. Figure 3 also exhibits the predictions for the several specific scenarios, i.e., U 1 LQ with real WC (solid line), S 1 LQ with real WC (dashed line), and R 2 LQ with C V R = 0 (gray region). It is seen that reducing the variable in the general LQ scenario provides the distinct prediction in particular for P D * τ and the correlation for P D τ -P D * τ becomes a useful tool to identify the LQ signature. Therefore, it is significant to restrict the LQ interactions by other processes or by a UV theory that realizes the LQ particle. The latter will be discussed in the next subsection for the U 1 LQ (corresponding to the cyan region in the figure). Regarding the S 1 LQ scenario, we comment that a part of the allowed parameter region is ruled out by the B → K * νν measurement and ∆M s (via LQ-ν τ box) [68]. in the LQ scenarios following the same procedure as in Fig. 2. The allowed regions are shown in dark (light) green, magenta, and yellow for the general U 1 , S 1 , and R 2 LQ scenarios, respectively. The specific scenarios; U 1 LQ with UV origin (cyan), real WC (solid line); S 1 LQ with real WC (dashed line); and R 2 LQ with C V R = 0 (gray), are also given. UV completion of U 1 LQ As the U 1 LQ provides a unique solution, not only to the b → cτ ν anomaly, but also to several flavor issues, UV completions of the U 1 LQ have been discussed [142][143][144][145][146][147][148][149][150]. A typical description is that the U 1 LQ is given as a gauge boson, embedded in a large gauge symmetry, such that the third-generation quarks and leptons are coupled to U 1 in the interaction basis. This means that the two LQ interactions of Eq. (A.1) are represented as a universal gauge coupling, x 33 L = x 33 R ≡ g U (see Appendix A). Moving to the mass basis leads to C S R (Λ LQ ) = −2β R × C V L (Λ LQ ) ,(3.19) where β R = e iφ denotes the relative complex (CP-violating) phase [149], which comes from the fact that the phases in the rotation matrices (to the mass basis) for quark and lepton are not necessarily identical. The LHC bound for this scenario has been studied and the typical scale of the constraint is obtained as Λ LQ 3.5 TeV [111]. The RGE running effect changes the above relation of Eq. (3.19) at the µ b scale of our interest. By taking Λ LQ = 4 TeV as a benchmark scale, we obtain where the first coefficient is the QCD two-loop RGE factor [140] and the second is the QCD one-loop matching correction [141] at the NP scale. Therefore, we have (3.21) in the case of the UV origin U 1 LQ scenario, applied to our fit analysis. The result of the best-fit point for the UV origin U 1 LQ scenario, with the definition of β R = e iφ , is shown as C V L (µ b ) = 1 × 1.11 × C V L (Λ LQ ) , C S R (µ b ) = 1.90 × 1.09 × C S R (Λ LQ ) ,C S R (µ b ) −3.7 β R × C V L (µ b ) ,(C V L , φ) (0.07 , ±0.54π) Pull = 4.1 . (3.22) One can see that this is consistent with the B c lifetime and LHC bounds. Predictions of the observables within ∆χ 2 ≤ 1, 4 are then given in Fig. 3. It is observed that the large complex phase is favored which suppress the interference. It should be also stressed that the τ polarizations are so unique that this scenario can be distinguished from the aforementioned LQ scenarios. The LFU violation in Υ decays The UV completed NP models contributing to b → cτ ν processes should also bring a related contribution to bb → τ + τ − or cc → τ + τ − interactions [28,151,152]. In this section, we show that U 1 and R 2 LQs predict a robust correlation between b → cτ ν and bb → τ + τ − via the LQ exchange. A definition of the LFU observable in the Υ(nS) decays is R Υ(nS) ≡ B(Υ(nS) → τ + τ − ) B(Υ(nS) → + − ) ,(4.1) with n = 1, 2, 3, where R Υ(nS) 1 holds in the SM. As for n ≥ 4, the leptonic branching ratios are significantly suppressed since a BB decay channel is open. #8 Since the short-and long-distance QCD corrections [154] are independent of the lepton mass, they are canceled in this ratio. One can also discuss the cc → l + l − LFU observable via ψ(2S) decays. However, we do not consider it because the present experimental error is relatively large. Recently, the BaBar collaboration has reported a precise result for measurement of R Υ(3S) [73]: R BaBar Υ(3S) = 0.966 ± 0.008 stat ± 0.014 syst , where = µ. Combing a previous measurement by the CLEO collaboration [155], an average for the Υ(3S) decay is [151] R exp Υ(3S) = 0.968 ± 0.016 . (4.2) #8 A novel method for the n = 4 mode has been proposed in Ref. [153] by using the inclusive di-leptonic channel Υ(4S) → ± τ ∓ X(νν), which could be probed in the Belle II experiment and is directly related to Γ(b → Xτ ν)/Γ(b → X ν). This value is consistent with the SM prediction [28] R SM Υ(3S) = 0.9948 ± O(10 −5 ) , (4.3) at the 1.7σ level. The SM prediction slightly deviates from 1 whose leading correction comes from the difference in the phase space factor between the τ / modes [156]. The next-to-leading contribution comes from the QED correction which depends on the lepton mass [157]; δ EM R Υ(nS) = +0.0002. The tree-level Z exchange also contributes, but its effect is O(10 −5 ) [28]. There is no Higgs boson contribution as one can see below. The other channels (n = 1, 2) still suffer from the current experimental uncertainty, and we do not utilize them in our presentation. The effective Hamiltonian which is relevant to the bottomonium decay into τ + τ − is described as −H NP eff = C bτ V LL (bγ µ P L b)(τ γ µ P L τ ) + C bτ V RR (bγ µ P R b)(τ γ µ P R τ ) + C bτ V LR (bγ µ P L b)(τ γ µ P R τ ) + C bτ V RL (bγ µ P R b)(τ γ µ P L τ ) (4.4) + C bτ T (bσ µν P R b)(τ σ µν P R τ ) + C bτ SL (bP L b)(τ P L τ ) + C bτ SR (bP R b)(τ P L τ ) + h.c. , at the scale µ = m Υ . Note that C bτ V LL , C bτ V RR , C bτ V LR and C bτ V RL are real coefficients, and C bτ SL and C bτ SR never contribute to the Υ(nS) → τ + τ − due to 0|bb|Υ = 0|bγ 5 b|Υ = 0. In this convention, the partial decay width is given by [28] Γ(Υ(nS) → τ + τ − ) = f 2 Υ 4πm Υ 1 − 4x 2 τ A 2 Υ (1 + 2x 2 τ ) + B 2 Υ (1 − 4x 2 τ ) + 1 2 C 2 Υ (1 − 4x 2 τ ) 2 + 1 2 D 2 Υ (1 − 4x 2 τ ) + 2A Υ C Υ x τ (1 − 4x 2 τ ) ,(4.5) with A Υ = 4πα 3 + m 2 Υ 4 C bτ V LL + C bτ V RR + C bτ V LR + C bτ V RL + 16x τ f T Υ f Υ Re C bτ T , (4.6) B Υ = m 2 Υ 4 C bτ V RR + C bτ V LR − C bτ V LL − C bτ V RL , (4.7) C Υ =2m 2 Υ f T Υ f Υ Re C bτ T , (4.8) D Υ =2m 2 Υ f T Υ f Υ Im C bτ T ,(4.9) and x τ = m τ m Υ . (4. 10) The f Υ and f T Υ are form factors for vector and tensor currents in Υ hadronic-matrix elements, and f Υ = f T Υ holds in the heavy quark limit, which is realized for the Υ decays [28]. Within the SM, this process is predominantly caused by the QED. Nevertheless, the photon-exchange QED contribution is suppressed by 1/m 2 Υ , and hence the NP contribution could be non-negligible [28,151,158]. In the SM, A Υ 4πα/3 and B Υ , C Υ , D Υ 0. Setting the light lepton mass to zero and m Υ = m Υ(3S) = 10.355 GeV, we obtain the following numerical formula R Υ(3S) R SM Υ(3S) = 1 + 1.64 × 10 −3 TeV 2 C bτ V LL + C bτ V RR + C bτ V LR + C bτ V RL + 6.37 × 10 −3 TeV 2 Re C bτ T + δ Υ , (4.11) with δ Υ = 5.22 × 10 −6 TeV 4 C bτ V LL + C bτ V RR + C bτ V LR + C bτ V RL Re C bτ T + 6.71 × 10 −7 TeV 4 C bτ V LL + C bτ V RR + C bτ V LR + C bτ V RL 2 + 5.59 × 10 −7 TeV 4 C bτ V RR + C bτ V LR − C bτ V LL − C bτ V RL 2 + 2.51 × 10 −5 TeV 4 Re C bτ T 2 + 1.79 × 10 −5 TeV 4 Im C bτ T 2 , (4.12) where the δ Υ term gives negligible contributions. Let us now look into a correlation between R Υ(3S) and R D ( * ) by using the specific examples of the U 1 and R 2 LQs. First, we exhibit the U 1 LQ case. The U 1 LQ interaction with the SM fermions is given in Eq. (A.1). Integrating the U 1 LQ out, as well as the charged current contributions (b → cτ ν) in Eq. (A.2), the neutral current ones (bb → τ + τ − ) are obtained as C bτ V LL (µ LQ ) = − |x bτ L | 2 m 2 U 1 , C bτ V RR (µ LQ ) = − |x bτ R | 2 m 2 U 1 , C bτ SR (µ LQ ) = 2x bτ L (x bτ R ) * m 2 U 1 . (4.13) The vector contributions do not change under the RGEs, while the scalar contribution does not affect the Υ decay. Here, an important point is that R Υ(3S) /R SM Υ(3S) is predicted to be less than 1 when NP contributions are dominated by vector interactions. It would lead to a coherent deviation with R D ( * ) . Setting m U 1 = 2.0 TeV and (V x L (µ LQ )) cτ = V cb x bτ L (µ LQ ), namely x sτ L (µ LQ ) = 0, we show a correlation between R Υ(3S) and R D ( * ) in Fig. 4. Here, favored parameter regions in the U 1 LQ model are exhibited on x bτ L -x bτ R plane at the renormalization scale µ LQ = m U 1 . The black contour represents the expected values of R Υ(3S) . The red shaded region is favored by R exp Υ(3S) in Eq. (4.2). It is noted that if we adopt the 2σ constraint of R exp Υ(3S) , the entire parameter region is allowed. The blue and green regions can explain the R D and R D * discrepancies within 1σ, respectively. The exclusion region by the LHC analysis (τ + missing search) is outside the blue line, while the future prospect of the High Luminosity LHC (HL-LHC) is shown by the red dashed line, see Table 3. #9 Furthermore, the orange dashed line #9 Note that a stronger collider bound would come from a non-resonant τ τ search [70,71], although it is model-parameter dependent. stands for a prediction in the case of the UV origin U 1 LQ with β R = −1 (φ = π). From the figure, it is found that the current R exp Υ(3S) overshoots favored parameter region from the R D ( * ) anomalies. The best fit points of Eq. (3.11) are shown by red crosses and predict R Υ(3S) = 0.991, distinct from the 0.99 contour line in the figure. Thus, it seems crucial to measure R Υ(nS) with less than 1% accuracy in order to distinguish the U 1 LQ signal. Next, we investigate the R 2 LQ scenario. The R 2 LQ interaction with the SM fermions is given in Eq. (A.6). The generated charged current contributions are given in Eq. (A.7), while the neutral current one is C bτ V LR (µ LQ ) = − |y bτ R | 2 2m 2 R 2 . (4.14) Since C bτ V LR < 0, R Υ(3S) /R SM Υ(3S) has to be less than 1 again. The result is shown in Fig. 5. Here, we set y cτ L (µ LQ ) = 1, |V cb | = 0.04, and m R 2 = 2.0 TeV, and take y bτ R (µ LQ ) as complex value. The color convention is the same as the U 1 LQ case. Furthermore, the gray shaded region is excluded by the B c lifetime, i.e., B(B c → τ ν) > 0.6. The best fit points in Eq. (3.15) are shown by red crosses, predicting R Υ(3S) = 0.992. Similar to U 1 LQ interpretation, 1% accuracy of the R Υ(nS) measurement is required as the R 2 LQ signature. At the current stage, the large experimental uncertainty in R Υ(3S) cannot allow a clearcut conclusion. One should note that the Belle and Belle II experiments have enough sensitivities to the R Υ(nS) measurements which would be more accurate than the existing BaBar measurement [159]. Conclusions and discussion In this work we revisited our previous phenomenological investigation and presented a statistical analysis of the LFU violation in R D ( * ) , including the new experimental data from the LHCb experiment. Starting with the re-evaluation of the generic formulae for R D ( * ) by Spin Charge Operators Table 6. Summary table for the single-mediator NP scenarios in light of the b → cτ ν anomaly. We add implications for the LHC searches and flavor observables in the last two columns, which is useful to identify the NP scenario. In the V R D R D * LHC Flavor H ± 0 (1, 2, 1 /2) O S L bτ ν B c → τ ν, F D * L , P D τ , M W S 1 0 (3, 1, 1 /3) O V L , O S L , O T τ τ ∆M s , P D τ , B → K ( * ) νν R ( 2 /3) 2 0 (3, 2, 7 /6) O S L , O T , (O V R ) bτ ν, τ τ R Υ(nS) , P D * τ , M W U 1 1 (3, 1, 2 /3) O V L , O S R bτ ν, τ τ R K ( * ) , R Υ(nS) , B s → τ τ V ( 1 /3) 2 1 (3, 2, 5 /6) O S R 2σ τ τ B s → τ τ , M W( 1 /3) 2 LQ scenario, 2σ for R D * implies that it can explain the R D * anomaly within the 2σ range (but not within 1σ). employing the recent development of the B → D ( * ) transition form factors, we examined the new physics possibility with the low-energy effective Lagrangian as well as the leptoquark models. In addition to the constraints from the low-energy observables and the high-p T mono-τ search at LHC, the predictions on the relevant observables of R Υ , R J/ψ , and the tau polarizations P D ( * ) τ are evaluated. To be precise, we performed the χ 2 fit to the experimental measurements of R D ( * ) and the D * polarization F D * L . This updated analysis shows that the present data deviates from the SM predictions at ∼ 4σ level. Our fit result is summarized in Table 4 with Eqs. (3.3)-(3.5) for the single-operator scenarios, and Table 5 with Eqs. (3.11)-(3.15), (3.22) for the single-mediator leptoquark scenarios. The NP fit improvements compared with the SM one are visualized by Pull as usual, and it was found that the SM-like vector operator still gives the best Pull. Due to the new LHCb result, the experimental world average has slightly come close to the SM predictions of R D and R D * . This change has affected the previous conclusions such that the scalar NP solutions to the b → cτ ν anomaly had been disfavored. Namely, the scalar NP interpretations have been revived now. On the other hand, it is found that the results of the LQ scenarios do not drastically change, compared with the previous fit. As it was pointed out in the literature, the precise measurements of the polarization observables P D ( * ) τ and F D * L have the potential to distinguish the NP scenarios. In Figs. 2 and 3, we show our predictions of P D τ and P D * τ for the possible NP scenarios. One can make sure that the single-operator NP scenario explaining the b → cτ ν anomaly can be identified by the P D ( * ) τ measurements, which may be available at the Belle II experiment. On the other hand, the general LQ scenarios are hard to be distinguished due to predicting wide ranges of P D ( * ) τ . Once the LQ model with restricted interactions is constructed, however, we see that the P D ( * ) τ measurement has significant potential to probe the LQ signature. The high energy collider search is also important since the high-p T lepton search at the LHC can directly probe the NP interactions affecting the LFU ratios. We also investigated the NP impacts on the LFU violation in the Υ(nS) decays. We found that the LFU ratio R Υ has to be correlated to R D ( * ) in the U 1 and R 2 LQ scenarios, while no correlation is expected in the S 1 LQ scenario. It is shown that an experimental accuracy of less than 1% for the R Υ(nS) measurement is necessary in order to identify the LQ scenario. We expect that this is possible in the Belle II experiment. In Table 6, we put a summary check sheet to find which single-mediator NP scenarios are viable and to see important observables in order to identify the NP scenario responsible for the b → cτ ν anomaly. Furthermore, it is known that the baryonic counterpart of R D ( * ) , namely R Λc ≡ B(Λ b → Λ c τ ν)/B(Λ b → Λ c µν), provides the independent cross check of the b → cτ ν anomaly [43,44]. Recently, R Λc has been measured for the first time as R Λc = 0.242 ± 0.026 ± 0.040 ± 0.059 by the LHCb experiment [160], where the last dominant uncertainty comes from an external branching fraction from the LEP measurement [161]. This result implies consistency with the SM prediction at 1.1σ level [27,162], while, instead, normalizing with the SM prediction of Γ(Λ b → Λ c µν) improves the accuracy and slightly up-lifts the central value, e.g., R Λc = |0.04/V cb | 2 (0.285 ± 0.073) [163]. Even though the current experimental uncertainty is not enough precise, it could already provide a nontrivial constraint on the NP parameter space which can explain the b → cτ ν anomaly. An implication of the measured R Λc for NP models is given in Ref. [164]. A Leptoquark interactions The LQ interactions are classified with the generic SU (3) c × SU (2) L × U (1) Y invariant form [165]. We leave details of the model constructions, and then just introduce the interactions relevant for b → cτ ν. As mentioned above, there are three viable candidates of leptoquark U 1 , S 1 , R 2 [166]. Their quantum numbers under SU (3) C , SU (2) L , U (1) Y are summarized in Table 6. First, the U 1 vector LQ interaction with the SM fermions, defined in the interaction basis, is given by L U 1 =x ij L Q i γ µ U µ 1 L j +x ij R d Ri γ µ U µ 1 Rj + h.c. . (A.1) Integrating out the U 1 LQ mediator particle, then, the Wilson coefficients (WCs) for the charged current of our interest (b → cτ ν) is obtained as C V L (µ LQ ) = 1 2 √ 2G F V cb (V x L ) cτ (x bτ L ) * m 2 U 1 , C S R (µ LQ ) = − 1 √ 2G F V cb (V x L ) cτ (x bτ R ) * m 2 U 1 , (A.2) where V is the CKM matrix and the couplings x L,R are in the mass basis. The relative sign and factor two in Eq. (A.2) come from the property of Fierz identity. In a typical UV completed theory [149], the U 1 LQ is realized as a gauge boson generated from a large gauge symmetry and only couples to the third-generation SM fermions. Namely, x bτ R =x bτ L ≡ g U , with the others to be zero, is indicated in the gauge interaction basis. Moving to the mass basis, then, generates a non-zero off-diagonal part such as x cτ L and also x bτ R = e iφ x bτ L , where the phase comes from those in the rotation matrices to the mass bases of the left-and right-handed quark and lepton fields that are not canceled in general. Therefore, the UV completion of U 1 LQ suggests C S R (µ LQ ) = −2e iφ C V L (µ LQ ) , (A.3) as introduced in the main text. We also comment that an extension of the fermion families with a nontrivial texture of the fermion mass matrices is necessary to construct a practical UV model [123]. The S 1 scalar LQ interaction in the mass basis is given by L S 1 = V * y L ij u C L i L j S 1 − y ij L d C L i ν L j S 1 + y ij R u C R i R j S 1 + h.c. . (A.4) In the scalar LQ scenario, the source of the generation violating couplings is off-diagonal element of Yukawa matrices. Then the four-fermion interactions of b → cτ ν are given by Finally, we introduce the R 2 scalar LQ interaction. R 2 is a SU(2) doublet and a component with 2/3 of the electromagnetic charge R ( 2 /3) 2 can contribute to b → cτ ν. The Yukawa interaction C S L (µ LQ ) = −4 C T (µ LQ ) = − 1 4 √ 2G F V cb y bτ L y cτ R * m 2 S 1 , C V L (µ LQ ) = 1 4 √ 2G F V cbL R 2 = y ij R d L i R j R ( 2 /3) 2 + y ij L u R i ν L j R ( 2 /3) 2 + h.c. , (A.6) gives C S L (µ LQ ) = 4 C T (µ LQ ) = 1 4 √ 2G F V cb y cτ L y bτ R * m 2 R 2 . (A.7) In contrast to the above two LQ scenarios, the R 2 LQ does not generate C V L but C V R . Thus we could expect solid predictions in polarization and related observables. To generate C V R , indeed, a large mixing between two distinct R 2 LQ doublet is required to induce a proper electroweak symmetry breaking. See details in Refs. [68,85]. 281 ± 0.018 ± 0.024 . . Summary of the SM predictions for the B → D ( * ) τ ν and related observables. The current combined results of the experimental measurements are also written in the last line. See the main text for the definitions of the observables. Figure 2 . 2τ from the SM predictions as shown with the P D * τ,exp = −0.38 ± 0.51 +0.Predictions of P D τ and P D * τ S 1 1LQ (C V L = 0) : C S L = −8.9 C T = 0.19 , Pull = 3.9 , (3.14) R 2 LQ (C V R = 0) : C S L = 8.4 C T = −0.07 ± i0.58 , Pull = 4.0 , (3.15) Figure 3 . 3Predictions of P D τ and P D * τ Figure 4 . 4The correlation between R Υ(3S) and R D ( * ) is exhibited in the U 1 LQ scenario with m U1 = 2 TeV by setting (V x L ) cτ = V cb x bτ L . The predicted values of R Υ(3S) are shown by black contours, and R exp Υ(3S) in Eq. (4.2) is shown in the red shaded area. R D and R D * anomalies can be explained in the blue and green regions, respectively. The LHC exclusion region (outside the blue line) and the HL-LHC sensitivity (red dashed line) are based on the result ofTable 3. The best fit points of Eq. (3.11) are shown by red crosses. The orange dashed line represents the U (2) flavor symmetry prediction with β R = −1. Figure 5 . 5Correlation between R Υ(3S) and R D ( * ) in the R 2 LQ scenario with m R2 = 2 TeV and y cτ L = 1. The color convention is the same as inFig. 4. The gray shaded region is excluded by the B c lifetime. The best fit points of Eq. (3.15) are shown by red crosses. }, one can see {3.2σ, 4.0σ, 4.1σ, 3.6σ} deviations corresponding to p-value = {1.2×10 −3 , 6.4×10 −5 , 4.8×10 −5 , 2.7×10 −4 } (∆χ 2 = {13.8, 19.3, 19.9, 16.4} for 2 degrees of freedom), respectively. Figure 1. A world average of the latest R D and R D * experimental results (red, 1, 2, 3σ contours), compared with the previous HFLAV 2021 average (dashed orange)[20] and with Ref.[21] (dashed blue) which includes the nontrivial D * * contribution. On the other hand, the several SM predictions are shown by crosses[22][23][24][25].HFLAV 2021 + Bernlochner, et al. + Iguro, Watanabe + Bordone, et al. 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[]
[ "Quantum mechanics is based on a relativity principle", "Quantum mechanics is based on a relativity principle" ]
[ "Léon Brenig \nUniversité Libre de Bruxelles. CP\n231. 1050BrusselsBelgium\n" ]
[ "Université Libre de Bruxelles. CP\n231. 1050BrusselsBelgium" ]
[]
Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schrödinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction PACS numbers: 03.65.Ta
null
[ "https://export.arxiv.org/pdf/quant-ph/0608025v1.pdf" ]
118,761,200
quant-ph/0608025
2f68ae831b0fda8d2f1884f74ee2bfbdc227811c
Quantum mechanics is based on a relativity principle 2 Aug 2006 Léon Brenig Université Libre de Bruxelles. CP 231. 1050BrusselsBelgium Quantum mechanics is based on a relativity principle 2 Aug 2006(Dated: 16 July 2006) Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schrödinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction PACS numbers: 03.65.Ta Unlike special relativity and the Einstein's theory of gravitation, quantum mechanics is not considered to be based on a principle of relativity. The formers require the invariance of the mathematical representation of the laws of Nature under transformations relating so-called observers or space-time frames of reference. The demand that classical mechanics becomes invariant under these transformations entails modifications of the fundamental laws of physics. Our claim in this Letter is that quantum mechanics is also a relativity theory in the sense that this theory emerges out of classical mechanics from the condition of invariance of the laws of mechanics under a new group of frame transformations. This group acts on the precision of position and momentum measurements of the observer. Observers or frames of references are characterized not only by the position of their space and time origins, direction of their axis and relative velocities but also by the accuracy or resolution of their instruments. Such an approach to quantum mechanics has not been contemplated often in modern physics. The most notorious exception is, to our knowledge, the interesting work of Nottale [1], [2], [3]. The theory reported here, however, differs fundamentally from that developed by Nottale. This difference is mainly due to the fact that we do not attribute a fractal character to space-time in contrast with this researcher. In Nottale's work, this fundamental fractality is inferred from the discovery by Abbott and Wise that the trajectory of a quantum free particle has fractal dimension 2 [4]. Nottale relates this characteristics to a fundamental fractal property of space-time. Due to this fractality, he is led to consider scale transformations. These transformations are, however, very different from ours. Moreover, the meaning of a fractal trajectory is somewhat vague in the sense that many aspects of quantum mechanics point to the vacuity of the concept of trajectory. In contrast, our work does not invoke such concept and remains in the usual Copenhague interpretation frame. There is, however, a possible articulation with Nottale's work in the sense that starting with a Galilean space-time and requiring only invariance of physics under a well-defined relativity group acting on measurements precision, we find a supplementary time dimension. Hence, "trajectories" are defined by two temporal real parameters and can be viewed as 2-dimensional objects. But this affects the topological dimension and not the fractal dimension. This is, perhaps, the limit between the two approaches. Due to the concision inherent to a Letter the demonstrations of many results reported here are only outlined, though, all of them can be retraced by the reader. A more complete article will be submitted elsewhere containing also generalization to a particle in an exterior potential and to systems of N particles interacting via a binary potential. Let us consider a non-relativistic free particle described by the Schrödinger equation. The transformations on the uncertainties we are proposing are the following ∆x ′ 2 = e −α ∆x 2 (1) ∆p ′ 2 = e −α ∆p 2 + 2 4 e α − e −α 1 ∆x 2 (2) where ∆x 2 is the Fisher dispersion of the position probability distribution (p.d.) ρ(x). It is given by ∆x 2 = 1 2 ∇ρ 1/2 (x) 2 d 3 x(3) where the denominator is the Fisher information associated to the p.d. ρ [5]. Its relation with the position mean square deviation σ x 2 is given by the Cramér-Rao inequality σ x 2 ≥ ∆x 2(4) a classical result in theoretical statistics obtained from the Schwartz inequality [6] . The quantity ∆p 2 is the mean square deviation of momentum calculated with the standard quantum algorithm for the expectation of any function of the momentum operator. The group property of the above transformations is easy to establish. From here on, we consider it as the relativity group relating the various observers and under which the laws of physics should be covariant. Multiplying equation (1) by equation (2) one gets ∆x ′ 2 ∆p ′ 2 = e −2α ∆x 2 ∆p 2 + 2 4 1 − e −2α(5) The parameter α is any real number. The oneparameter continuous group structure of the set of these transformations is easy to prove. Furthermore, when α→+∞, ∆x ′ 2 ∆p ′ 2 → 2 4 . If ∆x 2 ∆p 2 is already equal to 2 4 then the product ∆x ′ 2 ∆p ′ 2 keeps the value 2 4 for any value of α. For α→-∞, ∆x ′ 2 ∆p ′ 2 →+∞ for any value of ∆x 2 ∆p 2 ≥ 2 4 . Of course, since the Cramér-Rao inequality (4) guarantees that σ x 2 ∆p 2 ≥∆x 2 ∆p 2 , all these asymptotic results are lower boundaries for the values of σ x 2 ∆p 2 and its transformations. These remarkable properties bear some similarities with the Lorentz transformations. In the same way the velocity of light constitutes an upper value for the velocities of material bodies, the constant 2 4 represents a lower limit value for the product of uncertainties ∆x 2 ∆p 2 . The choice of the above transformations (1), (2) as relativity group for the laws of physics imposes a radical modification of the laws of dynamics that corresponds to the passage from classical to quantum mechanics as we prove now. Let us consider the classical mechanical description of a free non-relativistic particle of mass m. We describe it in a field canonical framework [7] by introducing at the initial time the p.d. ρ(x) of an ensemble of identical non-interacting particles. This function together with the classical action of the particle, s(x), are the basic field variables of the formalism. The time evolution of any functional of type A = d 3 xF (x, ρ, ∇ρ, ∇∇̺, ..., s, ∇s, ∇∇s, ...) (6) of the two variables ρ and s that is at least once functionally differentiable in terms of ρ and s is given by ∂ t A = {A, H cl }(7) where H cl = d 3 x ρ ∇s| 2 2m(8) is the classical Hamiltonian functional and {A, B} = d 3 x [ δA δρ(x) δB δs(x) − δB δρ(x) δA δs(x) ](9) where δ δρ(x) and δ δs(x) are functional derivatives. The above functional Poisson bracket endows the set of functionals of type (6) with an infinite Lie algebra structure G. Any functional of G, and H cl is one of them, generates a one-parameter continuous group of transformations . The time transformations are generated by H cl . Equation (7) when applied to ρ(x) and s(x) respectively, yields the continuity equation and the Hamilton-Jacobi equation ∂ t ρ = −∇. ρ ∇s m (10) ∂ t s = − |∇s| 2 2m(11) where the gradient ∇s is the momentum of the particle. Now let us consider the group of space dilatations and its action on ρ and s ρ ′ (x) = e 3α 2 ρ(e α 2 x), s ′ (x) = e −α s(e α 2 x)(12) where α is any real number. Note that these transformations preserve the normalization of the p.d. ρ(x) [5]. Clearly, they also keep the dynamical equations (10) and (11) invariant. Let us assume that the average momentum of the particle is vanishing. This corresponds to a particular choice of the frame of reference but, by no means, reduces the generality of our results. In this frame, the quadratic mean deviation of the momentum is given by ∆p cl 2 = d 3 x ρ ∇s| 2 = 2mH cl(13) and under transformations (12) Also, the Fisher dispersion of ρ(x), ∆x 2 , defined in equation (3) transforms as ∆x ′ 2 = e −α ∆x 2(15) Not surprisingly, it appears from equation (14) that the classical momentum uncertainty does not transform like prescribed by equation (2) above. In view of the physical dimensions of ∆p cl 2 as defined in equation (13), transformation (14) is expected under dilatations affecting the position coordinates. It corresponds to the first term in the right hand side of equation (2). Now, let us modify definition (13) of ∆p cl 2 by adding a new term proportional to the Fisher information, 2 2 F . This addition along with the use of equation (3) yields the following quantity ∆p q 2 = d 3 x ρ(x)|∇s(x)| 2 + 2 d 3 x ∇ρ(x) 1/2 2(16) We new prove that the above supplementary term restores the relativity transformation law (2). Let us apply the space dilatation (12) to this functional. This leads to ∆p ′ q 2 = e −α d 3 x ρ(x)|∇s(x)| 2 +e α 2 d 3 x ∇ρ(x) 1/2 2(17) Adding and subtracting an appropriate term, e −α 2 d 3 x|∇ρ(x) 1/2 | 2 , to the right hand side of equation (17) allows expressing this equation in terms of ∆p q 2 as follows ∆p ′ q 2 = e −α ∆p q 2 + 2 4 e α − e −α 1 ∆x 2(18) where we have used again equation (3). This equation is identical to the transformation law (2). We may, thus, identify ∆p q with ∆p , i.e. the quantal momentum uncertainty. Since the Hamiltonian functional for a free particle is its average kinetic energy, we have in this frame H q = ∆p q 2 2m(19) or H q = d 3 x ρ(x) ∇s(x)| 2 2m + 2 2m d 3 x ∇ρ(x) 1/2 2( 20) This is precisely the expected expression of the quantum average of the energy for a free particle. Clearly, the apparition of the Planck constant in this derivation is quite artificial. The constant multiplying the Fisher information in the added term in (16) is just arbitrary. The Planck value of that constant has been postulated in order to retrieve quantum mechanics. The functional H q generates the quantum time evolution of any functional A of the algebra G via equation (7) where H cl is to be replaced by H q . In particular it gives the Schrödinger equation when A is just the wave function ρ 1/2 e is/ . We leave this demonstration to the reader. One of the intermediate results is the apparition of the quantum potential [8] in the Hamilton-Jacobi equation (11) ∂ t s = − |∇s| 2 2m + 2 2m ∇ 2 ρ 1/2 ρ 1/2(21) while the continuity equation for ρ (10) is preserved. Let us summarize. We have derived the quantum evolution for a free particle from the requirement that the quadratic uncertainties on position and momentum should satisfy the relativity transformations laws (1) and (2). This result is easily generalized to a particle in an exterior potential or to N particles interacting via a binary potential. The form in which we obtain quantum mechanics is that of canonical field theory which has been introduced and studied from different points of view by various authors [9], [10], [11], [12], [13]. None of these authors, however, derives quantum mechanics from a relativity principle as we do here. They assume the existence of the quantum formalism based on Hilbert space and the algebra of observable operators acting on it, and show that this framework can be derived from a more general symplectic or canonical field theory and/or from a variational principle. One more important question has now to be investigated: That of the non-invariance of the Schrödinger equation under the transformation (1), (2). It is a wellknown fact that this equation is not invariant under the conformal group and our transformations are, indeed, dilatations of position coordinates. Let us first remark that the generator of transformations (1) and (2) in the algebra G is the functional S = d 3 x ρ(x)s(x)(22) This can readily be verified by exponentiating the infinitesimal transformation ∆p ′ q 2 = ∆p q 2 + δα ∆p q 2 , S(23) where the bracket is still the one defined in equation (9). So doing, one gets the same expression as equation (17) or (2). Let us now define the following new functional K q ≡ {S, H q } = d 3 x ρ ∇s| 2 2m − 2 2m d 3 x ∇ρ 1/2 2 (24) and let us apply the group generated by S on both H q and K q . An easy calculation yields H ′ q = coshα H q −sinhα K q , K ′ q = −sinhα H q +coshα K q(25) This is due to the fact that {S, K q } is equal to H q . These transformations are isomorphic to 2-D Lorentz transformations. Since K q only differs from H q by the sign of the quantum potential, the group it generates is parametrized by a new time parameter, τ . Any functional A of G can be considered as a function of both t and τ and its evolution in both times is given by ∂ t A = {A, H q } , ∂ τ A = {A, K q }(26) Note also that both generators tend to H cl for →0, i.e. both times variables become identical in the classical limit. For finite value of the transformations (25) induce Lorentz-like transformations in the plane (t, τ ) t ′ = coshα t + sinhα τ, τ ′ = sinhα t + coshα τ (27) Also, the remarkable property that H q + iK q is a holomorphic function of t+iτ is easily shown. The transformations generated by S mix the two time evolutions, hence, only system (26) is covariant but not the individual equations constituting it. Let us now consider the case where A is the wave function ψ given by ρ 1/2 e is/ . As stated above its evolution equation in variable t is linear and is Schrödinger's equation. In time τ , however, the equation is nonlinear i ∂ τ ψ = − 2 2m ∇ 2 ψ + 2 m ψ ∇ 2 |ψ| |ψ|(28) The nonlinear Schrödinger equation obtained here is not a newcomer in physics. It has been envisaged, though in the time t variable and in different contexts, by several authors [14], [15], [16]. It belongs to the class of Weinberg's nonlinear Schrödinger equations [17]. This equation admits a nonlinear superposition principle [18]. It has been studied, always in the usual time variable, as a member of the general class of nonlinear Schrödinger equations obtained under the so-called nonlinear gauge transformations introduced by Doebner and Goldin [19]. The evolution generated by this equation in our new time dimension is nonunitary as K q can not be reduced to the quantum average of a Hermitian operator. One easily shows also that together with the functionals generating translations, rotations and Galilean boosts, K q constitutes a field canonical representation of the Galilei algebra. A potentially important property is that equation (29) implies the continuity equation for the p.d. ρ. Moreover, the functional S is a Lyapunov-like function for this equation as ∂ τ S = {S, K q } = H q ≥ 0(29) This property is related to the fact that while ∂ t ∆x 2 ≥0 and ∂ t ∆p 2 =0, the product (∂ τ ∆x 2 )(∂ τ ∆p 2 ) is always negative. This is reminiscent of the process of state vector reduction in position measurement in which ∆x 2 → 0 while ∆p 2 → +∞, or conversely if one is measuring momentum. Would this evolution correspond to the nonunitary process that authors like R.Penrose [20] are invoking for the description of the collapse of the wave function? The difference with these approaches lies, at least, in the fact that they always consider the reduction process in the usual time. Several directions of generalization of our theory can be envisaged. One would be abandoning global invariance with respect to transformations (1) and (2) and requiring only local invariance. This could lead to the discovery of a new gauge field. Another orientation would be the extension of the above approach to the case of Klein-Gordon and Dirac equations, and more generally to quantum field theory with, perhaps, important conse-quences at the level of the general unified theory including gravitation. The author wants to dedicate this Letter to his master and friend Dr.R.Balescu who suddenly deceased during this work and who encouraged him in this approach. He wants also to thank Drs. C.George, J.Reignier, Y.Elskens, I.Veretennicoff, G.Barnich, R.Lambiotte, C.Schomblondt and Mr.F.Ngo for their many relevant comments on this work.Email address: [email protected] L Nottale, Fractal Space-Time and Microphysics. SingaporeWorld ScientificL. Nottale, Fractal Space-Time and Microphysics (World Scientific , Singapore, 1993). . L Nottale, Chaos, Solitons and Fractals. 7L. Nottale, Chaos, Solitons and Fractals, 7, 877, (1996). . M Célérier, L Notttale, J.Phys.A. 37M. Célérier and L. Notttale, J.Phys.A, 37, 931, (2004). . L F Abbott, M B Wise, Am.J.Phys. 49L.F. Abbott and M.B. Wise, Am.J.Phys., 49, 37, (1981). . M J Hall, Physical Review A. 62M. J. Hall, Physical Review A, 62, 012107-1, (2000). D R Cox, D V Hinkley, Theoretical Statistics (Chapman and Hall. LondonD.R. Cox and D.V. Hinkley, Theoretical Statistics (Chap- man and Hall, London, 1974). E Sudarshan, N Mukunda, Classical Dynamics: A Modern Perspective. MalabarRobert E.Krieger Pub.CompanyE. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (Robert E.Krieger Pub.Company, Malabar, 1983). . D Bohm, B J Hiley, Foundations of Physics. 14D. Bohm and B. J. Hiley, Foundations of Physics, 14, 255, (1984). . F Strocchi, Rev.Mod.Phys. 38F. Strocchi, Rev.Mod.Phys., 38, 36, (1966). . A Heslot, Phys.Rev.D. 311341A. Heslot, Phys.Rev.D, 31, 1341, (1985). . F Guerra, R Marra, Physical Review D. 28F. Guerra and R. Marra, Physical Review D, 28, 1916, (1983). ArXiv preprint grqc/9706069. A Ashtekar, T A Schilling, A. Ashtekar and T. A. Schilling, ArXiv preprint gr- qc/9706069, (1997). . M W Hall, M Reginatto, J.Phys.A. 35M. W. Hall and M. Reginatto, J.Phys.A, 35, 3289, (2002). . F Guerra, M Pusterla, Nuovo Cimento, 35F. Guerra and M. Pusterla, Nuovo Cimento, 35, 256, (1982). . J Vigier, Physics Letters A. 13599J. Vigier, Physics Letters A, 135, 99, (1989). . L Smolin, Physics Letters A. 113L. Smolin, Physics Letters A, 113, 408, (1986). . S Weinberg, Phys.Rev.Lett. 62485S. Weinberg, Phys.Rev.Lett., 62, 485, (1989). . G Auberson, P C Sabatier, J.Math.Phys. 35G. Auberson and P. C. Sabatier, J.Math.Phys., 35, 4028, (1994). . H Doebner, G A Goldin, P Nattermann, J.Math.Phys. 4049H. Doebner, G. A. Goldin and P. Nattermann, J.Math.Phys., 40, 49, (1999). The Road to Reality. R Penrose, Jonathan Cape, LondonR. Penrose, The Road to Reality (Jonathan Cape, Lon- don, 2004).
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[ "For Publisher's use SPONTANEOUS SUSY BREAKING IN N=2 SUPER YANG-MILLS THEORIES", "For Publisher's use SPONTANEOUS SUSY BREAKING IN N=2 SUPER YANG-MILLS THEORIES" ]
[ "Luzi Bergamin \nInstitute for Theoretical Physics\nUniversity of Bern\nSidlerstrasse 5CH-3012BernSwitzerland\n", "Peter Minkowski \nInstitute for Theoretical Physics\nUniversity of Bern\nSidlerstrasse 5CH-3012BernSwitzerland\n" ]
[ "Institute for Theoretical Physics\nUniversity of Bern\nSidlerstrasse 5CH-3012BernSwitzerland", "Institute for Theoretical Physics\nUniversity of Bern\nSidlerstrasse 5CH-3012BernSwitzerland" ]
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It is shown that the same essentially non-semiclassical mechanism, which generates in the nonsupersymmetric pure Yang-Mills theory the binary condensate of gauge field strengths, is responsible for the spontaneous breaking of the two supersymmetries in N=2 super Yang-Mills systems. A detailed discussion is presented in ref.1
null
[ "https://export.arxiv.org/pdf/hep-ph/0011041v1.pdf" ]
119,055,282
hep-ph/0011041
c23f06f57f7fa039f508025ab277142118c40176
For Publisher's use SPONTANEOUS SUSY BREAKING IN N=2 SUPER YANG-MILLS THEORIES 2 Nov 2000 Luzi Bergamin Institute for Theoretical Physics University of Bern Sidlerstrasse 5CH-3012BernSwitzerland Peter Minkowski Institute for Theoretical Physics University of Bern Sidlerstrasse 5CH-3012BernSwitzerland For Publisher's use SPONTANEOUS SUSY BREAKING IN N=2 SUPER YANG-MILLS THEORIES 2 Nov 2000 It is shown that the same essentially non-semiclassical mechanism, which generates in the nonsupersymmetric pure Yang-Mills theory the binary condensate of gauge field strengths, is responsible for the spontaneous breaking of the two supersymmetries in N=2 super Yang-Mills systems. A detailed discussion is presented in ref.1 Introduction Following the work of Seiberg and Witten 2 the semiclassical modifications due to multiinstanton configurations within N=2 theories were shown not to induce any spontaneous breaking of supersymmetry 3 , 4 . General Ward identities for N=1 Yabg-Mills systems with matter fields were derived in ref. 5 under the assumption of exactly unbroken susy. The above situation is widely interpreted as indication, not to say proof, that nonperturbative effects -as represented by instantonsdo not break supersymmetries. However the above configurations are semiclassical, nonperturbative, while binary condensate formation is nonsemiclassical, nonperturbative. The latter can in conjunction with spontaneous symmetry breaking also generate spontaneous symmetry restoration as is the case for CP in nonsupersymmetric QCD a We propose to discuss the extension of the effective potential, representing Greens functions of composite local operators to the theories under study. Short discussion A universal feature of susy in connection with spontaneous effects is revealed through the a For a detailed discussion we refer to the appendix in ref. 1 . 'once local' form of N=2 susy algebra j i µα (x) , Q kβ = δ i k ϑ µ αβ ( x ) ϑ µ αβ = ϑ µ ν σ ν αβ (1) In eq. 1 j i µα , i = 1, 2 , ϑ µ ν denote supercurrents and energy momentum tensor respectively. Now if we consider the spontaneous parameters to imply a nonvanishing expectation value for the energy momentum (density) operator -at this stage just a logical possibility -we obtain Ω | ϑ µν | Ω = ε g µν → Ω | j i µα (x) , Q kβ | Ω = = ε δ i k σ µαβ ; ε > 0(2) From eq. 2 the universal spontaneous breakdown of both (N=2) supersymmetries follows. In addition the spectral function of the two supercurrents exhibits the equally universal contribution from two massless goldstinos of the form Ω | j i µα (x) , j * k νβ (y) | Ω = δ ik ( 2π ) −3 d 4 q exp ( − i q z ) ε ( q 0 ) Γ µν̺ ( q ) σ ̺ αβ Γ µν̺ = δ ( q 2 ) γ µν̺ + · · · γ µν̺ = ε ( g µ̺ q ν + g ν̺ q µ − g µν q ̺ ) z = x − y (3) The Christoffel-symbol like structure of the quantity γ µν̺ in eq. 3 is not accidental. The spontaneous energy density ε in eqs. 2 and 3 has to be positive, as implied by susy and thus opposite to the same quantity in QCD. For further details we are forced here to refer to ref. 1 . This in order to focus on the essential features and to remain within the space requirements. Conclusions A universal connection between the ground state expected value of the energy momentum tensor and spontaneous breaking of supersymmetries in N=2 super Yang-Mills theories is demonstrated. Contrary to QCD the vacuum energy density ε is necessarily positive (nonnegative) as implied by susy. As a consequence the coupling of both goldstino modes to the supercurrents is completely determined by the vacuum energy density. . L Bergamin, P Minkowski, hep- th/hep-th/0003097L. Bergamin and P. Minkowski, hep- th/hep-th/0003097. . N Seiberg, E Witten, Nucl. Phys. 426484Nucl. Phys.N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19, Nucl. Phys. B431 (1994) 484. Multi-instanton contributions to gauge/string theory dynamics, see these proceedings. V V Khoze, V. V. Khoze, Multi-instanton contribu- tions to gauge/string theory dynamics, see these proceedings. Multi instanton calculus in supersymmetric theories, see these proceedings. F Fucito, F. Fucito, Multi instanton calculus in supersymmetric theories, see these pro- ceedings. . D Amati, K Konishi, Y Meurice, G C Rossi, G Veneziano, Phys.Rept. 162169D. Amati, K. Konishi, Y. Meurice, G.C. Rossi and G. Veneziano, Phys.Rept.162 (1988) 169.
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[ "The Implementation of Hadoop-based Crawler System and Graphlite-based PageRank-Calculation In Search Engine", "The Implementation of Hadoop-based Crawler System and Graphlite-based PageRank-Calculation In Search Engine" ]
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Nowadays, the size of the Internet is experiencing rapid growth. As of December 2014, the number of global Internet websites has more than 1 billion and all kinds of information resources are integrated together on the Internet , however,the search engine is to be a necessary tool for all users to retrieve useful information from vast amounts of web data.Generally speaking, a complete search engine includes the crawler system, index building systems, sorting systems and retrieval system. At present there are many open source implementation of search engine, such as lucene, solr, katta, elasticsearch, solandra and so on. The crawler system and sorting system is indispensable for any kind of search engine and in order to guarantee its efficiency , the former needs to update crawled vast amounts of data and the latter requires real-time to build index on newly crawled web pages and calculae its corresponding PageRank value. It is unlikely to accomplish such huge computation tasks depending on a single hardware implementation of the crawler system and sorting system,from which aspect, the distributed cluster technology is brought to the front. In this paper, we use the hadoop Map -Reduce computing framework to implement a distributed crawler system, and use the GraphLite , a distributed synchronous graph-computing framework, to achieve the real-time computation in getting the PageRank value of the new crawled web page.
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[ "https://arxiv.org/pdf/1506.00130v1.pdf" ]
13,190,079
1506.00130
ff13698939c7cad4bc501c8dab815b6d653e2170
The Implementation of Hadoop-based Crawler System and Graphlite-based PageRank-Calculation In Search Engine The Implementation of Hadoop-based Crawler System and Graphlite-based PageRank-Calculation In Search Engine 第 1 页/共 8 页 郭清沛 ,201428015029038, ISCAS 徐超 ,2014E8015061086, ISCAS, 宋扬 ,2014E8015061082,ISCASHadoopCrawler SystemGraphlitePageRankSearch Engine Nowadays, the size of the Internet is experiencing rapid growth. As of December 2014, the number of global Internet websites has more than 1 billion and all kinds of information resources are integrated together on the Internet , however,the search engine is to be a necessary tool for all users to retrieve useful information from vast amounts of web data.Generally speaking, a complete search engine includes the crawler system, index building systems, sorting systems and retrieval system. At present there are many open source implementation of search engine, such as lucene, solr, katta, elasticsearch, solandra and so on. The crawler system and sorting system is indispensable for any kind of search engine and in order to guarantee its efficiency , the former needs to update crawled vast amounts of data and the latter requires real-time to build index on newly crawled web pages and calculae its corresponding PageRank value. It is unlikely to accomplish such huge computation tasks depending on a single hardware implementation of the crawler system and sorting system,from which aspect, the distributed cluster technology is brought to the front. In this paper, we use the hadoop Map -Reduce computing framework to implement a distributed crawler system, and use the GraphLite , a distributed synchronous graph-computing framework, to achieve the real-time computation in getting the PageRank value of the new crawled web page. 1.Introduction Framework of Hadoop Map-Reduce The Map -Reduce is a programming model based on Hadoop. In the Map -Reduce distributed computing framework, the programmer only writes a serial program and ensure the correctness of the serial program and then the system will complete the execution in a parallel and distributed way, which is transparent for programmers. The hadoop-based distributed computing framework is as shown in the figure 1.1.1 below: Figure 1.1 .1 Hadoop Map-Reduce Framework The lowest layer is a cluster composed of many physical nodes, each Node in the cluster is divided in logic, and the implementation of each node is just a running process so that multiple nodes can be distributed in one or more physical hosts. HDFS and MAP -REDUCE tasks run on the cluster.HDFS defines a NameNode, usually with a Secondary NameNode for redundancy backup,which are commonly responsible for storing metadata and data backup, other DataNodes are responsible for the specific file operations such as reading and writing. The Map -Reduce tasks need to run on HDFS for sharing data between different physical host nodes and storing intermediate results. When a user submitts a Map-Reduce task ,the Map -Reduce framework can decompose a task into subtasks and assign them running on corresponding nodes in cluster.In such a way distributed computation is achieved by programmers without caring about any specific distributed implementation details. Framework of GraphLite Graphlite uses a called BSP (Bulk Synchronous Processing) programming model. As shown in figure 1.2: Figure 1.2.1 BSP programming model In graphLite framework, the computation on a graph will be divide into multiple supersteps . Between two supersteps are distributed computation without any reliance , in such a way , the goal of "general serial, parallel partial" is achieved. All dependencies of nodes operation in GraphLite are classified as "data dependence" and "temporal dependence", the former can be solved through the message-sending mechanism during the initial stage of a superstep , "temporal dependence" can be resolved by a serial of sequenced SuperSteps. At the end of each SuperStep, GraphLite will collect the messages sent by all the nodes, and sending them to the corresponding destination nodes before the next superstep begins ,then starts the next round of distributed computation. 2.Our Implementation Hadoop-based Crawler System How we use Map-Reduce In Our Crawler-System We use the Map-Reduce framework to implement the distributed crawler system as shown in figure 2.1.1: Figure2.1.1 CrawlerTask is a text file that stores the seed urls to be crawled, InputFormat is the pretreatment process before performing the of the Map operation, in which the data file will be cut into small shards, each we call it a InputSplit, defaulted with the size of 64M. Each of the InputSplits will be analyzed to a pair of <key,value>.The key of every <key,value> pair outputted by InputFormat is the starting offset of each line,while the value is the URL to be crawled. In Map process,The Mapper class input is formatted as a set of <offset,url> pairs that analyzed by InputFormat.In our system what implemented by Map function is swapping the value of key and value in input key-value pairs. We set the url of each input-pair as the output key, whose value dedicate the extracted crawling URL .While the output value is offset. The result will be written into the intermediate files, which exists in the HDFS. Given the thought that the overhead of communication between the nodes in Hadoop Cluster usually costs much in efficiency . We combine each Mapper output in Combiner stage of temporary files in the repeat key on the local merging, so we can reduce the amount of traffic between nodes and reduce the pressure of subsequent reducers. What implemented in the Partitioner process is partitioning the intermediate results. According to the value of the result-key, the results could be divided into R intermediate results after the combiner process, each will be sequently processed by a Reducer. The partition algorithm we use is aimed at calculating the hash value of each URL corresponding to the host so that the URLs belonging to the same host will be partitioned into the same bin, which will be then processed by the same Reducer. So that the same host URL will be crawled exactly on the same machine. In the Reduce phase,the URLs will be used to multithreaded downloading and the crawled web pages will be written in the HDFS. Our Running Results The crawled results based on Map-Reduce framework is stored in the HDFS , files of which are distributed saved at different host nodes. In order to display the results conveniently, we use a database visualization tool connected HDFS to display the crawled data. Distributed Crawler-System is running on the Hadoop cluster, each node in the cluster are definitely a centralized crawler, controlled by a master node to work together, so the efficiency of Distributed Crawler-System is much higher than the centralized Crawler-System. In our System ,We adopt three nodes, each node of the reducer crawling with 16 threads, 48 threads totally. Experimental results show that with the system running for 30 minutes, the size of web page stored on HDFS data is 872M (HDFS data redundancy backup number is set to 1). The single machine 16 threads run 30 minutes crawl web data of about 300M, which shows that the distributed crawler performance is much better than single node. Graphlite-based PageRank-Calculation our method to calculate PageRank In the search engine, each web page newly crawled by Crawler-System needs to be real-time calculated its weight among all web Pages in Ranking-System,namely PageRank, according to the number of up and down links . PageRank reflects the importance of web pages, which is critical for improving the user search-experience. Here we use GraphLite , a distributed System for large-scale graph processing to calculate PageRank. In our Crawler-System, after de-emphasis of each page, we will assign a unique Id for each of them. Each page is deemed as a vertex in graph computation System. The formula to calculate each vertex is as follows: Among them: Rv: PageRank * N of vertex N Lv: in-degree of vertex V B (u): out-degree of vertex u d: web links Probability N: the number of all pages In the program, we initialize the weight of all the pages as 1.0, and perform iterate computation until the result comes to convergence.We enabled four worker nodes in our program, each vertex is assigned to four worker nodes based on the result of its vertex Id mod 4, the organization format of the input file is shown as Figure 2.2.1.1 .Take the input data of Worker3 for example, the input data format is shown in Figure 2.2.1.2. In Figure 2.2.1.2 ,the number 1010 in the first line represents that this worker's entry page nodes is 1010, the 21037 in the second line represents that output edge associated with entry page node amount to 21,037. Each row of data from the beginning of the third line represents an edge, for example,"2 20" represents the page, the starting id of which is 2, and the end Id is 20. The main function of each worker is as follows: double acc = fabs(getValue() -val); accumulateAggr(0, &acc); } * mutableValue() = val; const int64_t n = getOutEdgeIterator().size(); sendMessageToAllNeighbors(val / n); } }; 第 6 页/共 8 页 Our Running Results During the cluster initializing state, the master node distribute all of the data to the 4 workers, as is shown in Figure 2.2.2.1. Computing 4039 nodes ,88,234 edges of a directed graph on Ubuntu 64bit 3.2 GHz dual-core 4 thread machine costs only 4.56s. It is almost single-node operation consuming 1 / 3! Though it seems still a little time consuming for real-time computation, but it's obvious that we can use more machines in our cluster to achieve faster computation. It is shown that the use of distributed computing that can improve the efficiency of PageRank efficiency greatly, and the use of a distributed computing architecture experiment can greatly reduce the requirements for in-memory of a single node during the computation process . 3.Conclusion In this paper, we design and implement two distributed systems to solve the real-time problem of big data processing in search engine.The Hadoop-based Crawler System and Graphlite-based PageRank-Calculation System running on a cluster are both proven highly effective than a single machine in big data process ing and can be used in real Industrial production environment as a solution. Figure 2 .Figure 2 . 3 2232 shows the results crawled by our distributed Crawler-System, andFigure 2.3 roughly display the specific content of a crawled web-page. The specific content of crawl pages Figure 2 2the organization format of the input file,Figure 2.2.1.2 shows the the input data format for each worker. Figure 2 . 22.2.1 initialization of the cluster Figure 2.2.2.2 shows that the program converges when proceeding to the 19th step. Figure 2 2Figure 2.2.2.2 Running Results . Vol 中国搜索研究中心, 18中国搜索研究中心,Vol.18,1999 Urs Hölzle, Web Search for a Planet: The Google Cluster Architecture. André 【2】 Luiz, Jeffrey Barroso, Dean, doi>10.1109/MM.2003.1196112IEEE Micro. 2【2】 Luiz André Barroso , Jeffrey Dean , Urs Hölzle, Web Search for a Planet: The Google Cluster Architecture, IEEE Micro, v.23 n.2, p.22-28, March 2003 [doi>10.1109/MM.2003.1196112] The Chubby lock service for loosely-coupled distributed systems. Howard Ghemawat, Gobioff, -Tak Shun, Sanjay Leung ; 【4】jeffrey Dean, Ghemawat ; 【7】paolo, Bruno Boldi, Massimo Codenotti, Sebastiano Santini, Vigna, doi>10.1145/945445.945450Proceedings of the 6th conference on Symposium on Opearting Systems Design & Implementation. the 6th conference on Symposium on Opearting Systems Design & ImplementationBolton Landing, NY, USA; San Francisco, CA 【5】Mike Burrows; Seattle, Washington 【6】Sergey Brin , Lawrence Page; Brisbane, Australia7Proc. AusWeb02. The Eighth Australian World Wide Web Conference. To appear in Software: Practice & ExperienceGhemawat , Howard Gobioff , Shun-Tak Leung, The Google file system, Proceedings of the nineteenth ACM symposium on Operating systems principles, October 19-22, 2003, Bolton Landing, NY, USA [doi>10.1145/945445.945450] 【4】Jeffrey Dean , Sanjay Ghemawat, MapReduce: simplified data processing on large clusters, Proceedings of the 6th conference on Symposium on Opearting Systems Design & Implementation, p.10-10, December 06-08, 2004, San Francisco, CA 【5】Mike Burrows, The Chubby lock service for loosely-coupled distributed systems, Proceedings of the 7th symposium on Operating systems design and implementation, November 06-08, 2006, Seattle, Washington 【6】Sergey Brin , Lawrence Page, The anatomy of a large-scale hypertextual Web search engine, Proceedings of the seventh international conference on World Wide Web 7, p.107-117, April 1998, Brisbane, Australia 【7】Paolo Boldi, Bruno Codenotti, Massimo Santini, and Sebastiano Vigna. Ubicrawler: A scalable fully distributed Web crawler. In Proc. AusWeb02. The Eighth Australian World Wide Web Conference, 2002. To appear in Software: Practice & Experience.
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[ "Improving the probabilistic quantum teleportation efficiency of arbitrary superposed coherent state using multipartite even and odd j-spin coherent states as resource", "Improving the probabilistic quantum teleportation efficiency of arbitrary superposed coherent state using multipartite even and odd j-spin coherent states as resource" ]
[ "Meryem El Kirdi \nFaculty of Sciences\nLPHE-Modeling and Simulation\nMohammed V University in Rabat\nRabatMorocco\n", "Abdallah Slaoui \nFaculty of Sciences\nLPHE-Modeling and Simulation\nMohammed V University in Rabat\nRabatMorocco\n\nCentre of Physics and Mathematics\nFaculty of Sciences\nCPM\nMohammed V University in Rabat\nRabatMorocco\n", "Hanane El Hadfi \nFaculty of Sciences\nLPHE-Modeling and Simulation\nMohammed V University in Rabat\nRabatMorocco\n", "Mohammed Daoud \nDepartment of Physics\nFaculty of Sciences\nUniversity Ibn Tofail\nKenitraMorocco\n" ]
[ "Faculty of Sciences\nLPHE-Modeling and Simulation\nMohammed V University in Rabat\nRabatMorocco", "Faculty of Sciences\nLPHE-Modeling and Simulation\nMohammed V University in Rabat\nRabatMorocco", "Centre of Physics and Mathematics\nFaculty of Sciences\nCPM\nMohammed V University in Rabat\nRabatMorocco", "Faculty of Sciences\nLPHE-Modeling and Simulation\nMohammed V University in Rabat\nRabatMorocco", "Department of Physics\nFaculty of Sciences\nUniversity Ibn Tofail\nKenitraMorocco" ]
[]
Quantum teleportation is one of the most important techniques for quantum information secure transmission. Using preshared entanglement, quantum teleportation is designed as a basic key in many quantum information tasks and features prominently in quantum technologies, especially in quantum communication. In this work, we provide a new probabilistic teleportation scheme for arbitrary superposed coherent states by employing the multipartite even and odd j-spin coherent states as the entangled resource connecting Alice (sender) and Bob (receiver). Here, Alice possesses both even and odd spin coherent states and makes repeated GHZ states measurements (GHZSMs) on the pair of spins, consisting of (1) the unknown spin state and (2) one of the two coherent spin states, taken alternately, until reaching a quantum teleportation with maximal average fidelity. We provide the relationship between the entanglement amount of the shared state, quantified by the concurrence, with the teleportation fidelity and the success probability of the teleported target state up to the n th repeated attempt. In this scheme, we show that the perfect quantum teleportation can be done even with a non-maximally entangled state. Furthermore, this repeated GHZSMs attempt process significantly increases both the average fidelity of the teleported state and the probability of a successful run of the probabilistic protocol. Also on our results, we show that the j-spin number, the target state parameter and the overlap between coherent states provide important additional control parameters that can be adjusted to maximize the teleportation efficiency.2 system is shown in[19]. Bell states are of paramount importance in quantum communication and are widely applied in quantum teleportation protocols. Existing Bell state analysis protocols typically focus on encoding the Bell state directly into the physical qubit. Zhou et al[20]proposed a comprehensive analysis of the Bell logic state using the helper single atoms in a low-quality cavity and discussed its application in quantum teleportation. Sheng and his colleagues[21]described an alternative approach to perform the almost complete logic analysis of the Bell state for the polarized concatenated GHZ state with two logic qubits. The analysis of hyperentangled Bell states is also discussed in[22], where a scheme to fully distinguish 16 hyperentangled Bell states is proposed.Subsequently, several theorists and experiments have proposed, discussed and reported this type of quantum communication. Saliman and his co-workers[23]proposed an efficient scheme to teleport an entangled state of two superconducting qubits from Alice's lab to Bob's lab. This kind of two-level system has recently attracted particular attention due to the possibility of tuning the coupling strength between them. They first generated the GHZ state as the chosen quantum channel, and then, by performing the appropriate gates and measurements in each lab, showed that the proposed protocol can be successfully executed with maximum values of fidelity and success probability. Sisodia et al.[26], provided an optimized controlled quantum teleportation protocol for a multi-qubit quantum state using only one GHZ state with 100% success probability. Furthermore, Jahanbakhch and his colleagues [25] presented a teleportation scheme from an unknown atomic state of a qubit (interacting with the quantized field in a cavity) to a second qubit (existing in another distant cavity field) beyond the rotating wave approximation and without the Bell state measurement method. Sisodia at al.[26]investigated an improved scheme for bidirectional quantum teleportation of two-two and two-three qubit quantum states using optimized quantum resource and less consumption of classical resources, where they found improved intrinsic efficiency and discussed the security of the protocol. Moreover, the exact dynamics of entanglement of two two-level atoms in a dissipative cavity and the entanglement protection approach in interacting two-level systems were investigated in[27]. Zhou et al.[28]proposed bidirectional controlled quantum teleportation of a three-qubit state using GHZ entangled states, where Alice transmits an unknown three-qubit entangled state to Bob, and Bob transmits an unknown three-qubit entangled state to Alice via the control of the supervisor Charlie.In realistic conditions, the impact of the environment exists and the spyder-environment coupling destroys the entanglement existing in the used maximally entangled state[29,30]. The condition of a maximally entangled quantum channel linking Alice and Bob is really hard to achieve or to maintain in its practical implementations due to the noise that reduces entanglement. Therefore, achieving the perfect quantum teleportation will be impossible. The standard figure of merit for quantum teleportation is given by the fidelity deviation [31], which has been the subject of special attention recently. This important quantity gives the idea of fluctuations associated with teleportation fidelity[32]. Hence one should consider situations with non maximally entangled state (NME). With a non-maximally entangled quantum channel, the fidelity of teleportation is always less than 1, so there are two available possibilities. First is to accept imperfect quantum teleportation with a maximal fidelity less than 1. The second possibility is to get perfect quantum teleportation with probabilities success which is called Probabilistic Quantum Teleportation (PQT) in the literature[33]. Indeed, the idea of probabilistic quantum teleportation was first proposed by Agrawal and Pati in[34], viewed as a generalized quantum teleportation with a non maximally entangled resource and they have shown that one can teleport an arbitrary state with unit fidelity but less than a unit probability. Furthermore, with the choice of probabilistic quantum teleportation, one has reliable quantum teleportation in two cases among four possible results with success probability goes from zero for an unentangled resource state to one-half for the maximally entangled state. The other one-half will come when the shared resource and joint measurement are maximally entangled states.Practically, the number of measurement repetitions depends on the parameter of the shared state, and it ranges from 1 for a maximally entangled state to infinity for an unentangled state. It should be noted that, in the probabilistic quantum teleportation, both Alice and Bob have to know the shared state. Otherwise, Bob cannot know what is the basis used by Alice to perform its task after receiving the classical communication. Furthermore, quantum teleportation via a partially entangled shared state was considered in[35], since a mixed state can be purified to a maximally entangled Bell state[36][37][38]. So, a mixed quantum channel could never give a reliable teleportation. On the other hand, if one selects an exact teleportation even with some probability, then one should use pure entangled pairs. Further, Fortes and Rigolin[39]showed a way to increase the fidelity of teleportation in the presence of noise, without decreasing the probability of success of a protocol execution, working only with probabilistic protocols. They showed that in some cases we cannot obtain a reliable quantum teleportation without using probabilistic protocols. The same team studied the probabilistic quantum teleportation when the entanglement needed to execute the protocol is given by the thermal entanglement[40].In this paper, we shall consider quantum teleportation of spin coherent states. Especially, we shall exploit the factorization property of su(2) coherent states introduced by Peremolov [41, 42] which describe many quantum systems with powerful applications. We focus on the probabilistic quantum teleportation using multipartite even and odd su(2) coherent states as a quantum channel. In fact, we shall employ the property according to which even and odd spin coherent states can contain two, three or more spin subsystems[43,44]. The idea is to teleport the superposed j-spin coherent states. We propose here a scheme involving repeated GHZ states measurements to improve both the average fidelity and the success probability of PQT[45,46]. Here, Alice keeps up with the two subsystems on its possession and performs repeated GHZ states measurements on the pair of systems, consisting of (1) the subsystem encoding the information she needs to send and (2) one of both spin of the entangled resource until reaching quantum teleportation with unit maximal fidelity. If perfect quantum teleportation is not attained, i.e.,
10.1007/s00340-023-08039-2
[ "https://export.arxiv.org/pdf/2202.08591v2.pdf" ]
246,904,371
2202.08591
16ef1a74199648fc95ff145ee8f59f45e869fa44
Improving the probabilistic quantum teleportation efficiency of arbitrary superposed coherent state using multipartite even and odd j-spin coherent states as resource Meryem El Kirdi Faculty of Sciences LPHE-Modeling and Simulation Mohammed V University in Rabat RabatMorocco Abdallah Slaoui Faculty of Sciences LPHE-Modeling and Simulation Mohammed V University in Rabat RabatMorocco Centre of Physics and Mathematics Faculty of Sciences CPM Mohammed V University in Rabat RabatMorocco Hanane El Hadfi Faculty of Sciences LPHE-Modeling and Simulation Mohammed V University in Rabat RabatMorocco Mohammed Daoud Department of Physics Faculty of Sciences University Ibn Tofail KenitraMorocco Improving the probabilistic quantum teleportation efficiency of arbitrary superposed coherent state using multipartite even and odd j-spin coherent states as resource (Dated: 12 mai 2023)Probabilistic quantum teleportationSpin Coherent StateGHZ States MeasurementsConcur- renceSuccess ProbabilityMaximal Average Fidelity PACS numbers: 0365Ta0365Yz0367Mn4250-p0365Ud Quantum teleportation is one of the most important techniques for quantum information secure transmission. Using preshared entanglement, quantum teleportation is designed as a basic key in many quantum information tasks and features prominently in quantum technologies, especially in quantum communication. In this work, we provide a new probabilistic teleportation scheme for arbitrary superposed coherent states by employing the multipartite even and odd j-spin coherent states as the entangled resource connecting Alice (sender) and Bob (receiver). Here, Alice possesses both even and odd spin coherent states and makes repeated GHZ states measurements (GHZSMs) on the pair of spins, consisting of (1) the unknown spin state and (2) one of the two coherent spin states, taken alternately, until reaching a quantum teleportation with maximal average fidelity. We provide the relationship between the entanglement amount of the shared state, quantified by the concurrence, with the teleportation fidelity and the success probability of the teleported target state up to the n th repeated attempt. In this scheme, we show that the perfect quantum teleportation can be done even with a non-maximally entangled state. Furthermore, this repeated GHZSMs attempt process significantly increases both the average fidelity of the teleported state and the probability of a successful run of the probabilistic protocol. Also on our results, we show that the j-spin number, the target state parameter and the overlap between coherent states provide important additional control parameters that can be adjusted to maximize the teleportation efficiency.2 system is shown in[19]. Bell states are of paramount importance in quantum communication and are widely applied in quantum teleportation protocols. Existing Bell state analysis protocols typically focus on encoding the Bell state directly into the physical qubit. Zhou et al[20]proposed a comprehensive analysis of the Bell logic state using the helper single atoms in a low-quality cavity and discussed its application in quantum teleportation. Sheng and his colleagues[21]described an alternative approach to perform the almost complete logic analysis of the Bell state for the polarized concatenated GHZ state with two logic qubits. The analysis of hyperentangled Bell states is also discussed in[22], where a scheme to fully distinguish 16 hyperentangled Bell states is proposed.Subsequently, several theorists and experiments have proposed, discussed and reported this type of quantum communication. Saliman and his co-workers[23]proposed an efficient scheme to teleport an entangled state of two superconducting qubits from Alice's lab to Bob's lab. This kind of two-level system has recently attracted particular attention due to the possibility of tuning the coupling strength between them. They first generated the GHZ state as the chosen quantum channel, and then, by performing the appropriate gates and measurements in each lab, showed that the proposed protocol can be successfully executed with maximum values of fidelity and success probability. Sisodia et al.[26], provided an optimized controlled quantum teleportation protocol for a multi-qubit quantum state using only one GHZ state with 100% success probability. Furthermore, Jahanbakhch and his colleagues [25] presented a teleportation scheme from an unknown atomic state of a qubit (interacting with the quantized field in a cavity) to a second qubit (existing in another distant cavity field) beyond the rotating wave approximation and without the Bell state measurement method. Sisodia at al.[26]investigated an improved scheme for bidirectional quantum teleportation of two-two and two-three qubit quantum states using optimized quantum resource and less consumption of classical resources, where they found improved intrinsic efficiency and discussed the security of the protocol. Moreover, the exact dynamics of entanglement of two two-level atoms in a dissipative cavity and the entanglement protection approach in interacting two-level systems were investigated in[27]. Zhou et al.[28]proposed bidirectional controlled quantum teleportation of a three-qubit state using GHZ entangled states, where Alice transmits an unknown three-qubit entangled state to Bob, and Bob transmits an unknown three-qubit entangled state to Alice via the control of the supervisor Charlie.In realistic conditions, the impact of the environment exists and the spyder-environment coupling destroys the entanglement existing in the used maximally entangled state[29,30]. The condition of a maximally entangled quantum channel linking Alice and Bob is really hard to achieve or to maintain in its practical implementations due to the noise that reduces entanglement. Therefore, achieving the perfect quantum teleportation will be impossible. The standard figure of merit for quantum teleportation is given by the fidelity deviation [31], which has been the subject of special attention recently. This important quantity gives the idea of fluctuations associated with teleportation fidelity[32]. Hence one should consider situations with non maximally entangled state (NME). With a non-maximally entangled quantum channel, the fidelity of teleportation is always less than 1, so there are two available possibilities. First is to accept imperfect quantum teleportation with a maximal fidelity less than 1. The second possibility is to get perfect quantum teleportation with probabilities success which is called Probabilistic Quantum Teleportation (PQT) in the literature[33]. Indeed, the idea of probabilistic quantum teleportation was first proposed by Agrawal and Pati in[34], viewed as a generalized quantum teleportation with a non maximally entangled resource and they have shown that one can teleport an arbitrary state with unit fidelity but less than a unit probability. Furthermore, with the choice of probabilistic quantum teleportation, one has reliable quantum teleportation in two cases among four possible results with success probability goes from zero for an unentangled resource state to one-half for the maximally entangled state. The other one-half will come when the shared resource and joint measurement are maximally entangled states.Practically, the number of measurement repetitions depends on the parameter of the shared state, and it ranges from 1 for a maximally entangled state to infinity for an unentangled state. It should be noted that, in the probabilistic quantum teleportation, both Alice and Bob have to know the shared state. Otherwise, Bob cannot know what is the basis used by Alice to perform its task after receiving the classical communication. Furthermore, quantum teleportation via a partially entangled shared state was considered in[35], since a mixed state can be purified to a maximally entangled Bell state[36][37][38]. So, a mixed quantum channel could never give a reliable teleportation. On the other hand, if one selects an exact teleportation even with some probability, then one should use pure entangled pairs. Further, Fortes and Rigolin[39]showed a way to increase the fidelity of teleportation in the presence of noise, without decreasing the probability of success of a protocol execution, working only with probabilistic protocols. They showed that in some cases we cannot obtain a reliable quantum teleportation without using probabilistic protocols. The same team studied the probabilistic quantum teleportation when the entanglement needed to execute the protocol is given by the thermal entanglement[40].In this paper, we shall consider quantum teleportation of spin coherent states. Especially, we shall exploit the factorization property of su(2) coherent states introduced by Peremolov [41, 42] which describe many quantum systems with powerful applications. We focus on the probabilistic quantum teleportation using multipartite even and odd su(2) coherent states as a quantum channel. In fact, we shall employ the property according to which even and odd spin coherent states can contain two, three or more spin subsystems[43,44]. The idea is to teleport the superposed j-spin coherent states. We propose here a scheme involving repeated GHZ states measurements to improve both the average fidelity and the success probability of PQT[45,46]. Here, Alice keeps up with the two subsystems on its possession and performs repeated GHZ states measurements on the pair of systems, consisting of (1) the subsystem encoding the information she needs to send and (2) one of both spin of the entangled resource until reaching quantum teleportation with unit maximal fidelity. If perfect quantum teleportation is not attained, i.e., I. INTRODUCTION Quantum teleportation [1] is a physical process via which it is possible to transfer the state of a quantum system from one location to another without knowing the state. In fact, it was one of the most profound discovery of quantum information theory. To accomplish this process, one can use quantum entanglement as shared resources [2][3][4]. In this respect, a sender Alice desires to send an unknown state of one qubit to a receiver called Bob across space without having to physically send it. Ideally, Alice and Bob need to share a maximally entangled two qubits state as resource (quantum channel) and a classical two bits classical channel. Then, Alice performs a Bell state measurement (all the orthogonal Bell states are maximally entangled too) on the combined tripartite state and communicates through the classical channel the result to Bob. Bob, after receiving the result from Alice, performs the suitable unitary transformation to reconstruct the desired state. In this original scheme, the pure maximally entangled state acts as an ideal noise-free quantum channel. Here it is easy to prove that teleportation can succeed faithfully via the maximally entangled state. In other words, the receiver can recover the sent pure state with the probability equal to 1. This results are extended to N -dimensional quantum states [5,6]. This protocol is a typical example of an entanglementassisted process that was first presented by Bennett and coworkers in 1993 [1]. Shortly after, this teleportation protocol was experimentally verified in 1997 [7][8][9]. Since then, various types of quantum teleportation protocols have been widely studied both theoretically [10][11][12][13] and experimentally [14][15][16]. Quantum teleportation protocols enable an unknown quantum state of an object to be transferred from one location to another further away without physically transferring the object itself. Transfer over large distances is necessary for protocols such as quantum networks and quantum computing. Practically, the distances for using optical fibers and free-space channels are restricted to about 100km. Recently, Ren et al. [17] reported the quantum teleportation of an independent single-photon qubit from a ground-based observatory to a satellite in low Earth orbit, via an uplink channel, verifying a new record up to 1400km. Moreover, Wang et al [18] proved multi-degree-of-freedom quantum teleportation of a single photon encoded in both degrees of freedom as a hyperentangled quantum channel and developed a method to project and discriminate hyper-entangled Bell states. Later, an experimental quantum teleportation of the multi-level state of a single photon in a three-dimensional six-photon fidelity less than 1, Alice repeats GHZ states measurement with the entangled subsystem not used in the previous GHZ states measurement replacing the one used. If perfect quantum teleportation is attained, Alice sends the subsystem not used in the last GHZ states measurement and the result of last GHZ states measurements to Bob, who then applies the suitable unitary transformations on his subsystem to get the exact information. The remaining of the paper is organized as follows. In Section II, we present a brief review of the multipartite even and odd spin coherent state that we use in the quantum teleportation protocol and discuss their entanglement degree using the concurrence concept in terms of the overlap parameter p and the spin number j. In Section III, we give a detailed description of our probabilistic quantum teleportation scheme, in which perfect transmission of the superposed coherent state is achieved with high efficiency. The primary attempt of the quantum teleportation of spin-j coherent state using GHZ states measurements is reported in Section IV. Then, we present successively the first and second repeated attempt of GHZ states measurements on the failure cases and we calculate the average fidelity and the success probability of the both attempts. This is done by analyzing their variations in terms of the different parameters characterizing the used teleported quantum state. Finally, we summarize our work in the last section. II. ENTANGLED SPIN COHERENT STATES AND QUBIT MAPPING A. Even and odd multipartite j-spin coherent states Broadly, coherent states (quasi-classical states associated to the Heisenberg-Weyl group) are regarded as the closest quantum counterpart to the classical radiation field states, as they are related to the classical wave characteristics of light [47,48]. Indeed, these states became very important in quantum optics thanks to Glauber [49] who proved that they are the eigenstates of the annihilation operator of the harmonic oscillator and minimizing the Heisenberg uncertainty relation [50]. Another concept widely used and applied in various quantum information processing and transmission tasks is the notion of spin coherent states introduced by Perelomov [41] (or su(2) coherent states), considered as the quantum states closest to the Glauber's coherent states. These states, also known as atomic coherent states, are non-orthogonal and their qubit states correspond to the spin-1/2 representations of su (2). Here, we shall strictly focus on the basic definition of odd and even su(2) coherent state. In particular, by using the property according to which a spin-j coherent state |j, η can be factorized as a tensor product of two su(2) coherent states |j, η ≡ |j 1 , η ⊗ |j 2 , η with j = j 1 + j 2 , it is possible to construct a picture where even and odd spin coherent states might be viewed as superpositions of two or more spin coherent systems. Firstly, the su(2) generators J ± and J Z satisfy the following structure relations [J Z , J ± ] = ±J ± , [J − , J + ] = −2J Z ,(1) where J + and J − represent the raising and lowering operators of su(2) Lie algebra, respectively. The (2j + 1)-dimensional Hilbert space H j is spanned by the irreducible tonsorial set {|j, m , m = −j, −j + 1, · · · , j − 1, j} characterizing the spin-j representations of the group su (2). On this basis {|j, m }, the su(2) generators acting on this irreducible unitary representation as follows J ± |j, m = (j ∓ m) (j ± m + 1) |j, m ± 1 , J Z |j, m = m |j, m .(2) The coherent state can be obtained by the action of the displacement operator D(ξ) = exp(ξJ + − ξ * J − ) on the extremal state |j, −j as |j, η = D(ξ)|j, −j = exp(ξJ + − ξ * J − )|j, −j = (1 + |η| 2 ) −j exp(ηJ + )|j, −j ,(3) with η = (ξ/|ξ|) tan |ξ|. The even and odd coherent states associated with the j-spin are defined by |j, η, m = N m |j, η + e imπ |j, −η ,(4) where the integer m takes the values m = 0 (mod 2) and m = 1 (mod 1), j is the quantum angular momentum which takes integer or half integer values and the spin coherent state |j, η is reduced to |j, η = (1 + |η| 2 ) −j j m=−j (2j)! (j + m)!(j − m)! 2 η j+m |j, m .(5) For a particle with spin-1 2 , the above equation (5) becomes |η = 1 1 + |η| 2 | ↓ + η 1 + |η| 2 | ↑ .(6) We set η here, the short for spin-1 2 coherent state | 1 2 , η with the basis states | ↑ ≡ | 1 2 , 1 2 and | ↓ ≡ | 1 2 , − 1 2 . In this context, any spin-j coherent states can be rewriting as a 2j tensorial product of spin-1 2 coherent states [42,43] |j, η = (| 1/2, η ) ⊗2j ≡ (| η ) ⊗2j .(7) The overlap between the coherent states |j, η > and |j, −η > writes as p 2j = j, η|j, −η = (< η| − η >) 2j = 1 − |η| 2 1 + |η| 2 2j ,(8) and the normalization N m in the equation (4) is given by N m = 2 + 2p 2j cos mπ − 1 2 .(9) Two asymptotic limits of spin coherent states arise when p → 0 and p → 1. In the first limiting case where p → 0, the logical qubits |j, η, 0 (even) and |j, η, 1 (odd) spin coherent states behave like a multipartite state of Greenberger-Horne-Zeilinger (GHZ) type [46]. In this case, the states |η and | − η are orthogonal and an orthogonal basis can be defined such that |0 ≡ |η and |1 ≡ | − η . Then, the state |j, η, m behave like a state of GHZ-type |j, η, m ∼ |GHZ 2j = 1 √ 2 |0 ⊗ |0 ⊗ · · · ⊗ |0 + e imπ |1 ⊗ |1 ⊗ · · · ⊗ |1 .(10) In the second limiting case where p → 1 (or η → 0), it is important to distinguish separately the situations when m = 0 (mod 2) and m = 1 (mod 1). Accordingly, the state |j, η, m = 0 (mod 2) reduces to ground state of the form |j, 0, 0 (mod 2) ∼ | ↓ ⊗ | ↓ ⊗ · · · ⊗ | ↓ ,(11) and for the second situation with odd m, the state |j, η, 1 (mod 1) becomes a multipartite state of W -type [51] |j, 0, 1 (mod 2) ∼ |W 2j = 1 √ 2j (| ↑ ⊗ | ↓ ⊗ · · · ⊗ | ↓ + | ↓ ⊗ | ↑ ⊗ . . . ⊗ | ↓ + · · · + | ↓ ⊗ | ↓ ⊗ · · · ⊗ | ↑ ) . (12) Also, it is interesting to note that even spin coherent states |j, η, m = 0 (mod 2) interpolate between the states of GHZ 2j (p → 0) and the separable state | ↓ ⊗| ↓ ⊗· · ·⊗| ↓ (p → 1). Alternatively, the odd case corresponding to |j, η, m = 0 (mod 2) can be observed as interpolating between GHZ (p → 0) and W (p → 1) type states. The physical advantage of these states is that are the robust states against the decoherence noises [52][53][54]. They are widely used and applied to implement and to achieve various quantum information processing tasks. Henceforth, due to the factorization property of su(2) coherent states (7), the state (5) can also be expressed as |j, η, m AB = N m (| l, η A ⊗| j − l, η B + e imπ l, −η A ⊗| j − l, −η B ) = N m (| η ⊗2l A ⊗ η ⊗2(j−l) B + e imπ − η ⊗2l A ⊗ − η ⊗2(j−l) B ,(13) where l = 0, 1/2, 1, ..., j − 1/2, j. In the following, we shall exploit this factorization property in a teleportation protocol involving multipartite spin coherent states. For this we will consider the expression (13), where A denotes the subsystem belonging to Alice and B denotes one of Bob. B. Target state : Input state (the message) By employing a state of the form (13) as the quantum channel for quantum communication between Alice and Bob, the input qubit that is teleported from Alice to Bob is assumed to be an arbitrary superposition coherent state. It is given by |I C = a |j − l, η C + b |j − l, −η C ≡ a |η ⊗2(j−l) C + b |−η ⊗2(j−l) C ,(14) where |a| 2 + |b| 2 = 1 and C denotes the spin coherent state at hand of a third party called Charlie. Using the notation adopted in reference [43], this subsystem can be rewritten in the orthogonal basis {|0 (j−l) , |1 (j−l) } defined as |0 (j−l) = |η ⊗2(j−l) + |−η ⊗2(j−l) 2(1 + p 2(j−l) ) , |1 (j−l) = |η ⊗2(j−l) − |−η ⊗2(j−l) 2(1 − p 2(j−l) ) ,(15) and the state (14) becomes |I C = N A j−l (a + b) |0 (j−l) + B j−l (a − b) |1 (j−l) ,(16) where A j−l = 1 + p 2(j−l) 2 , B j−l = 1 − p 2(j−l) 2 , N = (1 + 2abp 2(j−l) ) −1/2 ,(17) involving the overlap p which is related to the non-orthogonality of two spin coherent states of equal amplitude and opposite phase. In this scheme, the state |j, η, m can be expressed as a state of two logical qubits |j, η, m =N m (A l A j−l β + |0 ⊗2l A |0 ⊗2(j−l) B + A l B j−l β − |0 ⊗2(j−l) A |1 ⊗2(j−l) B + A j−l B l β − |0 ⊗2(j−l) A |1 ⊗2(j−l) B + B l B j−l β + |1 ⊗2l A |1 ⊗2(j−l) B ).(18) with β ± = 1 ± e imπ . In probabilistic quantum teleportation, which is based on the repeating quantum measurements over the parts that constitute the quantum channel, it is necessary that both Alice and Bob have the same j-spin number. For this purpose, we consider the splitting l = j − l = j 2 which arises from the decomposition of even and odd coherent states associated with the spin j 1 + j 2 = j. Then, the state |j, η, m writes as |j, η, m =N m (A j/2 A j/2 β + |0 ⊗j A |0 ⊗j B + A j/2 B j/2 β − |0 ⊗j A |1 ⊗j B + A j/2 B j/2 β − |0 ⊗j A |1 ⊗j B + B j/2 B j/2 β + |1 ⊗j A |1 ⊗j B ),(19) In fact, an odd or even spin-j coherent state holds an amount of intrinsic entanglement between its parts due to the division of the j-spin into two or more sub-parts. This can be understood as the existence of a single-particle entanglement that is emphasized by the authors of several works [55][56][57]. To examine the influence of this intrinsic entanglement on our probabilistic quantum teleportation protocol, we can use Wootters concurrence which is a most widely accepted entanglement measure for a two-party system [58]. Sure enough, it can be extended and adapted to multipartite coherent states. For the state under consideration |j, η, m (19), the amount of entanglement is given by the following expression C (|j, η, m ) = 1 − p 2j 1 + p 2j cosmπ .(20) The behavior of the concurrence C (|j, η, m ) for even spin coherent states versus the overlap p is plotted in the Fig.(1). It is shown that the degree of quantum correlations necessarily depends on the j-spin number. We can see that the concurrence increases with the increase of j-spin number, e.g., the entanglement for j = 31/2 is greater than other values of spin j and for any value of the overlap p. Also, for a given spin j, the maximal value of C is reached in the bipartition where j 1 = j 2 = j/2. Furthermore, for p → 0, the state (19) reduces to a bipartite state of GHZ-type which is maximally entangled (C = 1). Conversely, the even spin coherent state is separable when p → 1. In fact, we have C = 0 for m even (i.e., symmetric pure states). Contrary to the later case, for odd spin coherent states with m = 1 (mod 1), we have C (|j, η, m ) = 1 as it can be verified from the expression (20). In this case, the states (19) are maximally entangled states and include GHZ (10) and W (12) type states. III. SPIN COHERENT STATES IN PROBABILISTIC TELEPORTATION PROTOCOL The main objective of this technique is to teleport a state from A to B by sending two bits through a classical way. In the following, we describe a simple scheme to teleport an unknown spin state of the form Eq.(14) from Alice to Bob using a spin coherent state. In particular, we consider the GHZ states measurements on a pair of spin to improve success probability of the quantum teleportation. Alice and Bob initially share a state of the form Eq. (19) and apply the following steps : (i): Alice has initially in its possession both even and odd spin coherent state and she performs the GHZSMs on the pair of particles in her possession 1 (the unknown state of the particle to be teleported) and 2 (a part of the shared two spin coherent state taken alternatly). If perfect quantum teleportation is achieved, then she perform to the next steps. (ii): Alice will send the spin used in the last GHZSMs and the result of her last measurement to Bob by a classical way. (iii): Depending on the result of Alice's GHZSMs, Bob performs a unitary transformation allowing him to find the target state (state to be teleported). If perfect protocol is not attained, Alice repeats GHZSMs on a pair of particle states, 1 (the state to be teleported) and 2 (one of the two coherent state spin not used in the previous measure). Once Bob get the desired state, we evaluate the degree of fidelity of the state by comparing it with the starting state. In probabilistic quantum teleportation, Alice finds a success in only two cases (cases with unit maximum fidelity) and then, the probability of success will be the sum of the two probabilities for the two cases corresponding to a successful attempt. For the two cases where we find failure, Alice can accept half-success probability or, in order to get high success probability, she can continue with the exchange of the spins of the shared resource state, and the repetition of the GHZSM, which is called probabilistic quantum teleportation by repetitions. In the measurement process, each GHZ states measurements attempt indicates some results with probabilistic success and others with failure. For cases with an indicated success, Alice sends the particle not used in the GHZ state measurements to Bob, who then applies an appropriate unitary transformation to exactly reproduce the unknown information state. For the cases with failure indicated, another GHZ states measurements can be made but with the particle A and the particle not used during the previous attempt, because repeating GHZ states measurements with the same particles will give the same results, so nothing new. For primary attempt, Alice initially has a pair of particles C and A, where C is the particle to be teleported and A is a part of the shared entangled resource state between Alice and Bob, then Alice performs GHZ states measurements on particles in its possession (C and A). After GHZSMs, the output state of particle B is found and making particles C and A entangled (Panel 2.(a)). Practically in this primary attempt, success is indicated in two cases and failure in two cases. For the cases with indicated success, Alice sends Bob the unused particle and completes her task. For the cases with indicated failure, Alice repeat GHZ states measurements. The first repeated attempt of GHZSMs will be done when Alice faced with failure in the primary attempt of GHZSMs, i.e., the maximum fidelity is less than unity. At this step, the particles entangled are C and A, so Alice can perform the GHZSMs on particles C and B, finding the out state of particle A, and making particles C and B entangled (Panel 2.(b)). Practically in this first attempt of repetition, success is found in four cases and failure in others four cases. The second repeated attempt of GHZSMs can be performed when Alice find failure in the first repeated attempt. At this level, Alice can apply repeated GHZSMs on particles C and A, making them again entangled and getting the state of particle B (Panel 2.(c)). The third repeated attempt of GHZSMs will be applied when Alice finds that the second repeated GHZ measurement attempt fails, in this attempt and after the GHZSMs, Alice will find the state of particle A, which means that particles C and B are now entangled (Panel 2.(d)). In principle, this repeating process can be continued until success is indicated. IV. ENTANGLEMENT SWAPPING AND REPEATED GHZ STATES MEASUREMENTS To teleport a state of the form (14), we consider the combined tripartite state |ψ ABC = |I C ⊗ |j, η, m AB ,(21) where Alice keeps all the spins with her and makes GHZ states measurements on subsystems A and B which respectively contained the two spin coherent states of the entangled resource. A. Primary attempt of GHZ states measurements Alice chooses the following generalized orthogonal GHZ states [62] basis spanned by the four vectors |PA CA 0 ≡ |GHZ 0 = 1 √ 2 |0 ⊗j C ⊗ |0 ⊗j A + |1 ⊗j C ⊗ |1 ⊗j A , |PA CA 1 ≡ |GHZ 1 = 1 √ 2 |0 ⊗j C ⊗ |1 ⊗j A + |1 ⊗j C ⊗ |0 ⊗j A , |PA CA 2 ≡ |GHZ 2 = 1 √ 2 |0 ⊗j C ⊗ |1 ⊗j A − |1 ⊗j C ⊗ |0 ⊗j A , |PA CA 3 ≡ |GHZ 3 = 1 √ 2 |0 ⊗j C ⊗ |0 ⊗j A − |1 ⊗j C ⊗ |1 ⊗j A ,(22) and performs GHZ-states measurements on the pair of subsystems AC. After measurements, the corresponding states of the particle C become |T i for GHZSMs result i (with i = 0, 1, 2, 3). If the quantum teleportation is perfect, Alice sends GHZ measurement result |T i to Bob (see Table.(I)). Otherwise, she performs a new GHZ states measurements. In this picture, the probabilities P i are respectively given by : P 0 = (N N m ) 2 2 A j/2 A 2 j/2 (a + b)β + + B 2 j/2 (a − b) β − 2 + B j/2 A 2 j/2 (a + b) β − + B 2 j/2 (a − b) β + 2 , P 1 = (N N m ) 2 2 B j/2 A 2 j/2 ((a + b) β − + (a − b) β + ) 2 + A j/2 B 2 j/2 ((a + b) β + + (a − b) β − ) 2 , P 2 = (N N m ) 2 2 B j/2 A 2 j/2 ((a + b) β − − (a − b) β + ) 2 + A j/2 B 2 j/2 ((a + b) β + − (a − b) β − ) 2 , P 3 = (N N m ) 2 2 A j/2 A 2 j/2 (a + b) β + − B 2 j/2 (a − b) β − 2 + B j/2 A 2 j/2 (a + b)β − − B 2 j/2 (a − b) β + 2 .(23) To do better than classical teleportation scheme, we need the shared quantum state to be entangled. Furthermore, if two resource states are used, then it is necessary to find which resource state is the better one to teleport an unknown quantum state. In this context, the quantum teleportation efficiency through noisy quantum channels is quantified by the quantum fidelity. It is a measure of the overlap between a state to be teleported (14) and a teleported state |T i and was defined in [1] as follows F i = | T i |I C | 2 ,(24) with |I C is the input state and |T i is the output state. Therefore, one gets F 0 = (N 2 N m ) 2 2p 0 |(A 2 j/2 (a + b) + B 2 j/2 (a − b)) 2 + (A 2 j/2 (a + b) − B 2 j/2 (a − b)) 2 e imπ | 2 , F 1 = 2(N 2 N m ) 2 p 1 |B j/2 A 3 j/2 (a + b)(a − be imπ ) + A j/2 B 3 j/2 (a − b)(a + be imπ )| 2 , F 2 = 2(N 2 N m ) 2 p 2 |B j/2 A 3 j/2 (a + b)(b − ae imπ ) + A j/2 B 3 j/2 (a − b)(b + ae imπ )| 2 , F 3 = (N 2 N m ) 2 2p 3 |(A 4 j/2 (a + b) 2 − B 4 j/2 (a − b) 2 )(1 + e imπ )| 2 ,(25) and the average fidelity writes as F av = i F i P i .(26) This quantity characterizes the amount of fluctuations associated with teleportation fidelity [34,61]. We set N A j/2 (a + b) = cos(ω/2) and N B j/2 (a − b) = sin(ω/2), where ω ∈ [0, 2π] (look at cos 2 (ω/2) + sin 2 (ω/2) = 1). For m = 0 (mod 2), straightforward calculations leads to F av.2 = 1 4 |p j + cosω| 2 + |1 + p j cosω| 2 1 + p 2j + sin 2 ω .(27) To understand the interplay between the quantum teleportation fidelity and the quantum channel entanglement, we express the average fidelity associated with the primary attempt by GHZ states measurements in terms of the concurrence. Indeed, using the equations (20) and (27), one gets F av.2 = 1 8 | √ 1 − C + √ 1 + C cos ω| 2 + | √ 1 + C + √ 1 − C cos ω| 2 + 2 sin 2 ω ,(28) Here, one has to treat separately three cases ω = 0, π/2, π ; For ω = π/2, i.e. the teleported state is the superposed target state, the average fidelity (28) reduces to F av.2 = 1/2. When the teleported state is the separable target state (i.e. ω = 0 or ω = π), the formula (28) reads as F av.2 =        1 + √ 1 − C 2 2 , if ω = 0, 1 − √ 1 − C 2 2 , if ω = π(29) For antisymmetric spin coherent states with m = 1 (mod 1), the average fidelity (26) becomes F av.1 = 1 4 1 + |cosω| 2 + sin 2 ω .(30) Then, for this primary attempt of GHZSMs (see Table.I), it is clear that for m = 0, one has success if the GHZSMs results is associated to |GHZ 0 or |GHZ 3 , and for m = 1, the success is indicated with the results associated to |GHZ 1 or |GHZ 2 . Otherwise, the result is associated to |GHZ 1 or |GHZ 2 (for m = 0) and |GHZ 0 or |GHZ 3 (for m = 1) in the failure cases. In this attempt, the expression of probability success is the sum of the probabilities to get the states |GHZ 0 and |GHZ 3 , i.e., P success.2 = P 0 + P 3 . Also, it is important to notice that for the states encompassing m = 1, the probability success takes the form P success.1 = P 1 + P 2 = 1/2. Thus, we obtain the total probability of this successful teleportation as P success.2 = P success.1 + p j cosω 1 + p 2j .(31) In Fig.(3), we depict the variation of the maximal average fidelity with respect to the overlap p for various values of the j-spin. We can clearly see that for p approaching the unity, the maximum average fidelity approaches 1 too. In order to understand the interpretation of this behavior, we must consider the expression of the degree of entanglement of the initially shared state between Alice and Bob in terms of the overlap parameter p. If we go back to equation (20) and focus on figure (1) which represents the variation of the concurrence versus the overlap p for m = 0, we can observe that when p approaches 1, the shared state is non-maximally entangled. This indicates that for m = 0, the average fidelity reaches its maximal value if the shared state between Alice and Bob is a non-maximally entangled state. On the other hand, we can verify that when the number of spin j increases, the value of the maximum average fidelity decreases until reaching the classical fidelity 1/2 for a large number of spin j. In this way, for m = 0 and p = 0, i.e., the shared resource is a maximally entangled state (GHZ state (10)), the maximal average fidelity is one-half. For m = 0 and 0 < p < 1, i.e., the shared resource state is a non-maximally entangled state, the maximal average fidelity depends on the number of spin j and it is between 1 and 1/2 for an important number of spin j. For m = 0 and p = 1, i.e., the shared resource is a separable state (ground state (11)), the maximal average fidelity is 1 for different value of spin j. Now, for m = 1, the shared resource state is generally a maximally entangled state, for p = 0 (GHZ state) and for p = 1 (W state (12)), about maximal average fidelity is always 1/2. This proves that non-maximally entangled state can presents more interesting results in comparison with maximally entangled state. The plot in Fig. (3) shows the behavior of the success probability as a function of the overlap p for some specific values of the parameter ω of the input state (ω = 0 or ω = π) for which the average fidelity takes its maximal value. One can observe immediately that, for m = 0, the success probability depends on the spin number j ; more than this number increases more than the probability decreases until it will be achieved by its minimal value 1/2 for a large number of spin j. To successfully teleport an arbitrary superposition coherent state of type (14) by involving the even spin coherent states (m = 0) as a shared resource, we see clearly that one needs at least R repetitions number to get a quantum teleportation with unit probability, with R is the reciprocal of the success probability given in [34]. In our case we obtain R = 4(1 + p 2j ) p 2j + 2p j cosω + 1 .(32) For m = 1, the success probability is independent on the overlap parameter p (since P success.1 = 1/2) and it takes the value 1/2 for a maximally entangled state (both for W and GHZ type states). For m = 0, the success probability goes from zero (when the shared resource state is a ground state (p = 1) and ω = π, then the number of repetitions R tends to infinity) to one-half (when the shared resource state is GHZ state (p = 0) or when the shared resource state is a GHZ state (p = 1) and ω = π/2, then the minimal number of repetitions is R = 4 ) to one for a ground shared resource state (when p = 1 and ω = 0 then number of repetitions is at least R = 2). For indicated success, i.e., the results {|GHZ 0 or |GHZ 3 } and {|GHZ 1 or |GHZ 2 } obtained for m = 0 and m = 1 respectively, Alice sends to Bob the particle not used in GHZ state measurement (particle B) and Bob performs a suitable unitary transformation to get the exact information. If she faced with failure, then she repeats GHZ state measurement but with the systems pair CB not used in the previous GHZ states measurements and not with the pair of CA because a repeated GHZ states measurements with the same pair of particles will not give something new. In the following, Alice will perform new GHZ states measurements on failure cases by using the pair CB of even and odd spin coherent states. GHZSM State of particle B Fmax.2 Fmax.1 P |GHZ 0 |T0 = N Nm √ 2p0 ( [A 3 j/2 (a + b)β+ + A j/2 B 2 j/2 (a − b)β−] |0 ⊗j + [A 2 j/2 B j/2 (a + b)β− + B 3 j/2 (a − b)β+] |1 ⊗j )B 1 = 1 P0 |GHZ 1 |T1 = N Nm √ 2p1 ( B j/2 A 2 j/2 [(a + b)β− + (a − b)β+] |0 ⊗j + [B 2 j/2 A j/2 ((a + b)β+ + (a − b)β−)] |1 ⊗j )B = 1 1 P1 |GHZ 2 |T2 = N Nm √ 2p2 ( B j/2 A 2 j/2 [(a + b)β− − (a − b)β+] |0 ⊗j + A j/2 B 2 j/2 [(a + b)β+ − (a − b)β−] |1 ⊗j )B = 1 1 P2 |GHZ 3 |T3 = N Nm √ 2p3 ( A 3 j/2 (a + b)β+ − A j/2 B 2 j/2 (a − b)β− |0 ⊗j + A 2 j/2 B j/2 (a + b)β− − B 3 j/2 (a − b)β+ |1 ⊗j )B 1 = 1 P3 B. First repeated attempt of GHZ states measurements To carry out the process achieving the desired success, Alice makes first repeated attempt of GHZ states measurements. If Alice faced with failure in primary attempt of GHZSMs, then she can repeat GHZSMs on pair particles CB. For the states with m = 0, the two cases were GHZSMs results |GHZ 1 and |GHZ 2 . We examine here each of these two cases : Case -1 : In this first case, the input state after primary GHZ states measurements result |GHZ 1 is given by |ψ(1) m=0 ABC = |GHZ 1 ⊗ |T 1 = N N m 2 √ p 1 (|0 ⊗j C ⊗ |1 ⊗j A + |1 ⊗j C ⊗ |0 ⊗j A ) ⊗ (B j/2 A 2 j/2 [(a + b)β − + (a − b)β + ] |0 ⊗j B + B 2 j/2 A j/2 [(a + b)β + + (a − b)β − ] |1 ⊗j B ),(33) Here, Alice performs the 1 st repeated attempt on the couple of particles (C,B) and she uses the GHZ orthogonal basis of the subsystems C and B as follows |GHZ 10 = |PA CB 0 , |GHZ 11 = |PA CB 1 , |GHZ 12 = |PA CB 2 , |GHZ 13 = |PA CB 3 .(34) It should be noted that the first digit of states |GHZ 0,1,...,n th denotes GHZSMs result in primary attempt, second digit denotes GHZSM result in first attempt (see table 1) and so on n th digit denotes GHZSMs result in (n − 1) th attempt. The corresponding probabilities are found as P 10 = P 13 = (N N m ) 2 8p 1 A j/2 B 2 j/2 ((a + b)β + + (a − b)β − ) 2 + A 2 j/2 B j/2 ((a + b)β − + (a − b)β + ) 2 , and P 11 = P 12 = (N N m ) 2 8p 1 B j/2 A 2 j/2 ((a + b)β − + (a − b)β + ) 2 + B 2 j/2 A j/2 ((a + b)β + + (a − b)β − ) 2 . Then, the corresponding teleportation fidelity are evaluated as F 10 = 2(N 2 N m ) 2 p 1 p 10 |A 2 j/2 B 2 j/2 (a 2 + b 2 e imπ )| 2 , F 13 = (N 2 N m ) 2 2p 1 p 13 |A 2 j/2 B 2 j/2 (a 2 + b 2 )(1 + e imπ )| 2 . F 11 = (N 2 N m ) 2 2p 1 p 11 |B j/2 A 3 j/2 (a + b)(a − be imπ ) + A j/2 B 3 j/2 (a − b)(a + be imπ )| 2 , F 12 = (N 2 N m ) 2 2p 1 p 12 |B j/2 A 3 j/2 (a + b)(a − be imπ ) − A j/2 B 3 j/2 (a − b)(b + ae imπ )| 2 , Case -2 : The input state after primary GHZ states measurements result |GHZ 2 is : |ψ(2) m=0 ABC = |GHZ 2 ⊗ |T 2 = N N m 2 √ p 2 ( |0 ⊗j C ⊗ |1 ⊗j A − |1 ⊗j C ⊗ |0 ⊗j A ) ⊗ (B j/2 A 2 j/2 [(a + b)β − − (a − b)β + ] |0 ⊗j B + B 2 j/2 A j/2 [(a + b)β + − (a − b)β − ] |1 ⊗j B ),(35) and Alice may choose orthogonal GHZ basis of subsystems C and B as |GHZ 20 = |PA CB 0 , |GHZ 21 = |PA CB 1 , |GHZ 22 = |PA CB 2 , |GHZ 23 = |PA CB 3 .(36) The probabilities associated with these measures are given by P 20 = P 23 = (N N m ) 2 8p 2 A j/2 B 2 j/2 ((a + b)β + − (a − b)β − ) 2 + A 2 j/2 B j/2 ((a + b)β − − (a − b)β + ) 2 , and P 21 = P 22 = (N N m ) 2 8p 2 B j/2 A 2 j/2 ((a + b)β − + (a − b)β + ) 2 + B 2 j/2 A j/2 ((a + b)β + + (a − b)β − ) 2 .(37) Easily, one can work out the corresponding teleportation fidelity such as F 20 = (N 2 N m ) 2 2p 2 p 20 |A 2 j/2 B 2 j/2 (a 2 β − + 2abβ + − b 2 β + )| 2 , F 23 = 2(N 2 N m ) 2 p 2 p 23 |A 2 j/2 B 2 j/2 ab(1 + e imπ )| 2 , F 21 = (N 2 N m ) 2 2p 2 p 21 |B j/2 A 3 j/2 (a + b)(b − ae imπ ) − A j/2 B 3 j/2 (a − b)(b + ae imπ )| 2 , F 22 = (N 2 N m ) 2 2p 2 p 22 |B j/2 A 3 j/2 (a + b)(b − ae imπ ) + A j/2 B 3 j/2 (a − b)(b + ae imπ )| 2 . Therefore, the average fidelity (26) for the first repeated attempt of GHZ states measurements becomes F (1) av.2 = F 0 P 0 + F 3 P 3 + P 1 (P 10 F 10 + P 13 F 13 ) + P 2 (P 20 F 20 + P 23 F 23 ) . Just as primary attempt, we set N A j/2 (a + b) = cos(ω/2) and N B j/2 (a − b) = sin(ω/2), where ω ∈ [0, 2π]. Then the average fidelity at this stage proved to be F (1) av.2 = 1 8 3 1 + | cos ω| 2 + 2 1 − C 2 cos ω .(39) In the situation where the teleported state is the superposed target state (for ω = π/2), the average fidelity is F (1) av.2 = 3/4. For the separable target state, we have F (1) av.2 =        3 + √ 1 − C 2 4 , if ω = 0, 3 − √ 1 − C 2 4 , if ω = π.(40) For this attempt, it is easy to show that the average fidelity is minimal when ω = π/2 and its minimum value is F . It must be noticed that out of the eight cases of the first repeated attempt GHZSMs, Alice can achieve success in four cases. Thus success probability is increased by the quantity P (1) success.2 = P success.2 + P 1 (P 10 + P 13 ) + P 2 (P 20 + P 23 ) , due to the product of probability for failure in primary GHZSMs and the sum of probabilities for the results |GHZ 10 , |GHZ 13 , |GHZ 20 and |GHZ 23 . This remarkable improvement of the success probability due to increasing success cases. At this stage, the success probability takes the form P (1) success.2 = 1 4 3 + 2p j cosω 1 + p 2j .(43) For m = 0, the corresponding average fidelity and probabilities after the measurements are reported in Table.(II). Similarly, for m = 1, two failure cases of GHZ states measurements occur and are associated with the measurements |GHZ 0 and |GHZ 3 (see table.I). We consider these cases one by one : Case -3 : The input state after primary GHZ states measurements result |GHZ 0 is , Incidentally, we obtain |ψ(0) m=1 ABC = |GHZ 0 ⊗ |T 0 = N N m 2 √ p 0 ( |0 ⊗j C ⊗ |0 ⊗j A + |1 ⊗j C ⊗ |1 ⊗j A ) ⊗ (A j/2 [A 2 j/2 (a + b)β + + B 2 j/2 (a − b)β − ] |0 ⊗j B + B j/2 [A 2 j/2 (a + b)β − + B 2 j/2 (a − b)β + ] |1|GHZ 02 = |PA CB 2 |GHZ 03 = |PA CB 3 .(45)GHZSM State of particle A F(1) max.2 P |GHZ 10 |T10 = N Nm 2 √ 2p1p10 ( B 2 j/2 A j/2 [(a + b)β+ + (a − b)β−] |0 ⊗j + B j/2 A 2 j/2 [(a + b)β− + (a − b)β+] |1 ⊗j )A 1 P1P10 |GHZ 11 |T11 >= N Nm 2 √ 2p1p11 ( B j/2 A 2 j/2 [(a + b)β− + (a − b)β+] |0 ⊗j + B 2 j/2 A j/2 [(a + b)β+ + (a − b)β−] |1 ⊗j )A = 1 P1P11 |GHZ 12 |T12 = N Nm 2 √ 2p1p12 ( B j/2 A 2 j/2 [(a + b)β− + (a − b)β+] |0 ⊗j − B 2 j/2 A j/2 [(a + b)β+ + (a − b)−] |1 ⊗j )A = 1 p1p12 |GHZ 13 |T13 = N Nm 2 √ 2p1p13 ( −B 2 j/2 Aj[(a + b)β+ + (a − b)β−] |0 ⊗j + B j/2 A 2 j/2 [(a + b)β− + (a − b)β+] |1 ⊗j )A 1 P1P13 |GHZ 20 |T20 = N Nm 2 √ 2p2p20 ( A j/2 (B j/2 ) 2 [−(a + b)β+ + (a − b)β−] |0 ⊗j + A 2 j/2 B j/2 [(a + b)β− − (a − b)β+] |1 ⊗j )A 1 P2P20 |GHZ 21 |T21 = N Nm 2 √ 2p2p21 ( A 2 j/2 B j/2 [(a + b)β− − (a − b)β+] |0 ⊗j − A j/2 B 2 j/2 [(a + b)β+ − (a − b)β−] |1 ⊗j )A = 1 P2P21 |GHZ 22 |T22 = N Nm 2 √ 2p2p22 ( B j/2 (A j/2 ) 2 [(a + b)β− − (a − b)β+] |0 ⊗j + A j/2 (B j/2 ) 2 [(a + b)β+ − (a − b)β−] |1 ⊗j )A = 1 P2P22 |GHZ 23 |T23 = N Nm 2 √ 2p2p23 ( (A j/2 ) 2 B j/2 [(a + b)β− − (a − b)β+] |0 ⊗j + A j/2 (B j/2 ) 2 [(a + b)β+ − (a − b)β−] |1 ⊗j )A 1 P2P23P 00 = P 03 = (N N m ) 2 8p 0 A j/2 (A 2 j/2 (a + b)β + + B 2 j/2 (a − b)β − ) 2 + B j/2 (A 2 j/2 (a + b)β − + B 2 j/2 (a − b)β + ) 2 , and P 01 = P 02 = (N N m ) 2 8p 0 B j/2 A 2 j/2 (a + b)β − + B 2 j/2 (a − b)β + 2 + A j/2 A 2 j/2 (a + b)β + + B 2 j/2 (a − b)β − 2 . Thus, one can easily work out the following four fidelity F 00 = (N 2 N m ) 2 8p 0 p 00 |(A 2 j/2 (a + b) + B 2 j/2 (a − b)) 2 + (A 2 j/2 (a + b) − B 2 j/2 (a − b)) 2 e imπ | 2 , F 01 = (N 2 N m ) 2 2p 0 p 01 |B j/2 A 3 j/2 (a + b)(a − be imπ ) + A j/2 B 3 j/2 (a − b)(a + be imπ )| 2 , F 02 = N 4 N 2 m 2p 0 p 02 |B j/2 A 3 j/2 (a + b)(b − ae imπ ) + A j/2 B 3 j/2 (a − b)(b + ae imπ )| 2 , F 03 = N 4 N 2 m 8p 0 p 03 |(A 4 j/2 (a + b) 2 − B 4 j/2 (a − b) 2 )(1 + e imπ )| 2 . Case -4 : In this last case, we employ the result of |GHZ 3 where the input state becomes |ψ(3) m=1 ABC = |GHZ 3 ⊗ |T 3 = N N m 2 √ p 3 ( |0 ⊗j C ⊗ |0 ⊗j A − |1 ⊗j C ⊗ |1 ⊗j A ) ⊗ (A j/2 [A 2 j/2 (a + b)β + − B 2 j/2 (a − b)β − ] |0 ⊗j B + B j/2 [A 2 j/2 (a + b)β − − B 2 j/2 (a − b)β + ] |1 ⊗j B ).(46) Here, Alice may choose orthogonal GHZ basis of subsystems C and B as |GHZ 30 = |PA CB 0 , |GHZ 31 = |PA CB 1 , |GHZ 32 = |PA CB 2 , |GHZ 33 = |PA CB 3 .(47) The probabilities of the outcomes are given by P 30 = P 33 = (N N m ) 2 8p 3 A j/2 (A 2 j/2 (a + b)β + − B 2 j/2 (a − b)β − ) 2 + B j/2 (A 2 j/2 (a + b)β − − B 2 j/2 (a − b)β + ) 2 , and P 31 = P 32 = (N N m ) 2 8p 3 B j/2 A 2 j/2 (a + b)β − − B 2 j/2 (a − b)β + 2 + A j/2 A 2 j/2 (a + b)β + − B 2 j/2 (a − b)β − 2 .(48) We therefore explicitly derive the teleportation fidelities as follows F 30 = N 4 N 2 m 8p 3 p 30 |(A 2 j/2 (a + b) − B 2 j/2 (a − b)) 2 + (A 2 j/2 (a + b) + B 2 j/2 (a − b)) 2 e imπ | 2 , F 31 = N 4 N 2 m 2p 3 p 31 |B j/2 A 3 j/2 (a + b)(b − ae imπ ) − A j/2 B 3 j/2 (a − b)(b + ae imπ )| 2 , F 32 = N 4 N 2 m 2p 3 p 32 |B j/2 A 3 j/2 (a + b)(a − be imπ ) − A j/2 B 3 j/2 (a − b)(a + be imπ )| 2 , F 33 = N 4 N 2 m 8p 3 p 33 |(A 4 j/2 (a + b) 2 − B 4 j/2 (a − b) 2 )(1 + e imπ )| 2 . Consequently, for this first repeated attempt of GHZ states measurements based on failure cases when m = 1, the average fidelity takes the form F (1) av.1 = F 1 P 1 + F 2 P 2 + P 0 (P 01 F 01 + P 02 F 02 ) + P 3 (P 31 F 31 + P 32 F 32 ) = 3 8 1 + |cosω| 2 .(49) From this result, the minimum average fidelity of the teleported state with m = 1 occurs at the value ω = π/2 and it is F av.min.1 = 3/8. The maximal value of this quantity is given by F (1) av.max.1 = 2F (1) av.min.1 ,(50) for ω = (0, π). Indeed, the expression of probability success is P (1) success.1 = P success.1 + P 0 (P 01 + P 02 ) + P 3 (P 31 + P 32 ) = 3 4 .(51) The possible results in all the above cases are summarized in Table.(III). This teleportation scheme is not optimal and we need to make a second repeated attempt of GHZ states measurements. C. Second repeated attempt of GHZ states measurements Clearly, the same procedure may be applied when we achieve failure in the first attempt of repeated GHZSMs. This time 4 cases of failure in first attempt of repeated GHZSMs are considered case by case for m = 0 and m = 1, respectively. For m = 0, four failure cases were GHZ states measurements results |GHZ 11 , |GHZ 12 , |GHZ 21 and |GHZ 22 . In particular, out of 16 cases of second repeated attempt of GHZ states measurements, success is found in 8 cases. At this stage, the expression of average fidelity thus obtained as F (2) av.2 =F(1) av.2 + P 1 P 11 (P 110 F 110 + P 113 F 113 ) + P 1 P 12 (P 120 F 120 + P 123 F 123 ) + P 2 P 21 (P 210 F 210 + P 213 F 213 ) + P 2 P 22 (P 220 F 220 + P 223 F 223 ) . The explicit form of this quantity in terms of concurrence can be expressed as GHZSM State of particle A F(1) max.1 P |GHZ 00 |T00 = N Nm 2 √ 2p0p00 ( A j/2 [A 2 j/2 (a + b)β+ + B 2 j/2 (a − b)β−] |0 ⊗j + B j/2 [A 2 j/2 (a + b)β− + B 2 j/2 (a − b)β+] |1 ⊗j )A = 1 P0P00 |GHZ 01 |T01 = N Nm 2 √ 2p0p01 ( B j/2 [A 2 j/2 (a + b)β− + B 2 j/2 (a − b)β+] |0 ⊗j + A j/2 [A 2 j/2 (a + b)β+ + B 2 j/2 (a − b)β−] |1 ⊗j )A 1 P0P01 |GHZ 02 |T02 = N Nm 2 √ 2p0p02 ( B j/2 [A 2 j/2 (a + b)β− + B 2 j/2 (a − b)β+] |0 ⊗j − [A j/2 (A 2 j/2 (a + b)β+ + B 2 j/2 (a − b)β−)] |1 ⊗j )A 1 P0P02 |GHZ 03 |T03 = N Nm 2 √ 2p0p03 ( A j/2 [A 2 j/2 (a + b)β+ + B 2 j/2 (a − b)β−] |0 ⊗j − B j/2 [A 2 j/2 (a + b)β− + B 2 j/2 (a − b)β+] |1 ⊗j )A = 1 P0P03 |GHZ 30 |T30 = N Nm 2 √ 2p3p30 ( A j/2 [A 2 j/2 (a + b)β+ − B 2 j/2 (a − b)β−] |0 ⊗j − B j/2 [A 2 j/2 (a + b)β− − B 2 j/2 (a − b)β+] |1 ⊗j )A = 1 P3P30 |GHZ 31 |T31 = N Nm 2 √ 2p3p31 ( B j/2 [A 2 j/2 (a + b)β− − B 2 j/2 (a − b)β+] |0 ⊗j − A j/2 [A 2 j/2 (a + b)β+ − B 2 j/2 (a − b)β−] |1 ⊗j )A 1 P3P31 |GHZ 32 |T32 = N Nm 2 √ 2p3p32 ( B j/2 [A 2 j/2 (a + b)β− − B 2 j/2 (a − b)β+] |0 ⊗j + [A j/2 (A 2 j/2 (a + b)β+ − B 2 j/2 (a − b)β−)] |1 ⊗j )A 1 P3P32 |GHZ 33 |T33 = N Nm 2 √ 2p3p33 ( A j/2 [A 2 j/2 (a + b)β+ − B 2 j/2 (a − b)β−] |0 ⊗j + B j/2 [A 2 j/2 (a + b)β− − B 2 j/2 (a − b)β+] |1 ⊗j )A = 1 P3P33(52)F (2) av.2 = F (1) av.2 + 1 32 | 1 + C 2 − 1 − C 2 cos ω| 2 + | 1 − C 2 − 1 + C 2 cos ω| 2 ,(53) where for the superposed target state (ω = π/2), we find F (2) av.2 = 7/16, and for the separable target state, we get : F (2) av.2 =        7 + √ 1 − C 2 8 , if ω = 0, 7 − √ 1 − C 2 8 , if ω = π.(54) For this second repeated attempt of GHZSMs, we note that the average fidelity F av.2 increases monotonically to reach the unity independently of the spin j. In addition, the expression of success probability reads P (2) success.2 = P (1) success.2 + P 1 P 11 (P 110 + P 113 ) + P 1 P 12 (P 120 + P 123 ) + P 2 P 21 (P 210 + P 213 ) + P 2 P 22 (P 220 + P 223 ) = 1 8 7 + 2p j cosω 1 + p 2j .(55) Thus, we can say that our scheme gives an almost perfect quantum teleportation of GHZSMs and it becomes effective with higher values of the overllap p (i.e., for large coherent amplitudes) and for all values of ω. For m = 1, the 4 failure cases were GHZ states measurements results |GHZ 00 , |GHZ 03 , |GHZ 30 and |GHZ 33 . Similarly, success is achieved in 8 cases. At this stage, the expression of average fidelity becomes F (2) av.1 = F (1) av.1 + P 0 P 00 (P 001 F 001 + P 002 F 002 ) + P 0 P 03 (P 031 F 031 + P 032 F 032 ) + P 3 P 30 (P 301 F 301 + P 302 F 302 ) + P 3 P 33 (P 331 F 331 + P 332 F 332 ) = 7 6 F (1) av.1 ,(56) and the success probability is given by P (2) success.1 = P (1) success.1 + P 0 P 00 (P 001 + P 002 ) + P 0 P 03 (P 031 + P 032 ) + P 3 P 30 (P 301 + P 302 ) + P 3 P 33 (P 331 + P 332 ) = 7 8 . It is obvious that we repeats the same procedure in order to achieve a high success as desired. Thus considering third repeated attempt of GHZ states measurements. At this stage, there are 8 unsuccessful cases of second repeated attempt corresponding to 32 possibility of third repeated attempt out of which we get 16 successful cases. Therefore, the final success probability up to third repeated attempt, for m = 0, is derived as P (3) success.2 = P(3) success.1 + 1 8 p j cosω 1 + p 2j ,(58) and P success.1 = 15/16 for m = 1. From the success probabilities of the first attempt (43), second attempt (55) and third attempt (58), we can determine the success probability of the n th repeated. We easily obtain P (n) success.2 = P (n) success.1 + 1 2 n+1 1 − C 2 cos ω,(59) for the symmetric spin coherent state (mod 2) while the success probability of the n th repeated attempt for the antisymmetric spin coherent state (mod 1) takes the form P (n) success.1 = 2 n+1 − 1 2 n+1 ,(60) which is independent on the parameter ω of the target state. If the teleported state is the superposed target state with ω = π/2, Eq.(59) reduces to Eq.(60) and then the resource connecting Alice and Bob in both the symmetric and the antisymmetric state cases have the same success probability. The variations of the average fidelity in terms of Wootters concurrence when the shared resource state is a symmetric spin coherent state (m = 0), for different values taken by the w given in equation (54), are reported in Figure (4). As depicted in Fig.4(a), we observe that the average fidelity obtained within the n th repeated attempt of GHZ state measurements (with n = 0, 1, 2) displays an exponential behavior between the shared separable resource state (C = 0) and the shared maximally entangled resource state (C = 1). Hence, when the target state parameter ω = π, the maximum average fidelity is achieved if the resource connecting Alice and Bob is a maximally entangled state. On the contrary, for the target state parameter w is zero as in Fig.4(b), we get the opposite behavior of the one reported in Fig.4(a), where the maximum average fidelity is attained for the shared separable state and the minimum average fidelity is achieved for the shared maximally entangled state. Based on these results, we can confirm that the maximum average fidelity depends on the parameters of the teleported state which constitutes the information that Alice (sender) will transmit to Bob (receiver). Besides, in both cases and for any teleported state, we find that a higher average fidelity is obtained if GHZ state measurements are performed. This means that by increasing the number of repeated attempts n, the value of the average fidelity increases. In figure (5) we have plotted the evolution of the success probability against the shared resource state entanglement when the shared resource state is an symmetric spin coherent state (Fig.5(a) and Fig.5(c)) and also when it is an antisymmetric spin coherent state (Fig.5(b)). We notice that, as expected, the probability success improves by repeating the process of GHZ states measurements. As a matter of fact, as the number of attempts increases, the success probability increases. It is also worthwhile noticing that the behavior of success probability is almost similar to that observed in the average fidelity (see Fig.4) and that both are contingent on the target state parameter w. Hence we are led to the conclusion that we can achieve perfect teleportation with the highest average fidelity and a high success probability by repeating the GHZ state measurements even if the shared resource state is not a maximally entangled state. V. CONCLUDING REMARKS To summarize, we have provided a new scheme involving repeated GHZSMs to improve both the average fidelity and success probability of probabilistic quantum teleportation protocol, using the even and odd spin coherent states as entangled resource. This scheme allows to teleport an unknown state with unit fidelity and high success probability and to transmit quantum information more faithfully using non-maximally entangled states as a preshared quantum resource. Here, Alice initially has all the particles with her and performs repeated GHZ states measurements if she faced with failure, i.e., the maximal fidelity less than 1. Currently, Alice is looking at more faithful quantum teleportation with probabilistic success, i.e., maximum fidelities in the range of 1, which is significantly higher than the threshold value of 2/3 related to the non-quantum cloning theorem that underlies the security of such quantum communication protocol. In the primary attempt, she faced failure in two cases. That's gives in the first repeated GHZ states measurements four success and four failure cases, these four failure cases gives 8 success and 8 failure cases, which gives in the second repeated attempt of GHZ states measurements 16 success and 16 failure cases, etc. We show that repeated GHZ states measurements can, in principle, yield a quantum teleportation protocol with maximal fidelity (equal to 1). This gives more success cases with an improvement of the success probability and teleportation fidelity. In other words, we verify that a higher average fidelity is obtained, then the optimal successful teleportation is determined, if the repeated GHZ state measurements are performed. Therefore, the success probability and average fidelity are accordingly higher. Based on the variation of the concurrence that gives us for some specific values a maximally entangled shared resource state, a non-maximally entangled shared resource state or a separated shared resource state, we investigated the probabilistic quantum teleportation efficiency of an arbitrary superposed coherent state into each type of these three possible resources. Our study show that even spin coherent state resource gives more interesting average fidelity in comparison with odd spin coherent state. On the other hand, the repetition of GHZ state measurements in this protocol does not affect the security of the quantum teleportation and it remains completely secure. In practice, the number of repetition can be limited due to some reasons. For this case, the choice may be to (i) continue to use the GHZSMs in the n th repetition or (ii) to use the maximal entangled GHZ basis. With existing technology, this study can stimulate experimental studies to implement the probabilistic quantum teleportation protocol. Our new method can thus be directly applied to other teleported states and for any non-maximally entangled resources. Future work could focus on extending this present probabilistic teleportation scheme to other forms of resources beyond quantum entanglement, such as quantum discord-like [63,64] and quantum coherence [65]. Indeed, these types of quantumness are more robust to decoherence effects for dissipative systems and still go beyond quantum entanglement [66][67][68]. We hope to report on this subject in a forthcoming work. FIGURE 1 . 1Variation of the concurrence versus the overlap p for m = 0. FIGURE 2 . 2Measurement repetition process by GHZ states. Repetition takes place when failure is indicated. FIGURE 3 . 3(a) the maximal average fidelity, (b) the success probability cursus as a function of the overlap parameter p for different values of spin j when m = 0. If one studies average and maximizes the angle ω = (0; π), ⊗j B ). (44) Alice may use orthogonal GHZ basis of subsystems C and B as |GHZ 00 = |PA CB 0 , |GHZ 01 = |PA CB 1 FIGURE 4 . 4Variation of the average fidelity with the concurrence of the shared resource state with continuation of GHZSMs when : (a) m = 0 and ω = π, (b) m = 0 and ω = 0. 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[]
[ "Twisting on associative algebras and Rota-Baxter type operators", "Twisting on associative algebras and Rota-Baxter type operators" ]
[ "Kyousuke Uchino \nKeio University COE program. Science University of Tokyo\nWakamiya 26Shinjyuku TokyoJapan\n" ]
[ "Keio University COE program. Science University of Tokyo\nWakamiya 26Shinjyuku TokyoJapan" ]
[ "MSC" ]
We will introduce an operation "twisting" on Hochschild complex by analogy with Drinfeld's twisting operations. By using the twisting and derived bracket construction, we will study differential graded Lie algebra structures associated with bi-graded Hochschild complex. We will show that Rota-Baxter type operators are solutions of Maurer-Cartan equations. As an application of twisting, we will give a construction of associative Nijenhuis operators.
10.4171/jncg/59
[ "https://arxiv.org/pdf/0710.4309v8.pdf" ]
1,604,766
0710.4309
9a05d7f15705118781b08d240f143d1358a3284b
Twisting on associative algebras and Rota-Baxter type operators 2000 Kyousuke Uchino Keio University COE program. Science University of Tokyo Wakamiya 26Shinjyuku TokyoJapan Twisting on associative algebras and Rota-Baxter type operators MSC 2000deformation theorytwistingRota-Baxter operatorsReynolds operatorsNijenhuis operators We will introduce an operation "twisting" on Hochschild complex by analogy with Drinfeld's twisting operations. By using the twisting and derived bracket construction, we will study differential graded Lie algebra structures associated with bi-graded Hochschild complex. We will show that Rota-Baxter type operators are solutions of Maurer-Cartan equations. As an application of twisting, we will give a construction of associative Nijenhuis operators. 1 Introduction. In [7], Drinfeld introduced an operation "twisting", motivated by the study of quasi-Lie bialgebras and quasi-Hopf algebras. The twisting operations provide a method of analyzing Manin triples. In the context of Poisson geometry, they gave the detailed study of twisting operations (see Kosmann-Schwarzbach [12,14] and Roytenberg [21,22]). We shortly describe twisting operations. We consider a graded commutative algebra, known that the structure Θ is an invariant Lie algebra structure on V ⊕ V * . The structures are closely related with (quasi-)Lie bialgebra structures. A Lie bialgebra structure is defined as a pair of tensors (ν1, ν2) such that Θ12 := ν1 + ν2 is a structure in above sense, where ν1 ∈ ( V 2 V * ) ⊗ V and ν2 ∈ V * ⊗ V 2 V . When (ν1, ν2) is a structure of Lie bialgebra, the total space The aim of this note is to construct the theory of twisting on associative algebras along the philosophy and construction in [14] and [21]. At first, we will define a twisting operation in the category of associative algebras. The twisting operation is defined by using only a canonical bigraded system of the graded Poisson algebra V · (V ⊕V * ). Hence, given a suitable bigraded Lie algebra, one can define a twisting like operation on the bigraded Lie algebra. We consider a Hochschild complex C * (T ) := Hom(T ⊗ * , T ), where T is a vector space decomposed into two subspaces T := A1 ⊕ A2. In Section 2, we will introduce a canonical bigraded Lie algebra system on C * (A1 ⊕ A2). V · (V ⊕ V * ), where(V ⊕ V * , The graded Lie bracket is given by Gerstenhaber's bracket product. Our structures, θ, are defined as associative structures on A1 ⊕ A2, i.e., θ is a 2-cochain in C 2 (A1 ⊕ A2) and t1 * t2 := θ(t1 ⊗ t2) is associative for any t1, t2 ∈ A1 ⊕ A2. For a given 1-cochain H : A2 → A1, we define a twisting operation by the same manner with classical one, θ H := exp(X b H )(θ), where b H is the image of the natural map C * (A2, A1) ֒→ C * (A1 ⊕ A2) and X b H is an analogy of Hamiltonian vector field defined by X b H := {−, b H}, where C * (A2, A1) := Hom(A ⊗ * 2 , A1). We will see that θ is decomposed into the unique 4 substructures, θ =φ1 +μ1 +μ2 +φ2. The twisting operation is completely determined by transformation rules of the 4 substructures. In Section 4, we will give explicit formulas of the transformation rules (Theorem 4.5). We consider the case ofφ1 =φ2 = 0. In this case, A1 and A2 are both subalgebras of the associative algebra (A1 ⊕ A2, θ). Such a triple (A1 ⊕ A2, A1, A2) is called an associative twilled algebra, simply, twilled algebra (Carinena and coauthors [5]). When a Lie algebra is decomposed into two subalgebras, it is called a twilled Lie algebra ( [13]), or called a twilled extension in [10], or a double Lie algebra in [17]. This concept is used in order to construct integrable Hamiltonian systems (Adler-Kostant-Symes theorem). The notion of associative twilled algebra is considered as an associative version of the classical one. In [5], they studied associative twilled algebras from the point of view of quantization. In Section 3, we will give the detailed study for twilled algebras. By derived bracket construction in [13], a twilled algebra structure on A1 ⊕ A2 induces a differential graded Lie algebra (shortly, dg-Lie algebra) structure on C * (A2, A1) (see Proposition 3.3). So we can consider a deformation theory on the induced dg-Lie algebra. We consider a Maurer-Cartan equation in the dg-Lie algebra, dR + 1 2 [R, R] = 0. We can find a solution R in Rota-Baxter algebra theory. Let (A, R) be an arbitrary associative algebra equipped with an operator R : A → A. The operator R is called a Rota-Baxter operator, if R satisfies an identity (socalled Rota-Baxter identity), R(x)R(y) = R(R(x)y + xR(y)) + qR(xy), where q ∈ K is a scalar (called a weight). Rota-Baxter operators have been studied in combinatorics (see Rota [18,19]). In this note we do not study the combinatorial problem, because it is beyond our aim. Now A ⊕ A has a natural twilled algebra structure, and then C * (A, A) has a dg-Lie algebra structure. In Section 5.1, we will show that R is a Rota-Baxter operator if and only if R is a solution of the Maurer-Cartan equation. In Section 6, we will give an application of our construction. We recall the notion of associative Nijenhuis operator ( [5]). Let N : A → A be a linear map on an associative algebra A. The operator N is called an associative Nijenhuis operator, if it satisfies an associative version of classical Nijenhuis condition, N (x)N (y) = N (N (x)y + xN (y)) − N 2 (xy), where x, y ∈ A. They showed that a deformed multiplication, x ×N y := N (x)y + xN (y) − N (xy), is a new associative multiplication on A and it is compatible with original one. In this sense, an associative Nijenhuis operator induces a quantum bihamiltonian system (see [5]). We will give a construction of associative Nijenhuis operators by analogy with Poisson-Nijenhuis geometry. We recall a theorem of Vaisman [25]. Let (V, P ) be a Poisson manifold equipped with a Poisson structure tensor P , i.e., P is a solution of a Maurer-Cartan equation, 1 2 [P, P ] = 0, where the bracket product is a graded Lie bracket (called Schouten-Nijenhuis bracket). Since the Poisson structure is a (2, 0)-tensor, it is identified with a bundle map P : T * V → T V . The Poisson bundle map induces a Lie algebroid structure on the cotangent bundle T * V , i.e., the space of sections of V · T * V has a certain graded Lie bracket {, }P . He showed that if a 2-form ω is a solution of the strong Maurer-Cartan equation, dω = {ω, ω}P = 0, then the bundle map N := P ω : T V → T V is a Nijenhuis tensor and the pair (P, N ) is a compatible pair, or a Poisson-Nijenhuis structure in the sense of [11]. This compatibility implies that the bundle map N P : T * V → T V is a Poisson structure bundle map and P + tN P is a one parameter family of Poisson structures. We will show a similar theorem to Vaisman's theorem. First of all, we need Rota-Baxter type operators as substitutes for Poisson structures. Let A be an associative algebra and M an A-bimodule, and let π : M → A be a linear map. The linear map π is called a generalized Rota-Baxter operator of weight 0, or shortly GRB ( [23]), if π is a solution of π(m)π(n) = π(π(m) · n + m · π(n)), where m, n ∈ M and · is the bimodule action. When M = A as a canonical bimodule, (GRB) reduces to a classical Rota-Baxter identity of weight zero. We consider a semidirect product algebra (T := A ⋉ M,μ), whereμ is the associative multiplication of A ⋉ M . The Hochschild complex C * (A ⋉ M ) becomes a dg-Lie algebra by Gerstenhaber bracket and the coboundary map dμ := {μ, −}. We define, due to [13], a second bracket product on C * (A⋉M ) by [f, g]μ := (−1) |f |−1 {{μ, f }, g}. Here the new bracket is a graded Lie bracket on C * (M, A) ⊂ C * (A ⋉ M ). One can show that π is a generalized Rota-Baxter operator if and only if it is a solution of the Maurer-Cartan equation 1 2 [π,π]μ = 0, whereπ is the image of the natural map C 1 (M, A) ֒→ C 1 (A ⋉ M ), π →π. Now, given a generalized Rota-Baxter operator π : M → A, M becomes an associative algebra, where the associative multiplication on M is given by a structure {μ,π}. The associativity of {μ,π} is followed from [π,π]μ = 0. We denote the associative algebra by Mπ. One can show that Mπ ⊕ A has a twilled algebra structure. Thus a dg-Lie algebra structure, (dμ, [, ] Acknowledgements. The author wishes to thank very much the referees. He is greatly indebted to them for their numerous suggestion. Finally, he would like to thank very much Professors Jean-Louis Loday, Yoshiaki Maeda and Akira Yoshioka for helpful comments and encouragement. Cochain calculus. In this section, we will define a bigraded Lie algebra structure on Hochschild complex C * (A1 ⊕ A2). In the following, we assume that the characteristic of a ground field K is zero and that Q is included in K. Gerstenhaber brackets. We recall Gerstenhaber's bracket product. Let V be a vector space over K. We consider the space of cochains g(V ) : = L n∈N C n (V ), where C n (V ) = C n (V, V ) := Hom K (V ⊗n , V ). By definition, the degree of f ∈ g(V ) is |f |, if f is in C |f | (V ) . For any f ∈ C |f | (V ) and g ∈ C |g| (V ), we define a product, f•g := |f | X i=1 (−1) (i−1)(|g|−1) f •i g, where •i is the composition of maps defined by f •i g(b1, ..., b |f |+|g|−1 ) = f (b1, ..., bi−1, g(bi, ..., b i+|g|−1 ), b i+|g| ..., b |f |+|g|−1 ). The degree of f•g is |f | + |g| − 1. The Gerstenhaber bracket, or shortly, G-bracket on g(V ) is defined as a graded commutator, {f, g} := f•g − (−1) (|f |−1)(|g|−1) g•f. We recall two fundamental identities: (I) graded commutativity, {f, g} = −(−1) (|f |−1)(|g|−1) {g, f }, (II) graded Jacobi identity, (−1) (|f |−1)(|h|−1) {{f, g}, h} + (−1) (|h|−1)(|g|−1) {{h, f }, g}+ (−1) (|g|−1)(|f |−1) {{g, h}, f } = 0, where h ∈ C |h| (V ). The above graded Jacobi identity is equivalent with {f, {g, h}} = {{f, g}, h} + (−1) (|f |−1)(|g|−1) {g, {f, h}}. (II ′ ) (II ′ ) is called a graded Leibniz identity, or sometimes called, a graded Loday identity. Graded Lie algebras. Let g be a graded vector space equipped with a binary multiplication {, } of degree 0. When the bracket product satisfies the following two conditions (1) and (2), g is called a graded Lie algebra. {f, g} = −(−1) deg(f )deg(g) {g, f },(1){f, {g, h}} = {{f, g}, h} + (−1) deg(f )deg(g) {g, {f, h}},(2) where f, g, h, ∈ g and deg(−) is the degree of g. The cochain complex g(V ) is a graded Lie algebra of deg(f ) := |f | − 1. A graded Lie algebra g is called a differential graded Lie algebra (dg-Lie algebra), if g has a square zero derivation d of degree +1 satisfying, d{f, g} = {df, g} + (−1) deg(f ) {f, dg}.(3) Associative structures. It is well-known that S ∈ Derived brackets. Let g be a dg-Lie algebra. We define a new bracket product by [f, g] d := (−1) deg(f ) {df, g}. The new bracket is called a derived bracket ( [13]). It is well-known that the derived bracket is a graded Leibniz bracket, i.e., (2) holds up to degree shift. Remark that the derived bracket is not graded commutative in general. We recall a basic lemma. Lift and Bidegree. Let A1 and A2 be vector spaces, and let c : A ⊗n 2 → A1 be a linear map, or a cochain in C n (A2, A1). We can construct a cochainĉ ∈ C n (A1 ⊕ A2) bŷ c " (a1, x1) ⊗ ... ⊗ (an, xn) " := (c(x1, ..., xn), 0). In general, for a given multilinear map f : A i(1) ⊗ A i(2) ⊗ ... ⊗ A i(n) → Aj , i(1), ..., i(n), j ∈ {1, 2}, we define a cochainf ∈ C n (A1 ⊕ A2) bŷ f :=  f on A i(1) ⊗ A i(2) ⊗ ... ⊗ A i(n) , 0 all other cases. We call the cochainf a horizontal lift of f , or simply, lift. For instance, the lifts of α : A1 ⊗ A1 → A1, β : A1 ⊗ A2 → A2 and γ : A2 ⊗ A1 → A2 are defined by, respectively, α((a, x), (b, y)) = (α(a, b), 0),(4) β((a, x), (b, y)) = (0, β(a, y)), γ((a, x), (b, y)) = (0, γ(x, b)). Let H : A2 → A1 (resp. H : A1 → A2) be a 1 cochain. The lift is defined by b H(a, x) = (H(x), 0) (resp. b H(a, x) = (0, H(a))). For any (a, x) ∈ A1 ⊕ A2, we have b H b H(a, x) = b H(H(x), 0) = (0, 0). Lemma 2.2. b H b H = 0. This lemma will be used in Section 4. We denote by A l,k the direct sum of all l + k-tensor powers of A1 and A2, where l (resp. k) is the number of A1 (resp. A2). For instance, A 1,2 := (A1 ⊗ A2 ⊗ A2) ⊕ (A2 ⊗ A1 ⊗ A2) ⊕ (A2 ⊗ A2 ⊗ A1). The tensor space (A1 ⊕ A2) ⊗n is expanded into the direct sum of A l,k , l + k = n. For instance, (A1 ⊕ A2) ⊗2 = A 2,0 ⊕ A 1,1 ⊕ A 0,2 . We consider the space of cochains, C n (A1 ⊕A2) := Hom K ((A1 ⊕A2) ⊗n , A1 ⊕ A2). By the standard properties of Hom-functor, we have C n (A1 ⊕ A2) ∼ = X l+k=n C n (A l,k , A1) ⊕ X l+k=n C n (A l,k , A2),(7) where the isomorphism is the horizontal lift. Let f be a n-cochain in C n (A1 ⊕ A2). We say the bidegree of f is k|l, if f is an element in C n (A l,k−1 , A1) or in C n (A l−1,k , A2), where n = l + k − 1. We denote the bidegree of f by ||f || = k|l. In general, cochains do not have bidegree. We call a cochain f a homogeneous cochain, if f has the bidegree. We have k + l ≥ 2, because n ≥ 1. Thus there are no cochains of bidegree 0|0 or 1|0 or 0|1. If the dimension of A1 is finite and A2 = A * 1 is the dual space of A1, then a k|l-cochain is identified with an element in A ⊗k 1 ⊗ A * ⊗l 1 . Hence the definition above is compatible with the classical one. For instance, the lift of H : (4), (5) and (6). One can easily see ||α|| = ||β|| = ||γ|| = 1|2. Thus the sum ofα,β andγ, µ :=α +β +γ (8) is a homogeneous cochain with bidegree 1|2. The cochainμ is a multiplication of semidirect product type, A2 → A1, b H ∈ C 1 (A1 ⊕ A2), has the bidegree 2|0. We recallα,β,γ ∈ C 2 (A1 ⊕ A2) inµ((a, x), (b, y)) = (α(a, b), β(a, y) + γ(x, b)), where (a, x), (b, y) ∈ T . Remark thatμ is not lift (there is no µ), however, we will use this symbol, becauseμ is an interesting homogeneous cochain. Clearly, the lemma below holds. Lemma 2.3. Let f ∈ C n (A1 ⊕ A2) be a cochain. The bidegree of f is k|l if and only if the following 4 conditions hold. (deg1) k + l − 1 = n. (deg2-1) If x is an element in A l,k−1 , then f (x) is in A1. (deg2-2) If x is an element in A l−1,k , then f (x) is in A2. (deg3) All the other cases, f (x) = 0. Lemma 2.4. If ||f || = k|0 (resp. 0|k) and ||g|| = l|0 (resp. 0|l), then {f, g} = 0, or simply, {(k|0), (l|0)} = {(0|k), (0|l)} = 0. Proof. Assume that ||f || = k|0 and ||g|| = l|0. Then f and g are both horizontal lifts of cochains in C * (A2, A1). Thus, from the definition of lift, we have f •i g = g •j f = 0 for any i, j. Lemma 2.5. Let f ∈ C |f | (A1 ⊕ A2) and g ∈ C |g| (A1 ⊕ A2) homogeneous cochains with bidegrees k f |l f and kg|lg, respectively, where |f | and |g| are usual degrees of cochains f and g. The composition f •i g is again a homogeneous cochain, and the bidegree is k f + kg − 1|l f + lg − 1. Proof. We show the conditions (deg1)-(deg3). The condition (deg1) holds, because k f +kg −1+l f +lg −1 = |f |+|g| = |f •i g|+1. We show the condition (deg2). Take an element x ⊗ y ⊗ z in A l f +lg−1,k f +kg−2 . We consider f •i g(x, y, z) = f (x, g(y), z). (⋆) If (⋆) is zero, then it is in A1. Namely (deg2-1) is satisfied. So we assume (⋆) = 0. We consider the case of g(y) ∈ A1. In this case, y is in A lg ,kg −1 . and x ⊗ z is in A l f −1,k f −1 . Thus x ⊗ g(y) ⊗ z is an element in A l f ,k f −1 which implies f (x ⊗ g(y) ⊗ z) ∈ A1. When the case of g(y) ∈ A2, y is in A lg −1,kg and x⊗z is in A l f ,k f −2 . Thus x⊗g(y)⊗z is an element in A l f ,k f −1 which gives f (x ⊗ g(y) ⊗ z) ∈ A1. Similar way, when x ⊗ y ⊗ z is an element in A l f +lg−2,k f +kg−1 , the condition holds. We show (deg3). If x ⊗ y ⊗ z is an element in A l f +lg −1+i,k f +kg−2−i and g(y) = 0, then x ⊗ g(y) ⊗ z is in A l f +i,k f −1−i . When i = 0, from the assumption, f (x ⊗ g(y) ⊗ z) = 0. The proof is completed. Proposition 2.6. If ||f || = k f |l f and ||g|| = kg|lg, then the Gerstenhaber bracket {f, g} has the bidegree k f + kg − 1|l f + lg − 1. Proof. Straightforward. Remark. Given a bidegree k + 1|l + 1-cochain f , we define bideg(f ) := k|l. If bideg(f ) = k|l and bideg(g) = m|n, then bideg({f, g}) = bideg(f ) + bideg(g) = k + m|l + n. Thus the bidegree, bideg, of Gerstenhaber bracket is 0|0. Main objects. Notations. Let A1 and A2 be vector spaces. We denote any elements of A1 by a, b, c, ... and denote any elements of A2 by x, y, z, .... We sometimes use an identification (a, x) ∼ = a + x, where (a, x) ∈ A1 ⊕ A2. 3.1 Twilled algebras. Structures. Let T be an associative algebra equipped with an associative structure θ. We assume a decomposition of T into two subspaces, T = A1 ⊕ A2. The associative structure defines an associative multiplication by θ((a, x), (b, y)) := (a, x) * (b, y), for any (a, x), (b, y) ∈ T . Definition 3.1. ([5]) The triple (T , A1, A2), or simply T , is called an associative twilled algebra, if A1 and A2 are subalgebras of T . We sometimes denote a twilled algebra T by A1 1 A2. One can easily check that if A1 1 A2 is a twilled algebra, then A1 (resp. A2) is an A2-bimodule (resp. A1-bimodule). These bimodule structures are defined by the following decomposition of associative multiplication of T . For any a ∈ A1 and x ∈ A2, the multiplications a * x and x * a are decomposed into 4 multiplications, a * x = (a * 2 x, a * 1 x), x * a = (x * 2 a, x * 1 a), where a * 2 x and x * 2 a are A1-components of a * x and x * a respectively, and similar way, a * 1 x and x * 1 a are A2-components. One can easily check that the multiplication * 1 (resp. * 2) is the bimodule action of A1 to A2 (resp. A2 to A1). In general, the associative multiplication of A1 1 A2 has the form, (a, x) * (b, y) = (a * b + a * 2 y + x * 2 b, a * 1 y + x * 1 b + x * y). The total multiplication, * , is decomposed into two "associative" multiplications of semidirect product, (a, x) * 1 (b, y) := (a * 1 b, a * 1 y + x * 1 b), (a, x) * 2 (b, y) := (a * 2 y + x * 2 b, x * 2 y), where we put a * 1 b := a * b and x * 2 y := x * y. Hence the structure θ is also decomposed into two associative structures, θ =μ1 +μ2, whereμi is the structure associated with the multiplication * i for i = 1, 2. Recall (8). The cochainsμ1 andμ2 have the bidegrees 1|2 and 2|1 respectively. Under the assumption, the decomposition of θ is unique, i.e., if θ is decomposed into two substructures of bidegrees 1|2 and 2|1, then such substructures are uniquely determined. Lemma 3.2. The associativity of θ ({θ, θ} = 0) is equivalent with the compatibility conditions, 1 2 {μ1,μ1} = 0,(9) {μ1,μ2} = 0, 1 2 {μ2,μ2} = 0.(10) Proof. We will show a more generalized result in Lemma 3.9 below. The case of subalgebras in duality Given an arbitrary associative algebra A, we have a Lie algebra by the commutator, [a, b] := ab − ba on A. The induced Lie algebra is denoted by L(A). The correspondence L : A → L(A) is a functor (sometimes called a Liezation) from the usual category of associative algebras to the one of Lie algebras. In this short section, we assume that A1 =: A is a finite dimensional vector space and A2 is the dual space. In this case, T = A ⊕ A * has a nondegenerate symmetric bilinear form, (−|−), where (A|A * ) = (A * |A) is the dual pairing and (A|A) = (A * |A * ) = 0. We set a natural assumption, namely, the bilinear form is invariant (or associative) with respect to the associative multiplication of T , explicitly, (t1 * t2|t3) = (t1|t2 * t3) for any t1, t2, t3 ∈ T . Such a twilled algebra is called an invariant twilled algebra. If T is an invariant twilled algebra, then the triple (L(T ), L(A), L(A * )) is a Manin triple. It is a twilled Lie algebra with an invariant pseudo-Euclidean metric decomposed into two maximally isotropic subalgebras. In general, a pair of Lie algebras (g1, g2) becomes a Lie bialgebra if and only if a triple of Lie algebras (g1 1 g2, g1, g2) is a Manin triple. In this times, the total space g1 1 g2 is called a Drinfeld double. Thus the pair (L(A), L(A * )) becomes a Lie bialgebra and L(A) 1 L(A * ) is a Drinfeld double. If T is a quasi-twilled algebra in Definition 3.10 below, then the cocycle term φ1 (or φ2) is a cyclic cocycle, i.e., for any a, b, c ∈ A, φ1(a, b)(c) = φ1(b, c)(a) = φ1(c, a)(b). This fact is directly checked by the invariancy. And the commutator, Φ1(a, b) := φ1(a, b)−φ1(b, a), is identified with a skew symmetric 3-tensor in V 3 A * . This implies that if A ⊕ A * is a quasi-twilled algebra, then L(T ) is the double of quasi-Lie bialgebra (L(A), L(A) * ) (see [7], [12] for quasi-Lie bialgebras). The dual map of an associative multiplication on T becomes a coassociative multiplication T → T ⊗ T . Here T and T ⊗ T are identified with T * and (T ⊗ T ) * by the bilinear form. Sinceμi is associative, the dual map ofμi becomes a coassociative multiplication, ∆μ i : T → T ⊗ T , i = 1, 2. We rewrite the conditions (9), (10) and (11) by the comultiplications. (9) and (11) where (A|A) = 0 is used. From the invariancy, we have (s * 1 t|u * 2 v) = (s|t * 1 (u * 2 v)). By (10), we have t * 1 (u * 2 v) = (t * 2 u) * 1 v +(t * 1 u) * 2 v −t * 2 (u * 1 v). Thus (10) is equivalent with the condition, (∆μ 2 (s * 1 t)|u ⊗ v) = (s|t * 1 (u * 2 v)) = = (s|(t * 2 u) * 1 v) + (s|(t * 1 u) * 2 v) − (s|t * 2 (u * 1 v)).(12) The first term of the right-hand side of (12) is (s|(t * 2 u) * 1 v) = (v * 1 s|t * 2 u) = (u * 2 (v * 1 s)|t) = (u ⊗ (v * 1 s)|∆μ 2 (t)). We put ∆μ 2 (t) = P t1 ⊗ t2. Then we have (u⊗(v * 1s)|∆μ 2 (t)) = X (u|t2)(v * 1s|t1) = X (u|t2)(v|s * 1t1) = (u⊗v|s·∆μ 2 (t)).(A) And the second and third terms of the right-hand side of (12) are (s|(t * 1 u) * 2 v) − (s|t * 2 (u * 1 v)) = (∆μ 2 (s)|(t * 1 u) ⊗ v) − (s * 2 t|u * 1 v). We put ∆μ 2 (s) = P s1 ⊗ s2. Then we have (13) is equivalent with (14) below. Under the assumptions of this section, the identity (10) {μ1,μ2} = 0 is equivalent with (∆μ 2 (s)|(t * 1u)⊗v) = X (s1|v)(s2|t * 1u) = X (s1|v)(s2 * 1t|u) = (∆μ 2 (s)·t|u⊗v). (B) and (s * 2 t|u * 1 v) = (∆μ 1 (s * 2 t)|u ⊗ v) = (∆μ 1 •μ2(s, t)|u ⊗ v).Dμ 1 ∆μ 2 − ∆μ 1 •μ2 = 0.(14) Since {μ2,μ1} = 0, we have Dμ 2 ∆μ 1 −∆μ 2 •μ1 = 0. One can easily show that (14). Dμ i ∆μ i − ∆μ i •μi = 0 holds for i = 1, 2. Thus we have D θ ∆ θ − ∆ θ • θ = 0. From (14) we have Dμ 1 (∆1 •μ2) = 0. By direct computation, one can show that if A is unital (i.e. 1 * 1 A = A * 1 1), then Dμ 1 (∆1 •μ2) = 0 implies It is obvious that A is a sub-coalgebra of (T , ∆μ 2 ). Sinceμ2 is zero on A ⊗ A, ∆μ 2 is a derivation on A, i.e., for any a, b ∈ A, ∆μ 2 (a * 1 b) = ∆μ 2 (a) · b + a · ∆μ 2 (b). An associative and coassociative algebra (I, * , δ) is called an infinitesimal bialgebra ( [9]), if δ(a * b) = a · δ(b) + δ(b) · a for any a, b ∈ I. Thus the triple (A, * 1, ∆μ 2 ) is an infinitesimal bialgebra. We consider the converse. Given an infinitesimal bialgebra (I, * , δ), the multiplications * and δ are extended on I ⊕ I * by adjoint actions. However the compatibility condition (14) is not satisfied in general. This implies that the Liezation of an infinitesimal bialgebra is not a Lie bialgebra in general. For this problem, see the detailed study Aguiar [3]. Induced dg-Lie algebras. This short section is the heart of this article. The meaning of twilled algebra is given by the proposition below. From the associative condition (9), (C * (T ), dμ 1 (−) := {μ1, −}) becomes a dg-Lie algebra. The graded space C * (A2, A1) is identified with an abelian subalgebra of the dg-Lie algebra, via the horizontal lift. One can easily check that the derived bracket [f, g]μ 1 := (−1) |f |−1 {{μ1, f }, g} is closed on C * (A2, A1). From Lemma 2.1, C * (A2, A1) becomes a graded Lie algebra. Further, by (10) and (11), dμ 2 := {μ2, } becomes a square zero derivation on the induced graded Lie algebra C * (A2, A1). Proposition 3.3. If T = A1 1 A2 is a twilled algebra, then C * (A2, A1) has a dg-Lie algebra structure. The degree of dg-Lie algebra structure is the same as the usual degree of cochains. Proof. We show only a derivation property of dμ 2 . Sinceμ2 is an associative structure, dμ 2 is square zero. For any cochains f, g ∈ C * (A2, A1), we have dμ 2 [f, g]μ 1 := (−1) |f |−1 {μ2, {{μ1, f }, g}} = (−1) |f |−1 {{μ2, {μ1, f }}, g} − {{μ1, f }, {μ2, g}} = (−1) |f | {{μ1, {μ2, f }}, g} − {{μ1, f }, {μ2, g}} = [dμ 2 f, g]μ 1 + (−1) |f | [f, dμ 2 g]μ 1 . From Lemma 2.1, the derived degree is given by deg dμ 1 (f ) = deg(f ) + 1 = |f |, where deg(f ) = |f | − 1 is the degree of the canonical dg-Lie algebra (C * (T ), dμ 1 ) (recall Section 2.1). Thus dμ 2 satisfies the defining condition (3) of dg-Lie algebra. When we recall deformation theory, it is natural to ask: What is a solution of Maurer-Cartan equation in the dg-Lie algebra ? We will solve this question in Section 5. where · is the bimodule action of A on M . Examples A direct product algebra A × A is a twilled algebra. The following example is considered as a q-analogue of trivial extensions. Example 3.5. (q-trivial extensions.) Let A be an associative algebra. Define a multiplication on A ⊕ A by (a, x) * q (b, y) := (ab, ay + xb + qxy), where q ∈ K. Then (A ⊕ A, * q) becomes a twilled algebra. We denote the twilled algebra by A 1q A. If (T , θ) is an associative algebra, then C * (T ) becomes an associative algebra by a cup product, f ∨ θ g := θ(f, g), f, g ∈ C * (T ). Example 3.6. If T = A1 1 A2 is a twilled algebra, then C * (T ) = C * (T , A1 1 A2) ∼ = C * (T , A1) 1 C * (T , A2) is a twilled algebra, because the cup product is decomposed into ∨ θ = ∨μ 1 + ∨μ 2 . Proto-, Quasi-twilled algebras. A quasi-Lie bialgebra is known as a classical limit of a quasi-Hopf algebra. The notion of quasi-Lie bialgebra is generalized to proto-Lie bialgebras (see [12]). The latter is more complicated object than quasi-Lie bialgebras. The proto-Lie bialgebras provide a general framework of quantum-classical correspondence. In this section, we will study associative analogues of proto-, quasi-Lie bialgebras. Definition 3.7. Let (T , θ) be an associative algebra decomposed into two subspaces, T = A1 ⊕ A2. Here A1 and A2 are not necessarily subalgebras. We call the triple (T , A1, A2) a proto-twilled algebra. Proof. Recall the decomposition (7). The space of 2-cochains C 2 (T ) is decomposed into 4 subspaces, C 2 (T ) = (0|3) ⊕ (1|2) ⊕ (2|1) ⊕ (3|0), where (i|j) is the space of bidegree i|j-cochains, i, j = 0, 1, 2, 3. The decomposition is essentially unique. Thus θ is uniquely decomposed into homogeneous cochains of bidegrees 0|3, 1|2, 2|1 and 3|0. The 4 substructuresφ1,μ1, µ2 andφ2 in the lemma are given as the homogeneous cochains. The proof is completed. The multiplication (a, x) * (b, y) := θ((a, x), (b, y)) of T is uniquely decomposed by the canonical projections T → A1 and T → A2 into the 8 multiplications, a * b = (a * 1 b, a * 2 b), a * y = (a * 2 y, a * 1 y), x * b = (x * 2 b, x * 1 b), x * y = (x * 1 y, x * 2 y). We put bidegrees on the 4 cochains, φ 1 := 0|3, μ1 := 1|2, μ2 := 2|1 and φ 2 := 3|0. Then we obtain 2{μ1,φ1} + 2{μ2,φ2} = 0. φ1((a, x), (b, y)) = (0, a * 2 b), µ1((a, x), (b, y)) = (a * 1 b, a * 1 y + x * 1 b), µ2((a, x), (b, y)) = (a * 2 y + x * 2 b, x * 2 y),1 2 {μ1,μ1} + {μ2,φ1} = 0,(15){μ1,μ2} + {φ1,φ2} = 0,(16)1 2 {μ2,μ2} + {μ1,φ2} = 0,(17) The first two terms have 1|3-bidegree, the second two terms have 2|2-bidegree, the third two terms have 3|1-bidegree and the last two terms have 0|4 and 4|0 respectively. Thus we have {μ1,μ1} + 2{μ2,φ1} = 0 for 1|3-bidegree, and this is (16). Similarly, we obtain (15)- (19). 1 2 {μ1,μ1} + {μ2,φ1} = 0,(20) {μ1,μ2} = 0, 1 2 {μ2,μ2} = 0.(22) In Proposition 3.3, we saw C * (A2, A1) has a dg-Lie algebra structure. In the quasi-twilled algebra cases, from (23), dμ 2 is still a square zero derivation, but the derived bracket byμ1 does not satisfy the graded Jacobi identity in general. However the Jacobiator still satisfies a weak Jacobi identity in the sense of homotopy Lie algebras ( [6], [15]). The 3-cochain 1 2 {μ1,μ1} rises up to the graded Jacobiator via the derived bracket, (−1) |g|−1 1 2 {{{{μ1,μ1}, f }, g}, h} = [f, [g, h]μ 1 ]μ 1 − [[f, g]μ 1 , h]μ 1 − (−1) |f ||g| [g, [f, h]μ 1 ]μ 1 . From (21), the Jacobiator is also given by −{μ2,φ1}. We define a tri-linear bracket product (homotopy) on C * (A2, A1) by A1) is abelian with respect to {−, −}, the tribracket is skewsymmetric. We can show that the system, (dμ 2 , [·, ·]μ 1 , [·, ·, ·]φ 1 ), defines a strong homotopy Lie algebra structure of l n≥4 := 0 on C * (A2, A1). This assertion will be shown as a corollary of a more general result in [24]. [f, g, h]φ 1 := (−1) |g|−1 {{{φ1, f }, g}, h}. Since C * (A2, The complex plane, T := C, is a quasi-twilled algebra decomposed into the real part and the imaginary part. Given a R-algebra A, the complexifi- cation C ⊗ R A = A ⊕ √ −1A is a quasi-twilled algebra. Twisting by a 1-cochain Let h be a 1-cochain in C 1 (T ). By analogy with Hamiltonian vector field, we define an operator by X h := {·, h}, and by analogy with Hamiltonian flow, we put exp(X h )(·) := 1 + X h + 1 2! X 2 h + 1 3! X 3 h + ..., where X 2 h := {{·, h}, h} and X n h is defined by the same manner. Remark that exp(X h ) is not well-defined in general. Let (T = A1 ⊕ A2, θ) be a proto-twilled algebra, and letĤ ∈ C 1 (T ) be the lift of a linear map H : A2 → A1 (or H : A1 → A2). Then exp(X b H ) is always well-defined as an operator, because b H b H = 0 (recall Lemma 2.2). (24) is called a "twisting" of θ by H. Definition 4.1. A transformation θ H := exp(X b H )(θ).(24) It is clear that the result of twisting by H is again a 2-cochain. We can consider the twisting operations are special examples of gauge transformations in deformation theory (see [6]). The following Lemma 4.2 and Proposition 4.3 are followed from standard arguments in deformation theory. Lemma 4.2. θ H = e − b H θ(e b H ⊗ e b H ), where e ± b H = 1 ± b H. Proof. We have e − b H θ(e b H ⊗ e b H ) = θ(e b H ⊗ e b H ) − b Hθ(e b H ⊗ e b H ) = = θ+θ(1⊗ b H)+θ( b H⊗1)+θ( b H⊗ b H)− b Hθ− b Hθ(1⊗ b H)− b Hθ( b H⊗1)− b Hθ( b H⊗ b H) = θ+θ(1⊗ b H )+θ( b H⊗1)− b Hθ+θ( b H⊗ b H)− b Hθ(1⊗ b H)− b Hθ( b H⊗1)− b Hθ( b H⊗ b H). Since b H b H = 0, for any I ≥ 4, we have X I b H (θ) = 0. Thus we have exp(X b H )(θ) = θ + {θ, b H} + 1 2 {{θ, b H}, b H} + 1 6 {{{θ, b H}, b H}, b H}. One can directly check the three identities below. {θ, b H} = θ( b H ⊗ 1) + θ(1 ⊗ b H) − b Hθ, 1 2 {{θ, b H}, b H} = θ( b H ⊗ b H) − b Hθ( b H ⊗ 1) − b Hθ(1 ⊗ b H), 1 6 {{{θ, b H}, b H}, b H} = − b Hθ( b H ⊗ b H). The proof of the lemma is completed. As an example of Rota-Baxter operator, we know R(f )(x) := f (qx) + f (q 2 x) + f (q 3 x) + ... (convergent) where R is defined on a certain algebra of functions (see [19]). The cases ofμ 2 = 0. Consider the cases ofμ2 = 0. In this case, since dμ 2 = 0, the Maurer-Cartan equation simply has the form, [Ĥ,Ĥ]μ 1 /2 = 0, or equivalently, (29) reduces to the identity, H(x) * 1 H(y) = H(H(x) * 1 y + x * 1 H(y)). Further, if A2 = A1 as a canonical bimodule, then H is considered as a Rota-Baxter operator with weight zero. Definition 5.5. ([23]) Let A be an associative algebra and let M be an A-bimodule. A linear map π : M → A is called a generalized Rota-Baxter operator (of weight zero), if π is a solution of the identity, π(m)π(n) = π(π(m) · n + m · π(n)), or equivalently, [π,π]μ/2 = 0, where m, n ∈ M andμ is the associative structure of A ⋉ M . A generalized Rota-Baxter operator is obviously a (strong-)Maurer-Cartan operator. Given a generalized Rota-Baxter operator π : M → A, we have a twilled algebra A 1 Mπ by the twisting of A ⋉ M by π, where Mπ is an associative subalgebra given by Corollary 5.3. The associative structure of A 1 Mπ is the sum of two structures,μ + {μ,π}. Corollary 5.6. Under the assumptions above, if π1 is a second generalized Rota-Baxter operator on A ⋉ M , i.e., [π1,π1]μ = 0, then H := π1 − π is a Maurer-Cartan operator on A 1 Mπ. If H is strong, then π + tH is a one parameter family of generalized Rota-Baxter operators for any t ∈ K. Proof. From assumptions, we have [ b H, b H]μ/2 = −[π1,π]μ. On the other hand, since dμ 2 (·) = {{μ,π}, ·}, we have dμ 2 b H = {{μ,π},π1} = [π,π1]μ = [π1,π]μ. Simply, we obtain the condition (MC). Thus Maurer-Cartan operators on A 1 Mπ are given as the difference of π with generalized Rota-Baxter operators. If H is a strong Maurer-Cartan operator, then tH is also so for any t ∈ K. This implies the second part of the corollary. We recall in Section 3.1.2. Let A be a finite dimensional associative algebra and let A * the dual space. By a canonical adjoint action, A acts on the dual space. In this case, there are interesting similarities in between generalized Rota-Baxter operators and classical r-matrices. We recall classical Yang-Baxter equation (CYBE). There exists several equivalent definition of CYBE. We recall the one of them. CYBE is defined to be an operator identity in the category of Lie algebras, [r(x),r(y)] =r([r(x), y] + [x,r(y)]) where r is a two tensor in g ⊗ g (g is a finite dimensional Lie algebra), r : g * → g is the associated linear map, x, y are elements in the dual space g * and the brackets in the right-hand side are adjoint actions. The space of alternative tensors V * g has a graded Lie algebra structure of Schouten bracket. If r is an element in g ∧g, then the Schouten bracket [r, r] is in V 3 g, and [r, r] = 0 if and only ifr satisfies CYBE above. Such a matrix r is called a triangular r-matrix. When g is a Lie algebroid, a triangular r-matrix is a Poisson structure. The notion of generalized Rota-Baxter operator can be seen as an associative version of triangular r-matrices and Poisson structures. We believe that this picture is justified by the following example. Thus we have a twilled algebra A ⋉ A * . Define a tensor r by r := " 0 1 0 0 « ∧ " 1 0 0 0 « . The tensor r is identified with a mapr : A * → A. By direct computation, one can check that the map is a generalized Rota-Baxter operator. In general, if a 2-tensor r ∈ A ∧ A satisfies Aguiar's multiplicative equation (called an associative Yang-Baxter) in [1,2,3], r13r12 − r12r23 + r23r13 = 0, (AY BE) thenr : A * → A is a generalized Rota-Baxter operator (see [23] A Poisson structure is considered as a sheaf version of triangular matrices. It is natural to ask what is a sheaf version of Rota-Baxter operators. We do not yet have an interesting solution. We wish to find a Rota-Baxter operator on the universal enveloping algebra of a Lie algebroid. If there exists such a Rota-Baxter operator, it is considered as an example of the sheaf version. 5.2 The cases of φ 1 = 0 and φ 2 = 0. In this case, T = A1 ⊕ A2 is a quasi-twilled algebra. However TH = A1 ⊕ A2 is not necessarily a quasi-twilled algebra, because φ H 1 = φ1 = 0 and φ H 2 = dμ 2 b H + 1 2 [ b H, b H]μ 1 + 1 6 {{{φ1, b H}, b H}, b H} = 0. In general, the result of twisting have the forms, φ H 1 =φ1, µ H 1 =μ1 + {φ1, b H}, A Reynolds operator is used, in the study of turbulent flow, in order to induce a mean field model of Navier-Stokes equation (so-called Reynolds equation). One can easily verify that if R(f ) := f is the mean of f , then the operator satisfies the identity above, because an averaging operation satisfies the identities f g = f · g = f g and f = f in general. Unfortunately, we do not know an application of our construction to Rota's theory. 5.3 The cases of φ 1 = 0 and φ 2 = 0 In this case,φ1 =φ H 1 = 0, and thusμ1 andμ H 1 are both associative. The twisted 4 substructures have the forms, µ H 1 =μ1, µ H 2 =μ2 + dμ 1 b H, φ H 2 =φ2 + dμ 2 b H + 1 2 [ b H, b H]μ 1 . Similar with Corollary 5.1 and Corollary 5.3, we obtain the two corollaries below. dμ 2 b H + 1 2 [ b H, b H]μ 1 = −φ2, (QM C) We consider a case ofμ2 = 0. Then (QMC) and (33) reduce to the identities, respectively, 1 2 [ b H, b H]μ 1 = −φ2, and H(x) * 1 H(y) − H(H(x) * 1 y + x * 1 H(y)) = −φ2(x, y).(35) Recall the quasi-twilled algebra A ⊕Q A in Example 3.11. Claim. Define a linear map (a, x) → ( q 2 x, 0) on A ⊕ A. Then its integral e d q/2 is an algebra isomorphism, e d q/2 : A 1q A → A ⊕Q A, Q = q 2 4 . Proof. e d q/2 ((a, x) * q (b, y)) = (ab + q 2 ay + q 2 xb + q 2 2 xy, ay + xb + qxy) = ((a + q 2 x)(b + q 2 y) + q 2 4 xy, ay + xb + qxy) = (a + q 2 x, x) * Q (b + q 2 y, y), Q = q 2 4 . If Q = 0, then A⊕Q=0A is the semi-direct product algebra. Thus A 1q A is isomorphic with A ⋉ A modulus q 2 . Now, the claim says that A 1q A is the result of twisting of A ⊕Q A by q/2. Let (R(A), A) be the graph of R. One can easily verify that if R is a q-Rota-Baxter operator, then A 1q A = A 1 (R(A), A) is a second twilled algebra decomposition. By the twisting, we have a twilled algebra, The right-hand term q 2 /4xy := φ2(x, y) can be seen as the cocycle-term in (35). A 1 (R(A) + q 2 A, A), A 1 (R(A), A) = A 1q A e d q/2 → A ⊕ q 2 /4 A = A 1 (R(A) + q 2 A, A). 6 Application. In this section, we will give a construction of associative Nijenhuis operator. First we recall basic properties of Nijenhuis operator. A linear operator, N : A → A, is called an associative Nijenhuis operator, if N is a solution of N (x)N (y) = N (N (x)y + xN (y)) − N 2 (xy). In general, given a Nijenhuis operator, x ×N y := N (x)y + xN (y) − N (xy) is a second associative multiplication and it is compatible with the original multiplication. Namely, xy + tx ×N y is a one parameter family of associative multiplications for any t ∈ K ( [5]). In the following, we assume that A is an associative algebra, M is an Abimodule and we denote the multiplication of A by * A. Let π : M → A be a generalized Rota-Baxter operator, i.e., π satisfies the identity, π(m) * A π(n) = π(π(m) · n + m · π(n)). where · is the bimodule action of A on M and m, n ∈ M . We recall the twilled algebra A 1 Mπ in Section 5.1.3. The associative multiplication of A 1 Mπ has the form (a, m) * (b, n) = (a * A b + a ·π n + m ·π b, a · n + m · b + m ×π n), where ·π means the bimodule action of Mπ on A, explicitly, m ·π b := π(m) * A b − π(m · b), a ·π n := a * A π(n) − π(a · n), and m ×π n is the associative multiplication of Mπ, explicitly, m ×π n := π(m) · n + m · π(n). Simply, we have π(m ×π n) = π(m) * A π(n). We consider a linear map Ω : A → Mπ. The map Ω is a strong Maurer-Cartan operator on a twilled algebra Mπ 1 A if and only if Ω(a * A b) = a · Ω(b) + Ω(a) · b,(37)Ω(a) ×π Ω(b) = Ω(Ω(a) ·π b + a ·π Ω(b)),(38) or equivalently, Ω is a solution of dμΩ = 1 2 [Ω,Ω] {μ,π} = 0. We give the main result of this section. The pair of (π, N ) is compatible in the following sense. 2. The composition N π : M → A is a second generalized Rota-Baxter operator. 3. The operators π and N π are compatible, i.e., [π, d N π]μ = 0. This implies that N π is strong as a Maurer-Cartan operator and π + tN π t ∈ K is a one parameter family of generalized Rota-Baxter operators. Proof. 1. Applying π to (38), we have πΩ(a) * A πΩ(b) = πΩ(Ω(a) ·π b + a ·π Ω(b)). In the right-hand side, Ω(a) ·π b + a ·π Ω(b) = πΩ(a) * A b − π(Ω(a) · b) + a * A πΩ(b) − π(a · Ω(b)). From (37), we have Ω(a) ·π b + a ·π Ω(b) = πΩ(a) * A b + a * A πΩ(b) − πΩ(a * A b). Thus we obtain the desired condition, πΩ(a) * A πΩ(b) = πΩ(πΩ(a) * A b + a * A πΩ(b)) − πΩπΩ(a * A b). 2. We put a := π(m) and b := π(n) for any m, n ∈ M . Then, by the Nijenhuis condition of πΩ, we have πΩπ(m) * AπΩπ(n) = πΩ(πΩπ(m) * Aπ(n)+π(m) * AπΩπ(n))−πΩπΩ(π(m) * Aπ(n)). (39) From the identity (36), we have πΩπ(m) * A π(n) = π(πΩπ(m) · n + Ωπ(m) · π(n)), π(m) * A πΩπ(n) = π(π(m) · Ωπ(n) + m · πΩπ(n)), and from the derivation rule, we have πΩπΩ(π(m) * A π(n)) = πΩπ(Ωπ(m) · π(n) + π(m) · Ωπ(n)). Thus (39) has the form, πΩπ(m) * AπΩπ(n) = πΩπ(πΩπ(m)·n+Ωπ(m)·π(n)+π(m)·Ωπ(n)+m·πΩπ(n))− πΩπ(Ωπ(m) · π(n) + π(m) · Ωπ(n)) = πΩπ(πΩπ(m) · n + m · πΩπ(n)), this is the desired result. 3. It is obvious that d πΩπ =πΩπ. We have [π, d πΩπ]μ = {{μ,π},πΩπ} = {μ(π ⊗ 1) +μ(1 ⊗π) −πμ,πΩπ} = µ(π ⊗πΩπ) −πΩπμ(π ⊗ 1) +μ(πΩπ ⊗π) −πΩπμ(1 ⊗π) −πμ(πΩπ ⊗ 1) −πμ(1 ⊗πΩπ), (40) whereππ = 0 is used. From the generalized Rota-Baxter condition, [π,π]μ/2 = µ(π ⊗π) −πμ(π ⊗ 1) −πμ(1 ⊗π) = 0, we have (40) =μ(π⊗πΩπ)−πΩμ(π⊗π)+μ(πΩπ⊗π)−πμ(πΩπ⊗1)−πμ(1⊗πΩπ) = −πΩμ(π ⊗π) +μ(πΩπ ⊗π) −πμ(πΩπ ⊗ 1) +πμ(π ⊗Ωπ) = −πΩμ(π ⊗π) +πμ(Ωπ ⊗π) +πμ(π ⊗Ωπ). (41) SinceΩ is a derivation with respect toμ, the last equation of (41) is zero. Then a derivation from A to Mπ , Ω(f )(x) := ω(x) df dx (x) = ω(x)f ′ (x), ω(x) ∈ C 0 ([0, 1]) is a strong Maurer-Cartan operator. The induced Nijenhuis operator on A is N (f )(x) = Z x 0 ω(t)f ′ (t)dt. Proof. We only check the condition (38). For any f, g ∈ A, Ω(f ) ·π g = πΩ(f )g − π(Ω(f )g) = Z x 0 dtω(t)f ′ (t)g(x) − Z x 0 dtω(t)f ′ (t)g(t). We have Ω(Ω(f ) ·π g) = Z x 0 dtω(t)f ′ (t)ω(x)g ′ (x) , Ω(f ·π Ω(g)) = ω(x)f ′ (x) Z x 0 dtω(t)g ′ (t). On the other hand, Ω(f ) ×π Ω(g) = ω(x)f ′ (x) ×π ω(x)g ′ (x) = Z x 0 dtω(t)f ′ (t)ω(x)g ′ (x) + ω(x)f ′ (x) Z x 0 dtω(t)g ′ (t). Thus we obtain the desired condition. We consider two examples in noncommutative cases. In the proof of Example 6.2, we used the commutativity of only ω. Hence if ω is 1 or a central element, then the similar proof holds over noncommutative setting. where P is omitted. We define a formal integral operator, Z dνaiν i : = 1 i + 1 aiν i+1 , ai ∈ A. The integral operator is a Rota-Baxter operator with weight zero. The formal derivation operator is a strong Maurer-Cartan operator Ω(aiν i ) := z k ν k d dν (aiν i ) := iz k aiν i+k−1 , z k ∈ Z(A). Here Z(A) is the space of central elements. The induced Nijenhuis operator is N (aiν i ) := i i + k z k aiν i+k . Example 6.4. Let W x, ∂x be the Weyl algebra. Define a formal integral operator by, for the normal basis of the Weyl algebra, Z dx∂ i x * x j := 1 1 + j ∂ i x * x j+1 , i, j ≥ 0. Then the integral operator is a Rota-Baxter operator with weight zero (see [23]). We put Ω := i ∂x . Then Ω is a strong Maurer-Cartan operator. Thus the composition map N (u) := Z dxΩ(u) = Z dx[∂x, u] is a Nijenhuis operator on W x, ∂x . Since an arbitrary element u has the form of u := kij ∂ i x * x j(j =0) + ki∂ i x + k, we have N (u) = kij ∂ i x * x j(j =0) . Thus N is a projection onto the space of elements of the form kij ∂ i x * x j(j =0) . V is a vector space over R, V * is the dual space of V . The graded algebra has a graded Poisson bracket defined by {V, V } = {V * , V * } := 0 and {V, V * } := V, V * . By definition, a structure in the graded Poisson algebra is an element Θ in V 3 (V ⊕ V * ) satisfying a Maurer-Cartan equation {Θ, Θ} = 0. It is Θ12) is called a Drinfeld double. Let r be an element in V ∧ V . By definition, the twisting of a structure Θ by r is a canonical transformation;Θ r := exp(Xr)(Θ),where Xr is a Hamiltonian vector field Xr := {−, r} and Θ r is the result of twisting. Several interesting information is riding on the orbits of twisting operations. We recall a basic proposition. Let (ν1, 0) be a structure of Lie bialgebra such that ν2 = 0. Then the Drinfeld double is the space V ⊕ V * with the structure Θ1 := ν1.If r is a solution of a Maurer-Cartan equation (or classical Yang-Baxter equation) [r, r] = 0, then a pair (ν1, {ν1, r}) is a Lie bialgebra structure and the double, ν1 + {ν1, r}, is equal with the result of twisting Θ r 1 , where [r, r] := {{ν1, r}, r}. Conversely, the Maurer-Cartan condition of r is characterized by this proposition. {μ,π} ), is induced on C * (A, Mπ). By analogy with Vaisman's theorem, we assume that Ω : A → M is a solution of the strong Maurer-Cartan equation in C * (A, Mπ), dμΩ = [Ω,Ω] {μ,π} = 0, where dμ is the Hochschild coboundary on C * (A, M ) andΩ is defined by the similar manner withπ. Then we can show that a linear endomorphism N := πΩ : A → A is an associative Nijenhuis operator and the pair (π, N = πΩ) is compatible (see Proposition 6.1). This proposition can be considered as an associative version of Vaisman's result. C 2 ( 2V ) is an associative structure if and only if it is a solution of Maurer-Cartan equation, {S, S} = 0. If S is an associative structure, then dS(f ) := {S, f } is a coboundary map of Hochschild complex (C * (V ), dS), and then (g(V ), dS) becomes a dg-Lie algebra. are equivalent with coassociativity of ∆μ i , i = 1, 2, respectively. So we consider(10). We define a (T ,μ1)-bimodule structure on T ⊗ T byt · (T ⊗ T ) := (t * 1 T ) ⊗ T and (T ⊗ T ) · t := T ⊗ (T * 1 t) where t ∈ Tand * 1 is the associative multiplication ofμ1. For any s, t, u, v ∈ T , we have (∆μ 2 (s * 1 t)|u ⊗ v) = (s * 1 t|u * 2 v), where the pairing (−|−) is extended on T ⊗ T by the rule, (s ⊗ t|u ⊗ v) := (s|v)(t|u).The invariancy holds with respect toμi, i = 1, 2, for instance, (a * 1 x|b) = (a * x|b) = (a|x * b) = (a|x * 1 b), A),(B) and (C), we obtain a compatibility condition,(∆μ 2 (s * 1 t)|u⊗v) = (s·∆μ 2 (t)|u⊗v)+(∆μ 2 (s)·t|u⊗v)−(∆μ 1 •μ2(s, t)|u⊗v).(13) Since T ⊗T is a (T ,μ1)-bimodule, we have a Hochschild complex (C * (T , T ⊗ T ), Dμ 1 ), where Dμ 1 is a Hochschild coboundary map. The condition 3.4. (trivial extensions, semidirect product algebras.) Let A be an associative algebra and let M an A-bimodule. The trivial extension A ⋉ M is a twilled algebra of A = A1 and M = A2, where the structureμ2 is trivial andμ1 is defined by, for any (a, m), (b, n) ∈ A ⊕ M , µ1((a, m), (b, n)) := (a, m) * (b, n) := (ab, a · n + m · b), Lemma 3 . 8 . 38Let θ be an arbitrary 2-cochain in C 2 (T ). Then θ is uniquely decomposed into 4 homogeneous cochains of bidegrees 0|3, 1|2, 2|1 and 3|0, θ =φ1 +μ1 +μ2 +φ2. φ2((a, x), (b, y)) = (x * 1 y, 0). Remark thatφ1 andφ2 are lifted cochains of φ1(a, b) := a * 2 b and φ2(x, y) := x * 1 y. Lemma 3.9. The Maurer-Cartan condition {θ, θ} = 0 is equivalent with the following 5 conditions. {μ1,φ1} = 0, Definition 3 . 10 . 310Let T = A1 ⊕ A2 be a proto-twilled algebra equipped with the structures (μ1,μ2,φ1,φ2). We call the triple (T , A1, A2) a quasi-twilled algebra, if φ2 = 0, or equivalently, A2 is a subalgebra. Since A1 ⊕ A2 = A2 ⊕ A1, the definition is adapted in the case of φ2 = 0 and φ1 = 0.It is obvious that twilled algebras are special quasi-twilled algebras of φ1 = φ2 = 0. From Lemma 3.9, θ is the structure of a quasi-twilled algebra of φ2 = 0 if and only if {μ1,φ1} = 0, Example 3 . 311. (Quasi-trivial extension.) Let A be an associative algebra. Define a multiplication on A ⊕ A by (a, x) * Q (b, y) := (ab + Qxy, ay + xb),where Q ∈ K. Then A ⊕ A becomes a quasi-twilled algebra, where φ2(x, y) := Qxy. We denote the algebra by A ⊕Q A. From. above lemma, we have {θ H , θ H } = e −H {θ, The result of twisting θ H is an associative structure, i.e., {θ H , θ H } = 0.The following corollary is useful. Corollary 4 . 4 . 44The twisting by H induces an algebra isomorphism, e H : (T , θ H ) → (T , θ). Obviously, (T , θ H ) is also a proto-twilled algebra. Thus θ H is also decomposed into the unique 4 substructures. The twisting operations are completely determined by The bidegrees are {{μ1, b Proof. We have dμ 2 where we put R := H. Thus Rota-Baxter operators can be seen as examples of Maurer-Cartan operators. Example 5. 7 . 7Let A be a 2-dimensional algebra generated by The dual space A * is an A-bimodule by adjoint action. H (x) * 2 y + x * 2 H(y) + H(x) * 1 H(y) + φ2(x, y) = H(H(x) * 1 y + x * 1 H(y)) + H(x * 2 y). (33) Corollary 5.12. If H satisfied (QMC), thenμ H 2 is an associative structure and defines an associative multiplication on A2 byx × H,φ 2 y :=μ H 2 (x, y) = H(x) * 1 y + x * 1 H(y) + x * 2 y. Example 5 . 513. (Rota-Baxter operator mod q 2 [8]). Let (A, R) be a Rota-Baxter algebra. We define a linear map B : A → A by B(A) := R(A) + q 2 A. Then the graph of B, (B(A), A), is a subalgebra of the quasi-twilled algebra A ⊕ q 2 /4 A. This implies that B is a solution of B(x)B(y) − B(B(x)y + xB(y)) = − q 2 4 xy. Proposition 6. 1 . 1Let Ω : A → Mπ be a strong Maurer-Cartan operator.1. Then a composition map N := πΩ is an associative Nijenhuis operator on A. Namely N satisfies the conditionN (a) * A N (b) = N (N (a) * A b + a * A N (b)) − N N (a * A b)for any a, b ∈ A. Example 6. 2 . 2We put A := C 1 ([0, 1]) and M := C 0 ([0, 1]). We assume a canonical bimodule action of A on M . An integral operator is a Rota-Baxter operator with weight zero. π : M → A, π(f ) Example 6 . 3 . 63Let A be an associative algebra and let A[[ν]] an algebra of formal series with coefficients in A. The multiplication on A[[ν]] is defined by aiν i * bjν j := aibj ν i+j , ai, bj ∈ A, in Section 3.1.2. Thus the associative structureμ + {μ,r} satisfies the invariant condition in the sense of 3.1.2.). Con- versely, a skew symmetric generalized Rota-Baxter operator satisfies (AYBE) above. In non skewsymmetric cases, there is a delicate difference between AYBE and the generalized Rota-Baxter condition. When r is skewsymmetric, the twisting by r preserves the bilinear pairing (−|−) Corollary 5.11. The result of twisting TH = A1 ⊕ A2 is a usual twilled algebra, i.e.,φ H 2 = 0 if and only if H is a solution of the quasi-Maurer-Cartan equation, R(x)R(y) = R(R(x)y + xR(y)) + qR(xy). Theorem 4.5. Assume a decomposition of θ, θ :=μ1 +μ2 +φ1 +φ2. The unique 4 substructures of θ H have the following form: Proof. The first term of exp(X b H )(θ) is θ. From the bidegree calculus, we have {φ2, b H} = 0, because ||φ2|| = 3|0 and || b H|| = 2|0. Thus the second term of exp(X b H )(θ) has the form,H}, b H} = 0. Thus the third term has the form,H} which has the bidegree 3|0. Thus the sum of all 3|0-terms iŝwhich gives (28). In this way, the remaining 3 conditions hold.Maurer-Cartan equationsLet T = A1 ⊕ A2 be a proto-twilled algebra equipped with an associative structure θ and let (φ1,μ1,μ2,φ2) be the unique 4 substructures of θ. In this section, we discuss various examples of twisting operations.5.1The cases of φ 1 = 0 and φ 2 = 0.In this case, T = A1 1 A2 is a twilled algebra. However the result of twisting by H : A2 → A1, (TH, A1, A2), is a quasi-twilled algebra in general. The twisted structures have the forms,Thisφ H 2 is called a curvature. The derivation operator dμ 2 on the graded Lie algebra C * (A2, A1) is modified by H, dμHMaurer-Cartan operators.In Proposition 3.3, we saw that C * (A2, A1) has a dg-Lie algebra structure. We study a Maurer-Cartan equation in the dg-Lie algebra.The condition (MC) is equivalent withThis gives, for any (a, x), (b, y) ∈ T , H(x * 2 y) = x * 2 H(y) + H(x) * 2 y.In Liu and coauthors[16], a Maurer-Cartan equation in other dg-Lie algebra was studied. The concept of strong solution is due to their work. If H is strong, then the identity, H(x) * 1 H(y) = H(H(x) * 1 y + x * 1 H(y)), automatically holds. The strong Maurer-Cartan condition is equivalent withWe easily obtainCorollary 5.3. If H is a Maurer-Cartan operator, thenx ×H y := H(x) * 1 y + x * 1 H(y) + x * 2 y is an associative multiplication on A2.Proof. When H satisfies (MC), we haveφ H 2 = 0. By Lemma 3.9, we obtain {μ H 2 ,μ H 2 } = 0 which gives the associativity ofμ H 2 . The multiplication has the following form on A2,We recall Rota-Baxter operators in Introduction.Example 5.4. (Rota-Baxter operators of weight q.) Let A be an associative algebra. We recall the twilled algebra in Example 3.5. The multiplication ofwhere q ∈ K (weight). From (29), the Maurer-Cartan operators on A 1q A satisfy the Rota-Baxter identity of weight q,Sinceμ1 is not associative, the derived bracket [, ]μ 1 does not satisfy the graded Jacobi rule in general. However the space C * (A2, A1) still has a homotopy Lie algebra structure (dμ 2 , [·, ·]μ 1 , [·, ·, ·]φ 1 ) in Section 3.2. We consider a Maurer-Cartan equation in this homotopy Lie algebra. The following two corollaries are followed by the same manners with Corollary 5.1 and Corollary 5.3.or equivalently, for any x, y ∈ A2,, H(y))). (32) Corollary 5.9. If TH = A1 ⊕ A2 is a quasi-twilled algebra, thenx × H,φ 1 y :=μ H 2 (x, y) = H(x) * 1 y + x * 1 H(y) + x * 2 y + φ1(H(x), H(y)).is an associative multiplication on A2.Such an operator H is called a twisted Rota-Baxter operator (of weight zero).As an example of twisted Rota-Baxter operators, we know Reynolds operators in probability theory ([20]). Let A be a certain functional algebra. Define an operator R : A → A byThen R satisfies an identity, R(f )R(g) = R(R(f )g + f R(g)) − R(R(f )R(g)).Such an operator is called a Reynolds operator. The last term −R(R(f )R(g)) = Rφ(R(f ), R(g)) can be seen as the cocycle term of twisted Rota-Baxter identity. Thus a Reynolds operator can be seen as a homotopy version of Rota-Baxter operators of weight zero. Pre-Poisson algebras. M Aguiar, Lett. Math. Phys. 54M. Aguiar. Pre-Poisson algebras. Lett. Math. Phys. 54. (2000). no 4. 263-277. Infinitesimal Hopf algebras. M Aguiar, Contemporary Mathematics. 267M. Aguiar. Infinitesimal Hopf algebras. 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Loday-type algebras and the Rota-Baxter relation. K Ebrahimi-Fard, Lett. Math. Phys. 61K. Ebrahimi-Fard. Loday-type algebras and the Rota-Baxter relation. Lett. Math. Phys. 61. (2002). 139-147. Coalgebras and bialgebras in combinatorics. S A Joni, G.-C Rota, Stud. Appl. Math. 612S.A. Joni and G.-C. Rota. Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61. (1979). no. 2, 93-139. Poisson-Lie groups and complete integrability. I.Drinfeld bialgebras, dual extensions and their canonical representations. Y Kosmann-Schwarzbach, F Magri, Ann. Inst. H. Poincare Phys. Theor. 494Y. Kosmann-Schwarzbach and F. Magri. Poisson-Lie groups and com- plete integrability. I.Drinfeld bialgebras, dual extensions and their canon- ical representations. Ann. Inst. H. Poincare Phys. Theor. 49. (1988). no. 4, 433-460. Poisson-Nijenhuis structures. Y Kosmann-Schwarzbach, F Magri, Ann. Inst. Henri Poincare. 53Y. Kosmann-Schwarzbach and F. Magri. Poisson-Nijenhuis structures. Ann. Inst. Henri Poincare. 53. (1990). 35-81. Lie quasi-bialgebras and quasi-Poisson Lie groups. Y Kosmann-Schwarzbach, Contemp. Mathematics. 132Y. Kosmann-Schwarzbach. Lie quasi-bialgebras and quasi-Poisson Lie groups. Contemp. Mathematics. 132. (1992). 459-489. From Poisson algebras to Gerstenhaber algebras. Y Kosmann-Schwarzbach, Ann. Inst. Fourier (Grenoble). 46Y. Kosmann-Schwarzbach. From Poisson algebras to Gerstenhaber al- gebras. Ann. Inst. Fourier (Grenoble). 46. (1996). 1243-1274. Quasi-, twisted, and all that... in Poisson geometry and Lie algebroid theory. The Breadth of Symplectic and Poisson Geometry, Festschrift in honor of Alan Weinstein. Y Kosmann-Schwarzbach, Progress in Mathematics. 232.Y. Kosmann-Schwarzbach. Quasi-, twisted, and all that... in Poisson ge- ometry and Lie algebroid theory. The Breadth of Symplectic and Poisson Geometry, Festschrift in honor of Alan Weinstein, Progress in Mathemat- ics. 232. (2005). 363-389. Strongly homotopy Lie algebras. 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Baxter operators, an introduction. Gian-Carlo Rota on combinatorics, Contemp. Mathematicians, Birkhauser Boston, Boston, MA, (1995). 504-512. Gian-Carlo Rota on Analysis and Probability. Selected Papers and Commentaries. G.-C Rota, Contemporary Mathematicians. J. Dhombres, J.P.S. Kung and N. StarrBoston, MABirkhauser Boston, Incxxx+381 ppG.-C. Rota. Gian-Carlo Rota on Analysis and Probability. Selected Pa- pers and Commentaries. Edited by J. Dhombres, J.P.S. Kung and N. Starr editors. Contemporary Mathematicians. Birkhauser Boston, Inc., Boston, MA, (2003). xxx+381 pp. Quasi-Lie bialgebroids and twisted Poisson manifolds. D Roytenberg, Lett. Math. Phys. 61D. Roytenberg. Quasi-Lie bialgebroids and twisted Poisson manifolds. Lett. Math. Phys. 61. (2002). 123-137. On the structure of graded symplectic supermanifolds and Courant algebroid, in Quantization, Poisson Brackets and Beyond. D Roytenberg, Contemp. Math. 315D. Roytenberg. 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[ "Emergence, Construction, or Unlikely? Navigating the Space of Questions regarding Life's Origins", "Emergence, Construction, or Unlikely? Navigating the Space of Questions regarding Life's Origins" ]
[ "Stuart Bartlett \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n1200 E California Blvd91125PasadenaCAUnited States\n\nEarth-Life Science Institute\nTokyo Institute of Technology\nTokyoJapan\n", "Michael L Wong [email protected] \nCarnegie Institution for Science\n) Earth & Planets Laboratory\n5241 Broad Branch Rd NW20015WashingtonDCUnited States\n" ]
[ "Division of Geological and Planetary Sciences\nCalifornia Institute of Technology\n1200 E California Blvd91125PasadenaCAUnited States", "Earth-Life Science Institute\nTokyo Institute of Technology\nTokyoJapan", "Carnegie Institution for Science\n) Earth & Planets Laboratory\n5241 Broad Branch Rd NW20015WashingtonDCUnited States" ]
[]
We survey some of the philosophical challenges and pitfalls within origins research. Several of these challenges exhibit circularities, paradoxes, or anthropic biases. We present origins approaches in terms of three broad categories: unlikely (life's origin was a chance event), construction (life's origin was a stepwise series of synthesis and assembly processes), and emergence (life was always an amalgam of many parallel processes from which the living state emerged as a natural outcome of physical driving forces). We critically examine some of the founding and possibly misleading assumptions in these categories. Such assumptions need not be detrimental to scientific progress as long as their limits are respected. We conclude by attempting to concisely state the most significant enigmas still remaining in the origins field and suggest routes to solve them.
10.1002/9781119555568.ch3
[ "https://export.arxiv.org/pdf/2303.08018v1.pdf" ]
257,505,513
2303.08018
a8d95216ac43ce155bd905688cf4534ce8e756e9
Emergence, Construction, or Unlikely? Navigating the Space of Questions regarding Life's Origins Stuart Bartlett Division of Geological and Planetary Sciences California Institute of Technology 1200 E California Blvd91125PasadenaCAUnited States Earth-Life Science Institute Tokyo Institute of Technology TokyoJapan Michael L Wong [email protected] Carnegie Institution for Science ) Earth & Planets Laboratory 5241 Broad Branch Rd NW20015WashingtonDCUnited States Emergence, Construction, or Unlikely? Navigating the Space of Questions regarding Life's Origins *Corresponding author:Origins of lifeemergencecomplexityastrobiologyevolution We survey some of the philosophical challenges and pitfalls within origins research. Several of these challenges exhibit circularities, paradoxes, or anthropic biases. We present origins approaches in terms of three broad categories: unlikely (life's origin was a chance event), construction (life's origin was a stepwise series of synthesis and assembly processes), and emergence (life was always an amalgam of many parallel processes from which the living state emerged as a natural outcome of physical driving forces). We critically examine some of the founding and possibly misleading assumptions in these categories. Such assumptions need not be detrimental to scientific progress as long as their limits are respected. We conclude by attempting to concisely state the most significant enigmas still remaining in the origins field and suggest routes to solve them. How Can We Approach the Origins Quest(ion)? What does it mean to research the origins of life (OoL)? It is fast becoming a scientific discipline in its own right, partly driven by its very fundamental philosophical appeal . 1 And, indeed, why should we not consider the origins of life a scientific problem? It seems clear that science offers the best chance of resolving the story of life. However, we should bear in mind some strong constraints on how it can be investigated. The most glaring issue is that we only have one tree of life to study, and we cannot replay 2 the tape or generate ensembles. This imbues us with a degree of prejudice, narrowing our focus to a small region of the space of possible living systems. Second, we cannot study the OoL as we can study traditional scientific problems, through repeated iterations of experimentation, observation, and theorizing. In fact, the OoL is somewhat closer to a forensic detective problem. However, the crime was committed a long time ago by such a sophisticated criminal that there is little to no evidence of the deed. In the spirit of detective investigations, we could turn to the great master himself, Mr. Sherlock Holmes, who suggested that "If you eliminate the impossible, whatever remains, however improbable, must be the truth." Eliminating the impossible is itself impossible for us because we cannot effectively explore the space of possible origins (being astronomical in size due to the combinatorics of, e.g., chemical reaction networks). Instead, we have to carefully examine the lasting consequences of the crime and retrodict backward to the source. Much of modern origins research focuses on molecules, those that are deemed essential to life. The first issue with this perspective is that one could argue that life only uses essential molecules because there is a selection pressure against frivolous use of ancillary and "unnecessary" stuff (such stuff was perhaps useful in the past, but is now obsolete and, hence, absent). Animal counterexamples aside, we can, of course, point to highly simplified organisms, the prokaryotes, and "ancient" ones at that, and ponder their roots. They, of course, need genetic material, proteins, lipids, and sugars, among others. Hence, many origins researchers seek abiotic sources of amino acids, or the components of RNA, or lipids, or ideally all of the above. As a subproblem of organic synthesis, one can quite comfortably fill entire careers with this pursuit. But we should constantly remind ourselves where this pursuit might lead us. The day may well come (and soon) when we overcome this synthesis enigma and observe with some satisfaction a suite of organic molecules emerging from some kind of abiotic production line. But this molecular collection is not life, and if not prepared with a deeper understanding, we might be in for disappointment (see Construction section). We know from countless experimental endeavors that mixtures of organic molecules do not spring into life even when driven by various thermodynamic driving forces. We must have complementary theories and/or experiments that arm us with the knowledge of from where the dynamic organization of life comes, in particular the cryptic link between free energy dissipation and the emergence of information engines and systems capable of 3 learning. In this chapter, we take a frank look at the different approaches to the origins enigma in the context of similar scientific problems and some fundamental philosophical difficulties with the OoL. The interplay between chance and necessity is one of the most cryptic aspects of the biological world, and life and its history are full of surprises and frozen accidents that would not have been predictable a priori. Hence, we pose the following question: When the day comes that we "solve" the OoL, which news headline is more likely? "Scientists uncover origin of life: It was exactly what was predicted" or "Scientists uncover origin of life: Unconnected research stumbles upon completely unexpected answer." Avian Circularities Some of the most prominent discussions in the origins field relate to the sequential order-ing of the various functions of life (chicken-and-egg problems). The most common is the metabolism/proteins/hardware vs. reproduction/nucleic acids/software debate. Researchers argue ad nauseum about which of these processes preceded the other. Proteins are nonfunctional unless they are accurately assembled with the appropriate amino acid sequence. But the genetic sequences that are read out by translation machinery cannot replicate themselves without highly sophisticated protein molecular machines. There are two other permutations in this debate: both metabolism and replication were always present (this idea is increasingly appearing in the literature), and neither came first (this possibility is rarely discussed). Recently, researchers investigating the structure and history of the ribosome have indeed written strongly in favor of the presence of both protein and nucleic acids from the earliest phases. This is motivated by the inextricable link between ribosomal RNA and ribosomal proteins in the function of the ribosome. Because it is a nexus of biological information processing, it is a natural starting point for seeking the origins of that information processing. Given the poor ability of proteins to store information, the similarly poor ability of nucleic acids to act as catalysts , and the highly likely possibility 4 that prebiotic synthetic processes produce a broad range of biomolecular species (not just a narrow set), it seems most realistic to assume that the ancestors of proteins and nucleic acids existed alongside one another. The fourth option, that neither protein nor nucleic acid was present in the beginning, should also be considered. Many alternative molecular systems for early life have been considered, including α-hydroxy acids in place of amino acids [5], thioesters in place of ATP [8] [9], and green rust as an information engine [27]. It seems plausible that the earliest life used different components to perform familiar tasks, albeit less efficiently; one does not expect a scaffolding to be identical to the finished product. The philosophical question left over with such an approach is how would a protolife system eventually transform and discover the biomolecules we are familiar with? The most frequent answer comes under the heading of chemical evolution. Despite this being a discipline in its own right, with its own journal, there have been relatively few genuine demonstrations of chemical evolution (note, however, the recent example of [29]), especially in the absence of nucleic acids. In particular, comprehensive experimental demonstration of the emergence of composomes within the GARD model would be highly compelling [19]. Overall, the view that nucleic acids or protein, reproduction or metabolism must come first and the other second is likely an artificial question. Any molecules that can some-how replicate have to have been synthesized somehow. That synthesis requires free energy, and hence, one could argue also a rudimentary metabolic process. On the other hand, any plausible prebiotic system that processes free energy and produces molecules with catalytic ability will probably exhibit some form of positive feedback or network autocatalysis [14] [31]. This type of feedback could be considered a basic form of reproduction because the action of a set of molecules has a positive causal effect on the further synthesis of those molecules. So the answer to the chicken-and-egg paradox is arguably becoming clear with modern advances: It is likely that both metabolism and reproduction were present from the outset. Perhaps they were also preceded by as yet unknown processes that were neither metabolism nor reproduction but, when viewed after the fact, were necessary stepping stones for life's foundations to emerge. Assuming that... Science is designed to be axiomatic; a set of consequences and relations are derived from a basic set of assumptions. The history of science shows a consistent trend: At a given point in time, there is a set of assumptions and predictions that are tested and explored as far as possible. Eventually, scientific pursuits began to reach the limits of the knowledge status quo, and failures of existing theories began to accumulate. Some begin to question whether the basic assumptions might be too specific and overconstrained. With some combination of out-of-the-box thinking, imagination, creativity, experimentation, and luck, it is realized that a different, more general, or reduced set of assumptions are needed. This leads to a paradigm reset, an opening of new possibilities and new intellectual shores to be explored. In time, the limits of this new set of ideas will also be found. Whereas certainly providing considerable utility for a time, the new assumptions and theories will also eventually show their limitations, and those people seeking ultimate scientific truth will become dissatisfied and again consider disbanding the foundations on which that set of ideas are built. This is reminiscent of the idea of punctuated equilibrium: periods of incremental improvement, interspersed with dramatic jumps of innovation and change. For most of the history of origins research, a key assumption has been the association between life's identity and its molecular components: Life cannot exist in the absence of the nucleic acids, amino acids, lipids, and sugars that we see comprising life today (life likely used these molecules for most of its history). This assumption led to the quest for the prebiotic synthesis of biomolecules, which has had various highs and lows since the pioneering experiments of Urey and Miller. The assumption requires that any scenario for life's origins first explains the origins of life's known components and then explains the emergence of life's processes and functions. A potential corollary of this assumption is that the processes and functions of life are a natural and spontaneous consequence of the presence of biomolecules. But we have no guarantee nor any evidence of that. In the philosophical spirit, it is our duty to question our assumptions. Hence, is it only possible for life to originate with nucleic acids, amino acids, lipids, and sugars? If we deconstruct this assumption, we admit the possibility that the materials or components that are effective for maintaining and evolving life might be different from those that are effective for starting life. In many prebiotic chemistry experiments, there are significant quantities of "unwanted" synthesis products from side reactions. When such products are far from the "desired" molecular species, they are discarded as an unwanted distraction from the main show. Perhaps there should also be room for explorations of these less familiar species. In the spirit of novelty search [24], exploration of these molecules and systems may yield unexpected routes toward life. These routes may not follow our intuitions but instead follow the constraints of physics and the system boundary conditions. If those boundary conditions are aligned with prebiotic conditions, this approach would allow the systems themselves to tell us how life might have originated instead of following a storyline that we have come up with. Unlikely The history of life is full of unpredictable, strange twists and turns, exaptations, and frozen accidents. Given the huge influence of happenstance and history in life's story and the fact that such stochasticity would have been at its strongest during life's beginnings, it is possible that a logical deconstruction or retrodiction of life's origins is impossible. Is the scientific method really able to answer the question of how life began? Let us imagine that we know the answer, and it is that life began as a knock-on effect of an extremely unlikely fluctuation in a system that otherwise cares nothing for giving birth to life. This fluctuation was amplified by a series of downstream feedbacks that eventually led to life. If we were able to go back in time to damp out that fluctuation (as Q did with Picard in Star Trek: The Next Generation), the Earth would go on to remain sterile for its entire tenure. The sad consequence of this reality is that we would be lonely in the universe, the sole winners of a game for which almost no planet becomes victorious. Science generally studies typical behavior, events that happen near the center of nicely shaped distributions. Heavy-tailed distributions, extreme events, and systems that depend heavily on their histories tend to be tricky to analyze using conventional statistics and experiments or simulations that are expected to be consistent and convergent. So if life's emergence is in the extreme end of an extreme distribution, does it render the scientific study of life's origins futile? Perhaps not. Even if the OoL is unlikely in natural settings (making us quite lonely in the universe), one day we may be able to understand how to reliably create life artificially. Perhaps our experiments will discover ways of enhancing the "probabilities" of life's emergence by forcing the abiotic environment in one way or another (maybe there is a bottleneck in natural synthesis that we can overcome as we have done repeatedly in industrial projects, such as artificial nitrogen fixation to make fertilizer). Thus, in the space of all possibilities, there may be a corner in which the origin of life is both unlikely and scientifically explorable. Indeed, we might find that the spontaneous emergence of life in the majority of planetary locales is close to impossible, but being inventive beings, we may discover our own improvised settings that, in fact, give rise to life readily. If this were true, despite our own origins being both unlikely and largely off limits to the approaches of science, a separate, synthetic origins discipline would emerge (indeed the field of artificial life already thrives) and with it the possible task of filling the sterile universe with life. One intriguing example of this is the idea of directed panspermia (also explored in Star Trek: The Next Generation), whereby intelligent life forms "seed" or otherwise induce the emergence of life on other worlds. In this case, the universe could be teeming with abiogenesis events, but the vast majority of them would have been generated by preexisting life. Let us go one step further and imagine an even more pessimistic reality: that the OoL is both unlikely and not amenable to the scientific method, synthetic or otherwise. Is it possible to ever establish this as any kind of fact? Will we ever admit defeat? Say 200 years pass, and we still do not understand the emergence of life. Will we keep inventing new rabbit holes, each time thinking, "Ah, this one will finally lead us to the answer!" when in reality none can? Is there really a point at which the barren wasteland of OoL experiments will convince us that the OoL is not a scientific problem? In fact, it probably does not matter. Given all the fascinating and useful discoveries en route to an understanding of the OoL, we will still benefit greatly from this pursuit even if its ultimate objective is destined to be eternally elusive. Thus, we need not lie awake at night worrying about the possibility that the OoL can never be understood. Even if it cannot, we can still do valuable and fulfilling work as we seek this illusory goal. Construction "Before life was life, it did not know what life was, nor that it was on its way to becoming life." -Murthy Gudipati (2018) [10] Some storylines in origins research give the impression that, before life began, it knew where it was going and which steps would lead it there. This is a potentially misleading illusion that stems from viewing the problem through anthropic, engineer-like lenses. Pursuing the spontaneous synthesis of biomolecules goes all the way back to the founding experiments of Urey and Miller. Great progress since then, including analysis of meteoritic and cometary material, shows that amino acid synthesis is relatively common in a variety of scenarios [1] [18]. Nucleotide synthesis has proven to be dramatically less feasible with the most promising route at present relying on UV light and cyanide precursors [32]. Furthermore, large, unfolded polymers are extremely vulnerable to hydrolytic degradation, and difficult to ligate beyond a small number of monomeric units. Thus, the field focuses on the polymerization aspect intensely in recent decades, and these efforts have borne fruit in the form of thermal cycling mechanisms that can make polymerization favorable under certain conditions. Wet-dry cycles have become a popular method for defying the hydrolysis demon because it somewhat mimics the dehydration process that must accompany polymer growth [12] [22]. During the wet phase of the cycle, the requisite synthesis reactions produce monomers. As this system moves into the drying phase, evaporation removes water, making the mixture more concentrated and allowing the polymerization of the monomers to become more favorable. Rehydration causes a certain degree of hydrolytic damage, but the larger polymers can potentially resist through folding or self-shielding. Repeated cycles eventually produce longer molecules, perhaps up tõ 100 monomers. Note that other forms of thermodynamic cycling may also alleviate the polymerization problem. In particular, thermophoresis is shown to be effective at generating prebiotic polymers (e.g., [16]). In this process, convection cells in small, heated pores concentrate and polymerize due to thermal diffusivity effects as the molecules circulate between hot and cold regions. Whether it is wet-dry cycles, thermophoresis, or mineral molecular machines, we will likely solve the polymerization riddle in the near future. Synthesis of all the key molecules of life as we know it will also probably follow in due course. And, alas, the origins researchers will finally rest as their work will have come to a grand conclusion. Or will they? Let us imagine that at this advanced stage, the origins field has uncovered unequivocal methods for the synthesis and aggregation of all the key biomolecules: proteins, nucleic acids, lipids, sugars, and a few others for good measure. This appears to be the objective of a significant subset of the field, and let us envision the situation when this objective is entirely met. Will we have created life? We can approach the problem from the opposite direction, an experiment and mental exercise that has been hotly debated for decades if not centuries: Can a sterilized sample of biomatter come to life? The short answer to this question is no, but as with all aspects of life, the reality is likely more complicated. When an experimentalist autoclaves a sample, that sample is then rendered nonliving by all standard definitions. Furthermore, there is no trivial or known way to reverse the sterilization process. What if the sample was partially autoclaved? By this, we mean what if the system was sterilized, but there were significant quantities of recognizable biomolecular species and structure remaining? Is there any way that this system could be physically or chemically manipulated such that it comes back to life? Perhaps, and indeed we suggest a quantitative version of this experiment to be carried out just in case the forces of self-organization might actually be capable of recapitulating life from a pot of broken life pieces. Conventional wisdom states that a pot of broken life pieces, the target of origins synthesis experiments, cannot come to life. What then will we do when the constructionist dream comes true? Even with all the molecules of life at hand, it is highly unlikely that we will have achieved a second genesis. Most of physics is presented in the framework of driving forces and responses: An electromagnetic field compels the response of motion upon charged particles. Biology can similarly be cast in this light: competitive driving forces between species lead to the discovery of novel evolutionary innovations. Returning to the storyline above, if the constructionist dream comes true, we will have discovered driving forces in the form of precursor supplies and free energy sources, and responses in the form of the desired biomolecules. There is a large missing piece here though, namely, what are the driving forces that lead to the response of the formation of a living organism? Just the presence of molecular components is likely an insufficient driving force for the emergence of life. So, at this point, the question becomes what class of boundary conditions might compel a system rich in biomolecules to form a protocell or progenote? Indeed, at this point, it could be that the constructionist narrative reduces to the unlikely narrative: if making the molecules of life is straightforward but pushing them to form life itself is a next to impossible feat of extreme events. There is, of course, a more optimistic possibility: that in finding out how to make the material of life, the conditions for that material to form life naturally suggest themselves. In other words, perhaps the emergence of life was a by-product of what was initially an act of chemical synthesis. Perhaps the specific chemical routes to biomolecules are somehow predestined to produce the processes of life as well as the material of life. Whereas this may very well be true, overzealous hopefulness that function follows form may blind us to lifelike functionality that exists in different forms. An alternative school of thought considers the processes of life, irrespective of its components. To illustrate this point, we present the following analogy (also given in [3]). Figure 1 shows a JR Central SCMaglev Shinkansen. This is one of the most sophisticated locomotives ever built with a top speed of 603 km/h. It uses magnetic suspension and levitates approximately 10 cm off the ground. Imagine that distant aliens are bored and fancy a new game to play. They open a wormhole to Earth and steal one of these trains. The train is given to some friends who know nothing about Earth and humans. The game is simple: figure out what the first ever locomotive was like. Direct observation of the Shinkansen would reveal the various materials: carbon fiber, titanium, aluminum, silicon, etc. The control system uses silicon-based electronics, logic circuits, and perhaps a neural network or two. A not-so-clever alien might conclude that the first locomotive was built from less sophisticated artificial alloys and was run by a primitive electronic computer (both of which would be incorrect). A more clever alien would realize that the original locomotive would have been built in a time before Maglev technology and, therefore, skated along oiled surfaces or tracks made of ice (which would also be incorrect). An even cleverer alien might liberate their thoughts from the restrictive morphology of the Shinkansen and invoke alternative components, such as wheels and an engine that runs on liquid propellant (closer, but still not entirely correct). The cleverest alien would spend more time contemplating the purpose of the locomotive, remain skeptical about the prospects of knowing exactly the original train's components, and simply deduce that ancient locomotives needed a source of electrons and dissipated those electrons as heat via the generation of a motive force. Emergence On the other end of the chance-determinism scale, we have emergence. This paradigm might suggest that life was somehow destined to begin on Earth for reasons we do not yet understand, but are likely related to basic physical tendencies, such as the second law of thermodynamics and the principle of least action. As a strongly driven, open system, the Earth is teeming with gradients and free energy sources of myriad forms, and this has been true throughout its entire story. Emergence refers to the appearance of novel structures or processes in a system that either did not exhibit such features previously or did not exhibit them at a lower level (such as a smaller length scale or a smaller population of particles). Arguably the best understood example is the emergence of the ordered ferromagnetic phase in the Ising model of spin systems. The Ising model consists of a lattice of magnetic dipoles that interact with one another locally and are also influenced by a background magnetic field. Above the so-called critical temperature, the spins are completely uncorrelated due to the disordering effects of thermal noise. However, when the temperature passes below the critical temperature, a dramatic global shift occurs. Islands of correlated spins appear and grow with one domain eventually growing to the length scale of the whole system. This is the ferromagnetic phase. One of the most interesting aspects of this simple system is its behavior when the temperature is held at or infinitesimally close to the critical temperature. The name "critical" is used because, at this point, the system cannot decide whether to be in the paramagnetic (disordered) or ferromagnetic phase. As a result, it exhibits properties of both with ordered domains of finite sizes, but none dominating the whole system. In fact, the distribution of domain sizes is scale free, so the system also does not decide on a preferred length scale. The appearance of fractal domains at the critical temperature is another example of emergence. There is nothing in the Hamiltonian of the Ising model that relates to scale-free domains; this feature is emergent. However, another virtue of the Ising model is that it is exactly solvable and is even tractable in two dimensions (though not at higher dimensionalities). So this emergent feature is calculable. The mathematical approach that allows the prediction of emergent properties is known as renormalization group theory. Whereas this is a powerful technique, it is not straightforward to apply to many systems of interest. For this reason, emergence often remains shrouded in mystery. We associate emergence with nonlinear and networked systems, and often the emergent features can be described mathematically but not derived from the equations of motion of the system. Fluid turbulence is another classic example. There is no known violation of the Navier-Stokes equations for fluid flows; hence, we understand the lowest level of continuous fluid motion. Turbulent flows exhibit an immediately recognizable and deep level of structure (see Figure 2 for an example). Both time series and spatial fields from fluid flows show patterns that can be readily described mathematically. However, no one has succeeded at a universal theory for turbulence. Emergence plays a key role in our understanding of the living world. Alongside the origins of life, a related and similarly perplexing problem is the emergence of consciousness and thought patterns from the dynamics of neurons and synapses. It remains to be seen whether it is even possible to derive the higher level processes of the brain from the dynamics of its components (depending on the role of top-down causation and other compounding effects), but recent advances suggest that one day this goal might be achieved [21]. The deepest theme that permeates emergence questions such as these is causality: Why do nonequilibrium systems tend to form these complex patterns? Authors in the field note that all dynamical systems compute and process information [36] [37]. Of course, many dynamical systems compute in trivial ways: They either maintain the information through time (frozen systems with little or no dynamics), or they erase and generate information at equal rates (completely random systems or isolated systems at equilibrium). However, the natural world is also full of examples that combine these two extremes, and the biosphere is a case in point. It is capable of storing information over billions of years if not longer and also exhibits chaos. Furthermore, it can exploit encoded information in extraordinarily sophisticated ways, the extent of which is still incompletely explored. For the simpler examples of emergence, their causal relations can be traced back to fundamental physical principles. In the case of the Ising model, the ferromagnetic phase emerges from energy minimization. The behavior of the ideal gas can also be traced back to the kinetic behavior of its constituent particles. Moving toward the origins of life, driven chemical systems synthesize particular molecules due to a combination of thermodynamic and kinetic effects: Reactions bring a system toward the equilibrium state, subject to the constraints of energetic barriers to transition states. Even if someone were to achieve the constructionist dream of producing all the desired molecules in an integrated system, we may be able to map the entire reaction network and understand its steady states and kinetics. But what about the processes of life because we argued previously that the material of life is insufficient to constitute life itself? In our opinion, the most comprehensive theory for life would elucidate the causal relations between physical driving forces and the emergence of information processing that distinguishes life from nonlife. For many, this means understanding how and why the ribosome and the genetic code emerged [4] [30]. For others, this might mean a more basic emergence of learning behavior in physicochemical systems, such as an enzymatic reaction network capable of associative learning [13] [23] [33]. At its core, life is a learning system fueled by free energy sources. We see this at every level of its hierarchy from single molecules to the digital realm that connects human minds across the entire planet. Yet do we understand how and why a single molecule might process information? Significant advances in molecular biology, information thermodynamics, and stochastic thermodynamics reveal in quantitative detail how biological information engines can convert between forms of energy and information. Hence, the how and why of biomolecular information processing in extant life is gradually becoming understood. What remains a completely open question is how and why a nonequilibrium system with a given composition might produce emergent structures that spontaneously process information. Whereas real Maxwell demons at the scale of molecules to single electrons have been examined in laboratory settings [17] [25] [26], we still do not understand how natural information engines first arose. In closing this chapter, we speculate as to the general conditions that might need to be satisfied for such a transition to occur. At a minimum, the system in question would need to contain molecular components that can definitely form information-processing structures (peptides and nucleic acids are the obvious candidates, but other molecules may be better suited; this remains to be seen). Second, there would need to be signals or processes within the system or on its boundary that are learnable [2]. In other words, a system that is driven in a trivial way is unlikely to produce emergent learning systems because there has to be something to learn or a problem to solve [34]. Third, there needs to be a way for the candidate learning structures to interact with the learnable signals and with one another (communication channels and rudimentary sensors). Finally, there needs to be some kind of stabilizing feedback, such that systems exhibiting learning behavior are more dynamically stable as a result of their learning behavior. When these conditions are satisfied, we might witness emergent learners spontaneously learning, possibly en route to the living state. Fig. 1 : 1A JR Central Maglev Shinkansen during performance tests. Image due to Hisagi, Wikimedia Commons. Fig 2 . 2A turbulent flow field in a differentially heated fluid. Although bulk and statistical properties of such systems can be calculated, there is no universal theory that can predict the exact flow dynamics of such systems. Our tree could be the result of multiple abiogeneses, and we also don't have access to the original abiogenesis event to study; we have only its~4e9-year-old end products, plus a set of fossilized snapshots.1 Not to mention the highly interdisciplinary nature of the problem and the likelihood of applied avenues that emerge en route. Systems capable of making use of measurements or encoded information to convert one form of free energy into another. A prime example is protein-based molecular machines. Though it appears that DNA is a highly versatile material that can perform all manner of structural, mechanical, and informational tasks. Note that many of these novel functions are artificially engineered and do not appear in extant life's repertoire[11] [15] [20][28]. 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[ "Multi-Sideband RABBIT in Argon", "Multi-Sideband RABBIT in Argon" ]
[ "D Bharti \nMax-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany\n", "H Srinivas \nMax-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany\n", "F Shobeiry \nMax-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany\n", "K R Hamilton \nDepartment of Physics and Astronomy\nDrake University\n50311Des MoinesIAUSA\n", "R Moshammer \nMax-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany\n", "T Pfeifer \nMax-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany\n", "K Bartschat \nDepartment of Physics and Astronomy\nDrake University\n50311Des MoinesIAUSA\n", "A Harth \nMax-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany\n\nDepartment of Optics and Mechatronics\nHochschule AalenD-73430AalenGermany\n" ]
[ "Max-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany", "Max-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany", "Max-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany", "Department of Physics and Astronomy\nDrake University\n50311Des MoinesIAUSA", "Max-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany", "Max-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany", "Department of Physics and Astronomy\nDrake University\n50311Des MoinesIAUSA", "Max-Planck-Institute for Nuclear Physics\nD-69117HeidelbergGermany", "Department of Optics and Mechatronics\nHochschule AalenD-73430AalenGermany" ]
[]
We report a joint experimental and theoretical study of a three-sideband (3-SB) modification of the "reconstruction of attosecond beating by interference of two-photon transitions" (RABBIT) setup. The 3-SB RABBIT scheme makes it possible to investigate phases resulting from interference between transitions of different orders in the continuum. Furthermore, the strength of this method is its ability to focus on the atomic phases only, independent of a chirp in the harmonics, by comparing the RABBIT phases extracted from specific SB groups formed by two adjacent harmonics. We verify earlier predictions that the phases and the corresponding time delays in the three SBs extracted from angle-integrated measurements become similar with increasing photon electron energy. A variation in the angle dependence of the RABBIT phases in the three SBs results from the distinct Wigner and continuum-continuum coupling phases associated with the individual angular momentum channels. A qualitative explanation of this dependence is attempted by invoking a propensity rule. Comparison between the experimental data and predictions from an R-matrix (close-coupling) with time dependence calculation shows qualitative agreement in the observed trends.
null
[ "https://export.arxiv.org/pdf/2210.09244v1.pdf" ]
252,918,094
2210.09244
0c74530ce24b3f7399f1656d55be6d0aa0b253af
Multi-Sideband RABBIT in Argon (Dated: October 18, 2022) D Bharti Max-Planck-Institute for Nuclear Physics D-69117HeidelbergGermany H Srinivas Max-Planck-Institute for Nuclear Physics D-69117HeidelbergGermany F Shobeiry Max-Planck-Institute for Nuclear Physics D-69117HeidelbergGermany K R Hamilton Department of Physics and Astronomy Drake University 50311Des MoinesIAUSA R Moshammer Max-Planck-Institute for Nuclear Physics D-69117HeidelbergGermany T Pfeifer Max-Planck-Institute for Nuclear Physics D-69117HeidelbergGermany K Bartschat Department of Physics and Astronomy Drake University 50311Des MoinesIAUSA A Harth Max-Planck-Institute for Nuclear Physics D-69117HeidelbergGermany Department of Optics and Mechatronics Hochschule AalenD-73430AalenGermany Multi-Sideband RABBIT in Argon (Dated: October 18, 2022) We report a joint experimental and theoretical study of a three-sideband (3-SB) modification of the "reconstruction of attosecond beating by interference of two-photon transitions" (RABBIT) setup. The 3-SB RABBIT scheme makes it possible to investigate phases resulting from interference between transitions of different orders in the continuum. Furthermore, the strength of this method is its ability to focus on the atomic phases only, independent of a chirp in the harmonics, by comparing the RABBIT phases extracted from specific SB groups formed by two adjacent harmonics. We verify earlier predictions that the phases and the corresponding time delays in the three SBs extracted from angle-integrated measurements become similar with increasing photon electron energy. A variation in the angle dependence of the RABBIT phases in the three SBs results from the distinct Wigner and continuum-continuum coupling phases associated with the individual angular momentum channels. A qualitative explanation of this dependence is attempted by invoking a propensity rule. Comparison between the experimental data and predictions from an R-matrix (close-coupling) with time dependence calculation shows qualitative agreement in the observed trends. I. INTRODUCTION The reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) is a widely employed technique to measure attosecond time delays in photoionization processes [1][2][3]. The extraction of time information from the RABBIT measurements usually involves retrieving atomic phases encoded in the delay-dependent modulation of the sideband (SB) yield. These SBs are traditionally formed in the photoelectron spectrum by the interaction of two photons (one pump, one probe) with the target. Spectral harmonics from an attosecond pulse train (the pump photons) form discrete photoelectron signal peaks. The presence of a time-delayed infrared field (the probe photon) then creates a signal in between these main peaks that oscillates with the time delay. The so retrieved atomic phase (∆φ a ) from the RABBIT measurement can be separated into a single-photon ionization contribution (∆η, Wigner phase [4]) and a continuum-continuum (cc) coupling phase (∆φ cc ) by applying an "asymptotic approximation" [5][6][7]. Variations of the RABBIT scheme, such as 0-SB, 1-SB, and 2-SB, have been utilized to study the dipole transition phases and attosecond pulse shaping [8][9][10]. As the name suggests, in a 3-SB RABBIT scheme, three SBs are formed between two consecutive main photoelectron peaks [11,12]. The creation of these three SBs requires more than one transition in the continuum, i.e., the absorption or emission of several probe photons. To explain the phases of the oscillations in the yield of the three sidebands, we recently extended the asymptotic approximation and the decomposition scheme of the phases to include an arbitrary number of cc transitions. Specifically, we expanded the phase of the relevant N th -order transition matrix elements into a sum of the Wigner phase and N −1 cc phases, each describing a stepwise transition in the * [email protected]; [email protected] continuum [12]: arg[M (N ) ,λ ] ≈ − (N −2)π 2 − λπ 2 + η λ + φ cc k2,k1 + φ cc k3,k2 + ... + φ cc k N−2 ,k N−1 + φ cc k N ,k N−1 .(1) Here M (N,a/e) ,λ is the N th -order dipole matrix element corresponding to N − 1 absorptions (a) or emissions (e) of the probe photon. The magnitudes of the photoelectron's linear momentum in the intermediate states are indicated by k 1 , k 2 , . . ., k N−1 , respectively, k N is the final momentum, and λ labels possible orbital quantum numbers reached in the single photoionization step by the XUV pulse. Finally, is one of generally several allowed orbital angular momenta of the ejected electron. In a recent paper on atomic hydrogen [12], for which numerical calculations with high accuracy can be carried out by solving the time-dependent Schrödinger equation (TDSE) directly, we verified that the decomposition approximation explains the RABBIT phases in all three SBs qualitatively. As expected, its accuracy improves with increasing energy of the emitted photoelectron. On the other hand, assuming ∆φ cc to be independent of the orbital angular momenta of the continuum states involved leads to deviations from the analytical prediction, particularly in the lower and the higher SB of the triplet at low kinetic energies. Even though starting with a 3p electron still limits the information that can be extracted to the combined effect of the Wigner and cc phases, we decided to perform the present proof-of-principle study on argon due to its experimental advantages, including a significantly lower ionization potential than helium. In argon, λ = 0, 2, while λ = 1 in helium. For the latter target, therefore, the dependence on the Wigner phase would also drop out, and the 3-SB setup would provide direct access to the phase associated with higher-order cc transitions [11,12]. Nevertheless, a significant strength of our current setup already lies in the fact that the results within each group are independent of any atto-chirp in the XUV pulse, because the arXiv:2210.09244v1 [physics.atom-ph] 17 Oct 2022 XUV harmonic pair is common to all three SBs. This paper is organized as follows. We begin with a brief review of the basic idea behind the 3-SB setup in Sec. II. This is followed by a description of the experimental apparatus in Sec. III and the accompanying theoretical R-matrix (closecoupling) with time dependence (RMT) approach in Sec. IV. In section V, we first show angle-integrated data (Sec. V A) before focusing on the angle dependence of the RABBIT phases in the three SBs of each individual group in Sec. V B. We finish with a summary and an outlook in Sec. VI. II. THE 3-SB SCHEME In this section, we briefly review the 3-SB scheme introduced in [11] and the analytical treatment presented in [12] as applied to the 3-SB RABBIT experiment. and Hq+1) of the frequency-doubled fundamental probe frequency in the XUV pulse, while S q,l , Sq,c, and S q,h are the lower, central, and higher SBs, respectively. These SBs are formed by emission or absorption of probe photons by the quasi-free photoelectrons. |i denotes the initial state and Ip is the ionization potential. Figure 1 illustrates only the two most dominant transition paths for each SB contributing to the oscillation in their respective yields. The lowest-order transition dominates this yield, but its modulation requires interference between at least two distinct paths leading to the same energy. This involves two different XUV harmonics that are aided by absorption or emission of NIR photons. For the lower (l) and higher (h) SBs, S l and S h , the most important interfering paths are of 2 nd (one harmonic and one NIR) and 4 th order (one harmonic and three NIR), which results in a weak modulation of the yield. The lowest-order terms contributing to the build-up of the central (c) SB, S c , are both of 3 rd order (one harmonic and two NIR). Consequently, interference between them exhibits the delay-dependent oscillation most clearly. Mathematically, the angle-integrated yield in the three SBs, considering only two prominent transition paths, can be written as [12]: S q,l ∝ λ Ẽ q+1Ẽ * 3 ω M (4,e) ,λ (k l )+Ẽ q−1Ẽω M (2,a) ,λ (k l ) 2 = I l 0 + I l 1 cos(4 ωτ − φ l R + π); (2a) S q,c ∝ λ Ẽ q+1Ẽ * 2 ω M (3,e) ,λ (k c )+Ẽ q−1Ẽ 2 ω M (3,a) ,λ (k c ) 2 = I c 0 + I c 1 cos(4 ωτ − φ c R ); (2b) S q,h ∝ λ Ẽ q+1Ẽ * ω M (2,e) ,λ (k h )+Ẽ q−1Ẽ 3 ω M (4,a) ,λ (k h ) 2 = I h 0 + I h 1 cos(4 ωτ − φ h R + π) (2c) Here q labels the SB group. Furthermore,Ẽ Ω = E Ω e i φΩ andẼ ω = E ω e i ωτ (for absorption) are the complex electricfield amplitudes of the XUV-pump (Ω) and NIR-probe (ω) pulses, respectively. The yield of each SB is separated into an average part I 0 and an another term I 1 that oscillates at 4 ω with the delay. According to Eq. (1), every dipole transition adds a factor of i, i.e., a phase of π/2. Since the two dominant interfering terms in S l and S h are of different orders (2 nd and 4 th ), this leads to an additional π phase in S l and S h relative to S c , where both interfering terms are of the same (3 rd ) order. The RABBIT phase (φ R ) includes the spectral phase difference of the two harmonics and the weighted average of channel-resolved atomic phases of the transitions. Since the three SBs involve the same pair of harmonics, the contribution of the XUV group delay to the oscillation phase is the same in all three SBs. This is a key advantage of the 3-SB method, since it removes the influence of the XUV chirp when we compare the phases of the three SBs only within a particular group. Figure 2 shows the schematic design of our 3-SB RABBIT experimental setup. A commercial fiber-based laser delivers pulses with a duration of approximately 50 fs (FWHM) at a 49 kHz repetition rate with a pulse energy of 1.2 mJ and a center wavelength of 1030 nm. This pulse is split into two parts using a holey mirror (BS) that reflects ≈ 85% of the incoming beam in the pump arm, while the rest passes through the hole into the probe arm. The beam size of the reflected donut beam in the pump arm is reduced by a pair of lenses and passed through a 0.5 mm thick BBO crystal to double its frequency. III. EXPERIMENTAL SETUP The conversion efficiency for the Second-Harmonic Generation (SHG) by the BBO crystal is 25 − 30 %. A dichroic beam-splitter (DBS) filters out the fundamental beam, and a lens with a focal length of 12 cm focuses the second harmonic beam inside a vacuum chamber to a focal spot of 30 − 40 µ m on a jet of neon gas, which results in an XUV frequency comb through high-harmonic generation (HHG). The gas nozzle has a diameter of 100 µ m and is operated at a backing pressure of 1.2 bar with a chamber pressure of 5 × 10 −3 mbar. The generated XUV beam is spatially separated from the annular NIR with the help of an additional holey dumping mirror (DM). The co-propagating second harmonic is weak and does not generate any visible sidebands. The beam in the probe arm goes through a retro-reflector mounted on a piezoelectrictranslation stage that offers a step-resolution of 5 nm with closed-loop position control. Another holey mirror (RM) recombines the NIR (probe) and XUV (pump) beams, which are then focused inside a reaction microscope (ReMi) on a cold gas jet of argon. The efficiency of the ReMi made angular-resolved measurements with sufficiently high signal rates possible [13]. The setup was actively stabilized [14] to achieve a stability of ≈ 40 attoseconds over a data acquisition time of 7 hours. The stability of the interferometer was critical for the successful realization of the 3-SB scheme since the oscillation period was just 850 attoseconds. IV. THEORETICAL APPROACH In the theoretical part of this study, we employ the general R-matrix with time dependence (RMT) method [15] to generate theoretical predictions for comparison with our experimental data. In order to calculate the necessary timeindependent basis functions and dipole matrix elements, we set up the 2-state nonrelativistic model introduced by Burke and Taylor [16] to treat the steady-state standard photoionization process. In this model, multi-configuration expansions for the initial (3s 2 3p 6 ) 1 S bound state and the two coupled final ionic states (3s 2 3p 5 ) 2 P and (3s3p 6 ) 2 S were employed. We checked that the photoionization cross sections at the photon energies corresponding to the various HHG lines was reproduced properly (in agreement with Burke and Taylor [16] as well as experiment [17,18]) by our RMT model. The probe-pulse duration was chosen as about twice the length of the XUV pulse. We emphasize that the present calculation was meant as a supplement to the current experiment, with the hope of providing additional qualitative insights rather than quantitative agreement, which would require much more detailed information about the actual pulses than what was available. We purposely employed significantly lower NIR peak intensities (10 11 W/cm 2 ) than in the experiment (≈ 6×10 11 W/cm 2 ). This reduced the number of partial waves needed to obtain converged results and also led to clean spectra, which are easier to interpret. Specifically, we performed calculations for 11 delays in multiples of 0.05 NIR periods. For each delay, we needed about 5 hours on 23 nodes using all 56 available cores per node on the Frontera supercomputer hosted at the Texas Advanced Computing Center (TACC) [19]. V. RESULTS AND DISCUSSION Below we present our results. We start with the angleintegrated setup in Sec. V A before going into further detail with angle-resolved measurements and calculations in Sec. V B. delay-integrated signal. The delay-integrated photoelectron spectra (normalized to 1 at the highest peak) is plotted in panel (b). Due to the high NIR intensity, some of the main bands are depleted substantially and appear weaker than the SBs in their vicinity. The RABBIT phase (φ R ) is extracted by fitting a cosine function (cf. Eqs. (2)) to the delay-dependent oscillating signals of the sidebands, as seen in Fig. 4. Due to the large dataset available and the excellent stability of the interferometer, the phase retrieval resulted in error bars smaller than the symbol size in Fig. 3(c). This gives us confidence in the results obtained from our extraction procedure. A. Angle-integrated RABBIT phases As predicted by our generalized decomposition approximation (1), the lower and the higher SBs oscillate by π out of phase with the central SB. The retrieved RABBIT phases (φ R ) are plotted in Fig. 3(c) after removing the extra π from S l and S h to simplify the comparison. The time-delay axis on the right side of this panel was created via the conversion τ R = φ R /(4ω). Five SB groups are clearly identifiable in Fig. 3(c). While there are some irregularities in SB 8 and SB 16 , especially with the phase extracted from S l , groups SB 10 , SB 12 , and SB 14 show the expected trend: The RABBIT phases of the three SBs in each group are similar, although a clearly visible difference remains in SB 10 . That difference, however, essentially vanishes in SB 12 and SB 14 . The irregularity seen in the SB 8 group is due to a significant contribution of another 4 th -order transition in the absorption path of the lowest SB S l , which involves a transition from M 7 down to the Rydberg states and back up to S l . The Rydberg states enhance the strength of this transition and add a resonance phase that leads to a significant deviation in the RABBIT phase of S l compared to the other members of the SB 8 group. Furthermore, due to the low cutoff of the XUV spectrum based on HHG and the decreasing photoionization cross section of argon with increasing photon energy, the strength of the M 17 peak is very weak compared to the rest of the lower main peaks. As a result, higher-order transitions involving lower main bands also play a significant role in the oscillation of S l in the SB 16 group, which again affects the extracted phase. B. Angle-differential RABBIT phases We now further increase the level of detail by investigating angle-dependent RABBIT phases, which is possible due to the angle-resolving capability of the reaction microscope. For the reasons given above regarding the additional complexities associated with the SB 8 and SB 16 groups, we concentrate the remaining discussion on SB 10 , SB 12 , and SB 14 . Figure 5(a-c) shows the RABBIT phases extracted within these groups as a function of the photoelectron emission angle, which is defined relative to the (linear) laser polarization vector. The photoelectron signal is integrated over an angular window of 10 • for each data point. The angleresolved RABBIT phases are shifted to fix the starting phase of the central sideband in each group to zero. According to both our experiment and the calculation (Fig. 5, panels d-f), the phase of S h , in particular, exhibits a very strong angular dependence, while that of S l is nearly angleindependent. With increasing photoelectron energy, the differences diminish. The difference in the phases of the three SBs can possibly be explained by considering a propensity rule for transition amplitudes and the dependence of both the Wigner and φ cc phases on the orbital angular momenta involved. Similar to bound-continuum transitions [20], absorption (emission) within the continuum favors an increase (decrease) in the angular momentum of the outgoing photoelectron, especially for low kinetic energies [21][22][23][24][25]. The higher SB (S h ) of the group involves the absorption of three probe photons (H q−1 + 3 ω) that, according to the propensity rule, predominantly populate higher angular-momentum states. Along the other path (H q+1 − 1 ω) leading to S h , the emission of one probe photon mainly creates lower angularmomentum states. On the other hand, the emission of three probe photons (H q+1 − 3 ω) to the lower SB (S l ) of the group predominantly creates lower angular-momentum states and higher angular-momentum states in the other absorption path (H q−1 + 1 ω). It is well known that the Wigner phase depends on the angular-momentum channel. The cc phase has also been shown to slightly depend on whether there is an increase or decrease in the angular momentum, while it appears to remain independent of the target species [23,26]. The RABBIT phases in the individual -channels, therefore, are expected to differ as well. Since the retrieved angle-integrated RABBIT phase is the weighted average of all the channel-resolved RABBIT phases and the weights of the partial waves in the three SBs are different, the angle-integrated RABBIT phase in the three SBs also turns out different. For the angle-resolved RABBIT phase, the interplay of the propensity rule for transition amplitudes to each channel and the angle-dependent weights of the spherical harmonics determine the steepness of the angle-dependent phase curves of the three SBs. The higher states carry comparatively more weight in the higher SB and less in the lower SB. Also, the contributions of the associated spherical harmonics for larger vary rapidly with the angle. As a result, a steeper angle dependence of the RABBIT phase in the higher SBs compared to the central and lower SBs may be expected. This trend is, indeed, clearly seen in SB 10 . Since the difference in φ cc for different channels becomes smaller with increasing kinetic energy, the angle dependence of the RABBIT phases becomes flat, as can be seen in SB12 and SB14. Finally, we notice that the scale of variation in the angle dependence of the RABBIT phase is smaller in the calculation than in the experiment. Also, the position of S l and S h relative to S c is switched. In addition to always possible shortcomings in the theoretical model (as sophisticated as it might be) and unknown potential systematic errors in experiment, the difference in the probe intensities and the pulse details in general are likely responsible for at least some of the discrepancies seen here. VI. SUMMARY AND OUTLOOK In summary, we carried out a proof-of-principle 3-SB RABBIT experiment in argon. In contrast to more popular single-SB studies, our technique enables us to focus on the atomic phase of transitions without distortion from a possibly unknown or experimentally drifting XUV chirp. While we confirmed earlier predictions that the angleintegrated RABBIT phases extracted within a SB group become increasingly similar, we enhanced the analyzing power of the setup significantly by resolving the emission angle with a reaction microscope. By doing so, we could identify which of the three sideband phases within a group is most sensitive to a change in the detection angle. Our experimental efforts were supported by numerical calculations performed with the nonperturbative all-electron R-matrix with time dependence method. There is good qualitative agreement between experiment and theory regarding the general trends observed, but significant differences remain in the details. Given the remaining limitations and challenges faced in the present study, especially concerning the details of the pulse and the argon target, the remaining deviations between experiment and theory in the quantitative values of the phases are not too surprising. We hope to address these issues in future improvements of the setup. As the next step, we plan to repeat this experiment with helium, where the contribution of the Wigner phase for an s → p transition remains the same in all three sidebands. Any differences in the phases within the group then clearly indicate the influence of φ cc . This switch of targets would require extending the harmonic cut-off, which is by no means trivial in our scheme, as the cut-off in the HHG process decreases with the driving frequency. Using helium instead of argon also has the advantage of theory likely being more reliable due to the simplicity of the target. On the other hand, heavier quasi-two-electron targets with an (ns 2 ) 1 S outer-shell configuration (unfortunately, these are metals that would need to be vaporized rather than inert gases) would provide a larger short-range modification of the relevant interaction potential and, therefore, may be more suitable to investigate whether φ cc is indeed nearly universal. Undoubtedly, many open questions will need to be answered before the effect of the additional continuumcontinuum transitions in single-and multiple-SB RABBIT setups are fully understood. It would be interesting to analyze whether the SB phases always converge to each other with increasing energy, whether or not they cross in a predictable way with increasing emission angle, and how the behavior depends on the target investigated. While we cannot answer these questions at the present time, we hope that other groups will see the work reported in this paper as a worthwhile inspiration to carry out further studies in this field. SB RABBIT scheme. Mq−1 and Mq+1 label the main photoelectron peaks created directly by the odd harmonics (Hq−1 FIG. 2 . 2Experimental setup. A holey mirror (BS) splits the linearly polarized laser beam between the two arms of the interferometer. In the pump arm, the HHG process is driven by the second harmonic of the laser beam. The generated XUV and the fundamental probe beam are recombined and focused onto a supersonic gas jet of Argon. The interferometer is stabilized by tracking the movement of the fringes from the pump and the probe beams. Figure 3 3exhibits the results of our 3-SB RABBIT experiment after integrating the signal over all photoelectron emission angles. To highlight the oscillations, the RABBIT trace in panel (a) is plotted after subtracting the average SB RABBIT trace (a), normalized photoelectron spectra generated with the XUV pulse only (dark) and during the RABBIT measurement integrated over the delays (lighter) (b), and RABBIT phases extracted from all three sidebands (c). Note that the π phase difference between Sc and (S l , S h ), which is clearly seen in the position of the maxima in panel (a), has been removed for better visibility in panel (c). The error bars from the fitting procedure are smaller than the symbol size and hence not visible.ℎ FIG. 4. The delay-dependent normalized signal (dots) of the three sidebands in the SB12 group and fits to the cosine function (lines). FIG. 5 . 5Top row: Angle-dependent RABBIT phases extracted from the measurements in group SB10 (a), SB12 (b) and SB14 (c). Bottom row: Corresponding RMT predictions. . 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[ "FROM BLACK HOLES TO POMERON: Tensor Glueball and Pomeron Intercept at Strong Coupling a", "FROM BLACK HOLES TO POMERON: Tensor Glueball and Pomeron Intercept at Strong Coupling a" ]
[ "Richard C Brower ", "Samir D Mathur ", "Chung-I Tan " ]
[]
[]
We briefly review the approach for strong coupling calculation of glueball masses based on the duality between supergravity and Yang-Mills theory. Earlier work is extended to non-zero spin. Fluctuations in the gravitational metric lead to the 2 ++ tensor glueball state on the leading Pomeron trajectory with a mass relation: m(0 ++ ) < m(2 ++ ). In particular, for QCD 4 , a strong coupling expansion for the Pomeron intercept is obtained.
null
[ "https://export.arxiv.org/pdf/hep-ph/0003153v1.pdf" ]
18,080,247
hep-ph/0003153
095a1d668effe1055b137ab479745b5256f2fd94
FROM BLACK HOLES TO POMERON: Tensor Glueball and Pomeron Intercept at Strong Coupling a 15 Mar 2000 Richard C Brower Samir D Mathur Chung-I Tan FROM BLACK HOLES TO POMERON: Tensor Glueball and Pomeron Intercept at Strong Coupling a 15 Mar 2000 We briefly review the approach for strong coupling calculation of glueball masses based on the duality between supergravity and Yang-Mills theory. Earlier work is extended to non-zero spin. Fluctuations in the gravitational metric lead to the 2 ++ tensor glueball state on the leading Pomeron trajectory with a mass relation: m(0 ++ ) < m(2 ++ ). In particular, for QCD 4 , a strong coupling expansion for the Pomeron intercept is obtained. Introduction The Maldacena conjecture 1 and its further extensions allow us to compute quantities in a strongly coupled gauge theory from its dual gravity description. In particular, Witten 2 has pointed out if we compactify the 4-dimensional conformal super Yang Mills (SYM) to 3 dimensions using anti-periodic boundary conditions on the fermions, then we break supersymmetry and conformal invariance and obtain a theory that has interesting mass scales. This approach has been used to calculate a discrete mass spectrum for0 ++ states associated with T r[F 2 ] at strong coupling by solving the dilaton's wave equation in the corresponding gravity description. 3,4 Although the theory at strong coupling is really not pure Yang-Mills, since it has additional fields, some rough agreement was claimed with the pattern of glueball masses. Here we report on the calculation of the discrete modes for the perturbations of the gravitational metric. 5,6 A complete description for all discrete fluctuations has also been carried out, both for QCD 3 and QCD 4 . 7 For simplicity, we shall discuss here mostly QCD 3 . For QCD 4 , from the mass of the 2 ++ state and the calculated QCD string tension, we obtain a strong coupling expansion for the Pomeron intercept: α P (0) = 2 − 0(1/g 2 N ). In this approach, the Pomeron corresponds to a "massive graviton". Other results will be reported elsewhere. 7 2 AdS/CFT Duality at Finite β Let us review briefly the proposal for getting a 3-d Yang-Mills theory dual to supergravity. One begins by considering Type IIB supergravity in Euclidean 10-dimensional spacetime with the topology M 5 × S 5 . The Maldacena conjecture asserts that IIB superstring theory on AdS 5 × S 5 is dual to the N = 4 SYM conformal field theory on the boundary of the AdS space. ds 2 /R 2 ads = r 2 (dτ 2 + dx 2 1 + dx 2 2 + dx 2 3 ) + dr 2 r 2 + dΩ 2 5 ,(1) where the radius of the AdS spacetime is given through R 4 AdS = g s N α ′2 (g s is the string coupling and l s is the string length, l 2 s = α ′ ). The Euclidean time is τ = ix 0 . To break conformal invariance, following ref. 2 , we place the system at a nonzero temperature described by a periodic Euclidean time τ = τ + β, β = 2πR 0 . The metric correspondingly changes, for small enough R 0 , to the non-extremal black hole metric in AdS space. For large black hole temperatures, the stable phase of the metric corresponds to a black hole with radius large compared to the AdS curvature scale. To see the physics of discrete modes, we may take the limit of going close to the horizon, whereby the metric reduces to that of the black 3-brane. This metric is, (where f (r) = r 2 − 1 r 2 , and we have scaled out all dimensionful quantities), ds 2 = f dτ 2 + f −1 dr 2 + r 2 (dx 1 1 + dx 2 2 + dx 2 3 ) + dΩ 2 5 ,(2) On the gauge theory side, we would have a N = 4 susy theory corresponding to the AdS spacetime, but with the S 1 compactification with antiperiodic boundary conditions for the fermions, supersymmetry is broken and massless scalars are expected to acquire quantum corrections. Consequently from the view point of a 3-d theory, the compactification radius acts as an UV cut-off. Before the compactification the 4-d theory was conformal, and was characterized by a dimensionless effective coupling (g (4) Y M ) 2 N ∼ g s N . After the compactification the theory is not conformal, and the radius of the compact circle provides a length scale. Let this radius be R 0 . Then a naive dimensional reduction from 4-d Yang-Mills to 3-d Yang-Mills, would give an effective coupling in the 3-d theory equal to (g (3) Y M ) 2 N = (g (4) Y M ) 2 N/(2πR 0 ). The 3-d YM coupling has the units of mass. If the dimensionless coupling (g (4) Y M ) 2 N is much less than unity, then the length scale associated with this mass is larger than the radius of compactification, and we may expect the 3-d theory to be a dimensionally reduced version of the 4-d theory. Unfortunately the dual supergravity description only applies at (g (4) Y M ) 2 N >> 1, so that the higher Kaluza-Klein modes of the S 1 compactification have lower energy than the mass scale set by the 3-d coupling. Thus we do not really have a 3-d gauge theory with a finite number of additional fields. One may nevertheless expect that some general properties of the dimensionally reduced theory might survive the strong coupling limit. Moreover, we expect that the pattern of spin splittings might be a good place to look for similarities. In keeping with earlier work, we ignore the Kaluza-Klein modes of the S 1 and restrict ourselves to modes that are singlets of the SO(6), since non-singlets under the S 1 and the SO(6) can have no counterparts in a dimensionally reduced QCD 3 . Wave Equations We wish to consider fluctuations of the metric of the form, g µν =ḡ µν + h µν (x) ,(3) leading to the linear Einstein equation, h µν;λ λ + h λ λ ;µν − h µλ;ν λ − h νλ;µ λ − 8h µν = 0 .(4) Our perturbations will have the form h µν = ǫ µν (r)e −mx3(5) where we have chosen to use x 3 as a Euclidean time direction to define the glueball masses of the 3-d gauge theory. We fix the gauge to h 3µ = 0. ¿From the above ansatz and the metric, we see that we have an SO(2) rotational symmetry in the x 1 − x 2 space, and we can classify our perturbations with respect spin. Spin-2: There are two linearly independent perturbations which form the spin-2 representation of SO (2): h 12 = h 21 = q T (r)e −mx3 , h 11 = −h 22 = q T (r)e −mx3 with all other components zero. The Einstein equations give, (r 2 − 1 r 2 )q ′′ T + (r + 3 r 3 )q ′ T + ( m 2 r 2 − 4 − 4 r 4 )q T = 0.(6) Defining φ T (r) = q T (r)/r 2 , this is the same equation as that satisfied by the dilaton (with constant value on the S 5 ). Spin-1: The Einstein equation for the ansatz, h iτ = h τ i = q V (r)e −mx3 , i = 1, 2 gives (r 2 − 1 r 2 )q ′′ V + (r − 1 r 3 )q ′ V + ( m 2 r 2 − 4 + 4 r 4 )q V = 0 .(7) Spin-0: Based on the symmetries we choose an ansatz where the nonzero components of the perturbation are h 11 = h 22 = q 1 (r)e −mx3 h τ τ = −2q 1 (r) f (r) r 2 e −mx3 + q 2 (r)e −mx3 h rr = q 3 (r)e −mx3 where f (r) is defined above in the metric. The field equation for q 3 ≡ q S (r), is p 2 (r)q ′′ S (r) + p 1 (r)q ′ S (r) + p 0 (r)q S (r) = 0,(8) where p 2 (r) = r 2 (r 4 − 1) 2 [3(r 4 − 1) + m 2 r 2 ], p 1 (r) = r(r 4 − 1)[3(r 4 − 1)(5r 4 + 3) + m 2 r 2 (7r 4 + 5)] and p 0 (r) = 9(r 4 − 1) 3 + 2m 2 r 2 (3 + 2r 4 + 3r 8 ) + m 4 r 4 (r 4 − 1). Numerical Solution To calculate the discrete spectrum for our three equation, one must apply the correct boundary conditions at r = 1 and r = ∞. The result is a Sturm-Liouville problem for the propagation of gravitational fluctuations in a "wave guide". Using this shooting method we have computed the the first 10 states given in Table 4. The spin-2 equation is equivalent to the dilaton equation 3,4 , so the excellent agreement with earlier values validates our method. We used a standard Mathematica routine with boundaries taken to be x = r 2 − 1 = ǫ and 1/x = ǫ reducing ǫ gradually to ǫ = 10 −6 . Note that since all our eigenfunctions must be even in r with nodes spacing in x = r 2 − 1 of O(m 2 ), the variable 1/x is a natural way to measure the distance to the boundary at infinity. For both boundaries, the values of ǫ was varied to demonstrate that they were near enough to r = 1, and ∞ so as not to substantially effect the answer. As one sees in the accompanying figure, they match very accurately with the leading order WKB approximation. Simple variational forms also lead to very accurate upper bounds for the ground state (n = 0) masses. Strong coupling Expansion for Pomeron Intercept Our current exercise has been extended to 4-d QCD using a scheme involving the finite temperature version of AdS 7 × S 4 . As has been suggested elsewhere, one goal is to find that background metric that has the phenomenologically best strong coupling limit. This should provide an optimal starting point for approaching the continuum weak coupling regime. Here, we shall report briefly the key constraint provided by the Pomeron intercept. The Pomeron is the leading Regge trajectory passing through the lightest glueball state with J P C = 2 ++ . In a linear approximation, it can be parameterized by α P (t) = 2 + α ′ P (t − m 2 T ),(9) where we can use the strong coupling estimate for the lightest tensor mass b , m T ≃ [9.86 + 0( 1 g 2 N )] β −1 .(10) Moreover if we make the standard assumption that the closed string tension is twice that between two static quark sources, we also have a strong coupling expression for the Pomeron slope, α ′ P ≃ [ 27 32πg 2 N + 0( 1 g 4 N 2 )] β 2 .(11) Putting these together, we obtain a strong coupling expansion for the Pomeron intercept, α P (0) ≃ 2 − 0.66 ( 4π g 2 N ) + 0( 1 g 4 N 2 ) .(12) Turning this argument around, we can estimate a crossover value between the strong and weak coupling regimes by fixing α P (0) ≃ 1.2 at its phenomenological value. In fact this yields for QCD 4 at N = 3 a reasonable value for α strong = g 2 /4π = 0.176 for the crossover. Much more experience with this new approach to strong coupling must be gained before such numerology can be taken seriously. However, similar crude argument have proven to be a useful guide in the crossover regime of lattice QCD. One might even follow the general strategy used in the lattice cut-off formulations. Postpone the difficult question of analytically solving the QCD string to find the true UV fixed point. Instead work at a fixed but physically reasonable cut-off scale (or bare coupling) to calculate the spectrum. If one is near enough to the fixed point, mass ratios should be reliable. After all, the real benefit of a weak/strong duality is to use each method in the domain where it provides the natural language. On the other hand, clearly from a fundamental point of view, finding analytical tools to understand the renormalized trajectory and prove asymptotic scaling within the context of the gauge invariant QCD string would also be a major achievement -an achievement that presumably would include a proof of confinement itself. Results on these computations will be reported in a future publication. 7 The metric of this a Talk presented by R. C. Brower at ISMD99: QCD and Multiuparticle Production. This work was supported in part by the Department of Energy under Contract No. DE-FG02/91ER40688 and DE-FG02-91ER40676spacetime is Table 1 . 1Glueball Excitation Spectrumlevel 0 ++ 1 −+ 2 ++ n= 0 5.4573 18.676 11.588 n= 1 30.442 47.495 34.527 n= 2 65.123 87.722 68.975 n= 3 111.14 139.42 114.91 n= 4 168.60 203.99 172.33 n= 5 237.53 277.24 241.24 n= 6 317.93 363.38 321.63 n= 7 409.82 461.00 413.50 n= 8 513.18 570.11 516.86 n= 9 628.01 690.70 631.71 . J Maldacena, hep-th/9711200Adv. Theor. Math. Phys. 2231J. Maldacena, Adv. Theor. Math. Phys. 2:231, 1998, hep-th/9711200. . E Witten, hep-th/9803131Adv. Theor. Math. Phys. 2505E. Witten, Adv. Theor. Math. Phys.2: 505, 1998, hep-th/9803131. . C Csáki, H Ooguri, Y Oz, J Terning, hep-th/9806021C. Csáki, H. Ooguri, Y. Oz and J. Terning, hep-th/9806021. . R. De Mello Koch, A Jevicki, M Mihailescu, J Nunes, hep-th/9806125R. De Mello Koch, A. Jevicki, M. Mihailescu and J. Nunes, hep-th/9806125. . R Brower, S Mathur, C-I Tan, hep-th/9908196Nucl. Physics. R. Brower, S. Mathur and C-I Tan, (to be published in Nucl. Physics), hep-th/9908196. . N R Constable, R C Meyers, hep-th/9908175N. R. Constable and R.C. Meyers, hep-th/9908175. R Brower, S Mathur, C-I Tan, hep-th/0003115Glueball Spectrum for QCD from AdS Supergravity Duality. R. Brower, S. Mathur and C-I Tan, "Glueball Spectrum for QCD from AdS Supergravity Duality", hep-th/0003115.
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[ "\nSIDNEY DONATO\n\n" ]
[ "SIDNEY DONATO\n" ]
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On compact 2-manifolds with non-empty strictly convex boundary, we prove a regularity result for integral 1-varifolds V that are stationary with free boundary and Z 2 -almost minimizing in small annuli. That regularity says that V is a free boundary finite geodesic network. Next, using that regularity, we compute the first p-widths of the unit closed ball B 2 , for p = 1, ..., 4.
10.1007/s12220-022-00913-3
[ "https://arxiv.org/pdf/2002.06724v2.pdf" ]
211,132,984
2002.06724
76aecad2642beceea9754ec7a6fc98d3fcf99db3
20 Sep 2021 SIDNEY DONATO 20 Sep 2021THE FIRST p-WIDTHS OF THE UNIT DISK On compact 2-manifolds with non-empty strictly convex boundary, we prove a regularity result for integral 1-varifolds V that are stationary with free boundary and Z 2 -almost minimizing in small annuli. That regularity says that V is a free boundary finite geodesic network. Next, using that regularity, we compute the first p-widths of the unit closed ball B 2 , for p = 1, ..., 4. Introduction For n > 0, let (M n+1 , g) be a compact Riemannian manifold with (possibly empty) boundary. Gromov [8,7], Guth [10], Marques and Neves [16] introduced the notion of volume spectrum {ω p (M )} ∞ p=1 for the area functional in the space of relative mod 2 cycles Z n (M, ∂M, Z 2 ). The volume spectrum is a sequence of positive numbers that satisfies similar properties to the spectrum of the Laplacian: 0 < ω p (M ) ≤ ω p+1 (M ) and ω p (M ) → ∞ as p → ∞. Where ω p (M ) is called the p th min-max width of M. The Liokumovich-Marques-Neves-Weyl [13] law gives the asymptotic behavior of this spectrum, precisely: In contrast with the Weyl law for eigenvalues of the Laplacian, the constant above is almost totally unknown. Obviously, if we have a full description of the values of the volume spectrum, we can deduce the constant a(n). So far this full description seems to be very hard. In fact, the results on this direction got only to compute some initial widths (Aiex [1], Gaspar and Guaraco [6], and Nurser [17]). More recently, Chodosh-Mantoulidis [3] gives a description of this spectrum for the round two-sphere, and they found that a(n) = √ π in this case. The main objective of this article is to compute the first widths of the unit closed ball (unit disk) B 2 ⊂ R 2 and of planar full ellipses E 2 closed to B 2 (see Theorem 4.3). The ideas to prove our results are similar to what was done by Aiex [1] for the 2-sphere S 2 and for ellipsoids close to S 2 . In this case, it is used a regularity result due to Allard and Almgren [2] which says that stationary integral 1-varifolds on closed Riemannian manifolds are finite geodesic networks. This means that the varifold is a finite union of geodesic segments such that the singularities are given by their possible stationary junctions. This regularity is an important tool, because it holds for the 1-varifolds obtained in the Min-Max Theorem, so for each p ∈ N there exist geodesic networks sufficiently close to achieve the p-width. In our case, we had to extend this regularity result for two dimensional manifolds M 2 with non-empty boundary and we did this supposing the boundary strictly convex. In this hypothesis we get (see Theorem 3.12): If V is a stationary integral 1-varifold which is Z 2 -almost minimizing in small annuli, then V is a free boundary finite geodesic network. This means that V restricted to the interior of M, int(M ), is a finite geodesic network, each geodesic segment has its interior in int(M ), and each point p ∈ ∂M that is on the support of V is given by the intersection of boundaries of geodesic segments from int(M ) such that: if each of these segments at p are parameterized to start at p, then the resultant of the unit tangent vectors of the segments (and its multiplicities) is perpendicular to ∂M at p. The extra hypothesis that V is Z 2 -almost minimizing in small annuli is a classical hypothesis to get regularity for the codimension one case and for 3 ≤ n + 1 ≤ 7 (see Pitts [20], Simon [23] and Li and Zhou [12]). Essentially, the regularity comes from the fact that almost minimizing varifolds are locally stable almost everywhere. The hypothesis of strictly convex boundary follows the ideas from [12], where they prove a regularity result for strictly convex boundary and 3 ≤ n + 1 ≤ 7. As in [1], the regularity is an important step to calculate the p-widths. In fact, by that regularity the varifolds obtained in our adapted version of the Min-Max Theorem (see Theorem 2.3) are free boundary finite geodesic networks. We did a classification (Theorems 3.5 and 3.6) of these varifolds which have low mass in B 2 and E 2 and then, we get candidates for the first p-widths. Finally, to compute the p-widths of B 2 we use p-sweepouts whose image are given by real algebraic varieties restricted to B 2 . We estimate these p-sweepouts and we combine with the classification to deduce the first widths. For E 2 we do similarly and using continuity. This article is organized in the following way: in Section 2 we remember some basic theory and we give some definitions, also we explain how we adapt the Min-Max Theorem for our case (Theorem 2.3 and Corollary 2.4); in Section 3 we talk about free boundary geodesic networks and its properties, also we classify the free boundary finite geodesic networks which are Z 2 -almost minimizing in small annuli and have low mass in B 2 and E 2 , and we conclude proving our regularity result (Theorem 3.12); in the Section 4 we compute the first p-widths of B 2 and E 2 (Theorem 4.3) using the regularity, classification and the estimates obtained for the p-sweepouts; and in the Section 5 (Appendix) we prove the sharp estimate for these p-sweepouts. Preliminaries Throughout this section M denotes a compact Riemannian (n + 1)-manifold, n ≥ 0, with smooth and possibly empty boundary ∂M. We can always assume that M is isometrically embedded in some Euclidean space R Q for some Q ∈ Z + . We denote by B r (p) as the open Euclidean ball of radius r centered at p ∈ R Q , and A s,r (p) the open annulus B r (p)\B s (p) for 0 < s < r. When M has non-empty boundary, the embedding above is obtained in the following way: we can extend M to a closed Riemannian manifold M with the same dimension such that M ⊂ M (see ), and so, by the Nash's Theorem, we get the isometric embedding M ֒→ R Q . We denote by B r (p) as the open geodesic ball in M of radius r centered at p. We consider the following spaces of vector fields: Definition 2.3. (Fermi coordinates) Given p ∈ ∂M and suppose that the coordinates (x 1 , · · · , x n ) are the geodesic normal coordinates of ∂M in a neighborhood of p. Take t = dist M ( · , ∂M ), which is a smooth map well-defined in a relatively open neighborhood of p in M. The Fermi coordinates system of (M, ∂M ) centered at p is given by the coordinates (x 1 , · · · , x n , t). Also, the Fermi distance function from p on a relatively open neighborhood of p in M is defined by Xr := r p (q) = |(x, t)| = x 2 1 + · · · + x 2 n + t 2 . Definition 2.4. Given p ∈ ∂M, we define the Fermi half-ball and half-sphere of radius r centered at p respectively by B + r (p) := {q ∈ M : r p (q) < r}, S + r (p) := {q ∈ M : r p (q) = r}. Also we consider the following open annular neighborhood in the Fermi coordinates: A s,t (p) := B + t (p)\Clos( B + s (p)) for p ∈ ∂M, and 0 < s < t. Where Clos( B + s (p)) denotes the closure of B + s (p) on M. Also, when p ∈ int(M ), we require that t < dist M (p, ∂M ). The geometric properties of the Fermi half-ball and half-sphere can be summarized in the following proposition: Proposition 2.1. [12, Lemma A.5] There exists a small constant r F ermi > 0, depending only on the isometric embedding M ⊂ R Q , such that for all 0 < r < r F ermi (i) S + r (p) is a smooth hypersurface meeting ∂M orthogonally; (ii) B + r (p) is a relatively strictly convex 1 domain in M ; (iii) B r/2 (p) ∩ M ⊂ B + r (p) ⊂ B 2r (p) ∩ M. Relative Flat Cycles We recall some definitions that can be found in [15,Section 2] or in [5,Section 4]. For each 0 ≤ k ≤ n + 1, R k (M ; Z 2 ) denotes the set of k-dimensional rectifiable mod 2 flat chains in R Q whose support lies in M. Given T ∈ R k (M ; Z 2 ), we denote by F (T ) and M(T ) the flat norm and the mass of T, respectively. Also, the support of T is denoted by spt(T ). Consider the following sets: I k (M ; Z 2 ) = {T ∈ R k (M ; Z 2 ) : ∂T ∈ R k−1 (M ; Z 2 )}, Z k (M ; Z 2 ) = {T ∈ I k (M ; Z 2 ) : ∂T = 0}, Z k (M, ∂M ; Z 2 ) = {T ∈ I k (M ; Z 2 ) : spt(∂T ) ⊂ ∂M }, and Z k,rel (M, ∂M ; Z 2 ) = Z k (M, ∂M ; Z 2 )/I k (∂M ; Z 2 ). The set Z k (M ; Z 2 ) is the space of mod 2 (integral) flat k-cycles in M and we call the quotient space Z k,rel (M, ∂M ; Z 2 ) as the space of relative (mod 2) flat cycles. When ∂M = ∅, we have that Z k,rel (M, ∂M ; Z 2 ) is identical to Z k (M ; Z 2 ). The support spt( [T ]) of a class [T ] ∈ Z k,rel, (M, ∂M ; Z 2 ) is defined by spt([T ]) = T ∈[T ] spt(T ) . Also, the mass norm and flat norm in the space of relative cycles are defined, respectively, by M([T ]) = inf T ∈[T ] M(T ), F ([T ]) = inf T ∈[T ] F ([T ]), for [T ] ∈ Z k,rel, (M, ∂M ; Z 2 ). We consider the space of relative flat cycles Z k,rel (M, ∂M ; Z 2 ) endowed with the flat norm F . When it is endowed with the topology of the mass norm, we denote it by Z k (M, ∂M ; M; Z 2 ). Note that each [T ] ∈ Z k,rel (M, ∂M ; Z 2 ) has a unique canonical representative k-chain T 0 ∈ [T ] such that T 0 ∂M = 0, in particular, M([T ]) = M(T 0 ) and spt([T ]) = spt(T 0 ), see [12,Lemma 3.3]. Also, it follows that F ([T ]) ≤ M([T ]). This canonical representative is obtained taking T 0 = S (M \∂M ) for any S ∈ [T ]. To keep the notation simple we denote [T ] by T. Varifolds in manifolds with boundary The following definitions can be found in [20] and [23]. We denote by RV k (M ) the set of k-dimensional rectifiable varifolds in R Q with support contained in M and equipped with the weak topology. Also V k (M ) is the closure of RV k (M ) in the weak topology. Given a varifold V ∈ V k (M ), the weight and the support of V are denoted by V and spt V , respectively. Also, for x ∈ spt V , we denote by VarTan(V, x) ⊂ V(R Q ) as the set of the varifold tangents of V at x, which is a natural generalization of tangent planes for smooth surfaces. Given V, W ∈ V k (M ), the Pitts' F-metric is denoted by F(V, W ). This metric induces precisely the usual weak topology on the set {V ∈ V k (M ) : V (M ) ≤ L}, for each constant L > 0. If R ⊂ M is a k-rectifiable set and θ is a H k -integrable non-negative function on R, we denote by υ(R, θ) ∈ V k (M ) as being the rectifiable k-varifold associated to R with multiplicity function θ. If θ assumes only positive integers values, we say that υ(R, θ) is an integral varifold. We denote by IV k (M ) the space of k-dimensional integral varifolds in M. Given T ∈ R k (M ; Z 2 ), we denote by |T | ∈ V k (M ) the varifold induced by the support of T and its coefficients. And for T ∈ Z k,rel, (M, ∂M ; Z 2 ), we take |T | = |T 0 |. Given V ∈ V k (M ), let X ∈ X tan (M ) be a generator of a one-parameter family of diffeomorphisms φ t of R Q with φ 0 (M ) = M, we have that the first variation of V along the vector field X is given by [23, 39.2]). δV (X) := d dt t=0 M((φ t ) ♯ V ), where (φ t ) ♯ V is the pushfoward varifold of V (seeDefinition 2.5. Let U ⊂ M be a relatively open subset. A varifold V ∈ V k (M ) is said to be stationary in U with free boundary if δV (X) = 0 for any X ∈ X tan (M ) compactly supported in U. Note that a free boundary minimal submanifold is also stationary with free boundary. However, the reverse may not be true. By the relative topology we consider the k-dimensional density, Θ k (V, x), of a varifold V ∈ V k (M ) as the density restricted to M, that is, given x ∈ M, we take the limit, if it exists, Θ k (V, x) := lim ρ→0 V (B ρ (x) ∩ M ) ρ k |B k | , where |B k | is the volume of the k-dimensional unit Euclidean ball B k . For stationary varifolds the limit above always exists. For a fixed x, define the function Θ k x (V, ρ) := V (B ρ (x) ∩ M ) ρ k |B k | . In the case ∂M = ∅, we have B ρ (x) ⊂ M and it is known that the function above for stationary varifolds satisfies the monotonicity formula [23, Sections 17 and 40]: Θ k x (V, ρ) is non-decreasing in ρ. Also, it is well known that any tangent varifold of a stationary varifold is a stationary Euclidean cone and Θ k x (C, ρ) = Θ k (V, x) for any C ∈ VarTan(V, x) and for all ρ > 0. We write this fact as Θ k x (C, ∞) = Θ k (V, x). Min-Max Definitions In the following we use the notions of homotopy as in [16,Section 2], just replacing Z n (M ; M; Z 2 ) by Z n,rel (M, ∂M ; M; Z 2 ) in those definitions. Here we are taking k = dim(M ) − 1 = n, in the notations of the previous sections. The set [X, Z n,rel (M, ∂M ; M; Z 2 )] ♯ denotes the set of all equivalence classes of (X, M)-homotopy classes of mappings into Z n,rel (M, ∂M ; M; Z 2 ). Given an equivalence class Π ∈ [X, Z n,rel (M, ∂M ; M; Z 2 )] ♯ , each S ∈ Π is given by S = {φ i } i∈N for some (X, M)-homotopy sequence of mappings {φ i } i∈N into Z n,rel (M, ∂M ; M; Z 2 ). We define L(S) = lim sup i→∞ max{M(φ i (x)); x ∈ dmn(φ i )}. Definition 2.6. The width of Π is defined by L(Π) = inf{L(S) : S ∈ Π}. We say that S ∈ Π is a critical sequence for Π if L(S) = L(Π), and the critical set C(S) of a critical sequence S is given by C(S) = K(S) ∩ {V ∈ V n (M ) : V (M ) = L(S)}, where K(S) = V ∈ V n (M ) : V = lim j→∞ |φ ij (x j )| as varifolds, for some subsequence {φ ij } ⊂ S and x j ∈ dmn(φ ij ) .sup{ Φ(x) (B r (p)\∂M ) : x ∈ dmn(Φ), p ∈ M } = 0. The set of all p-sweepouts with no concentration of mass is denoted by P p (M ). Definition 2.9. The p-width of M is given by Roughly speaking, it means that we can approximate V by a varifold induced from a current T such that for any deformation of T by a discrete family supported in U, and with the mass not increasing too much (parameter δ), then at the end of the deformation the mass cannot be deformed down too much (parameter ǫ). ω p (M ) = inf Φ∈Pp(M) sup{M(Φ(x)) : x ∈ dmn(Φ)}. Min-Max Theorem A varifold V ∈ V k (M ) is said to be Z 2 -almost minimizing in annuli with free boundary if for each p ∈ spt V there exists r > 0 such that V is Z 2 -almost minimizing in the annuli M ∩ A s,r (p) = M ∩ B r (p)\B s (p) for all 0 < s < r. If p / ∈ ∂M, we require that r < dist(p, ∂M ). By Proposition 2.1 (iii), this definition with respect to A s,r (p) or A s,r (p) is equivalent. When ∂M = ∅, we do not need use the expression 'with free boundary'. The next result is a tightening process to a critical sequence S ∈ Π so that every V ∈ C(S) becomes a stationary varifold with free boundary. Theorem 2.2. Let Π ∈ [X, Z n,rel (M, ∂M ; M; Z 2 )] ♯ . For each critical sequence S * ∈ Π, there exists another critical sequence S ∈ Π such that C(S) ⊂ C(S * ) and each V ∈ C(S) is stationary in M with free boundary. If V ∈ V k (M ) is Z 2 -almost Proof. The proof of this result is essentially the same as [14,Prop. 8.5]. The only modifications are the use of Th. 13.1 and 14.1 of [14], as noted in [12,Th. 4.17]. In place of [14,Th. 14 With the tightening process above we can prove the existence of a Z 2 -almost minimizing varifolds with free boundary such that it reaches the width of a chosen (X; M)-homotopy class Π ∈ [X, Z n,rel (M, ∂M ; M; Z 2 )] ♯ . When ∂M = ∅, it was first proved by Pitts [20,Th. 4.10] with maps in cubical domains for 1 ≤ k ≤ n, and later by Marques and Neves [16,Th. 2.9] for cubical subcomplex domains when k = n. For the case with boundary, a version for cubical domains was proved by Li and Zhou [12,Th. 4.21]. We present below a version for the case ∂M = ∅ and take maps in cubical subcomplex domains when k = n. Theorem 2.3. For any Π ∈ [X, Z n,rel (M, ∂M ; M; Z 2 )] ♯ , there exists V ∈ IV n (M ) such that (i) V (M ) = L(Π); (ii) V is stationary in M with free boundary; (iii) V is Z 2 -almost minimizing in small annuli with free boundary. Proof. Using the previous theorem, we can follow the same procedure in the proof of [20,Th. 4.10] (see also [12,Th. 4.21]). To prove that V is Z 2 -almost minimizing in small annuli with free boundary on ∂M, just do as in the proof of [12,Th. 4.21]. We present now an important result that we use in the last section. Corollary 2.4. For p ∈ N and each ǫ > 0, we can find V ∈ IV n (M ) such that (i) ω p (M ) ≤ V (M ) ≤ ω p (M ) + ǫ; (ii) V is stationary in M with free boundary; (iii) V is Z 2 -almost minimizing in small annuli with free boundary. Proof. Note that the results in Section 3.3 of [16] can be extended for compact manifolds (with or without boundary) from the results in Section 2 of [13]. So we can use the results from Section 3.3 of [16]. By definition we can find Φ : X → Z n,rel (M, ∂M ; Z 2 ) a p-sweepout with no concentration of mass such that sup{M(Φ(x)) : x ∈ dmn(Φ)} ≤ ω p (M ) + ǫ. From Th. 3.6 of [16] there exists an (X, M)-homotopy sequence of mappings S = {φ i } i∈N ∈ Π associated. By Th. 3.7 and Cor. 3.9 (ii) of [16] we can extended this sequence to a sequence {Φ i } i∈N of maps continuous in the mass norm and homotopics to Φ in the flat topology for large i. Moreover L(Π) ≤ L(S) = lim i→∞ sup sup{M (Φ i (x)) : x ∈ X} ≤ sup x∈X M (Φ(x)). As Φ is a p-sweepout and Φ i is flat continuous and homotopic to Φ for large i, then Φ i is also a p-sweepout for large i with no concentration of mass by Lemma 3.5 of [16]. Also from Cor. 3.9 (i) of [16] we have that { Φ i } i∈N ∈ P p (M ) for each S = { φ i } i∈N ∈ Π and for large i, where Φ i is the Almgren extension of φ i . Together with the above inequality we conclude that ω p (M ) ≤ L(Π) ≤ sup x∈X M (Φ(x)) ≤ ω p (M ) + ǫ. The remaining items are deduced from the above theorem. One Dimensional Stationary Varifolds In this section we prove some results related to stationary integral 1-varifolds. In particular, we prove some properties of free boundary geodesic networks. When M is the unit disk B 2 = B 1 (0) ⊂ R 2 , or a planar full ellipse E 2 ⊂ R 2 sufficiently close to B 2 , we classify the free boundary finite geodesic networks, provided they are Z 2 -almost minimizing in annuli and have low mass. Also we prove our main theorem about regularity (Theorem 3.12). Free Boundary Geodesic Networks Here we define certain stationary integral 1-varifolds whose support is given by geodesic segments. We follow the notations of Aiex [1]. Definition 3.1. Let U ⊂ M be a relatively open set. A varifold V ∈ IV 1 (M ) is called a (finite) geodesic network in U if there exist geodesic segments {α 1 , . . . , α l } ⊂ int(M ) and {θ 1 , . . . , θ l } ⊂ Z + such that (i) V U = l i=1 v(α i ∩ U, θ i ); (ii) The set of junctions is the set Σ V = l i=1 (∂α i ) ∩ U. Each p ∈ Σ V belongs to a set {α i1 , . . . , α im } for some m = m(p) ∈ Z + , with m ≥ 3 if p ∈ int(M ). If each of those geodesic segments is parameterized by arc-length with initial point p, then m k=1 θ i kα i k (0) = 0, if p ∈ Σ V ∩ int(M ).(1) The varifold V above is called a (finite) free boundary geodesic network in U, if additionally holds m k=1 θ i kα i k (0) ⊥ ∂M, if p ∈ Σ V ∩ ∂M.(2) A junction p ∈ Σ V ∩ int(M ) is said to be singular in int(M ) if there exist at least two geodesic segments with θ i kα i k (0) = −θ i k ′αi k ′ (0), and regular in int(M ) otherwise. In other words, an interior regular junction belong to the interior of each segment that contains it. When p ∈ Σ V ∩ ∂M, we said that it is regular iḟ α i k (0) ⊥ ∂M for every α i k such that p ∈ ∂α i k . A triple junction is a point p ∈ Σ V such that it belongs to exactly three geodesic segments with multiplicity one each. Obviously a triple junctions is not regular in int(M ). We can deduce the following properties as did in [1]: Proposition(i) V is stationary in U ; (ii) Θ 1 (V, x) = m k=1 θ i k 2 for x ∈ m k=1 v(α i k ∩ U, θ i k ); (iii) If Θ 1 (V, x) < 2 for all x ∈ spt V ∩ int(M ), then every p ∈ Σ V ∩ int(M ) is a triple junction; (iv) If Θ 1 (V, x) ≤ 2 for all x ∈ spt V ∩ int(M ), then either Σ V ∩ int(M ) contains a triple junction, or each junction of Σ V ∩ int(M ) is regular and the geodesic segments that define a such junction have multiplicity one each; (v) If Θ 1 (V, x) ≤ 1 for x ∈ spt V ∩ ∂M, then a junction on x is given by a geodesic segment with multiplicity one or two and orthogonal to ∂M, or by two geodesic segments with multiplicity one each and with the same angles with respect to ∂M. Upper Bound for the Density Now we get an upper bound for the density in free boundary (finite) geodesic networks. This is similar to Prop. 3.6 and Th. 3.7 from [1], but with a different approach. Following the notations of the Definition 3.1, Let V ∈ V 1 (M 2 ) and suppose that V U = i,j v(α ij ∩ U, θ ij ) , not necessarily satisfying (1) or (2), and here the density θ ij of each geodesic segment α ij is a positive real number. Supposing that the number of geodesic segments is finite, we call such varifold a generalized finite geodesic network. Denote by J i the i-th junction of V. For M 2 ⊂ R 2 , , , each segment α ij of V is determined by two junctions J i and J j such that |a ij | = |J j −J i .| In each J i we see thatα ij (0) = (J j − J i )/|J j − J i |. Also, in these notations, we have α ij = α ji ,α ij (0) = −α ji (0) and |α ij (0)| = 1. Let x ∈ R 2 such that i θ ijαij (0), J j − x = 0, ∀J j ∈ Σ V ∩ int(M ).(3) Obviously, if V satisfies (1) then it satisfies (3) for all x ∈ R 2 . Lemma 3.2. Let M 2 be a compact region in R 2 with non-empty boundary and V ∈ V 1 (M ) be a generalized finite geodesic network such that it satisfies (3) for some x ∈ R 2 . At each J l ∈ Σ V ∩ ∂M, J l = x, such that i θ ilαil (0) = 0, define F l := i θ ilαil (0) and |F x l | := |F l | cos(φ x l ), where φ x l = ∠(F l , J l − x). Then V (M ) = l |F x l ||J l − x|. In particular, given R > 0, holds (i) V (M ) = R l |F x l | if M 2 = B R (x). (ii) R l |F x l | → V (M ) if the convergence M 2 → B R (x) is smooth. Pre- cisely: given ε > 0, C > 0, then for M 2 sufficiently C ∞ -close to B R (x), we have that V (M ) − R l |F x l | < ε for every free boundary geodesic network V ∈ IV 1 (M ) with V (M ) < C. Proof. Following the above notations, consider the index l such that J l ∈ ∂M. We have that V (M ) = 1 2 i,j θ ij |α ij | = 1 2 i,j θ ij |J j − J i | = 1 2 i,j θ ij ( α ij (0), J j − x + α ji (0), J i − x ) = i,j θ ijαij (0), J j − x = i,l θ ilαil (0), J l − x = l F l , J l − x = l |F x l ||J l − x|. Where we use (3) in the step to restrict the sum to junctions on the boundary. If M = B R (x), then |J l − x| = R for all l such that J l ∈ ∂M. So, V (B R (x)) = R l |F x l |. For M close to B R (x), we have |J l − x| ≈ R for all l such that J l ∈ ∂M. So, V (M ) = l |F x l |(R ± ε l ). Where 0 < ε l < ε 1 and ε 1 → 0 as M → B R (x). Since V (M ) < C, we see by the above expression that l |F x l | < C 1 for some constant C 1 > 0. Therefore, for ε 1 < ε/C 1 , we obtain V (M ) − R l |F x l | ≤ ε 1 l |F x l | < ε 1 C 1 < ε. Theorem 3.3. Let V ∈ IV 1 (B 2 ) be a free boundary geodesic network. Suppose that V (B 2 ) ≤ µ for some positive real number µ, then (i) Θ 1 (V, x) ≤ µ 2 for all x ∈ int(B 2 ) ∩ spt V . (ii) Θ 1 (V, x) < µ 2 √ 2 for all x ∈ ∂B 2 ∩ spt V . Futhermore, let V ∈ IV 1 (M 2 ) be a free boundary geodesic network and 0 < δ < µ such that V (M 2 ) ≤ µ − δ, where M 2 is a compact region of R 2 with strictly convex boundary. Then we can take M sufficiently close to B 2 , whose approximation depends of µ and δ, such that the conclusions (i) and (ii) above are still true for M 2 in place of B 2 . Proof. Consider the case M 2 close to B 2 . We follow the notations above and we fix x ∈ spt V . If x ∈ int(M ), we extend V to a varifold V ∈ V 1 (R 2 ) adding at each J l ∈ Σ V ∩ ∂M the semi-straight line r x l starting at J l with directionα x l (0) := (J l − x)/|J l − x| and multiplicity θ x l := F l , −α x l (0) (see Fig. 1 (a)). By the convexity of M, we note that F l , −α x l (0) > 0, ∀l. If x ∈ ∂M, then x = J k for some J k ∈ Σ V ∩ ∂M. In these case, we extend V to a varifold V ∈ V 1 (R 2 ) adding at each J l ∈ Σ V ∩ ∂M, l = k, the semi-straight line r x l with multiplicity θ x l as before, and at J k we add the semi-straight lines r x ki starting at J k with directions −α ki and multiplicities θ ki , respectively (see Fig. 1 (b)). (a) x 0 M J l r x l B R (x) (b) x r x l J l r x ki α ki M B R (x) 0 Figure 1.U ⊂ R 2 \{J l ∈ Σ V ∩ ∂M : J l = x}, therefore holds (3) for all J j ∈ {Σ V ∩ R 2 }\{J l ∈ Σ V ∩ ∂M : J l = x}. And for each J l ∈ Σ V ∩ ∂M, J l = x, we have i θ ilαil (0) + θ x lα x l (0), J l − x = F l , J l − x − F l , J l − x = 0. Take the smallest R 0 > 0 such that B R (x) ⊃ M for all R ≥ R 0 . For R ≥ R 0 , let r x l (R) = r x l B R (x) and r x ki (R) = r x ki B R (x) . Note that, r x l (R) and r x ki (R) do not intersect M in R 2 \{J l } and in R 2 \{J k }, respectively, since M has strictly convex boundary. (i) Suppose x ∈ spt V ∩ int(M ). For R ≥ R 0 , the monotonicity formula 2 is given by Θ 1 x ( V , R) = V (B R (x)) 2R = V (M ) + l r x l (R)|F x l | 2R . Note that r x l (R)/R → 1 as R → ∞. Thus Θ 1 x ( V , R) → l |F x l |/2 as R → ∞. For C = µ − δ, we can apply the above lemma taking M close to B 2 such that l |F x l | − V (M ) < δ. Therefore, for R large Θ 1 ( V , x) ≤ Θ 1 x ( V , R) < V (M ) + δ 2 ≤ µ 2 . 2 We can define Θ 1 ( V , x) and Θ 1 ( V , x), since these varifolds have first null variation for radial directions with respect to the point x. Also, Θ 1 x ( V , R) and Θ 1 x ( V , R) are non-decreasing (see [23,17]). Where we used the fact that the function Θ 1 x ( V , R) is non-decreasing for x fixed. As x ∈ spt V ∩ int(M ), we have that Θ 1 (V, x) = Θ 1 ( V , x) < µ/2. For the case M 2 = B 2 , we know that l |F x l | = V (B 2 ) and taking C = µ above, we get Θ 1 (V, x) ≤ µ/2. (ii) Suppose now that x ∈ spt V ∩ ∂M. In this case the monotonicity formula is given by Θ 1 x ( V , R) = V (B R (x)) 2R = V (M ) + l =k r x l (R)|F x l | + i r x ki (R)|F x ki | 2R . Where |F x ki | = θ ki . Note that r x ki (R) = R. As above, we taking R → ∞ and we get Θ 1 ( V , x) ≤ l =k |F x l | + i |F x ki | 2 . As i |F x ki | = Θ 1 ( V , x) = 2Θ 1 (V, x), we see that l =k |F x l | ≥ 2Θ 1 (V, x).(4) By the above lemma V (M ) = l =k |F x l ||J l − x|. In the case M = B 2 , note that |J l − x| = 2 cos(φ x l ), since x ∈ ∂M. For M close to B 2 we get that |J l − x| = 2 cos(φ x l ) ± δ l , for all x ∈ ∂M, where 0 ≤ δ l ≤ δ 0 and δ 0 → 0 as M → B 2 . This follows from the compactness of M and the fact that the convergence M → B 2 is smooth. Therefore, for M close to B 2 , V (M ) = l =k |F x l | 2 cos(φ x l ) ± δ l = l =k 2 |F x l | 2 |F l | ± δ l |F x l | ≥ 2 l =k |F x l | 2 l =k |F l | ± δ l l =k |F x l |. Where the inequality follows from the Cauchy-Schwarz inequality. As V is bounded, l =k |F x l | is also bounded by the above lemma. Taking |F 0 l | = cos(φ 0 l )|F l |, we note that cos(φ 0 l ) ≈ 1 for M ≈ B 2 = B 1 (0). By the above lemma, we can take M close to B 2 = B 1 (0) such that l |F 0 l | − V (M ) < δ. Therefore, using (4) in the last inequality, 2 2Θ 1 (V, x) 2 < V (M ) V (M ) + δ ± δ. Where δ ≥ 0 is such that δ → 0 as M → B 2 . Since V (M ) ≤ µ − δ, we can take M close to B 2 , depending of µ and δ, such that Θ 1 (V, x) < µ 2 √ 2 . For the case M 2 = B 2 , note that l |F l | = l |F 0 l | = V (B 2 ) and δ 0 , δ = 0 in the above expressions. Note that the inequality (b) above is not sharp, since in the above proof we use that l =k |F l | < l |F l |. The sharp inequality seems to be Θ 1 (V, x) ≤ µ/4 for x ∈ ∂M ∩ spt V . In fact, we can prove this for µ < 6 assuming that V is Z 2 -almost minimizing in small annuli with free boundary. We do not know any counterexample and we not discuss about this result in this article. Free Boundary Geodesic Networks with Low Mass In the following, we describe the free boundary geodesic networks with low mass and Z 2 -almost minimizing in annuli on the unit ball B 2 , and on full ellipses E 2 sufficiently close to B 2 . We need the following theorem: Theorem 3.4. ([1], Th. 4.13) Given V ∈ IV 1 (M ) a geodesic network with free boundary and p ∈ Σ V ∩ int(M ). If V is Z 2 -almost minimizing in annuli with free boundary at p, then Θ 1 (V, p) ∈ N. For k ≥ 3, let P k be a regular k-sided polygon inscribed in the unit circle. We consider P 2 as a diameter of the unit ball B 2 . Note that two regular k-sided polygons P k and P k inscribed in the unit circle are distinguished by a rotation. More generally, a k-polygon inscribed in a domain Ω is a k-periodic billiard trajectory in Ω, which is a periodic (billiard) trajectory obtained by k reflexions at points of ∂Ω (the angle of incidence equals the angle of reflection). Theorem 3.5. Let V ∈ IV(B 2 ) be a free boundary geodesic network and Z 2 -almost minimizing in annuli with free boundary in B 2 . If 0 < V (B 2 ) < 3 √ 2, then V = P 2 or V = P 2 + P 2 . Proof. From Theorem 3.3 we know that Θ 1 (V, x) < 3 √ 2/2 for x ∈ int(B 2 ) , and Θ 1 (V, x) < 1.5 for x ∈ ∂B 2 . Now using Proposition 3.1 (ii) and Theorem 3.4, we deduce that Θ 1 (V, x) = 1 or 2 for x ∈ int(B 2 ), and Θ 1 (V, x) = 0.5, or 1 for x ∈ ∂B 2 . Therefore, Proposition 3.1 (iv) says that all junctions of V in int(B 2 ) are regular and the geodesic segments from each junction have multiplicity one. Also, Proposition 3.1 (iv) and (v) say that each segment of V has multiplicity one or two and touches ∂B 2 orthogonally, or has multiplicity one and touches ∂B 2 making a reflexion and generating another segment with multiplicity one also. As V (B 2 ) < 3 √ 2, we note that V touches ∂B 2 orthogonally at some point, and we have that V is a diameter (V = P 2 ) or two diameters (V = P 2 + P 2 ) of B 2 . Indeed, if V does a reflexion at some point of ∂B 2 , then it contains a closed k-polygon inscribed in B 2 . A closed k-polygon in B 2 has all the sides with the same length and it is tangent to some circle C k concentric with ∂B 2 (see Fig. 2 (a), (b) and (c)), then the perimeter is at least |C k |. Each polygon P k gives a unique turn around C k . From five reflexions, we can have non-convex closed polygons as in the Fig. 2 (b). If P 3 ⊂ V, then we have that V (B 2 ) > 3 √ 2, and if P k ⊂ V, for k ≥ 4 , we observe that: if the radius of C k is bigger than 0.7, then the perimeter of a closed k-polygon is bigger than 2 · 0.7π > 3 √ 2. Otherwise, if the radius of C k is less or equal to 0.7 (see Fig. 2 (d)), then each side of the closed k-polygon is bigger than 1.4, and so the perimeter is bigger than 4 · 1.4 > 3 √ 2. Therefore, V does not contain a closed k-polygon. Let E 2 be a planar full ellipse E 2 . We denote by P E k , for k ≥ 3, the closed convex k-polygon (not necessary regular) inscribed in E 2 . Here we consider P E 2 as the smallest or the largest diameter of E 2 . As we see below, the smallest and the largest diameter of E 2 are the only 2-polygons inscribed in E 2 . The polygons P E k are examples of closed periodic billiard trajectories in ellipses (Poncelet polygons). We see more properties of these polygons in the proof below. For instancy, given a point A ∈ ∂E 2 and k ≥ 3, there exists a unique P E k that passes through A (see the proof below). So, for a fixed integer l ≥ 3, is not difficult (a) (b) (c) (d) Figure 2. to see that if E 2 is close to B 2 , then a k-polygon P E k is close to some P k , for 3 ≤ k ≤ l. In fact, the boundary of E 2 is given by an ellipse x 2 /a 2 + y 2 /b 2 = 1 and a point A 0 ∈ ∂E 2 is given in polar coordinates by A 0 = (a cos(t 0 ), b sin(t 0 )), for some t 0 ∈ [0, 2π). For t 0 fixed and E 2 → B 2 , we take the k-polygons P E k that pass through A 0 , defined by the points A 0 , A 1 , · · · , A k−1 ∈ ∂E 2 , and such that |P E k | = |A 0 A 1 | + · · · + |A k−2 A k−1 | + |A k−1 A 0 |. Let O = (0, 0) ∈ R 2 , the segments OA i tend to be perpendicular to ∂E 2 , as a, b → 1. Since at A i the angle of incidence is equals the angle of reflection, we see 3 that ∠(OA i A i−1 ) ≈ ∠(OA i A i+1 ). Moreover, |A j O| ≈ 1 for all j, so |∠(A i OA i−1 ) − ∠(A i OA i+1 )| < ε, for small ε > 0 such that ε → 0 as E 2 → B 2 . In particular, |2π/k − ∠(A j OA j+1 )| = | k−1 i=0 ∠(A i OA i+1 )/k − ∠(A j OA j+1 )| < kε for all j. Finally, as ∂E 2 → ∂B 2 , we conclude that these k-polygons P E k tend to the regular k-polygon P k that passes through the point (cos(t 0 ), sin(t 0 )) ∈ ∂B 2 . Theorem 3.6. Let E 2 be a planar full ellipse and 0 < R < 3 √ 2 be a real number. For E 2 sufficiently close to B 2 , depending only on the parameter R, the following is true: if V ∈ IV 1 (E 2 ) is a free boundary geodesic network such that it is Z 2almost minimizing in annuli with free boundary in E 2 and 0 < V (E 2 ) < R, then V = P E 2 or V = P E 2 + P E 2 . Proof. Consider E 2 a planar full ellipse which boundary is given by an ellipse x 2 /a 2 + y 2 /b 2 = 1 for a > b with foci F 1 , F 2 ∈ Ox (see Fig. 3 (a)). Let d and D be the values of the smallest and largest diameters of E 2 , respectively. So, d = 2b and D = 2a. Also, here we are always considering E 2 sufficiently close to B 2 , so d ≈ D ≈ 2, for example. Let C = R = 3 √ 2 − δ, for some δ > 0. We take E 2 ≈ B 2 as in the Theorem 3.3 and, as in the proof of the theorem above, applying Proposition 3.1 (iv) and Theorem 3.4 to get: all junctions of V in int(E 2 ) are regular and the geodesic segments from each junction have multiplicity one; each segment of V has multiplicity one or two and touches ∂E 2 orthogonally, or has multiplicity one and touches ∂E 2 making a reflexion and generating another segment with multiplicity one also. Therefore, V can be the smallest or the largest diameters of E 2 , since they touch ∂E 2 orthogonally (see Fig. 3 (a)). Also, V can be P E 2 + P E 2 , and then V (E 2 ) = 2d, d + D or 2D, since d ≈ D ≈ 2 and V < R < 3 √ 2. We could have V as in the Fig. 3 (b): a segment touching ∂E 2 orthogonally at A 1 , making a reflexion at (0, b) ∈ ∂E 2 with respect to ∂E 2 and generating another segment, which touches orthogonally ∂E 2 at A 2 = (−x(A 1 ), y(A 1 )). This can happen for a >> b. However, for E 2 close to B 2 we have a, b ≈ 1, and the cases V = P E 2 or V = P E 2 + P E 2 are the only possibilities such that V touches ∂M orthogonally in some point with V (E 2 ) < R. Indeed, let (a cos(t), b sin(t)) be the polar coordinates on ∂E 2 for t ∈ [0, 2π), and take without loss of generality (by symmetry) A ∈ ∂E 2 such that A = (a cos(t A ), b sin(t A )) for t A ∈ (3/4π, 2π). We claim that if a segment AB ⊂ E 2 touches ∂E 2 orthogonally at A, then AB is not orthogonal to ∂E 2 at B ∈ ∂E 2 , and the segment BC, reflexion of AB at B, is also not orthogonal to ∂E 2 at C (see Fig. 3 (c)). F 1 F 2 a b F 1 F 2 O O F 1 F 2 A B C I D (a) (b) (c) A 1 A 2 Figure 3. In fact, the equation of the straight line which is perpendicular to ∂E 2 at A is given by y = a tan(t A ) b x + sin(t A ) b − a 2 b . If AB is orthogonal to ∂E 2 at B = (a cos(t B ), b sin(t B )), the equation of the straight line through B is similar to above, this implies that tan(t A ) = tan(t B ) and sin(t A ) = sin(t B ), which is impossible, since t A = t B . So, AB is not orthogonal to ∂E 2 at B and there exists BC, reflexion of AB at B. Note that a 2 ≤ 2b 2 , since a, b ≈ 1. So, for x = 0 in the equation above, we see that 0 < y(I) < b, where I is the intersection of AB with Oy (Fig. 3 (c)). In an ellipse we have the following fact: if AB is orthogonal to ∂E 2 at A, then AB bisects the angle ∠F 1 AF 2 . In particular, AB passes through F 1 F 2 and, since 0 < y(I) < b, we have t B ∈ (π/2, π). Remember from billiard theory in ellipses that, if a segment in E 2 passes through F 1 F 2 , then all the segments in that billiard trajectory (segments reflected at ∂E 2 ) pass through F 1 F 2 (see for example [11,Th. 4]). So BC passes through F 1 F 2 . Supposing that BC is orthogonal to ∂E 2 at C, the same argument applied for AB can be applied to BC to get that t C ∈ (3/4π, 2π) and t C = t A , where C = (a cos(t C ), b sin(t C )). Taking the equations of the straight lines that are perpendicular to A and C, respectively, we would have that they intersect at B = (a cos(t B ), b sin(t B )), then a 2 b cos(t B )(tan(t A ) − tan(t C )) + b 2 − a 2 b (sin(t A ) − sin(t C )) = 0. As t A , t C ∈ (3/4π, 2π), t A = t C and cos(t B ), (b 2 − a 2 ) < 0, the left side of the last expression above is not equal to zero. Then, BC is not perpendicular to ∂E 2 at C and there is another reflexion CD at C (see Fig. 3 (c)). Consider E 2 ≈ B 2 such that each segment in E 2 through F 1 F 2 has length at least R/3, since the length of each of these segments tending to 2 as E 2 tends to B 2 and R < 3 √ 2. By the above arguments, if V = P E 2 and V = P E 2 + P E 2 , then V has at least three segments, none of them is orthogonal to ∂E 2 and neither passes through F 1 F 2 . So, V contains a closed k-polygon P k , and moreover each segment is tangent to the same ellipse ∂(E k ), where E k is a planar full ellipse inside of P k and with the same foci of E 2 (see [11,Th. 4]). For simplicity, we just say that P k is tangent to ∂(E k ) The Poncelet theorem (see for instance [22,Th. 4]) says that if a closed kpolygon P k is tangent to ∂(E k ), then any other polygon Q that is tangent to ∂(E k ) is also a closed k-polygon with the same perimeter of P k . Moreover, for each k ≥ 3 there exists a unique E k such that all the convex closed k-polygons P E k have its trajectory tangent to ∂(E k ) (see for example [19,Section 4]). In particular for a fixed k ≥ 3, all the polygons P E k have the same perimeter. Note that, given A ∈ ∂E 2 there is a unique P E k through A for each k ≥ 3. Indeed, just take the billiard trajectory starting at A and tangent to ∂E k . Also, note that |∂(E k )| < |∂(E k+1 )| since the tangency property of the polygons and the strict convexity of the ellipse ∂(E) (see Fig. 4 (a)). We can take E 2 ≈ B 2 such that |P E 3 | ≈ |P 3 |, in particular V does not contain F 1 F 2 F 1 F 2 F 1 F 2 (a) (b) (c) A E k E k+1P E 3 , since |P 3 | > 3 √ 2. And if V contains a closed k-polygon for k ≥ 4, we argue as in the proof of the theorem above. Indeed, the estimates in the accounts of the theorem above are strict, so for E 2 ≈ B 2 and replace C k by E k with average radius approximately 0.7, we conclude that the perimeters are bigger than 3 √ 2. Compare the Fig. 2 (d) and 4 (c). In the Fig. 4 (b) we have an example of a closed non-convex 5-polygon. Replacement and Regularity The regularity of stationary integral 1-varifolds for open sets was proven by Allard and Almgren ([2], Section 3). As noted by Aiex ([1], Th. 3.5), the regular structure described in [2] is exactly our definition of geodesic network. Precisely: By Proposition 2.1 (iii), the definition above is equivalent if we take Fermi halfballs B + rp (p) instead of Euclidean balls B rp (p) restricted to M. The following theorem is about replacements of almost minimizing varifolds, which is one of the most important properties of this kind of varifolds. Roughly speaking, we can replace an almost minimizing varifold V by another almost minimizing varifold V * , which has better regularity properties. [20] to get (iv) from (iii). V * ∈ V k (M ) such that (i) V * (M \K) = V (M \K); (ii) V * (M ) = V (M ); (iii) V * is Z 2 -almost minimizing in U with free boundary; (iv) V * ∈ IV(U ∩ int(M )); (v) V * U = lim i→∞ |T i | as varifolds for some {T i } ∈ Z k,rel (M, (M \U ) ∪ ∂M ; Z 2 ) such that each T 0 i is locally mass minimizing in int M (K The varifold V * in the above theorem is called of a replacement of V in K. See Section 2.1 to remember the notation of T 0 i above. In the next lemma we prove a weak regularity of V * ∈ V 1 (M ) for manifolds with strictly convex boundary. Lemma 3.9. (Weak Regularity of Replacements) Under the same hypotheses of Theorem 3.8, assume that ∂M is strictly convex and take V a one-dimensional varifold. Then spt V * ∩ int M (K) is a free boundary geodesic network (possibly infinite) without junctions in (K ∩ int(M ))\∂ rel K, such that each geodesic segment has endpoints in ∂ rel K ∪ ∂M, and they can touch ∂M ∩ int M (K) only orthogonally. Proof. From [1,Prop. 4.6] we know that if T is a one-cycle that is locally mass minimizing in an open set W ⊂ int(M ) and Z ⊂ W is compact, then T Z is a geodesic network (finite) such that each geodesic segment has endpoints in W \Z and those segments do not intersect each other. So, for a relatively compact K ⊂ M and T 0 i locally mass minimizing in int M (K) (as in Theorem 3.8, (v)), we have that T 0 i int M (K) is given by geodesic segments not intersecting each other, all segments have endpoints in U \K, and each segment that touches ∂M ∩int M (K) is orthogonal to ∂M, in particular |T 0 i | int M (K) is a free boundary geodesic network (possibly infinite). Indeed, as T 0 i is locally mass minimizing, each segment of T 0 i that touches ∂M is locally the shortest path, so it is orthogonal to ∂M. In the proof of Theorem 3.8 we have that M(T i ) is uniformly bounded, so we can use Th. 6.1 from [9] to get that the limit in the Theorem 3.8 (v) is smooth (after passing to a subsequence) for this lemma, then V * is given by geodesic segments such that each segment has endpoints in ∂ rel K ∪ ∂M, and they can touch ∂M ∩ int M (K) only orthogonally. The last one follows from the fact that ∂M is strictly convex, thus geodesic segments can touch ∂M only in its endpoints. Also, as each T i only can touch ∂M ∩ int M (K) orthogonally, therefore the same happens in the limit. Finally, as the segments of each T i do not intersect each other, we have that in the limit we do not have junctions. We called the result above as weak regularity, because we do not know if the number of geodesic segments could be infinite. However, the above lemma is true for any codimension. Let p ∈ R 2 and let C ∈ V 1 (R 2 ) be a varifold such that C = l i=1 v(r i , m i ) for some l, m 1 , · · · , m l ∈ N, and each r i is some semi-straight line from p. We say that C is a cone with vertex at p. The next Lemma is very important to prove our main result about regularity. Essentially, we use it to glue replacements on overlapping annuli (see Step 2 in the proof of Theorem 3.12). Lemma 3.10. Let C ∈ IV 1 (R 2 ) be a stationary cone with vertex at the origin 0 ∈ R 2 , and such that it is Z 2 -almost minimizing in B 2 (0) ⊂ R 2 . Then C = v(r, m), for some r a straight line passing through the origin 0, and for some m ∈ N. Proof. We use the following fact: if C is Z 2 -almost minimizing in B 2 (0), then each varifold tangent is also a stationary integral varifold on T x R 2 ≡ R 2 such that it is By Theorem 3.4 we have that Θ 1 (C, 0) = k for some k ∈ N. We prove the result by induction on Θ 1 0 (C, ∞). Indeed, the result is obvious for Θ 1 0 (C, ∞) ≤ 1. Suppose that Θ 1 0 (C, ∞) = k + 1, and that the result is true for Θ 1 0 (C, ∞) ≤ k, k ≥ 1. Let C * be a replacement of C on B 1 (0), we know that C * is integral, stationary and Z 2 -almost minimizing in B 2 (0). Also, C * (B 2 (0)) = C (B 2 (0)), C * (B 2 (0)\B 1 (0)) = C (B 2 (0)\B 1 (0)), and together with the monotonicity formula we get Θ 1 y (VarTan(C * , y), ∞) = Θ 1 (C * , y) ≤ lim ρ→∞ Θ 1 y (C * , ρ) = Θ 1 0 (C, ∞), where y ∈ ∂B 1 (0) ∩ spt C * . We have two cases: Θ 1 (C * , y) = lim ρ→∞ Θ 1 y (C * , ρ) for some y ∈ ∂B 1 (0) ∩ spt C * , or lim ρ→∞ Θ 1 (C * , y) < Θ 1 y (C * , ρ) for any y ∈ ∂B 1 (0) ∩ spt C * . In the first case, C * is a cone with vertex at y. This implies that C = v(r y , m), for some m ∈ N and r y is the straight line that passes through y and the origin, since C * (B 2 (0)\B 1 (0)) = C (B 2 (0)\B 1 (0)). In the second case, Θ 1 x (VarTan(C * , y), ∞) ≤ k for any y ∈ ∂B 1 (0) ∩ spt C * , since Θ 1 0 (C, ∞) = k + 1. So, as VarTan(C * , y) is Z 2 -almost minimizing in B 2 (0), we can use the induction hypothesis for each y to get that VarTan(C * , y) = v(r y , m y ) for some m y ∈ N and r y is the straight line that passes through y and the origin. Using that C * (B 2 (0)\B 1 (0)) = C (B 2 (0)\B 1 (0)), we conclude C = v(r, m) for some m ∈ N, and for some straight line r through the origin. The next result is a boundary maximum principle for stationary varifolds with free boundary in codimension one case. (i) ∂ rel N meets ∂M orthogonally, if ∂ rel N ∩ ∂M = ∅; (ii) N is relatively strict convex in M ; (iii) spt V ⊂ N . Then we have spt V ∩ ∂ rel N = ∅. Now we prove our main theorem about regularity of stationary Z 2 -almost minimizing varifolds with free boundary. Theorem 3.12. Let M 2 be a compact Riemannian manifold with non-empty strictly convex boundary. If V ∈ IV 1 (M ) is a stationary varifold with free boundary such that it is integral in M and Z 2 -almost minimizing in small anulli with free boundary, then V is a free boundary geodesic network. Proof. Here we follow similarly to the proof of [12,Th. 5.2] and [4,Prop. 6.3], with the necessary modifications. Given p ∈ spt V ∩int(M ), we know by the Theorem 3.7 that in a small compact neighborhood around p we have that V is a geodesic network. So, assume that p ∈ spt V ∩ ∂M and fix r > 0 such that r < 1 4 min{r F ermi , r am (p), r ort (p)},(5) where r am (p) > 0 is such that V is Z 2 -almost minimizing in A s,t (p) with free boundary for all 0 < s < t < r am , and r ort (p) > 0 is such that two distinct geodesics that are orthogonal to ∂M ∩ B + δ (p) do not intersect each other in B + δ (p) for all 0 < δ < r ort (p). Note that, as a consequence of the maximum principle (Theorem 3.11), we have the following: if W ∈ V 1 (M ) is stationary in B + r (p) with free boundary for p ∈ spt W ∩ ∂M and r as above, then spt W ∩ S + t (p) = ∅ for all 0 < t ≤ r. In fact, there exists t 0 ∈ (0, r] the smallest number such that spt W B + r (p) ⊂ Clos B + t0 (p) . By the maximum principle we have that spt W B + r (p) ∩ S + t0 (p) = ∅, then spt W B + r (p) ⊂ Clos B + t1 (p) for some 0 < t 1 < t 0 , which is contradiction. Using the same argument and suppose only that W = 0 in B + r (p) for some p ∈ spt W ∩ ∂M, we conclude that there exists 0 < t < r such that spt W ∩ S + t (p) = ∅ for all 0 < t < t ≤ r.(7) Step 1: Constructing successive replacements on two overlapping annuli. Fix any 0 < s < t < r. As r < (1/4)r am and V is Z 2 -almost minimizing in A s,ram/2 (p) with free boundary for all 0 < s < t < r am /2, we can use the Theorem 3.8 to get a first replacement V * of V on K = A s,t (p). The Lemma 3.9 says that Σ 1 := spt V * ∩ A s,t (p) is a free boundary geodesic network (possibly infinite). By Theorem 3.8 (iii) we have that V * is still Z 2 -almost minimizing in A s,ram/2 (p) with free boundary for all 0 < s < t < r am /2, so we can apply again the Theorem 3.8 to get a second replacement V * * of V * on K = A s1,s2 (p) for 0 < s 1 < s < s 2 < t. Again, Σ 2 := spt V * * ∩ A s1,s2 (p) is a free boundary geodesic network (possibly infinite). Let us consider the following choices: we fix any s 1 ∈ (0, s), and we choose s 2 ∈ (s, t) such that VarTan(Σ 1 , x) is a straight line transversal to S + s2 (p) for all x ∈ ( S + s2 (p)\∂M ), and (α∩ S + s2 (p))\∂M = ∅ for every geodesic segment α ∈ Σ 1 . Indeed, fixing s 2 ∈ (s, t), we know by the regularity of replacements (Lemma 3.9) that VarTan(Σ 1 , x) is a straight line for any x ∈ A s,t (p). Also, we have only a finite number of geodesic segments {α i } ⊂ Σ 1 in A s, t (p) for any 0 < s < t < t. To see the last one, note that any geodesic segment (with possible multiplicity) α i ∈ Σ 1 ∩ A s,t (p) has to touch S + t (p). Indeed, by the Lemma 3.9 each α i has to touch S + s (p) ∪ S + t (p) ∪ (∂M ∩ A s,t (p)) and it can touch ∂M ∩ A s,t (p) only orthogonally. Using that any two orthogonal geodesic segments to ∂M do not intersect each other in B + r (p), together with the fact that S + s (p) is strictly convex and orthogonal to ∂M, we conclude that if α i ∈ Σ 1 touches ∂M ∩A s,t (p), then α i ∩ S + s (p) = ∅ only if α i touches S + s (p)∩∂M (see Fig. 5). Also, if α i does not touch S + t (p), then its endpoints cannot be on ∂M ∩ A s,t (p), because α i would be a stationary varifold with free boundary, contradicting (7). Then, any α i that touches S + s (p) or ∂M ∩ A s,t (p), should touch S + t (p). Therefore, if there is an infinite number of geodesic segments {α i } ⊂ Σ 1 in A s, t (p), then there are an infinite number of geodesic segments from S + t (p) to S + t (p), contradicting the fact that Σ 1 has finite mass. Thus the set {α i } is finite. Finally, using again the strict convexity of S + s2 (p), each geodesic segment that is tangent to S + s2 (p) cannot touch S + s2 (p) for all 0 < s 2 < s 2 . So, by the finiteness of the geodesic segments and by (6), we can choose s 2 ∈ (s, t) as requested (see Fig. 5). p S + t (p) S + s2 (p) S + s1 (p) S + s (p) ∂M Σ 1 Figure 5. Note that each α i ⊂ Σ 1 has to touch S + t (p) at points in int(M ), since S + t (p) is orthogonal to ∂M. Step 2: Gluing Σ 1 and Σ 2 across S + s2 (p). As before, any geodesic segment (with possible multiplicity) β i ∈ Σ 2 ∩ A s,s2 (p) has to touch S + s2 (p) in points belonging to int(M ). Since V * * is stationary and integral in A s1,t (p), we have by the interior regularity (Theorem 3.7) that each x ∈ spt V * * ∩ int(M ) ∩ A s,t (p) belongs to a finite number of geodesic segments (including multiplicity). In particular, if x ∈ spt V * * ∩ int(M ) ∩ S + s2 (p) then x belongs to Σ 1 ∩ Σ 2 , since each geodesic segment of Σ 1 touches S + s2 (p) transversally. So, Σ 1 and Σ 2 glue continuously across S + s2 (p). Note that spt V * * ∩ S + s2 (p) = Σ 1 ∩ S + s2 (p) = Σ 2 ∩ S + s2 (p) ⊂ int(M ). Moreover, as VarTan(V * * , x) is a cone satisfying Lemma 3.10, we see that the gluing is actually C 1 (smooth) since VarTan(V * * , x) is a straight line (with possible multiplicity). Step 3: Unique continuation up to the point p. By Step 2 and property (i) of Theorem 3.8, we can extend Σ 2 to Σ 2 in A s1,t (p) such that Σ 2 = Σ 1 on A s,t (p), Σ 2 is given by geodesic segments possibly with multiplicity and without interior junctions that can touch A s1,t (p) ∩ ∂M only orthogonally, Σ 2 A s,s2 (p) has a finite number of geodesic segments, and each geodesic segment of Σ 2 has to touch S + t (p). Using (6), we can continue to take replacements in this way for all 0 < s 1 < s. For each 0 < s 1 < s as before, denote Σ 2 by Σ s1 . If 0 < s ′ 1 < s 1 < 0, then we have that Σ s ′ 1 = Σ s1 on A s1,t (p). Thus Σ := 0<s1<s Σ s1 in B + t (p) is given by geodesic segments possibly with multiplicity and without interior junctions that can touch ∂M ∩ ( B + t (p)\{p}) orthogonally only, and each geodesic segment of Σ has to touch S + t (p). Moreover, Σ B + t (p) has a finite number of segments for all 0 < t < t (see Fig. 6). Claim: spt V = Σ in the punctured ball B + s (p)\{p}. Proof of Claim: Consider the set T V p = y ∈ spt V : VarTan(V, y) is a straight line or a semi-straight line transversal to S + rp(y) (p) . As in [12,Claim 3,p. 42 ], we can use the convexity of small Fermi half-balls to apply a first variation argument as in [4,Lemma B.2], getting that the set T V p is a dense subset of spt V ∩ B + s (p). Note that the Lemma B from [4] holds, in the respective hypotheses, for V ∈ IV n (M n+1 ), ∀n > 0. Given y ∈ T V p ∩ ( B + s (p)\{p}), let ρ = r p (y). Take V * the replacement of V in A s,t (p) and V * * the replacement of V * in A ρ,s2 (p) for s 2 ∈ (s, t) chosen as in Step 1. By the property (i) from Th. 3.8, we have V * * = V * = V in B + ρ (p), then y ∈ spt V ∩ B + ρ (p) ∩ S + ρ (p) = spt V * * ∩ B + ρ (p) ∩ S + ρ (p). Since spt V * * = Σ in A ρ,t (p) and VarTan(V * * , y) is transversal to S + ρ (p), we have by (6) and above that y ∈ Σ. Thus, T V p ∩ ( B + s (p)\{p}) ⊂ Σ, and hence spt V ∩ ( B + s (p)\{p}) ⊂ Σ. The last one is deduced using that T V p is a dense subset of spt V ∩ B + s (p), and the fact that Σ is compact in B + s (p). To see the converse inclusion Σ ⊂ spt V in B + s (p), note that by the Constancy Theorem [23, Th. 41.1], we have spt V ∩ ( B + s (p)\{p}) = Σ in M \∂M. For y ∈ Σ ∩ ∂M ∩ ( B + s (p)\{p}), we know that VarTan(Σ, y) is a straight line perpendicular to T y (∂M ), which implies that y is a limit point of Σ ∩ int(M ) and thus y ∈ spt V . Therefore, spt V ∩ ( B + s (p)\{p}) = Σ. Step 4: V is a free boundary geodesic network From the interior regularity (Theorem 3.7) and the Step 3, V is a geodesic network (finite) in B + s (p) and a free boundary geodesic network (finite) in ( B + s (p)\{p}). In particular, Θ 1 (V ∂M, p) = 0. So, if there exist geodesic segments at p, as in the Fig. 6, then those segments must satisfy (2), and then V is a free boundary geodesic network (finite) in B + s (p). Varying p ∈ spt V ∩∂M, we see that V is a free boundary geodesic network (not necessarily finite) on M. Given any compact K ⊂ int(M ), the interior regularity says that V K has a finite number of geodesic segments. So, we only need to find a compact K ⊂ int(M ) such that V (M \K) has also a finite number of geodesic segments. Indeed, take a cover of spt V ∩ ∂M, by small open balls B + s (p) as in the previous steps, extract a finite cover { B + j (p j )} l j=1 , and define K := M \ l j=1 B + j (p j ) . Note that K can be empty. This finishes the proof. The Width of a Full Ellipse In this section we prove our main theorem about p-widths: we calculate the first p-widths of B 2 and E 2 , where E 2 is a planar full ellipse C ∞ -close to B 2 . As in [1], we take the p-sweepouts from Guth [10, Section 6]. We consider some adaptations to get a convenient upper bound for the mass of the cycles. Also, we need to take a better estimate than that given by the Cauchy-Crofton Formula. Indeed, to calculate the widths of the unit sphere in [1], the Cauchy-Crofton Formula gives a sharp estimate, which does not happen in our case. A Sweepout for B 2 The sweepout that we use to calculate the p-widths is obtained by a map whose image is given by real algebraic varieties. The properties of this map can be found in Guth [10,Section 6]. Let Q i : R 2 → R denote the following polynomials for i = 1, . . . , 4 : Q 1 (x, y) = x, Q 2 (x, y) = y, Q 3 (x, y) = x 2 and Q 4 (x, y) = xy. Also, put A p = span ({1} p i=1 Q i ) \{0} and define the relation Q ∼ λQ, for λ = 0 and Q ∈ A p . The quotient (A p , ∼) can be identified with RP p and by this identification we can define the map F p : RP p → Z 1,rel (B 2 , ∂B 2 ; Z 2 ), which send a class [Q] to the real algebraic variety defined by Q(x, y) = 0 restricted to B 2 , considered as a mod 2 relative Lipschitz cycle. As proved in [10, Section 6], F p is a flat continuous map and it defines a p-sweepout. In the next lemma we use the Cauchy-Crofton formula to prove that F p has no concentration of mass, thus F p ∈ P p (B 2 ). Proof. Without loss of generality, consider P 0 = (p 0 , 0) ∈ B 2 for p 0 ≥ 0, and the ball B s (P 0 ) for s > 0 sufficiently small. Fixing [Q] ∈ RP p and recall that every straight line r in the plane can be parameterized by the equation x cos(θ) + y sin(θ) = ρ, where ρ is the distance from r to the origin and θ ∈ [0, 2π) is the angle between the axis Ox and the straight line that is perpendicular to r and passes through the origin. Denote a such straight line by r ρ,θ and let n(ρ, θ) be the number of intersection points (with multiplicity) of the straight line r ρ,θ with F p ([Q]) in B s (P 0 ). If p 0 > 0, note that for θ ∈ [0, π/2 − sin −1 (s/p 0 )] the straight line r ρ,θ intersects B s (P 0 ) if and only if ρ ∈ [p 0 cos(θ) − s, p 0 cos(θ) + s] (see Fig. 7 (a)). On the other hand, if θ ∈ (π/2 + sin −1 (s/p 0 ), π], then r ρ,θ does not intersect B s (P 0 ) ∩ B 2 for all ρ (see Fig. 7 (b)). And for θ ∈ (π/2 − sin −1 (s/p 0 ), π/2 + sin −1 (s/p 0 )), the straight line r ρ,θ does not intersect B s (P 0 ) if ρ > 2s. For p = 1, . . . , 4 note that if F p ([Q]) is degenerate or the intersection to B 2 of a straight line, or two straight lines, then F p ([Q]) ≤ 4. Also, this estimate holds when F p ([Q]) is the restriction to B 2 of a hyperbola H such that each branch intersects B 2 . Indeed, if we take B r (0) for r → ∞, we note that this hyperbola intersects ∂B r (0) in exactly four distinct points for all r > 1. In particular, as the diagonally opposite arms tend to the respective asymptote of those arms, we see that for r large the intersection of the two asymptotes is inside of B r (0) and each asymptote intersects ∂B r (0) in two points z, z ′ (see Fig. 8). Given a point w ∈ H ∩ ∂B r (0) in a arm of a branch, take the respective half asymptote and consider the distance |wz|, where z is the intersection between this half asymptote and ∂B r (0). Let ε be the sum of the four distances given by the four points in H ∩∂B r (0). Note that the length L(H ∩B r (0)) of this hyperbola restricted to B r (0) is less than the length of the two asymptotes restricted to B r (0) added with ε, so L(H ∩ B r (0)) < 4r + ε. Now, decreasing r to r − s, s ∈ (0, r − 1], we note that the total reduction of length of the branches is at least 4s, since there exist four points in H ∩ ∂B 2 during the reduction r → 1 + . Therefore L(H ∩ B 2 ) < 4 + ε. Note that ε → 0 as r → ∞, since each arm of the branches tends to their respective half asymptote. So, starting the reduction for r as large as we want, we get that L(H ∩ B 2 ) < 4 + ε for all ε > 0, that is, L(H ∩ B 2 ) ≤ 4. In the other cases, hyperbolas with a unique branch intersecting B 2 or parabolas intersecting B 2 , we prove in the Appendix that the maximum length of these curves Proof. (i) Let p = 1, 2 and take the p-sweepout F p ∈ P p (B 2 ). By Lemma 4.2 we know that F p ([Q]) ≤ 2 for all [Q] ∈ RP p , thus ω 1 (B 2 ), ω 2 (B 2 ) ≤ 2. Now, given ǫ > 0 we can find by the Corollary 2.4 a special varifold V such that 0 < ω p (B 2 ) ≤ V (B 2 ) ≤ ω p (B 2 ) + ǫ ≤ 2 + ǫ. By Theorems 3.12 and 3.5 we actually have that V is a diameter of B 2 and V (B 2 ) = 2. Therefore, ω 1 (B 2 ) = ω 2 (B 2 ) = 2. (ii) We observe that ω 3 (B 2 ) > 2 as a consequence of the Lusternik-Schnirelmann theory (see for instancy Guth [10], p. 1923-24). Indeed, we can take three disjoint closed balls B i in B 2 \∂B 2 with radius 0.4 each ball. Each 3-sweepout Φ of B 2 is also an 1-sweepout of B 2 , in particular it is an 1-sweepout of each B i . The Lusternik-Schnirelmann theory says that Φ contains a cycle such that its mass is at least the sum of the first width of each B i . By the item (i) above we know that the first width of a ball is equal to the diameter of that ball, so ω 3 (B 2 ) ≥ 3 × 0.8 > 2. Using Theorem 3.5 we get that ω 3 (B 2 ) ≥ 4 (two diameters). Now, Lemma 4.2 says that 4 ≤ ω 3 (B 2 ), ω 4 (B 2 ) < 4.003, and so by Corollary 2.4, Theorems 3.12 and 3.5, we actually have that ω 3 (B 2 ), ω 4 (B 2 ) = 4. (iii) For p = 1, 2 and E 2 close to B 2 , we deduce by continuity and Lemma 4.2 that ω p (E 2 ) ≤ 2 + δ for some small δ > 0. Therefore, by Corollary 2.4, Theorems 3.12 and 3.6, we conclude that the only possible values for ω 1 (E 2 ) and ω 2 (E 2 ) are d or D. Suppose that ω 1 (E 2 ) = D for some E 2 . As a 2 ≤ 2b 2 (notation in the proof of Theorem 3.6), we can take two small ellipses defined by scaling E 2 by half, taking a π/2 rotation and translating the variable x by +b/2 and −b/2, respectively. These two ellipses are inside of int(E 2 ) and, as ω 1 (E 2 ) = D, we have that the 1-width of each small ellipse is equal to D/2. Using the Lusternik-Schnirelmann theory as before and the fact these two ellipses are inside of int(E 2 ), we conclude that ω 2 (E 2 ) > D/2 + D/2 = D, which contradicts the fact that ω 1 (E 2 ), ω 2 (E 2 ) ∈ {d, D}. So, ω 1 (E 2 ) = d. Applying the same argument for ω 1 (E 2 ) = d, we get that ω 2 (E 2 ) > d. Therefore, ω 2 (E 2 ) = D. (iv) We use again the continuity, Corollary 2.4, Theorems 3.12 and 3.6 to conclude that the only possible values to ω 3 (E 2 ) and ω 4 (E 2 ) are 2d, d + D or 2D. As γ is convex downward and ∂B 2 ∩ {(x, y) ∈ R 2 : y > 0} is convex upward, we conclude that there exist at most two points A, D ∈ γ ∩ ∂B 2 such that y(A), y(D) > 0. So, consider two cases: there exist two points A, D ∈ γ ∩ ∂B 2 such that y(A), y(D) > 0; or there exists at most one such point. In the first case, as in the examples of the Fig. 9 (a), take B = (x(A), −y(A)), C = (x(D), −y(D)) ∈ ∂B 2 , and the circular arcBC. The length L(γ ∩ B 2 ) is at most |AB| + |BC| + |CD|. Let α (resp. β) be the angle between OA (resp. OD) and x-axis for α, β ∈ (0, π/2], then L(γ ∩ B 2 ) ≤ |AB| + |CD| + |BC| ≤ 2 sin(α) + 2 sin(β) + π − (α + β). In the second case, as in the example of the Fig. 9 (b), where do not exist A or D as in the first case, we take α = 0 or β = 0 in the above estimate, respectively. Without loss of generality suppose β = 0. As α ∈ [0, π/2], we get L(γ ∩ B 2 ) ≤ 2 sin(α) + π − (α) ≤ 2 √ 3 2 + 2π 3 < 4.(8) So, from now consider that there exist two points at the intersection between γ ∩ B 2 and the upper half plane H + := {(x, y) ∈ R 2 : y > 0}. Suppose that V / ∈ B 2 and γ ∩ B 2 is connected. Let A ∈ ∂B 2 be first point of contact between γ and B 2 , and D ∈ ∂B 2 be last point of contact, x(A) < x(D). Thus, γ ⊂ B 2 between the points A and D. In particular, x(V ) < x(A) or x(V ) > x(D). Because of symmetry, we can assume without loss of generality that x(V ) < x(A). So γ ∩ B 2 is strictly increasing, in particular y(A) < y(D). Note that γ is contained within the triangle △ADE between the points A and D, where E = (x(D), y(A)). Also, by the conditions on the points A and D, we see that the intersection △ADE ∩ γ ∩ B 2 ∩ H + is empty, or is the point D, or are the points A and D. Therefore, by the assumption of the previous paragraph, we necessarily have that y(A) > 0, y(D) > 0 (see Fig. 10 (a)). Let β be the counterclockwise angle between the x-axis and OD, and let α be the angle between OA and the x-axis. Note that α ∈ (0, π/2), β ∈ (0, π). As L(γ ∩ B 2 ) is bounded by |AE| + |DE|, we get L(γ ∩ B 2 ) ≤ |AE| + |DE| = cos(α) + cos(β) + sin(β) − sin(α) = (cos(α) − sin(α)) + (cos(β) + sin(β)) < 3. In the following arguments we see that to get the maximum length of γ ∩ B 2 by translations, it is necessary that V ∈ B 2 . Suppose now that V / ∈ B 2 and γ∩B 2 is not connected. As the intersection γ∩∂B 2 has at most four points, γ∩B 2 has at most two connected components γ 1 and γ 2 such that L(γ 1 ) > 0 and L(γ 2 ) > 0. If there exists only one such connected component, which goes inside B 2 at A ∈ ∂B 2 and it goes outside B 2 at D ∈ ∂B 2 , then we can use the last estimate if y(A), y(D) > 0, or the estimate (8) in all other cases to get that L(γ ∩ B 2 ) < 4. If there exist two such connected components, then there exist two points A, C ∈ ∂B 2 where the curve γ goes inside B 2 , and two points B, D ∈ ∂B 2 where the curve goes outside B 2 . Supposing that x(A) < x(B) < x(C) < x(D), we claim that x(B) < x(V ) < x(C). Otherwise, as V / ∈ B 2 , we have that x(V ) < x(A), or x(D) < x(V ). By symmetry, it is enough to verify that the second inequality cannot be true (see Fig. 10 (b)). Indeed, as γ goes outside B 2 at B and goes inside B 2 at C, there exists x ′ ∈ (x(B), x(C)) such that κ γ (x ′ ) > κ ∂B 2 = 1. In particular, κ γ (x) > 1 for all x ∈ [x ′ , x(V )), since the curvature κ γ (x) of the γ(x) is increasing for x < x(V ), this contradicts the fact that γ goes inside B 2 at C and going outside B 2 at D, since x(D) < x(V ) and κ ∂B 2 = 1. Therefore, x(B) < x(V ) < x(C), also y(A) > y(B) and y(C) < y(D). As we are supposing that γ ∩ B 2 ∩ H + has two points, it is not difficult to see that y(A), y(D) > 0 and y(B), y(C) < 0. It follows as in the previous triangle argument. Now we have (by symmetry) two cases: CD is on the left of O; or AB is on the left of O, and CD is on the right of O (see Figs. 11 and 12). Here, a segment is said on the left of O (resp. on the right of O) if it intersects the x-axis for x ≤ 0 (resp. x > 0). In the first case, we have necessarily (see Fig. 11 (a)): y(D) > y(A) > 0, y(B), y(C) < 0 and x(A) < x(B) < x(C) < 0. Taking a short translation of γ, we get news points A ′ , B ′ , C ′ , D ′ ∈ γ ∩ ∂B 2 . To keep the properties (9) for the news points, we translate γ such that V → C ′ , and |C ′ D ′ | = |CD| is constant during the translation. This is possible considering the following map: γ is increasing for x ≥ x(V ) and |(x, γ(x))| → ∞ as x → ∞, the choice of D ′ always exists and it is unique. This monotonicity also implies that F is one-toone. Consider Dom(F ) : F : C ′ = γ(x) → D ′ ∈ γ.= {γ(x) : x ∈ [x(V ), x(C)} and Im(F ) := {γ(x) : x ∈ [x(D 0 ), x(D)]}, where D 0 is the unique point in γ such that |V D 0 | = |CD| and x(D 0 ) ≥ x(V ). Again, the monotonicity of γ implies that F is onto. Moreover, by the continuity of the distance function, we have that F : C ′ → D ′ is continuous. Take E ′ the unique point such that |E ′ C ′ | = |E ′ D ′ | = 1 and y(E ′ ) ≤ y, for all y ∈ C ′ D ′ . Consider the unity disk through the points C ′ and D ′ with center at E ′ . So we can take the inverse function of F and move continuously the unity disk such that C ′ → V, keeping |C ′ D ′ | = |CD|. Note that E ′ moves continuously to down and to left, since γ is increasing. This is equivalent to translate continuously γ such that V → C ′ , keeping |C ′ D ′ | = |CD|, the curve gamma moves continuously to up and to right, and C ′ D ′ keeps on the left of O. The last two implies that y(A ′ ), x(A ′ ), x(B ′ ) increase, y(B ′ ) decreases, and the properties (9) still hold during the move. Indeed, this is obvious for a short move and it holds during the translation because C ′ D ′ keeps on the left of O, which implies that y(A ′ ) < 1, x(A ′ ), x(B ′ ), x(C ′ ), y(B ′ ) < 0. Also, as x(A) ′ < x(B ′ ) < x(V ), we have that y(C ′ ) < 0. By the triangle argument and the fact that C ′ D ′ keeps on the left of O, we see that γ D ′ C ′ ⊂ B 2 during the translation. Also, A ′ , C ′ , D ′ are distinct and V / ∈ B 2 during the move, thus there exists exact more one point B ′ ∈ γ ∩ ∂B 2 . In particular, γ B ′ A ′ ⊂ B 2 during the translation. Here we use the fact that A ′ cannot be tangent to ∂B 2 , since x(A ′ ) < 0, y(A ′ ) > 0 and the monotonicity properties of γ. Note that |A ′ B ′ | is increasing, B ′ is approaching to V, and then L(γ) B ′ A ′ is increasing. We also note that L(γ) D ′ C ′ increases because C ′ is approaching to V, and then the curvature of γ is increasing between C ′ and D ′ . In the end of the translation we get that L(γ ∩ B 2 ) increases, and V ∈ B 2 (see Fig. 11 (b)). In the second case, we just know that x(A), y(B), y(C) < 0 and y(A), y(D), x(D) > 0 (see Fig. 12 (a)). In this case, we take a short translation to up of γ to get news points A ′ , B ′ , C ′ , D ′ with the same previous properties. We take this translation as long as A ′ B ′ and C ′ D ′ do not pass through the origin O, and V / ∈ ∂B 2 . In particular, the last properties hold for the news points. Note that y(A ′ ), y(D ′ ) increase and B ′ , C ′ are approaching to V, so we have that |A ′ B ′ |, |C ′ D ′ | increase and therefore L(γ) B ′ A ′ and L(γ) D ′ C ′ are increasing, since the curvature of γ ∩ B 2 is increasing. We stop this translation when V touches ∂B 2 , or when A ′ B ′ or C ′ D ′ pass through the origin, in the last case we continue with the translation as in the previous case until V touches ∂B 2 . In the end, we get again that L(γ ∩ B 2 ) increases and V ∈ B 2 (see Fig. 12 (b)). By the above arguments we simplify our analysis to the case such that there exist two points A, D ∈ γ ∩ ∂B 2 with y(A), y(D) > 0, and V ∈ B 2 . If we suppose that x(A) < x(D), then by the triangle argument we have that x(A) < x(V ) < x(D). For a fix γ in this case, the maximum length of γ ∩ B 2 is reached when x(V ) = 0, that is, when we translate γ horizontally such that x(V ) → 0. Indeed, suppose for now that γ ∩ B 2 is connected and consider a such translation so that x(V ) → 0 and A → A ′ , D → D ′ . If x(D) < 0, then during the translation and as long as x(D ′ ) < 0, we have that the curve goes inside B 2 , and in particular L(γ ∩ B 2 ) increases (see Fig. 13 (b)). Now, consider the case x(D) ≥ 0 and x(A) < 0 as in the Fig. 13 (a). In this case, let |x(V )| = ǫ > 0. Take E = (−(x(D) + ǫ), y(D)), F = (−x(D), y(D)), G = (x(A) + ǫ, y(A)), H = (x(A ′ ), y(D)), and I = (x(A ′ ), y(A)), we claim that |EH| < |IG|. To see this, we take the tangent line r(x) to γ(x) at A ′ , and we take J = (r −1 (y(D)), y(D)), L = ((r −1 (y(A)), y(A)). Note that HI is orthogonal to JF and to AG, also E, H ∈ JF , I, L ∈ AG and ǫ = |EF | = |AG| (see Fig. 13 (a) and 14 (a)). As |EF | = |AG|, if |EH| ≥ |IG| then we would have |HF | ≤ |AI| and, therefore, |HA ′ | < |A ′ I|, because A ′ ∈ B 2 and x(A ′ ) < 0, y(A ′ ) > 0. Using this and the fact that γ is convex we see that |EH| < |JH| < |IL| < |IG|, which is a contradiction. Let γ be the curve γ after the translation. The inequality |EH| < |IG| means that L(γ) E A ′ < L(γ) G A ′ , since the curvature of γ is strictly increasing in the direction of the vertex V. Thus, the length of γ ∩ B 2 increases after the translation A → A ′ , D → D ′ , V → V ′ because L(γ) A ′ E is the amount of the curve that goes outside B 2 , and L(γ) G A ′ is the amount that goes inside B 2 . Here, we are using the fact that γ ∩ B 2 has at most four points, the symmetric of γ, y(A ′ ) = y(D ′ ) > 0 and the fact that V ′ ∈ B 2 to conclude that γ For the case γ ∩B 2 not connected, we have similarly that the length of γ ∩B 2 also increases after the above translation x(V ) → 0. Indeed, as before we are supposing that there exist A, D ∈ ∂B 2 such that y(A), y(D) > 0 and V ∈ B 2 . In particular, x(A) < x(V ) < x(D) if x(A) < x(D). In this case it is enough suppose that there exist two connected components and the points A, B, C, D ∈ γ ∩ B 2 such that γ is inside B 2 between A and B, and between C and D; otherwise it is outside. Note that during the translation y(A) increases. In the end, we get new points A ′ , D ′ such that y(D ′ ) = y(A ′ ) > 0, and V ′ ∈ B 2 , since the vertex V is the global minimum of γ. Finally, as before γ V ′ A ′ ⊂ B 2 and γ D ′ V ′ ⊂ B 2 . In particular, L(γ ∩ B 2 ) increases, since the curve goes inside B 2 for γ(x) < y(A), and the previous paragraph for γ(x) ≥ y(A) (see Fig. 14 (b).) By the last two paragraphs, we need to find an upper bound for L(γ ∩ B 2 ), when x(V ) = 0, γ ∩ B 2 is connected, and {γ ∩ B 2 }\{V } is given by two points A, D such that −x(A) = x(D), and y(A) = y(D) > 0. In this situation, if y(V ) > −1, we can translate γ to down such that y(V ) → −1, then L(γ ∩ B 2 ) increases as long as y(A ′ ) = y(D ′ ) > 0. So, we consider the last hypothesis above with V ′ = (0, −1), in other words, the curve is tangent to ∂B 2 at V ′ (see Fig. 15 (a)). Here we are using that the length L(γ ∩ B 2 ), for V ′ = (0, −1) and γ passes through A ′ , D ′ is bigger than the length of L( γ ∩ B 2 ), if γ passes through A ′ , D ′ and V ∈ int(B 2 ). Figure 15. V ′ A ′ ⊂ B 2 and γ D ′ V ′ ⊂ B 2 . Remember, the curve γ(x) can be a hyperbola H(x) with a unique branch intersecting B 2 , or a parabola P (x). To satisfy our situation, the equations become, respectively H(x) = c d d 2 + x 2 − (1 + c) and P (x) = ax 2 − 1, where a, c, d > 0, H(1) = (c/d) √ d 2 + 1 − (1 + c) > 0, and P (1) = a − 1 > 0. Thus, H(x) ∩ B 2 ∩ H + and P (x) ∩ B 2 ∩ H + have two points, also H(x) and P (x) pass through V = (0, −1). Suppose that H(x) and P (x) pass through the same points A, D ∈ B 2 ∩ H + , then H(x) ∩ P (x) = {A, D, V }. This is because of the symmetry, and the fact that the intersection H(x) ∩ P (x) has at most four points. Note that H(x) and P (x) are tangent at V, so A (and then D, by symmetry) cannot be a point where H(x) and P (x) has a common tangent, since two distinct conics are tangent at most at two points. As P (x) > H(x) for x → ∞, we conclude that the graphic of H(x) is above of P (x) for x(A) ≤ x ≤ x(D). In particular, L(H(x) ∩ B 2 ) < L(P (x) ∩ B 2 ), so we only need to bound L(P (x) ∩ B 2 ) for a > 1 (see Fig. 15 (b)). As a > 1, the points A, D can be determined uniquely by the value of the parameter a. In fact, −x(A) = x(D) = x(a) = √ 2a − 1/a, where x(a) is the positive solution of x 2 + (ax 2 − 1) 2 = 1. Then, we can calculate L(P (x) ∩ B 2 ) in the parameter a : (M ) := {X ∈ X(R Q ) : X(p) ∈ T p M for all p ∈ M } and X tan (M ) := {X ∈ X(M ) : X(p) ∈ T p (∂M ) for all p ∈ ∂M }. Definition 2.1. (Relative Topology) Given any subset A ⊂ M, where A is equipped with the subspace topology, the interior relative of A, int M (A), is defined as the set of all p ∈ M such that there exists a relatively open neighborhood U ⊂ A of p. The exterior relative of A is denoted by int M (M \A). And the relative boundary of A, ∂ rel A, is the subset of M such that is neither in the relative interior nor exterior of A. Definition 2.2. (Relative Convexity) A subset Ω ⊂ M is said to be a relatively convex (respect. relatively strictly convex ) domain in M if it is a relatively open connected subset in M whose relative boundary ∂ rel Ω is smooth and convex (respect. strictly convex) in M. From [ 14 , 14Lemma 15.1] (see also[20, 4.1 (4)]) we know that there exist critical sequences for each class Π, and from[20, 4.2 (2)], C(S) is compact and non-empty. Definition 2.7. [13, Section 2.5] Let X ⊂ I m be a cubical subcomplex. We say that a continuous map in the flat topology Φ : X → Z n,rel (M, ∂M ; Z 2 ) is a psweepout if the p-th cup power of Φ * (λ) is nonzero in H p (X; Z 2 ), where λ is the generator of H 1 (Z n,rel (M, ∂M ; Z 2 ); Z 2 ). Definition 2.8. A flat continuous map Φ : X → Z n,rel (M, ∂M ; Z 2 ) has no concentration of mass if lim r→0 Definition 2 . 10 . 210Let U ⊂ M be a relatively open subset, we say that a varifold V ∈ V k is Z 2 -almost minimizing in U with free boundary if for every ǫ > 0 we can find δ > 0 and T ∈ Z k,rel (M, ∂M ; Z 2 ) with F(V, |T |) < ǫ and such that the following property holds true: if T = T 0 , T 1 , . . . , T m ∈ Z k,rel (M, ∂M ; Z 2 ) with • spt(T − T i ) ⊂ U for i = 1, . . . , m; • F (T i − T i−1 ) ≤ δ for i = 1, . . . , m and • M(T i ) ≤ M(T ) + δ for i = 1, . . . , m then M(T m ) ≥ M(T ) − ǫ. minimizing in a relatively open set U ⊂ M with free boundary, then V is stationary in U with free boundary ([20], Th. 3.3). By construction, V and V satisfy (3) for this x. In fact, V and V are stationary in any relatively open set Figure 4 . 4Figure 4. . Let M be a Riemannian manifold, U ⊂ int(M ) open and K ⊂ U compact. If V ∈ IV 1 (M ) is a stationary varifold in U, then V K is a geodesic network. Definition 3 . 2 . 32Let T ∈ Z k (M ; Z 2 ) and U ⊂ M be a relatively open subset. We say that T is locally mass minimizing in U if for every p ∈ spt(T ) ∩ U there exists r p > 0 such that B rp (p) ∩ M ⊂ U and for all S ∈ Z k (M ; Z 2 ) with spt(T − S) ⊂ B rp (p) ∩ M we have M(S) ≥ M(T ). Theorem 3 . 8 . 38Let U ⊂ M be a relatively open set, K ⊂ U compact and V ∈ V k (M ) be an Z 2 -almost minimizing varifold in U with free boundary. There exists Z 2 -almost minimizing in any bounded open subset of R 2 [20, Th. 3.11 and 3.12(1)]. Theorem 3.11. (Boundary maximum principle [12, Th. 2.5]). Let U ⊂ M n+1 be a relatively open subset and V ∈ V n (M ) be stationary with free boundary in U. Suppose N ⊂⊂ U is a relatively open connected subset in M such that Figure 6 . 6Figure 6. Lemma 4 . 1 . 41The map F p : RP p → Z 1,rel (B 2 , ∂B 2 ; Z 2 ) has no concentration of mass for p = 1, . . . , 4. Figure 7 . 7For p = 1, . . . , 4, we have that F p ([Q]) is an algebraic variety of degree at most 2, so F p ([Q]) intersects r ρ,θ at most two times. By the Cauchy-Crofton Formula weobtain F p ([Q]) (B s (P 0 )\∂B 2 ) ≤ F p ([Q]) (B s (P Similarly we have F p ([Q]) (B s (P 0 )\∂B 2 ) ≤ 4sπ, when p 0 = 0. Then, in all the cases we conclude that F p ([Q]) (B s (P 0 )\∂B 2 ) → 0 as s → 0.In the following, we estimate an upper bound for F p ([Q]) , p = 1, . . . , 4. In other words, we estimate the maximum length of the algebraic variety F p ([Q]). By the definitions above, F p ([Q]) is degenerate or is the restriction to B 2 of a straight line, or of two straight lines, or of a parabola, or of a hyperbola. In other words, F p ([Q]) is a quadratic curve which is not an ellipse, since we excluded the monomial Q 5 (x, y) = y 2 . Lemma 4 . 2 . 42For any [Q] ∈ RP p we have that F p ([Q]) ≤ 2, p = 1, 2, and F p ([Q]) < 4.00267, p = 3, 4. Proof. Clearly, for p = 1, 2 the algebraic variety F p ([Q]) is degenerate or the restriction to B 2 of a straight line, thus F p ([Q]) ≤ 2 for p = 1, 2 and for all [Q] ∈ RP p . Figure 8 . 8restricted of the unity disk is bounded from above by approximately 4.00267, which concludes the lemma.4.2. The First Widths of B 2 and E 2Now, we prove our main result about p-widths: we calculate the low p-widths of the unit ball B 2 , and of full ellipses C ∞ -close to B 2 . Theorem 4 . 3 . 43For B 2 we have (i) ω 1 (B 2 ) = ω 2 (B 2 ) = 2; (ii) ω 3 (B 2 ) = ω 4 (B 2 ) = 4.Also, if E 2 is a full ellipse C ∞ -close to B 2 with small diameter d and large diameter D, then (iii) ω 1 (E 2 ) = d and ω 2 (E 2 ) = D;(iv) ω 3 (E 2 ), ω 4 (E 2 ) ∈ {2d, d + D, 2D}. Figure 9 . 9Figure 9. Figure 10 . 10Figure 10. Figure 11 . 11Where x ≥ x(V ) and D ′ is chosen such that x(D ′ ) ≥ x(C ′ ), |C ′ D ′ | = |CD|.Let's see that the above map is well defined with respect the choice of D ′ . Indeed, as Figure 12 . 12Figure 12. Figure 14 . 14Figure 13. .1] we use [13, Th. 2.11]; and a compatible version of [14, Th. 13.1] follows from [13, Lemma A. 1] in the same way that the [14, Th. 13.1] follows from [14, Lemma 13.4]. ). Proof. The proof follows as in Prop. 5.3 from [12], replacing Lemmas 3.10 and 3.7 by Th. 2.3 and Prop. 2.4 from [13], respectively. See also Th. 3.11 and 3.13 from The convexity in[12] is assumed to be strict convexity. For i = 0, we consider A i−1 = A k−1 , and for i = k − 1 we consider A i+1 = A 0 . Acknowledgements: I would like to deeply thank to Professor F. Marques for suggesting me to work on this problem and your support while visiting him at Princeton University. I am very thankful to Professors F. Vitório (PhD adviser), R. Montezuma and T. Rivière by comments and suggestions. Also I am grateful to Department of Mathematic of Princeton University by its hospitality and where part of this work was done.Therefore, there exists a unique a 0 > 1 such that L ′ (a 0 ) = 0. Moreover, L(a) is strictly increasing for 1 < a < a 0 , and it is strictly decreasing for a > a 0 . In particular, L(a 0 ) > 4 and L(a 0 ) is the global maximum of L(a), since L(a) → 4 as a → ∞. We can estimate a 0 such that (10) becomes zero, and we obtain a 0 ≈ 94.091282, and then L 1 < L 0 = L(a 0 ) ≈ 4.00267.AppendixIn this appendix we prove the following result:Theorem 5.1. Let L 0 the maximum length of a parabola restricted to B 2 and L 1 the maximum length of a hyperbola restricted to B 2 . Then L 1 < L 0 ≈ 4.00267. Moreover, there exists a unique parabola P 0 such that L(P 0 ∩ B 2 ) = L 0 .In Rack[21]was proved that L 0 ≈ 4.00267. Since we do not have direct access to[21], we give a geometric proof, and in our case we include the estimate of L 1 .The length of a real algebraic curve C restricted to B 2 can be bounded in terms of its degree using the Cauchy-Crofton Formula (seeLemma 4.1). In fact, if that curve has degree d, then it intersects a straight line at most d times, so by the Cauchy-Crofton Formula the length of C restricted to B 2 is at most d · area(B 2 ) = πd. Obviously, this upper bound is not sharp. For example, if d = 1 we have that C is a straight line and the length of the intersection of a straight line with B 2 is at most 2. It is intuitive, and it was conjectured in Guth[10, p. 1974], that the general sharp upper bound is similar to the case d = 1, that is,Our counterexample is the parabola P 0 from the above theorem.Proof of Theorem 5.1. By the proof of Lemma 4.2, we know that if γ is a hyperbola such that each branch intersects B 2 , then L(γ ∩ B 2 ) ≤ 4. Thus, from now consider γ a branch of a hyperbola or a parabola. We choose an orientation such that the axis of symmetry of that curve is orthogonal to x-axis, and such that γ is convex downward. So, γ is a function of x with a global minimum at the vertex V, it is strictly increasing for x > x(V ), and strictly decreasing for x < x(V ). Moreover, the curvature is strictly increasing in the direction of the axis of symmetry, and there exist at most four points in the intersection γ ∩ ∂B 2 . We fix a such curve γ such that L(γ ∩ B 2 ) > 0, and by translation we find the positions such that the length L(γ ∩ B 2 ) increases, next we change the parameters of that curve to get the maximum of L(γ ∩ B 2 )..By the expression above we have that L(P (x) ∩ B 2 ) → 4 as a → ∞ (P (x) ∩ B 2 becomes two diameters). We see below that L(a) has a global maximum point at a 0 < ∞, and then L(a 0 ) > 4. Indeed,Taking z = 2a − 1, the denominator above becomesSo, the sign of L ′ (a) is the sign ofNote that the expression above starts positive for a > 1 and tends to −∞ when a → ∞, moreover it is strictly decreasing for a > 1/2. The latter is because the derivative of the last expression is given by − z + z 2 z 5/2 √ 4z + 1 . The width of ellipsoids. Aiex Nicolau Sarquis, Comm. Anal. Geom. 272Nicolau Sarquis Aiex. The width of ellipsoids. Comm. Anal. Geom., 27(2):251-285, 2019. The structure of stationary one dimensional varifolds with positive density. W K Allard, F J AlmgrenJr, Invent. Math. 342W. K. Allard and F. J. Almgren, Jr. The structure of stationary one dimensional varifolds with positive density. Invent. Math., 34(2):83-97, 1976. The p-widths of a surface. Otis Chodosh, Christos Mantoulidis, Otis Chodosh and Christos Mantoulidis. The p-widths of a surface. https://arxiv.org/abs/ 2107.11684, 2021. The min-max construction of minimal surfaces. 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Marques and André Neves. Morse index and multiplicity of min-max minimal hypersurfaces. Camb. J. Math., 4(4):463-511, 2016. Existence of infinitely many minimal hypersurfaces in positive Ricci curvature. C Fernando, André Marques, Neves, Invent. Math. 2092Fernando C. Marques and André Neves. Existence of infinitely many minimal hypersurfaces in positive Ricci curvature. Invent. Math., 209(2):577-616, 2017. Low min-max widths of the round three-sphere. Charles Nurser, Imperial College LondonPhD thesisCharles Nurser. Low min-max widths of the round three-sphere. PhD thesis, Imperial College London, October 2016. The smooth Riemannian extension problem. Stefano Pigola, Giona Veronelli, Ann. Sc. Norm. Super. Pisa Cl. Sci. 205Stefano Pigola and Giona Veronelli. The smooth Riemannian extension problem. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 20(4):1507-1551, 2020. Non-persistence of resonant caustics in perturbed elliptic billiards. Ergodic Theory Dynam. Sônia Pinto-De Carvalho, Rafael Ramírez-Ros, Systems. 336Sônia Pinto-de Carvalho and Rafael Ramírez-Ros. Non-persistence of resonant caustics in perturbed elliptic billiards. Ergodic Theory Dynam. Systems, 33(6):1876-1890, 2013. Jon T Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton, N.J.; TokyoUniversity of Tokyo PressMathematical NotesJon T. Pitts. Existence and regularity of minimal surfaces on Riemannian manifolds, vol- ume 27 of Mathematical Notes. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. Der quadratische durchmesser eines kreises. Heinz-Joachim Rack, Praxis der. Mathematik. 26Aulis VerlagCologne/Köln)Heinz-Joachim Rack. Der quadratische durchmesser eines kreises. Praxis der. Mathematik (Aulis Verlag, Cologne/Köln), 26:109-111, 1984. On Cayley conditions for billiards inside ellipsoids. Rafael Ramírez-Ros, Nonlinearity. 275Rafael Ramírez-Ros. On Cayley conditions for billiards inside ellipsoids. Nonlinearity, 27(5):1003-1028, 2014. Lectures on geometric measure theory. Leon Simon, Proceedings of the Centre for Mathematical Analysis. 3Australian National University. Australian National University, Centre for Mathematical AnalysisLeon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. . Matemática Instituto De, Maceió, AL, BrazilUniversidade Federal de AlagoasEmail address: [email protected] de Matemática, Universidade Federal de Alagoas, Maceió, AL, Brazil. Email address: [email protected]
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[ "Thermal and chemical equilibration of hadronic matter *", "Thermal and chemical equilibration of hadronic matter *" ]
[ "E L Bratkovskaya \nInstitut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany\n", "W Cassing \nInstitut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany\n", "C Greiner \nInstitut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany\n", "M Effenberger \nInstitut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany\n", "U Mosel \nInstitut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany\n", "A Sibirtsev \nInstitut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany\n" ]
[ "Institut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany", "Institut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany", "Institut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany", "Institut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany", "Institut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany", "Institut für Theoretische Physik\nUniversität Giessen\n35392GiessenGermany" ]
[]
We study thermal and chemical equilibration in 'infinite' hadron matter as well as in finite size relativistic nucleus-nucleus collisions using a BUU cascade transport model with resonance and string degrees-of-freedom. The 'infinite' hadron matter is simulated within a cubic box employing periodic boundary conditions. The various equilibration times depend on baryon density and energy density and are much shorter for particles consisting of light quarks then for particles including strangeness. For kaons and antikaons the chemical equilibration time is found to be larger than ≃ 40 fm/c for all baryon and energy densities considered. The inclusion of continuum excitations, i.e. hadron 'strings', leads to a limiting temperature of T s ≃ 150 MeV. * Work supported by BMBF, GSI Darmstadt and DFG.
10.1016/s0375-9474(00)00486-3
[ "https://export.arxiv.org/pdf/nucl-th/0007019v1.pdf" ]
119,496,741
nucl-th/0007019
02b85b5560e071bfe822dcdea69f6b79e795bff2
Thermal and chemical equilibration of hadronic matter * Jul 2000 E L Bratkovskaya Institut für Theoretische Physik Universität Giessen 35392GiessenGermany W Cassing Institut für Theoretische Physik Universität Giessen 35392GiessenGermany C Greiner Institut für Theoretische Physik Universität Giessen 35392GiessenGermany M Effenberger Institut für Theoretische Physik Universität Giessen 35392GiessenGermany U Mosel Institut für Theoretische Physik Universität Giessen 35392GiessenGermany A Sibirtsev Institut für Theoretische Physik Universität Giessen 35392GiessenGermany Thermal and chemical equilibration of hadronic matter * Jul 2000arXiv:nucl-th/0007019v1 10 1 We study thermal and chemical equilibration in 'infinite' hadron matter as well as in finite size relativistic nucleus-nucleus collisions using a BUU cascade transport model with resonance and string degrees-of-freedom. The 'infinite' hadron matter is simulated within a cubic box employing periodic boundary conditions. The various equilibration times depend on baryon density and energy density and are much shorter for particles consisting of light quarks then for particles including strangeness. For kaons and antikaons the chemical equilibration time is found to be larger than ≃ 40 fm/c for all baryon and energy densities considered. The inclusion of continuum excitations, i.e. hadron 'strings', leads to a limiting temperature of T s ≃ 150 MeV. * Work supported by BMBF, GSI Darmstadt and DFG. INTRODUCTION Nucleus-nucleus collisions at relativistic and ultrarelativistic energies are studied experimentally and theoretically to obtain information about the properties of hadrons at high density and/or temperature as well as about the phase transition to a new state of matter, the quark-gluon plasma (QGP). In the latter deconfined partons are the essential degrees of freedom that resolve the underlying structure of hadrons [1]. Whereas the early 'bigbang' of the universe most likely evolved through steps of kinetic and chemical equilibrium, the laboratory 'tiny bangs' proceed through phase-space configurations that initially are far from an equilibrium phase and then evolve by fast expansion. These 'specific initial conditions' -on the theoretical side -have lead to a rapid development of nonequilibrium quantum field theory and nonequilibribrium kinetic theory [2,3]. Presently, semiclassical transport models are widely used as approximate solutions to these theories and practically are an essential ingredient in the experimental data analysis. For recent reviews we refer the reader to Refs. [4][5][6]. On the other hand, many observables from strongly interacting systems are dominated by many-body phase space such that spectra and abundances look 'thermal'. It is thus tempting to characterize the experimental observables by global thermodynamical quantities like 'temperature', chemical potentials or entropy [7][8][9][10][11]. We note, that even the use of macroscopic models like hydrodynamics [12,13] employs as basic assumption the concept of local thermal and chemical equilibrium. The crucial question, however, how and on what timescales a global thermodynamic equilibrium can be achieved, is presently a matter of debate. Thus nonequilibrium approaches have been used in the past to address the problem of timescales associated to global or local equilibration [14][15][16][17][18][19][20][21]. In view of the increasing 'popularity' of thermodynamic analyses a thorough microscopic reanalysis of this questions appears necessary especially for nucleus-nucleus collisions at ultrarelativistic energies that aim at 'detecting' a phase transition to the QGP. In this contribution we discuss equilibration phenomena in 'infinite' hadronic matter using a microscopic transport model that contains both hadron resonance and string degrees-of-freedom. With this investigation we want to provide insight into the reaction dynamics by the use of cascade-like models and also point out some of their limitations. The 'infinite' hadronic matter is modelled by initializing the system solely by nucleonic degrees of freedom through a fixed baryon density and energy density, while confining it to a cubic box and imposing periodic boundary conditions during the propagation in time. EQUILIBRATION AND LIMITING TEMPERATURE To investigate the equilibration phenomena addressed above we perform microscopic calculations using the Boltzmann-Uehling-Uhlenbeck (BUU) model of Refs. [23,24]. This model is based on the resonance concept of nucleon-nucleon and meson-nucleon interactions at low invariant energy √ s [22], adopting all resonance parameters from the Manley analysis [25]. The high energy collisions -above √ s = 2.6 GeV for baryon-baryon collisions and √ s = 2.2 GeV for meson-baryon collisions -are described by the LUND string fragmentation model FRITIOF [26]. This aspect is similar to that used in the HSD approach [6,[27][28][29] and the UrQMD code [5]. For a detailed description of the underlying model at low energy we refer the reader to Ref. [24]. A box with periodic boundary conditions In order to study 'infinite' hadronic matter problems we confine the particles in a cubic box with periodic boundary conditions for their propagation similar to a recent box calculation within the UrQMD model [18]. We specify the initial conditions, i.e. baryon density ρ, strange particle density ρ S and energy density ε as follows: first the initial system is fixed to N p = 80 protons and N n = 80 neutrons, which are randomly distributed in a cubic box of volume V . The 3-momenta p i of the nucleons in a first step are randomly distributed inside a Fermi-sphere of radius p F = 0.26 GeV/c (at ρ 0 ) and in a second step boosted by ±β cm by a proper Lorentz transformation. Thus the initial baryon density ρ is fixed as ρ = A/V , A = N p + N n . The strange particle density is set to zero as in related heavy-ion experiments while the energy density is defined as ε = E/V , where E is the total energy of all nucleons E = A i p 2 i + m 2 N .(1) The boost velocity β cm is related to the initial energy density ε (excluding Fermi motion) as β cm = 1 − ρ 2 m 2 N ε 2(2) using ε = γ cm ρm N with γ cm = 1/ 1 − β 2 cm . Recall that ρ 0 m N ≃ 0.15 GeV/fm 3 so that an energy density ε ≃ 1.5 GeV/fm 3 at density ρ 0 corresponds to γ cm ≃ 10, i.e. the SPS energy T lab ≃ 185 A·GeV. We thus start with a 'true' nonequilibrium situation in order to mimique the initial stage in a relativistic heavy-ion collision. The initial phase represents two interpenetrating, (ideally) infinitely extended fluids of cold nuclear matter. We now propagate all particles in the box in the cascade mode (without mean-field potentials) using periodic boundary conditions, i.e. particles moving out of the box are reinserted at the opposite side with the same momentum. The phase-space distribution of particles then can change due to elastic collisions, resonance and string production and their decays to mesons and baryons again. We recall that we include all baryon resonances up to an invariant mass of 2 GeV and meson resonances up to the φ-meson. According to the initial conditions for ε and ρ the factor γ cm determines if strings are excited in the very first collisions. This is the case for γ cm > 1.4 where the early equlibration stages are dominated by string formation and decay. Figure 1 shows the time evolution of the various particle abundances (nucleons N, ∆, Λ, π and K + mesons) for density ρ = ρ 0 (left panel) at energy density ε = 0.22 GeV/fm 3 and for ρ = 3ρ 0 (right panel) at ε = 0.66 GeV/fm 3 . These initial conditions correspond to bombarding energy T lab per nucleon of roughly 2 A·GeV. In Fig. 1 we count all particles which are 'hadronized', i.e. produced by string decay after a formation time of τ F = 0.8 fm/c in their rest frame. After several fm/c the number of nucleons decreases due to inelastic collisions that produce either baryon resonances or additional mesons. The number of ∆-resonances grows up to a maximum in a few fm/c, since a lot of ∆'s are produced in the first NN collisions; their number subsequently decreases with time due to their decay and excitation of further resonances or due to reabsorption. The numbers of π's and η's increase very fast and reach the equilibrium value within a few fm/c whereas the strange particles (K + , K − , Λ) require a much longer time for equilibration. Chemical equilibration For the higher energies the initial particle production proceeds via the formation and decay of string excitations. This leads in particular to a very early onset of strange particles (mainly kaons and hyperons) within the first fm/c either due to the initial strings or due to secondary or ternary baryon-baryon, meson-baryon and meson-meson induced string-like interactions. In Ref. [29] it was shown that these early secondary and ternary reactions can contribute up to about 50 % of the total strange particles obtained in a Pb + Pb reaction at CERN SPS energies and thus explain the factor of 2 in the observed relative strangeness enhancement compared to p+p reactions. This, however, does not imply that chemical equilibrium for the dominant strange particles has been achieved in this reaction, as our analysis clearly shows. In the later stages, when the system has become, more or less, isotropic in momentum space, strange particles can only be further produced by low energy hadronic reactions, which, however, have a considerable threshold and are thus strongly suppressed. This explains the long chemical equilibration times for the strange particles first demonstrated by Koch, Müller and Rafelski [14]. In order to define an overall chemical equilibration time we perform a fit to the particle abundances N(t) for pions and kaons as N(t) = N eq (1 − exp(−t/τ eq )) ,(3) where N eq is the equilibrium limit. The equilibration time τ eq thus corresponds to the time t when ≃ 63% of N eq is achieved. Figure 2 shows the equilibration time τ eq versus energy density for π and K + mesons at different baryon densities of 1/3ρ 0 , ρ 0 , 3ρ 0 and 6ρ 0 . We find that the equilibration time for pions scales as τ π eq ∼ 1/ρ or Γ π ∼ ρ, thus we present the curve only for baryon density ρ 0 . Whereas τ π eq slowly grows with energy-density, τ K eq falls steeply with ε. This marked difference is due to the fact that, on one hand, the kaon production rate increases dramatically with √ s whereas that of the pions, on the other hand, is more flat. With increasing energy thus more strange particles are produced through strings especially from the primary collisions with high √ s and the chemical equilibration is achieved faster. In Fig. 2 we have considered an 'ideal' situation, i.e. hadron matter at fixed energy and baryon density. In realistic heavy-ion collisions the system goes through the different stages due to interactions and expansion. However, as follows from Fig. 2, the equilibration time for strangeness is larger than 40 fm/c for all energy and baryon densities. Thus in realistic nucleus-nucleus collisions the chemical equilibration of strange particles requires also a time above 40 fm/c which is considerably larger than the actual reaction time of a few 10 fm/c or less [30]. The particle abundances used to extract τ eq in Fig. 2 have been calculated without any in-medium potentials. In fact, the introduction of attractive potentials (especially for K − ) will lower the hadronic thresholds and thus increase the scattering rate between strange and nonstrange hadrons, whereas the K + feels some repulsive potential and the trend goes in the opposite way. According to our calculations such in-medium modifications (in line with Ref. [6]) give a correction to the K + equilibration times by atmost 10 % and shortens the K − equilibration times up to 20 % at density ρ 0 . Figure 2. Equilibration time τ eq versus energy density ε for π and K + mesons at different baryon densities 1/3ρ 0 , ρ 0 , 3ρ 0 and 6ρ 0 . Thermal equilibration and limiting temperature In this subsection we investigate the approach to thermal equilibration. For the equilibrated system we can extract a temperature T by fitting the particle spectra with the Boltzmann distribution d 3 N i dp 3 ∼ exp(−E i /T ),(4) where E i = p 2 i + m 2 i is the energy of particle i. We note that at the temperatures of interest here, the Bose and Fermi distributions are practically identical to a Boltzmann distribution. We find that in equilibrium the spectra of all particles can be characterized by one single temperature T [30]. In the left panel of Fig. 3 we display the temperature T versus energy density ε for different baryon densities ρ: 1/3ρ 0 (open down triangles), ρ 0 (full squares), 3ρ 0 (full dots), 6ρ 0 (full up triangles). In order to compare calculations for different baryon densities we have subtracted the baryon energy density at rest, i.e. ≃ m N ρ (except for Fermi motion). As seen from Fig. 3 the temperature grows with energy density up to a limiting value reminiscent of a 'Hagedorn' temperature [31]. From our detailed investigations we obtain for the limiting temperature T s ≃ 150 ± 5 MeV which practically does not depend on baryon density. Such a singular behavior of ε(T ) for T ≃ T s has also been found in the box calculations in Ref. [18] for ρ = ρ 0 . Our limiting temperature is slightly higher than that in Ref. [18] (T s = 130±10 MeV) due to the different number of degrees of freedom; the model [18] contains more resonances and uses a different threshold for string excitations. Thus, there is some phenomenological sensitivity to the hadronic zoo of particles and string thresholds employed in the model. In order to investigate the equilibrium behavior of hadron matter we also compare our transport (box) calculations with a simple Statistical Model (SM) for an Ideal Hadron Gas where the system is described by a grand canonical ensemble of non-interacting fermions and bosons in equilibrium at temperature T . All baryon and meson species considered in the transport model [23] also have been included in the statistical model [30]. Within the SM we find that the temperature increases continuously with energy density since the continuum excitations, i.e. the string degrees of freedom, are not included (full dots in the right panel of Fig. 3), whereas the box calculation with strings gives the limiting temperature (full squares in Fig. 3). Both curves in Fig. 3 have been calculated for density ρ 0 . To reproduce qualitatively our box result within the SM we have to include continuum excitations in the statistical model, i.e. a Hagedorn mass spectrum for strings [31] (for details see [30]). For the 'Hagedorn' temperature T H we use the temperature T s as obtained from the box calculations, i.e. T H = T s ≃ 150 MeV. As seen in right panel of Fig. 3 we achieve agreement of the extended SM and our box calculations. However, we point out that the limiting temperature T s from our string model involves somewhat different physics assumptions than the Hagedorn model at temperature T H . T s should not really be identified with the 'Hagedorn' temperature T H , though close similarities exist. In the Hagedorn picture and for temperatures close to T H the abundance of 'normal' hadrons or known resonances stays constant with increasing energy density whereas the number and energy density of the (hypothetical) bootstrap excitations diverges for T → T H . The Hagedorn model thus assumes 'particles' of mass m → ∞ to be populated for T → T H , that dynamically can be formed in collisions of high mass hadrons for t → ∞. In contrast, our string model does not include energetic string-string interactions that might produce more massive strings. SUMMARY In this contribution we have performed a systematic study of equilibration phenomena and equilibrium properties of 'infinite' hadronic matter as well as of relativistic nucleus-nucleus collisions using a BUU transport model that contains resonance and string degrees-of-freedom. The 'infinite' hadron matter is modelled by initializing the system at fixed baryon density, strange density and energy density by confining it in a cubic box with periodic boundary conditions [30]. We have shown that the equilibration times τ eq for different particles depend on baryon density and energy density. The time τ eq for non-strange particles is much shorter than for particles including strangeness; for kaons and antikaons the equilibration time is found to be larger than ≃ 40 fm/c for all baryon and energy densities considered. The overall abundance of the dominant strange particles (kaons and Λ's) being produced and obtained within the BUU cascade model for heavy-ion collisions can therefore not be described by assuming a perfect chemical equilibrium as strangeness is typically still undersaturated to a quite large extent. We mention that transport model calculations like ours can describe the yield and spectra of the produced nonstrange hadrons as well as K + , K − , Λ yields quite well at SPS energies [6,29]. On the other hand, at AGS energies the measured K + /π + ratio in central Au + Au collisions is underestimated by about 30% [32]. However, we have to point out that the more exotic strange particles (like the measured antihyperon yields of Ref. [33]) can by far not be explained within such standard hadronic multiple channel reactions. These hadronic data possibly point towards new physics. We have, furthermore, shown that thermal equilibrium is established quickly, within about 5 fm/c at SIS energies and samewhat larger times at high energies. The inclusion of continuum excitations, i.e. hadron 'strings', leads to a limiting temperature of T s ≃ 150 MeV in our transport approach which practically does not depend on the baryon density and energy. We have compared our results with the statistical model (SM), which contains the same degrees of freedom and the same spectral functions of particles as our transport model. We found that the limiting temperature behaviour can be reproduced in the statistical model only after including continuum excitations of the Hagedorn type, otherwise the fireball temperature extracted from the particle abundances and spectra is overestimated substantially. Close to the critical temperature T s , the hadronic energy densities can increase to a couple of GeV/fm 3 . From lattice QCD calculations one expects that a phase transition to a potentially deconfined QGP state should occur. Referring to the limiting temperature T s ≈ 150 MeV obtained, a QGP should be revealed and clearly distinguished from a hadronic state of matter if one can unambiguously prove the existence of an equilibrated and thermal phase of strongly interacting matter with temperatures exceeding, e.g., 200 MeV. The best candidates are electromagnetic probes, either direct photons or dileptons. On the other hand these are also 'contaminated' by hadronic background and/or preequilibrium physics. So far no thermal electromagnetic source with temperatures larger or equal than 200 MeV has been clearly identified. This might happen at RHIC energies in central Au + Au collisions which are expected to be studied soon. REFERENCES Figure 1 . 1Time evolution of the various particle abundances (nucleons N, ∆, Λ, π and K + mesons) for density ρ = ρ 0 (left panel) at energy density ε = 0.22 GeV/fm 3 and for ρ = 3ρ 0 (right panel) at ε = 0.66 GeV/fm 3 . Figure 3 . 3Left panel: equilibrium temperature T versus the energy density ε − m N ρ for different baryon densities ρ: 1/3ρ 0 (open down triangles), ρ 0 (full squares), 3ρ 0 (full dots), 6ρ 0 (full up triangles). Right panel: equilibrium temperature T versus the energy density for baryon density ρ = ρ 0 . 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[ "THE DIFFICULTY OF SYMPLECTIC ANALYSIS WITH SECOND CLASS SYSTEMS", "THE DIFFICULTY OF SYMPLECTIC ANALYSIS WITH SECOND CLASS SYSTEMS", "THE DIFFICULTY OF SYMPLECTIC ANALYSIS WITH SECOND CLASS SYSTEMS", "THE DIFFICULTY OF SYMPLECTIC ANALYSIS WITH SECOND CLASS SYSTEMS" ]
[ "A Shirzad [email protected]:[email protected] \nIntroduction\n\n", "M Mojiri ", "\nDepartment of Physics\nInstitute for Studies in Theoretical Physics and Mathematics\nIsfahan University of Technology Isfahan\nP. O. Box: 553119395TehranIRAN, IRAN\n", "A Shirzad [email protected]:[email protected] \nIntroduction\n\n", "M Mojiri ", "\nDepartment of Physics\nInstitute for Studies in Theoretical Physics and Mathematics\nIsfahan University of Technology Isfahan\nP. O. Box: 553119395TehranIRAN, IRAN\n" ]
[ "Introduction\n", "Department of Physics\nInstitute for Studies in Theoretical Physics and Mathematics\nIsfahan University of Technology Isfahan\nP. O. Box: 553119395TehranIRAN, IRAN", "Introduction\n", "Department of Physics\nInstitute for Studies in Theoretical Physics and Mathematics\nIsfahan University of Technology Isfahan\nP. O. Box: 553119395TehranIRAN, IRAN" ]
[]
Using the basic concepts of chain by chain method we show that the symplectic analysis, which was claimed to be equivalent to the usual Dirac method, fails when second class constraints are present. We propose a modification in symplectic analysis that solves the problem. 1
10.1063/1.1828588
[ "https://arxiv.org/pdf/hep-th/0405052v2.pdf" ]
10,206,235
hep-th/0405052
b71e3846faa3214ab811555e462f8836ed8175f8
THE DIFFICULTY OF SYMPLECTIC ANALYSIS WITH SECOND CLASS SYSTEMS 11 Oct 2004 A Shirzad [email protected]:[email protected] Introduction M Mojiri Department of Physics Institute for Studies in Theoretical Physics and Mathematics Isfahan University of Technology Isfahan P. O. Box: 553119395TehranIRAN, IRAN THE DIFFICULTY OF SYMPLECTIC ANALYSIS WITH SECOND CLASS SYSTEMS 11 Oct 2004 Using the basic concepts of chain by chain method we show that the symplectic analysis, which was claimed to be equivalent to the usual Dirac method, fails when second class constraints are present. We propose a modification in symplectic analysis that solves the problem. 1 Introduction There are some attempts to study a constrained system in the framework of first order Lagrangian [1,2]. The coordinates appearing in a first order Lagrangian are in fact the phase space coordinates. The Euler-Lagrange equations of motion of a first order Lagrangian in an ordinary (non-constrained) system are the same as the canonical equations of motion. The kinetic term in a first order Lagrangian constitutes of a one-form whose exterior derivative appears in the equations of motion. The resulted two-form, called the symplectic tensor, is singular for a constrained system. If the system is not constrained, usually the inverse of the symplectic tensor exists and provides the fundamental Poisson brackets (we exclude degenerate systems discussed in [17,18] in which the symplectic tensor may have a lower rank in some regions of the phase space). The properties of a constrained system can be determined by trying to overcome the singularity of the symplectic tensor. Faddeev and Jackiw [3] used the Darboux theorem to separate canonical and non-canonical coordinates.They solved the equations of motion for non-canonical coordinates either to decrease the degrees of singularity of the symplectic tensor or to find the next level constraints. Then using a special system of coordinates, the authors of [4] showed that the Faddeev-Jackiw approach is essentially equivalent to the usual Dirac method [5]. In a parallel approach, known as symplectic analysis [6,7,8,9] one extends the phase space to include the Lagrange multipliers. In this approach the consistency of constraints at each level adds some additional elements to the symplectic tensor. In other words, the kinetic part of the (first order) Lagrangian is responsible to impose the consistency conditions. The important point in most papers written in Faddeev-Jackiw method or symplectic analysis is that they often show their results for the constraints in the first level and then deduce that the same thing would be repeated at any level. However, the whole procedure of studying the singularities of symplectic tensor, demonstrates some global aspects. For example, some questions that may arise are as follows: What happens, after all, to the symplectic tensor? Is it ultimately singular? How many degrees of singularity may it have? What is the relation of ultimate singularities with the gauge symmetries of the system? and so on. In [10] we showed that the symplectic analysis gives, at each step, the same results as the traditional Dirac method (in the framework of level by level approach). The symplectic analysis may also be studied in the framework of chain by chain approach [11] to obtain the Dirac constraints. Meanwhile, some recent observation [12] shows that in some examples the result of symplectic analysis and the well-established method of Dirac are not the same. This creates serious doubt about the validity of the symplectic analysis. Therefore, it is worth studying the origin of the difference between this approach and that of Dirac [5]. This is the aim of this paper. In the next section we first review the basic concept of symplectic approach as given in [10]. As we will show the symplectic analysis is equivalent to a special procedure in Dirac approach in which one uses the extended Hamiltonian at each level of consistency. In section(3) we will show that in the framework of Dirac method one is not allowed to use an extended Hamiltonian when there exist second class constraints. The important point to be emphasized is that this result can be understood more clearly in the framework of chain by chain method. In section(4) we show that for a one chain system with second class constraints the symplectic analysis as proposed in the literature fails. This result can be simply generalized to the general case of a multi-chain system. When recognizing the origin of the problem, we give our prescription to solve it in section (5). Finally in section(6) we give an example. The last point to be noticed is that the problem would not show itself for systems with two levels of constraints. As we will show, this is the case for second class systems with at least four levels of constraints. That is the reason for the fact that the problem does not appear if one considers just first level of constraints. Review of symplectic approach Consider a phase space with coordinates y i (i = 1, . . . , 2K) specified by the first order Lagrangian L = a i (y)ẏ i − H(y)(1) where H(y) is the canonical Hamiltonian of the system. The equations of motion read f ijẏ j = ∂ i H(2) where ∂ i ≡ ∂ ∂y i and the presymplectic tensor f ij is defined as f ij ≡ ∂ i a j (y) − ∂ j a i (y).(3) We denote it in matrix notation as f . This matrix is invertible for a regular system. Let f ij be the components of the inverse, f −1 . From (2) we havė y i = y i , H ,(4) where the Poisson bracket { , } is defined as {F (y), G(y)} = f ij ∂ i F ∂ j G.(5) If f is singular, then using the Darboux theorem, as shown in [3], one can choose the independent coordinates (y ′α , λ l ) such that L = a ′ αẏ ′α − λ l Φ l (y ′ ) − H(y ′ ) (6) where f ′ αβ = ∂ α a ′ β − ∂ β a ′ α is invertible. This shows that one can consider a system with a singular tensor f ij , as a regular one described by L = a ′ αẏ ′α − H(y ′ )(7) together with by the primary constraints Φ l (y ′ ). In other words, without losing the generality one can assume that one is at first given the first order Lagrangian (1) with a regular presymplectic two-form (3), and then the set of primary constraints Φ (1) µ (µ = 1, · · · , M) are applied to the system. In this way the system is described by the Lagrangian L = a iẏ i − λ µ Φ (1) µ − H(y)(8) in the extended space (y i , λ µ ). The equations of motion (2) should be replaced in matrix form by f 0 0 0 ẏ λ = ∂H Φ (1)(9) which is equivalent to Eq. (2) together with the constraint equations Φ (1) µ = 0 (µ = 1, · · · , M). Now one should impose the consistency conditionsΦ (1) µ = 0. To do this, one should extend the space to include new variables η µ and add the term η µΦ (1) µ (or equivalently −η µ Φ (1) µ ) to the Lagrangian (8). This leads in the extended space (y, λ, η) to the equations    f 0 A 0 0 0 −à 0 0      ẏ λ η    =    ∂H Φ (1) 0   (10) where the elements of the rectangular matrix A are given by A µi = ∂ i Φ (1) µ .(11) However, nothing would be lost if one forgets about the variables λ µ and reduces the system to the Lagrangian L (1) = a iẏ i −η µ Φ (1) µ − H(y).(12) This leads to the symplectic two-form F = f A −à 0 (13) in the (2K + M) dimensional space of variables Y ≡ (y i , η µ ). It should be noted that the Lagrangian L (1) in Eq. (12) is the same as Eq. (8) in which λ µ is replaced byη µ . This means that the derivativesη µ have the same role as Lagrangian multipliers λ µ corresponding to primary constraints in the total Hamiltonian H T = H + λ µ Φ (1) µ .(14) In other words, if some ofη µ 's are found by the dynamical equations of the system, then the corresponding Lagrange multipliers are obtained. In Dirac approach [14] this would be the case if there exist some second class constraints. The equations of motion due to the Lagrangian L (1) can be written in matrix notation as FẎ = ∂H.(15) Using operations that keep the determinant invariant, it is easy to show that det F = det f A 0Ãf −1 A = (det f )(detÃf −1 A).(16) Since det f = 0, F would be singular if C ≡Ãf −1 A is singular. Using (5) and (11) we have C µν = Φ (1) µ , Φ (1) ν .(17) Suppose rank(C) = M ′′ where M ′′ ≤ M. This means that F possesses M ′ = M − M ′′ null-eigenvectors. One can, in principle, divide Φ (1) µ 's in two sets Φ (1) µ ′ and Φ (1) µ ′′ such that Φ (1) µ ′ , Φ (1) ν ≈ 0 Φ (1) µ ′′ , Φ (1) ν ′′ ≈ C µ ′′ ν ′′ , det C µ ′′ ν ′′ = 0.(18) where the weak equality symbol ≈ means equality on the surface of the constraints already known (here, the primary constraints). The matrix A can be decomposed to A ′ and A ′′ such that A µ ′ i = ∂ i Φ (1) µ ′ A µ ′′ i = ∂ i Φ (1) µ ′′ .(19) Accordingly the symplectic tensor F can be written as F =    f A ′′ A ′ −à ′′ 0 0 −à ′ 0 0    .(20) Consider the rectangular matrix à ′ f −1 , 0, 1(21) which has M ′ rows and 2K + M columns. Using (18) one can show that its rows are left null-eigenvectors of F . Multiplying (21) with the equations of motion (15) gives the second level constraints as Φ (2) µ ′ ≈ Φ (1) µ ′ , H = 0.(22) On the other hand, F in (20) has an invertible sub-block F inv = f A ′′ −à ′′ 0(23) with the inverse F −1 inv = f −1 − f −1 A ′′ C ′′−1Ã′′ f −1 −f −1 A ′′ C ′′−1 C ′′−1Ã′′ f −1 C ′′−1 .(24) This can solve the equations of motion (15) for variablesη µ ′′ to givė η µ ′′ = −C µ ′′ ν ′′ Φ (1) ν ′′ , H(25) where C µ ′′ ν ′′ is the inverse of C µ ′′ ν ′′ . Inserting this in the Lagrangian (12) gives L (1) = a i (y)ẏ i −η µ ′ Φ (1) µ ′ − H (1) (y)(26) where H (1) = H − H, Φ (1) µ ′′ C µ ′′ ν ′′ Φ (1) ν ′′ .(27) In this way a number of Lagrange multipliers corresponding to the second class constraints are derived whose effect is only replacing the canonical Hamiltonian H with H (1) . Now we can forget about them and suppose that we are given the primary constraints Φ (1) µ and the second level constraints φ (2) µ . Next, we should consider the consistency of Φ (2) µ and add the term −η µ 2 Φ (2) µ to the Lagrangian L (1) . Renaming the previous η µ ′ 's as η µ 1 , the new Lagrangian would be L (2) = a i (y)ẏ i −η µ 1 Φ (1) µ −η µ 2 Φ (2) µ − H (1) (y)(28) this gives the symplectic two-form F (2) =    f A (1) A (2) −à (1) 0 0 −à (2) 0 0   (29) in the space (y, η 1 , η 2 ). Assuming that the composed matrix A ≡ A (0) , A (1) , F (2) has the same from as (13). One should again proceed in the same way to find the null-eigenvectors as well as the invertible sub-block of F (2) . The process goes on in this and the subsequent steps as explained in more detail in [10]. The important point to be emphasized is that the Lagrangian L (n) = a i (y)ẏ i − n k=1η µ k Φ (k) µ − H (n) (y)(30) at the n-th level, say, is equivalent to a system with extended Hamiltonian H (n) E = H (n−1) + n k=1 λ µ k Φ (k) µ(31) at that level. In other words, the symplectic analysis is equivalent to the Dirac approach in the context of level by level method provided that at each level one adds the new constraints with the corresponding Lagrange multipliers to the Hamiltonian. In fact this slight difference with the standard Dirac method may lead to some difficulties as we will see in the following section. The problem with extended Hamiltonian The extended Hamiltonian formalism is well-known in the context of first class constraints [13,14]. In fact, it can be shown that the dynamical equatioṅ g = {g, H E } ,(32) leads to the correct equation of motion provided that g is a gauge invariant quantity. In Eq. (32) the extended Hamiltonian H E is defined as H E = H + λ m Φ m(33) where Φ m are only first class constraints (primary or secondary). For a first class system, the extended Hamiltonian can also be used step by step during the process of producing the constraints . In other words, when all of the constraints are first class, there is no difference whether one usesΦ = {Φ, H T } orΦ = {Φ, H E }. Now we show that the extended Hamiltonian formalism in Dirac approach is not suitable when second class constraints are present. We show this point for a system with only one primary constraint, i. e. a one-chain system in the language of chain by chain method. We remember that for such a system level by level and chain by chain methods coincide. Consider a system with the canonical Hamiltonian H(y) and one primary constraint Φ (1) . The total Hamiltonian reads H T = H + λΦ (1) .(34) Suppose the consistency of Φ (1) leads to Φ (2) = Φ (1) , H . Then Φ (3) emerges as Φ (2) , H , and so on. The iterative process that produces the constraints is described by Φ (n+1) = Φ (n) , H .(35) The above procedure progresses unless Φ (N ) , H T ≈ 0 or Φ (N ) , Φ (1) = 0 at the last step N. In the former case the constraints in the chain are first class, i.e. commute with each other [11]; while in the latter all the constraints are second class which means that the matrix C nm = Φ (n) , Φ (m)(36) is invertible. In this case the Lagrange multiplier λ would finally be determined as λ = Φ (N ) , H {Φ (N ) , Φ (1) } .(37) Using the Jacobi identity, it is shown in [11] that the matrix C nm in Eq. (36) has the following form C ≈          0 0 · · · 0 C 1N 0 0 · · · C 2(N −1) C 2N . . . . . . . . . . . . 0 C (N −1)2 · · · C (N −1)(N −1) C (N −1)N C N 1 C N 2 · · · C N (N −1) C N N          .(38) In other words Φ (i) , Φ (j) ≈ 0 if i + j ≤ N.(39) Moreover using the Jacobi identity one can show from (35)that Φ (1) , Φ (N ) ≈ − Φ (2) , Φ (N −1) ≈ · · · ≈ (−1) ( N 2 −1) Φ ( N 2 ) , Φ ( N 2 +1) = 0. (40) Remember that N is the number of second class constraints and necessarily should be even. Now suppose that in order to define the dynamics of the system at some level n, one wishes to use the extended Hamiltonian H (n) E = H + n k=1 λ k Φ (k) .(41) If n ≤ N 2 then from (38) the consistency of the constraint Φ (n) giveṡ Φ (n) = Φ (n) , H (n+1) E ≈ Φ (n) , H(42) which by (35), is the same as Φ (n+1) . However at level N 2 + 1 the consistency of Φ ( N 2 +1) , using H ( N 2 +1) E giveṡ Φ ( N 2 +1) = Φ ( N 2 +1) , H + λ N 2 Φ ( N 2 +1) , Φ ( N 2 ) .(43) As is apparent from (40) the above equation solves the Lagrange multiplier λ N 2 . There is no justification to keep Φ ( N 2 +1) , H as the next constraint Φ ( N 2 +2) . In order to knit the second class chain up to the last element Φ (N ) , one is just allowed to use the total Hamiltonian (34). In other words, the second half of the chain can be derived if only the primary constraint Φ (1) is present in the corresponding Hamiltonian. As explained in the previous section, using the standard symplectic analysis is equivalent to using the extended Hamiltonian formalism described above. So one should expect some contradiction in symplectic analysis when second class constraints are present. In the next section we will show the essence of this contradiction for a one chain system and propose a method to resolve it. Second class one -chain in symplectic analysis According to the algorithm given in section 2, given the canonical Hamiltonian H(y) and the primary constraint Φ (1) µ , at the first step of consistency one should consider the Lagrangian (see 12) L (1) = a iẏ i −η 1 Φ (1) − H(y).(44) The equations of motion can be written in matrix form as f A (1) −à (1) 0 ẏ η 1 = ∂H 0 .(45) It is easy to see that u 1 ≡ à (1) f −1 , 1(46) is the null-eigenvector of the matrix F = f A (1) −à (1) 0 .(47) Implying u 1 on both sides of (45) and using (5) gives the new constraint Φ (2) = Φ (1) , H .(48) Adding the term −η 2 Φ (2) to the Lagrangian (to perform consistency) gives L (2) = a iẏ i −η 1 Φ (1) −η 2 Φ (2) − H(y).(49) The equations of motion are    f A (1) A (2) −à (1) 0 0 −à (2) 0 0      ẏ η 1 η 2    =    ∂H 0 0   (50) Assuming Φ (1) , Φ (2) ≈ 0, one can find the new null eigenvector u 2 ≡ à (2) f −1 , 0, 1 .(51) Multiplying u 2 by (50) gives the new constraint Φ (3) = Φ (2) , H , and so on. Suppose one wishes to proceed in this way to find the constraints of the chain discussed in the previous section, i.e. the second class chain Φ (1) , · · · , Φ (N ) with the algebra given in (38-40). Suppose the above procedure has been proceeded up to the step N 2 + 1 where the equations of motion are        f A (1) · · · A ( N 2 +1) −à (1) 0 · · · 0 . . . . . . . . . . . . −à ( N 2 +1) 0 · · · 0              ẏ η 1 . . . ηN 2 +1        =       ∂H 0 . . . 0       .(52) Clearly no more null-eigenvector can be find. In fact adding the column and row corresponding to the constraint Φ ( N 2 +1) has increased the rank of the matrix F by two. This means that the equations of motion can be solved to findη ( N 2 ) andη ( N 2 +1) . There is no way in the context of symplectic analysis to proceed further to find the remaining constraints Φ ( N 2 +2) , · · · , Φ (N ) of the chain. This is really the failure of traditional symplectic analysis. In fact this is the reason why the symplectic analysis has failed in the example given in [12] (Particle in hyper sphere). We will discuss this example in section (6). What we showed here is the failure of symplectic analysis for a second class system with only one primary constraint (i.e. a one chain system). However, one can easily observe that for an arbitrary system with several primary constraints again the symplectic analysis would fail. The reason is that for such a system some of the constraints driven at level n, i.e. Φ (n) µ , may have non vanishing Poisson brackets with constraints of previous levels while commuting with primary constraints. As we know from Dirac approach, in such a case the Poisson brackets of these constraints with Hamiltonian give the next level constraints. Meanwhile, a little care on symplectic analysis shows that in this case a number of Lagrange Multipliers corresponding to non-primary constraints would be determined and there is no way to find the next level constraints. In this way, we conclude that the symplectic analysis would fail whenever second class constraints emerge at third level or higher. How to solve the problem In this section we try to find a way to maintain the symplectic analysis by imposing some modifications. The origin of the problem is the fact that Φ ( N 2 +1) has non-vanishing Poisson bracket with Φ ( N 2 ) . As a result, the symplectic two-form on the left hand side of Eq. (52), i.e. F =        f A (1) · · · A ( N 2 +1) −à (1) 0 · · · 0 . . . . . . . . . . . . −à ( N 2 +1) 0 · · · 0        ,(53) does not possess a new null-eigenvector. If one could consider the vector u ( N 2 +1) ≡ à ( N 2 +1) f −1 , 0, , · · · , 0, 1 ,(54) as a null-eigenvector, then by multiplying u ( N 2 +1) on the right hand side of (52), one would obtain the next constraint as Φ ( N 2 +2) = Φ ( N 2 +1) , H .(55) To reach this goal one should truncate those columns of F which are located after A (1) . In other words, instead of F in Eq. (53) one should consider the rectangular matrix F =        f A (1) −à (1) 0 . . . . . . −à ( N 2 +1) 0        .(56) Clearly u ( N 2 +1) in Eq. (54) is the null-eigenvector ofF . It is obvious that if one does the same thing in the subsequent steps, one can produce all the remaining constraints of the chain, i.e. Φ ( N 2 +1) , · · · , Φ (N ) . In the last step the chain terminates, since Φ (N ) , Φ (1) = 0. But what is the justification to find the null-eigenvectors ofF , i.e. the truncated F . In fact using Eq. (5) the set of equations        f A (1) −à (1) 0 . . . . . . −à (N ) 0        ẏ η 1 =       ∂H 0 . . . 0       .(57) is equivalent toẏ i = y i , H +η 1 Φ (1) i = 1, · · · , 2K Φ (j) = 0 j = 1, · · · , N.(58) Remembering thatη 1 has the same role as the Lagrange multiplier λ 1 corresponding to the primary constraint Φ (1) , we see that Eq. (58) is the correct equation of motionẏ i = {y i , H T } .(59) On the other hand, it is easy to see that the equations of motion resulting from Eq. (52) can be written asẏ i = {y i , H E }(60) where H E contains all derived constraint(including second class ones). In fact as we explained before, the correct equations of motion are (58) and not (60). Therefore, if one wishes to proceed in the context of symplectic analysis, one should consider Eq.(57) instead of Eq.(52) Example Consider the Lagrangian L = 1 2q 2 + v q 2 − 1(61) where q ≡ (q 1 , · · · , q n ). The primary constraint is P v . The corresponding Hamiltonian is H = 1 2 p 2 − v q 2 − 1(62) where p ≡ (p 1 , · · · , p n ). In the usual Dirac approach, using the total Hamiltonian H T = H + λP v , the consistency of Φ (1) = P v gives the following chain of constraints Φ (1) = P v Φ (2) = q 2 − 1 Φ (3) = 2q.p Φ (4) = 2 (p 2 + 2vq 2 )(63) As is apparent, Φ (4) and Φ (3) are conjugate to Φ (1) and Φ (2) respectively. It is worth remembering that although Φ (3) is second class, when reaching at third level, the process of consistency should not stop, i. e. it should be proceeded one level more to find Φ (4) which is conjugate to the primary constraint Φ (1) . In the symplectic approach the corresponding first order Lagrangian is L = pq + P vv − 1 2 p 2 + v q 2 − 1 − λP v .(64) This gives the singular presymplectic tensor F = f 0 0 0 (65) where f is the usual (2n + 2) × (2n + 2) symplectic tensor: f = 0 −1 1 0 .(66) The equations of motion for y i = (q, v, p, P v , λ) are f ijẏ j = ∂ i H T where H T = H + λP v . Clearly this gives the canonical equation of motion with Hamiltonian H T , together with the constraint equation P v = 0. Adding the consistency term −η 1 P v to the Lagrangian (64), where η 1 is a new variable and forgetting about the term proportional to λ (which just reproduces the primary constraint) one finds L (1) = pq + P vv −η 1 P v − 1 2 p 2 + v q 2 − 1 .(67) This gives the equations of motion F (1) ijẎ j = ∂ i H(68) where Y i ≡ (q, v, p, P v , η 1 ). In the matrix form we have F (1) = f A (1) −à (1) 0(69) whereà (1) = (0, 0, 0, 1). Here, bold zero(0) means a row vector with n zero components. Clearly u (1) = (0, −1, 0, 0, 1) is the left null-eigenvector of F (1) . Multiplying the equations of motion (68) from the left by u (1) gives the constraint Φ (2) = q 2 − 1. In the next level we have the Lagrangian L (2) = L −η 1 P v −η 2 (q 2 − 1)(70) written in the space Y i ≡ (q, v, p, P v , η 1 , η 2 ). The corresponding symplectic tensor reads F (2) =    f A (1) A (2) −à (1) 0 0 −à (2) 0 0   (71) whereà (2) = (2q, 0, 0, 0). Clearly u (2) = (0, 0, 2q, 0, 0, 1) is the null-eigenvector of F 2 . Multiplying the equations of motion F (2) ijẎ j = ∂ i H T from the left by u (2) gives the next level constraint Φ (3) = 2q.p. Again considering another variable η 3 , the third level Lagrangian would be L (3) = L −η 1 P v −η 2 (q 2 − 1) −η 3 (2q.p).(72) This gives the following symplectic tensor F (3) =       f A (1) A (2) A (3) −à (1) 0 0 0 −à (2) 0 0 0 −à (3) 0 0 0      (73) whereà (3) = (2p, 0, 2q, 0). Now the crucial point appears. That is, F (3) has no new null-eigenvector. In fact one expects that multiplying u (3) = (−2q, 0, 2p, 0, 0, 0, 1) by the equations of motion due to L (3) gives the next constraint Φ (4) = 2 (p 2 + 2vq 2 ). However, it can be easily checked that u (3) F (3) = 0. Moreover, u (2) (with one additional zero as the last element) is no more the null-eigenvector of F (3) . This means that adding the (2n + 5)th row and columns to F (2) has led to increasing the rank of F (3) by two. In other words, the equations of motion forη 2 andη 3 can be solved. Unfortunately without any modification there is no way to find the Lagrangian L (4) = L −η 1 P v −η 2 q 2 − 1 −η 3 (2q.p) −η 4 2 p 2 + 2vq 2 . If we could find L (4) , then we would be able to have F (4) =         f A (1) A (2) A (3) A (4) −à (1) 0 0 0 0 −à (2) 0 0 0 0 −à (3) 0 0 0 0 −à (4) 0 0 0 0        (75) whereà (4) = (8vq, 4q 2 , 4p, 0). If we had somehow derived (74) and (75), then the singularity of symplectic tensor would completely disappear anḋ η 1 , · · ·η 4 would be obtained. However, using the truncated symplectic tensor at the second step asF (2) =    f A (1) −à (1) 0 −à (2) 0   (76) and similarlyF (3) at the third level as F (3) =       f A (1) −à (1) 0 −à (2) 0 −à (3) 0      (77) makes it possible to introduce again u (2) and u (3) as the corresponding left null-eigenvectors ofF (2) andF (3) , respectively. This makes us able to find Φ (4) as explained before. It should be noted that one can after all write the complete symplectic tensor F (4) . This example has also been discussed in [12], where some other reason is proposed as the origin of failure of the symplectic analysis. The same results as what we derived here can be found in every second class system possessing at least four levels of constraints. For example, one can study the simpler Lagrangian L =ẋẏ − z(x + y) as well as the more complicated example of bosonized Schwinger model in (1 + 1) dimensions [15,16] given by L = 1 2 ∂ µ φ∂ µ φ + (g µν − ε µν )∂ µ φA ν − 1 4 F µν F µν + 1 2 A µ A µ .(78) One can see that the main feature of the above calculations will more or less appear in all such examples. AcknowledgmentWe thank Esmaeil Mosaffa for reading the manuscripts. . M Gotay, J M Nester, G Hinds, J. Math. Phys. 192388M. Gotay, J. M. Nester and G. Hinds, J. Math. Phys. 19, 2388(1978) . S Hojman, F Urrutia, J. Math. Phys. 221896S. Hojman and F. Urrutia, J. Math. Phys. 22, 1896(1981) . L Faddeev, R Jackiw, Phys. Rev. Lett. 601692L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692(1988). . J A Garcia, J M Pons, Int. J. Mod. Phys. 12451J. A. Garcia and J. M. Pons, Int. J. Mod. Phys. A12, 451(1997). P A M Dirac, Lectures on Quantum Mechanics. New YorkYeshiva University PressP. A. M. Dirac, "Lectures on Quantum Mechanics" New York: Yeshiva University Press (1964). . J Barcelos-Neto, C Wotzasck, Mod. Phys. Lett. 71737J. Barcelos-Neto and C. Wotzasck, Mod. Phys. Lett. A7, 1737(1992). . J Barcelos-Neto, C Wotzasck, Int. Mod. Phys. Lett. 74981J. Barcelos-Neto and C. Wotzasck, Int. Mod. Phys. Lett. A7, 4981(1992). . J Barcelos-Neto, N R F Beraga, J. Math. Phys. 3573497J. Barcelos-Neto and N. R. F. Beraga, J. Math. Phys. 35, No. 7, 3497(1994). . H , Montani Int, Mod. Phys. 84319H. Montani Int. Mod. Phys. A8, 4319(1993). . A Shirzad, M Mojiri, Mod. Phys. Lett. 16382439A. Shirzad and M. Mojiri, Mod. Phys. Lett. A16, No. 38, 2439(2001). . F Loran, A Shirzad, Int. J. Mod. Phys. 17625F. Loran and A. Shirzad, Int. J. Mod. Phys. A17, 625(2002). . H J Rothe, K D Rothe, J. Phys. 361671H. J. Rothe and K. D. Rothe, J. Phys. A36, 1671(2003). Quantization of Gauge Systems. M Henneaux, C Teitelboim, Princton Univ. PressM. Henneaux and C. Teitelboim, "Quantization of Gauge Systems", Princton Univ. Press (1992). . C Batlle, J M Pons, N Roman-Roy, J. Math. Phys. 272953C. Batlle, J. M. Pons and N. Roman-Roy, J. Math. Phys. 27, 2953(1986). . P Mitra, R Rajaraman, Phys. Lett. 225267P. Mitra and R. Rajaraman, Phys. Lett. B225, 267(1989). . R Jackiw, R Rajaraman, Phys. Rev. Lett. 541219R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54, 1219(1985). . J Saavedra, R Troncoso, J Zanelli, J. Math. Phys. 424383J. Saavedra, R. Troncoso and J. Zanelli, J. Math. Phys. 42, 4383(2001). . O Miskovic, J Zanelli, J. Math. Phys. 443876O. Miskovic and J. Zanelli, J. Math. Phys. 44, 3876(2003).
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[ "Differentially Private Diffusion Auction: The Single-unit Case", "Differentially Private Diffusion Auction: The Single-unit Case" ]
[ "Fengjuan Jia \nSchool of Computer Science and Engineering\nUniversity of Electronic Science\nTechnology of China\n", "Mengxiao Zhang \nSchool of Computer Science and Engineering\nUniversity of Electronic Science\nTechnology of China\n", "Jiamou Liu \nSchool of Computer Science\nThe University of Auckland\n\n", "Bakh Khoussainov \nSchool of Computer Science and Engineering\nUniversity of Electronic Science\nTechnology of China\n" ]
[ "School of Computer Science and Engineering\nUniversity of Electronic Science\nTechnology of China", "School of Computer Science and Engineering\nUniversity of Electronic Science\nTechnology of China", "School of Computer Science\nThe University of Auckland\n", "School of Computer Science and Engineering\nUniversity of Electronic Science\nTechnology of China" ]
[]
Diffusion auction refers to an emerging paradigm of online marketplace where an auctioneer utilises a social network to attract potential buyers. Diffusion auction poses significant privacy risks. From the auction outcome, it is possible to infer hidden, and potentially sensitive, preferences of buyers. To mitigate such risks, we initiate the study of differential privacy (DP) in diffusion auction mechanisms. DP is a well-established notion of privacy that protects a system against inference attacks. Achieving DP in diffusion auctions is non-trivial as the welldesigned auction rules are required to incentivise the buyers to truthfully report their neighbourhood. We study the single-unit case and design two differentially private diffusion mechanisms (DPDMs): recursive DPDM and layered DPDM. We prove that these mechanisms guarantee differential privacy, incentive compatibility and individual rationality for both valuations and neighbourhood. We then empirically compare their performance on real and synthetic datasets.
10.48550/arxiv.2302.07072
[ "https://export.arxiv.org/pdf/2302.07072v2.pdf" ]
256,901,265
2302.07072
c45cedacab1ddcf310fdb636cc275ac6c2beb5f7
Differentially Private Diffusion Auction: The Single-unit Case Fengjuan Jia School of Computer Science and Engineering University of Electronic Science Technology of China Mengxiao Zhang School of Computer Science and Engineering University of Electronic Science Technology of China Jiamou Liu School of Computer Science The University of Auckland Bakh Khoussainov School of Computer Science and Engineering University of Electronic Science Technology of China Differentially Private Diffusion Auction: The Single-unit Case Diffusion auction refers to an emerging paradigm of online marketplace where an auctioneer utilises a social network to attract potential buyers. Diffusion auction poses significant privacy risks. From the auction outcome, it is possible to infer hidden, and potentially sensitive, preferences of buyers. To mitigate such risks, we initiate the study of differential privacy (DP) in diffusion auction mechanisms. DP is a well-established notion of privacy that protects a system against inference attacks. Achieving DP in diffusion auctions is non-trivial as the welldesigned auction rules are required to incentivise the buyers to truthfully report their neighbourhood. We study the single-unit case and design two differentially private diffusion mechanisms (DPDMs): recursive DPDM and layered DPDM. We prove that these mechanisms guarantee differential privacy, incentive compatibility and individual rationality for both valuations and neighbourhood. We then empirically compare their performance on real and synthetic datasets. INTRODUCTION New technological shift in AI and data science has given rise to an imminent need to address data privacy issues in online platforms. Indeed, a Gartner survey shows that 41% of the surveyed organisations have experienced a privacy breach or security incident 1 . Data privacy issues have been especially serious and impactful around the use of social commerce platforms such as Instagram and Facebook. As users of such a platform find, browse and buy products through the social network, they are also exposed to a significant risk of privacy leakage. A recent PCI Pal survey shows that fewer than 7% of users are confident about their data security on social commerce sites 2 . Thus designing new tools to facilitate safe and private use of social commerce platforms is of crucial importance. Auction is important in facilitating online commerce. Auctions have been applied in many contexts, e.g., radio spectrum, sponsored search ads, virtual resource allocation. In an auction, buyers submit their (private) valuations in bids to the auctioneer. The bids often imply buyers' preferences and confidential business strategies, and competitors may exploit them to gain an advantage. Hence, there is a need to protect the privacy of bid information. The privacy issues in auctions have recently been studied in [McSherry and Talwar, 2007, Jian et al., 2018, Zhu et al., 2014, Ni et al., 2021. To mitigate privacy risks, these studies employ the wellestablished notion of differential privacy (DP) [Dwork et al., 2006]. Here, DP is used to protect individual's bid information when the auction outcome is published. To achieve DP on bids, the work of McSherry and Talwar [2007] proposed exponential mechanism. The mechanism randomises auction results so that a change in a buyer's bid does not significantly affect the auction outcome. In this way, the mechanism prevents the bid from being inferred from the auction outcome. This mechanism has so far been a predominant method to protect privacy in auctions. Diffusion auction is an emerging form of auction. In this setting, a seller is able to harness the power of social network to diffuse auction information, inviting friends, friends-offriends, etc., to join the auction, thereby attracting a large number of potential buyers. This differs from a standard auction (without social network) where the participants are fixed beforehand. Thus, diffusion auction are especially suitable for facilitating online social commerce platforms where the social network plays a prominent role. A challenge in diffusion auctions lies in resolving the conflict between Figure 1: A social network with a seller s and seven buyers. The number beside each node is the valuations of the buyer. The seller s has an item to sell, and initially knows only a, b, c. The mechanism will construct a probability distribution over potential buyers which determines how likely a buyer is to win the item. the seller who wants to attract more participants for better revenue and the buyers who are reluctant to invite their friends to avoid competition. Thus there is a need to extend incentive compatibility (IC) for hidden valuations in classical auctions, to diffusion IC for hidden valuation as well as social ties. Numerous studies, e.g., [Li et al., 2017, Zhang et al., 2020b, have proposed mechanisms for diffusion auction that achieve diffusion IC. Diffusion auctions are prone to all aforementioned privacy risks for auctions in general. However, no study has focused on the privacy issues for diffusion auctions. Here we close this gap by investigating the following question: How do we design a differentially private diffusion mechanism (DPDM) that guarantees desirable properties and preserves valuation privacy? Answering this question is not a trivial task. As mentioned above, the exponential mechanism is the main approach to ensure DP for auctions. An exponential mechanism firstly creates a probability distribution over all possible auction results such that more preferable result is associated with a higher probability, and then outputs an auction result according to the distribution. However, this mechanism can not be directly extended to diffusion auctions as it fails to ensure diffusion IC property. For instance, run the exponential mechanism to the scenario in Figure 1 (See Example 4.1 for a detailed implementation). Assume that all buyers except buyer b reveal their neighbours truthfully. From b's perspective, revealing her neighbour f means getting a lower probability of winning the auction, as the exponential mechanism would distribute the winning probabilities over 7 buyers instead of 5. Therefore, the buyers are not incentivised to diffuse auction information to their friends. Contribution. In this paper, we design a DPDM for the single-unit auction case where a single seller sells one indivisible item to multiple potential buyers. The seller and the buyers are assumed to be nodes in a social network with their connections represented as edges. The seller initially only has access to her direct neighbours, and must incentivise the buyers to truthfully report both their valuations of the item, and their neighbourhood. At the same time, the DPDM should ensure the DP property for buyers' bids. We design two DPDMs: recursive DPDM and layered DPDM. The idea for these two mechanisms is market division that partitions the buyers into sub-markets. The mechanism then associates a probability with each sub-market. To ensure diffusion IC, the probability should be monotonic on the size of the sub-markets: • The recursive DPDM maps the network into a tree that captures information flow among buyers. Then it recursively divides the market such that each sub-tree is a submarket and its probability is non-decreasing on the size of the sub-tree. • The layered DPDM also relies on the tree above, except the market is not partitioned by sub-trees, but rather by buyers' distances from the seller. In this way, each layer is a sub-market and its probability is fixed. These two mechanisms are proven to meet all the desirable properties. The layered DPDM has a lower bound on expected social welfare. The recursive DPDM achieves a better social welfare empirically. We demonstrate this using a series of experiments that simulate diffusion auctions over three real-world social network datasets. Our experiments reveal that in most cases, the recursive DPDM reaches comparable social welfare as the theoretical upper bound. We now highlight our contributions: 1. We expand diffusion mechanisms adding the DP condition. This builds a bridge between diffusion auctions and privacy preservation. See Section 3.2. 2. Using the idea of market division, we design recursive DPDM (Section 4) and layered DPDM (Section 5). These mechanisms are IC and differentially private. 3. We empirically evaluate our two mechnaisms on realworld network datasets. See Section 6. RELATED WORK Differentially private mechanism. Differential privacy (DP) is proposed to protect individual data from inference attacks on an aggregate query over a database [Dwork et al., 2006]. The notion has since been extended to various domain such as statistical data inference [Dwork, 2008], decision trees [Fletcher and Islam, 2019], and unstructured data [Zhao and Chen, 2022]. McSherry and Talwar [2007] extend DP to auctions and propose exponential mechanism. This mechanism ensures a weaker version of IC, namely approximate IC, which ensures that any user can only gain a bounded extra utility from misreporting. This solution concept is adopted in subsequent studies [Zhu et al., 2014] and [Diana et al., 2020] on multi-item auctions and one-shot double auctions. As approximate IC allows bidders to have non-zero incentives to lie, these methods would not meet the requirements in our problem. Many works design DP auctions that ensure traditional version of IC [Huang and Kannan, 2012, Xiao, 2013, Zhu and Shin, 2015, Jian et al., 2018. Specifically, Huang and Kannan [2012], Xiao [2013] propose general methods to transform a classical IC mechanism to a privacy preserving counterpart that is still IC. The method of [Xiao, 2013] works only when the valuation space is small and can not be applied to general problems, including ours. In contrast, the transformation method in [Huang and Kannan, 2012] can be applied to more general problems. The transformed mechanism can be seen as a generalisation of Vickrey-Clarke-Groves (VCG) mechanism [Groves, 1973], which is paired with a carefully designed payment rule. However, when the mechanism is applied to multi-item auctions, it is approximately IC rather than IC. Later, Zhu and Shin [2015] and Xu et al. [2017], Jian et al. [2018] propose mechanisms that combine the exponential mechanism with the payment rule in [Archer and Tardos, 2001], applying to combinatorial auctions and reverse auctions, respectively. No mechanism above can be applied to our problem of designing DPDM because they fail to ensure diffusion IC. We next introduce existing diffusion auction mechanisms. Diffusion auction. Diffusion auction is an emerging topic in mechanism design. Li et al. [2017] are the first to investigate diffusion auction and propose information diffusion mechanism (IDM), a mechanism for single-unit auction in a social network. The basic idea is to give monetary reward to buyers who are critical to diffusion, and it ensures diffusion IC. Following this idea, Li et al. [2019], Zhang et al. [2020b,a] further study single-unit diffusion auction from different aspects. Later, Zhao et al. [2018], Kawasaki et al. [2020] extend single-unit diffusion auctions to multiple-unit cases and propose generalised IDM (GIDM) and DNA-MU, resp. However, all of these mechanisms are deterministic and suffer from privacy leakage risks. PROBLEM FORMULATION PRELIMINARIES Consider the following setup: There is a seller, denoted by s, and n buyers, denoted by N = {1, 2, 3, ..., n}. Seller s has a single indivisible item to sell. Each buyer i ∈ N is willing to buy the item and attaches a valuation v i to the item. Valuation v i is the maximum amount of money that i is willing to pay. This value is private to the seller. The seller and the buyers form a social network, represented by a graph G = (V, E), where V = N ∪ {s} is the vertex set and E ⊆ V 2 is the edge set. Each node i ∈ V has a neighbour set, denoted by r i := {j ∈ V | (i, j) ∈ E}. We assume that only the seller's neighbours know the auction information initially. The seller would like to attract more buyers to participate in the auction and spread the auction information. Each buyer is able to deliver the auction information to her neighbours. The set r i is also a private information of buyer i. Each buyer i ∈ N , once informed with the auction, can participate in the auction. Also, for each buyer a pair consisting of her valuation and neighbour set is called the profile of the buyer. We use θ i := (v i , r i ) to denote this profile. The profile is known to the buyer and it is hidden to anyone else. Let Θ denote the set of all possible profiles. Also, we let θ := (θ 1 , . . . , θ n ) be the global profile of all buyers and θ −i := (θ 1 , . . . θ i−1 , θ i+1 , . . . θ n ) be the global profile of all buyers except for i. In the auction, each buyer is asked to report her profile θ i = (v i , r i ), which is not necessarily the true one. We define θ ∈ Θ n as the reported global profile of all buyers. Given θ , we construct a directed graph G θ = (V θ , E θ ): add a directed edge (i, j) if j is reported by i as a neighbour. We call such graph profile digraph. Diffusion auction has two forms of information asymmetry: (1) Valuation asymmetry. The buyers' true valuations are private information and hidden from the seller. Thus buyers have an advantage over the seller as they can misreport their valuations. The auction should prevent misreporting of valuation through appropriate allocation and pricing strategies. (2) Neighbourhood asymmetry. By Bulow-Klemperer theorem, the revenue of an auction increases as the number of buyers grows [Bulow and Klemperer, 1996]. However, as buyers' neighbours on the social network are hidden, the seller would hope the buyers to diffuse the auction information to their neighbours to allow more participants to join. However, being rational, the buyers are not necessarily willing to disseminate the auction information as this may hinder their own chance of winning. Here, we follow the standard convention and assume that the reported neighbour set r i is a subset of r i . Diffusion auction mechanisms are designed to address these two challenges. Now we give the definition of a mechanism. A mechanism, denoted by M , takes the reported global profile θ of all buyers as input, and determines who is allocated the item and how much to pay. Definition 3.1. A mechanism M consists of two functions (π(·), p(·)), where π : Θ n → {0, 1} n is an allocation function and p : Θ n → R n is a payment function. The allocation function determines whether the buyers get the item while the payment function determines the amount of money that the buyers need to pay. Given a reported global profile θ of all buyers, we write the allocation result π(θ ) as (π 1 (θ ), . . . , π n (θ )) and the payment result p(θ ) as (p 1 (θ ), . . . , p n (θ )), where π i (θ ) and p i (θ ) is buyer i's allocation and payment. The utility of buyer i with profile θ i = (v i , r i ) is u i (θ ) = v i π i (θ ) − p i (θ ) when reported global profile is θ . The social welfare of mechanism M on θ , denoted by sw M (θ ), is defined as the sum of the seller and the buyers' utility, i.e., sw M (θ ) = i∈V u i (θ ). We aim to maximise the social welfare. PRIVACY-AWARE DIFFUSION AUCTION In addition to Challenges (1) and (2) above, we consider a third challenge in diffusion auction when the buyers are privacy-aware. (3) Valuation privacy. Once the auction result is annouced, an attacker with certain background information may infer the bid information from the published auction result. This is known as the inference attack [Li et al., 2017]. This disadvantages the buyer(s) whose private valuation is diclosed. Therefore, the buyers require the guarantees that their private valuations are protected. To achieve privacy preservation, we apply a randomised mechanism M to implement an auction on the reported global profile. Definition 3.2. A randomised mechanism M is one that, given a global profile θ, outputs a pair (π, p) such that π is a randomised allocation function and p is a randomised payment function. Given a global profile θ, the randomised mechanism M outputs π(θ) and p(θ) such that π(θ) is a random variable with possible values {0, 1} n and p(θ) is a random variable with possible values (R + ) n . We use the concept of differential privacy to define the privacy protection of a mechanism. Basically, differential privacy requires that the distributions over the outcomes are nearly identical when the global profiles are nearly identical. The privacy protection level is measured by a privacy parameter ∈ R + . Definition 3.3. A randomised mechanism M isdifferential privacy ( -DP) if for any two global profiles θ, θ ∈ Θ n that differ on a single buyer's valuation, and for any possible outcome o ∈ O, Pr[M (θ) = o] ≤ exp( )Pr[M (θ ) = o](1) Eqn. (1) shows if any buyer i changes her reported profile from θ i = (v i , r i ) to θ i = (v i , r i ) , the auction outcome does not change too much. Therefore, no one could infer the valuation of any buyer from the randomised outcome. Exponential mechanism [McSherry and Talwar, 2007] is an existing mechanism that ensures -DP for valuation privacy. Given a global profile, an exponential mechanism creates a distribution over all possible auction outcomes, and outputs an outcome according to the distribution. Intuitively, the higher a reported valuation is, the more likely the corresponding buyer is selected as a winner. Specially, given a global profile θ, define a score function σ : Θ n × O → R that assigns a real valued score to each pair (θ, o) from Θ n × O. The more preferable an outcome is, the higher the score of the outcome is. An exponential mechanism M (θ) outputs a result o * ∈ O with probability exp( σ(θ, o * )) o∈O exp( σ(θ, o)) In our problem, a result corresponds to that a certain buyer i wins, and we use o i to denote this result. In randomised mechanisms, we assume that the buyers are risk-neutral and care about their utilities in expectation. We use E M [u i (·)] to denote i's expected utility in M and redefine the standard IC and IR properties by expected utility. Definition 3.4. Let M be a randomised mechanism, • The mechanism M is IC if for all i ∈ N , all θ i , θ i ∈ Θ and for all θ −i , θ −i ∈ Θ n−1 , we have the following, E M [u i ((θ i , θ −i ))] ≥ E M [u i ((θ i , θ −i ))]. • The mechanism M is IR if for all i ∈ N and all θ −i ∈ Θ n−1 , we have E M [u i ((θ i , θ −i ))] ≥ 0. The IR and IC properties ensure that buyers are willing to participate in the auction and to reveal their true valuations and neighbours, as they are rational and doing so leads to the best expected utilities. Hence, information asymmetry issues can be addressed. The social welfare of M is also in expectation, i.e., E M [sw M (θ)] = i∈V E M [u i (θ)]. We aim to design a randomised mechanism that is IC, IR, -DP (for reasonable ) while maximising social welfare. RECURSIVE DPDM Preserving valuation privacy in diffusion auctions is not a trivial task. On one hand, existing diffusion auctions, including IDM [Li et al., 2017], CMD [Li et al., 2019], and FDM [Zhang et al., 2020b], are deterministic, and thus fail to preserve privacy. On the other hand, existing differential private mechanisms, including exponential mechanism, fail to incentivise truthful report of neighbours, which is illustrated in Example 4.1. Example 4.1. We apply exponential mechanism paired with score function σ(θ, o i ) = v i to the scenario in Figure 1. That is, the score of the result that i wins is i's reported valuation v i . We assume that the buyers truthfully report their valuations. Then buyer i wins with probability exp( v i )/ κ∈N exp( v κ ). Now if buyer b reports her neighbour f , b wins with probability exp(8 )/ κ∈N exp( v κ ), whereas she wins with probability exp(8 )/ κ∈N \{f,j} exp( v κ ) had she chose not to report f . In the latter case, the winning probability is even higher, and thus b has incentive to hide her neighbours. To incentivise buyers to diffuse auction information, we need to ensure each buyer's utility of reporting her neighbours should be no less than that of non-reporting. We now propose recursive DPDM REC to achieve this condition. The basic idea is "market division", i.e., treat the social network as a market, partition the market into multiple sub-markets and assign each sub-market a probability with which buyers in this sub-market win, as shown in Eqn. (2). In this case, each buyer would report as many neighbours as possible in order to maximise the probability of the submarket she belongs to. Then the buyers in a sub-market share the probability of the sub-market in such a way that the winning probability of any buyer is independent from her children, as shown in Eqn. (3). Therefore, the buyers have no competition with their children and have no incentive to misreport them. We now describe REC in detail: Fix a score function σ(·) that is non-decreasing in reported valuation v i . Given a reported global profile θ , a privacy parameter and the score function σ(·) as input, REC works as follows: (1) Construction of diffusion critical tree. Given a profile digraph G θ , REC first constructs a diffusion critical tree, denoted by T θ . When the context is clear, we write the tree as T . The idea of diffusion critical tree is originally introduced by [Zhao et al., 2018]. For any buyers i, j ∈ V θ , we say that i is θ -critical to j, denoted by i θ j, if all paths from s to j in G θ go through i. A diffusion critical tree is a rooted tree, where the root is seller s and the nodes V θ are the buyers who are connected to s, and for each j ∈ V θ , her parent is the node i θ j who has the closest distance to j. When there are more than one parents, only one node is randomly selected as the parent. The depth of buyer i, denoted by d i , is the distance from s to i. (2) Assignment of winning probabilities. This step determines the probabilities that buyers win the item. This is a recursive process. This process starts with the constructed T rooted by s. Given a (sub-)tree rooted by i ∈ V , REC assigns a probability to each sub-tree rooted by j ∈ r i , and a winning probability to each j ∈ r i . This operation is repeated for j's children, children of j's children and so on until there is no more children. Now we define Pr T [j] for each j ∈ r i as Pr T [j] = Pr T [i] − Pr i × Exp(T [j]) Exp(T (i))(2) (b) Assignment of winning probabilities to buyers within a sub-market. In a sub-tree T [i], REC assigns the winning probability Pr j to each j ∈ r i as Pr j = Pr T [i] − Pr i × Exp(j) Exp(T (i) \ T (j))(3) At the very beginning, REC starts with the tree T rooted by s. We label s as node 0 and set Pr T [0] = 1 and Pr 0 = 0. REC ends with the leaves. For a sub-tree T [i] where each j ∈ r i are leaves, REC assigns the winning probability to each j as Pr j = Pr T [i] − Pr i × Exp(j) Exp(T (i)) . (3) Allocation and payment. Randomly select a buyer w as a winner according to the constructed distribution in Step (2). Set w's allocation π w = 1, and payment as p w = v w − v w 0 Pr w ((x, r w ))dx/Pr w (θ w )(4) We present the details of REC in Algorithm 1 and give a running example of Step (2) in Example 4.2. Algorithm 1 Recursive DPDM REC Input: Reported global profile θ , privacy parameter and score function σ Output: Allocation result π(θ ) and payment result p(θ ) 1: Initialise π(θ ) = 0, p(θ ) = 0 2: Construct a profile digraph G θ = (V θ , E θ ) 3: Construct a critical diffusion tree T θ 4: Run GetPro(T θ [0], 1, 0) 5: Randomly select a buyer w with the distribution 6: Set π w = 1 and p w by Equation (4) Algorithm 2 Calculate Pr j of buyer j by Equation (3) 4: Run GetPro(T [j], Pr T [j] , Pr j ) 5: end for Example 4.2. We apply REC paired with score function σ(θ, o i ) = v i to the scenario in Fig. 1. Firstly, Pr[T ] = 1 and Pr s = 0. Next we calculate the probabilities of s's children. The probability for T [a] is Pr T [a] = (exp(10 ) + exp(9 ) + exp(12 ))/Exp(T ). Buyer a wins with probability Pr a (10) = exp(10 )/(Exp(T )−(exp(9 )+exp(12 ))). Similarly, we can get the probabilities for T [b], T [c] and b, c. Consider buyer d. d wins with probability Pr d (9) = (Pr(T [a]) − Pr a ) × exp(9 )/(exp(9 ) + exp (12 )). Similarly, we can also get the probabilities for e, f, g. Next we show that recursive DPDM satisfies IC, IR and DP. The next classical result is important for IC. Proof. We first show REC is IC in terms of valuations. By Equation (2), the probability for any sub-tree T [i] is proportional to the score, which is non-decreasing in v i . Hence, Pr T [i] in non-decreasing in v i . Similarly, by Equation (3), given a sub-tree T [i], the winning probability Pr i is nondecreasing in v i , which meets the condition (1) in Thm. 4.3. Also, by Equation (4), the expected payment 1. Pr i (v i ) is monotonically non-decreasing in v i ; 2. E[p i ] = v i Pr i (v i ) − v i 0 Pr i (x)dxE[p i ] = p i × Pr i = v i Pr i (θ i ) − v i 0 Pr i ((x, r i ))dx, which meets the condition (2) in Theorem 4.3 when r i is fixed. Therefore, REC is IC in terms of valuations. Next we show REC is IC in terms of neighbours. By the definitions of expected utility and payment function (4), we know that i's expected utility is only determined by the winning probability Pr i . Let a be an ancestor of i with distance . When i reports truthfully as θ i and the reported global profile is θ −i , then i's winning probability is Pr i = Exp(i) Exp(T (a 1 ) \ T (i)) × (Pr T [a 1 ] − Pr a 1 ) = Exp(i) Exp(T (a 1 ) \ T (i)) × Pr T [a 2 ] − Pr a 2 × Exp(T [a 1 ]) Exp(T (a 2 )) − Exp(a 1 ) Exp(T (a 2 ) \ T (a 1 )) = Exp(i) Exp(T (a 1 ) \ T (i)) × (Pr T − Pr s ) × di−1 =1 Exp(T [a ]) Exp(T (a +1 )) − Exp(a ) Exp(T (a +1 ) \ T (a )) (5) If i hides some of her neighbours and reports any θ i where r i ⊆ r i , instead, and the others report θ −i . Then in Equation (5), Pr T , Pr s and Exp(i) Exp(T (a 1 )\T (i)) does not change. Also, for each , Exp(a ) Exp(T (a )\T (a +1 )) remains intact, but Exp(T [a ]) Exp(T (a +1 )) decreases. So we can know that Pr i decreases when i misreports her neighbourhood. Therefore, we have E REC [u i (((v i , r i ), θ −i ))] ≥ E REC [u i (((v i , r i ), θ −i ))]. Lemma 4.5. Recursive DPDM REC is individually rational in terms of both valuations and neighbours. Proof. Given a global profile θ, for each buyer i with (v i , r i ), E REC [u i (θ)] = (v i − p i (θ))Pr i (θ i ) = vi 0 Pr i ((x, r i ))dx ≥ 0. Therefore, the lemma holds. In following lemma, we use the following terminologies: • d max denotes the maximum depth of the diffusion critical tree, • ∆σ denotes the largest possible difference in the score function σ when applied to two global profiles that differ only on a single user's valuation, for all possible outcome o i ∈ O. Lemma 4.6. Given a reported global profile θ , recursive DPDM REC is d max ∆σ-differential privacy, where is the privacy parameter of REC. Proof. Given two reported global profiles θ and θ that differ in an arbitrary buyer i's reported valuation such that i reports v i in θ and v i in θ , we consider the probabilities that REC(θ) and REC(θ ) return a winner w. In a critical diffusion tree T θ , let d w denote the depth of w, a w be an ancestor of w with distance . Also, let Exp θ (T (a 1 w ) − T (w)) and Exp θ (T (a 1 w ) − T (w)) denote the value derived from θ and θ , respectively. Then by Equation (3), we have Pr[REC(θ) = o w ] Pr[REC(θ ) = o w ] = Exp(w) Exp θ (T (a 1 w )−T (w)) Exp θ (w) Exp θ (T (a 1 w )−T (w)) × Pr θ T [a 1 w ] − Pr θ a 1 w Pr θ T [a 1 w ] − Pr θ a 1 w We repeatedly replace Pr θ T [a w ] , Pr θ a w , Pr θ T [a w ] , Pr θ a w by expressions of a +1 w until we get an expression of s. For each distance 0 ≤ < d w , we denote Exp(T [a w ]) Exp(T (a +1 w )) as A θ , Exp(a w ) Exp(T (a +1 w )\T (a w )) as B θ . For θ , we have similar notations as A θ and B θ . Then the above ratio can be written as Pr[REC(θ) = o w ] Pr[REC(θ ) = o w ] = B θ 0 B θ 0 × dw−1 =1 A θ − B θ A θ − B θ Next we proof the lemma through that for each 0 ≤ < d w , A θ −B θ A θ −B θ is bounded by exp( ∆σ). Here we skip the proof for this due to space limitation. See details in App. B. Then we have Pr[REC(θ) = o w ] Pr[REC(θ ) = o w ] ≤ exp( ∆σ) × dw−1 =1 exp( ∆σ) ≤ exp( d w ∆σ) ≤ exp( d max ∆σ) Next theorem easily follows from Lemmas 4.4, 4.5 & 4.6. Theorem 4.7. Recursive DPDM REC is IC, IR and d max ∆σ-DP. LAYERED DPDM Following the same idea of market division, we propose layered DPDM LAY in this section. Different from REC, LAY divides the market by the buyers' distances to the seller. Specifically, given a constructed critical diffusion tree, LAY allocates a certain probability to each layer of the tree, which will be shared by the buyers on this layer. For any buyer, once she is invited by her parent(s), her layer is fixed. Also, the buyer(s) whom she invites will be on the next layer, and thus has no competition with her. LAY executes the same operations as in REC, where the only difference is in Step (2) "Assignment of winning probabilities". Below we describe Step (2) of LAY in detail: (2) Assignment of winning probabilities. In this step, given a critical diffusion tree T θ , LAY assigns a probability to each layer of the tree and then assigns a winning probability to buyers on each layer. (a) Assignment of probability to layers. Now we give the definition of layer. Given a tree, the buyers with the same distance d i form a layer of a tree. The distance d i ∈ {1, . . . , d max }. We use L to denote the set of buyers with distance , i.e., L := {i | d i = }. For each layer L , 1 ≤ ≤ d max , LAY assigns a probability, denoted by Pr θ L . We write it as Pr L when there is no ambiguity. Given an infinite decreasing sequence γ = (γ 1 , γ 2 , . . .), where γ i = 1, we define the probability for layer L as Pr L = γ(6) (b) Assignment of winning probability to the buyers on a layer. On the th layer, LAY assigns buyer i with θ i on layer d i = with probability Pr i (θ i ) = Pr L × Exp(i) Exp(L )(7) Once the probability distribution over all possible outcomes is determined, LAY computes the payment and randomly selects a winner w, following Step (3) of REC. The complete process of layered DPDM is shown in Alg. 3. Example 5.1 provides a running example of Step (2). Example 5.1. Apply LAY paired with score function σ(θ, o i ) = v i and sequence γ = 1 2 κ+1 κ∈N to the scenario in Figure 1. Then in this graph, three layers, L 1 = {a, b, c}, L 2 = {d, e, f }, L 3 = {g} correspond to probabilities 1 2 , 1 4 , 1 8 , resp. In L 1 , buyer a wins with probability exp(10 )/(2(exp(10 ) + exp(8 ) + exp (14 ))). Similarly, we get the probabilities for b and c. Then in L 2 , d wins with probability exp(9 )/(4(exp(9 ) + exp(12 ) + exp(15 ))). The probabilities for e, f can be obtained in a similar way. Lastly, in L 3 , buyer g wins with probability 1 8 . Algorithm 3 Layered DPDM LAY Input: Reported global profile θ , privacy parameter and score function σ Output: Allocation result π(θ ) and payment result p(θ ) 1: Initialise π(θ ) = 0, p(θ ) = 0 2: Construct a profile digraph G θ = (V θ , E θ ) 3: Construct a critical diffusion tree T θ 4: for 1 ≤ ≤ d max do 5: Calculate the probability of layer by Equation (6) 6: for i ∈ L do 7: Calculate winning probability Pr i by Eqn. (7) 8: end for 9: end for 10: Randomly select a buyer w with the distribution 11: Set π w = 1 and p w by Equation (4) Next we show that layered DPDM LAY has the desirable properties, including IC, IR and DP. Lemma 5.2. Layered DPDM LAY is incentive compatible in terms of both valuations and neighbours. Proof. The IC property in terms of valuations can be proved in a similar way for Lemma 4.4. What we need to show is Pr i is non-decreasing in her reported valuation v i . By Eqn. (7), Pr i ((v i , r i )) is proportional to σ(θ, o i ), which is non-decreasing in v i . Then we show IC in terms of neighbours. For an arbitrary buyer i, her expected utility is E LAY [u i (θ)] = (v i − p i (θ))Pr i when the global profile is θ. We plug in Eqn. (4) (7) into u i (θ). Then we can see Pr i is determined by d i and d i is determined by her ancestors. Therefore, her utility will not be effected if she misreports her neighbours, i.e., E LAY [u i (((v i , r i ), θ −i ))] = E LAY [u i (((v i , r i ), θ −i ))]. Lemma 5.3. Layered DPDM LAY is individually rational in terms of both valuations and neighbours. The proof of Lemma 5.3 follows the same reasoning as Lemma 4.5. See details in Appendix C. Lemma 5.4. Given a reported global profile θ , layered DPDM LAY is ∆σ-differential private, where is the privacy parameter of LAY. Eqn. (7), the change on a single buyer's valuation is bounded by ∆σ. Due to space limit, the proof of Lem. 5.4 is deferred to App. D. The next thm. then easily follows from Lem. 5.2,5.3 and 5.4. Theorem 5.5. Layered DPDM LAY is IC, IR and ∆σ-DP. Lem. 5.4 is proved by showing in Next we analyse the expected social welfare of LAY. We consider a hypothetical scenario where the exponential mechanism is applied to the whole social network where the seller knows all buyers. In this scenario, the auction information is diffused to all buyers without any incentive. We call such a mechanism as exponential mechanism with diffusion (EMD). EMD has the optimal expected social welfare than all DPDMs and thus is used as the benchmark. Theorem 5.6. Given a global profile θ, the expected social welfare of layered DPDM LAY is at least γ dmax E EMD [sw EMD (θ)]. Proof. Given a global profile θ, the expected social welfare E LAY [sw LAY (θ)] of LAY is i∈V v i × Pr LAY i (θ i ) = i∈V v i exp( , σ(θ, o i )) j∈L d i 1 γ d i exp( , σ(θ, o j )) = γ dmax E LAY [sw LAY (θ)] See full derivation in Appendix E. The next result is an easy corollary. Corollary 5.7. For γ = ( a−1 a , a−1 a 2 , . . . ), where a > 1, layered DPDM achieves an expected social welfare ≥ a−1 a dmax E EMD [sw EMD (θ)]. EXPERIMENT We evaluate the performances of REC and LAY, in terms of social welfare under different privacy levels and valuations on three real world social network datasets. We also analyse the effect of sequence γ = ( a−1 a , a−1 a 2 , . . .) on the performance of LAY. For each setup, we run 5000 times and get average social welfare. Dataset. We use three real world network datasets, including Hamsterster friendships with 1, 858 nodes and 12, 534 edges [Kunegis, 2013], Facebook with 4, 039 nodes and 88, 234 edges [McAuley and Leskovec, 2012] and Email-Eu-core network 1, 005 nodes and 25, 571 edges [Yin et al., 2017]. For each dataset, the seller s is randomly selected. Valuation. The network datasets contain no information about buyers' valuations. We generate random numbers as the valuations. We consider two commonly used distributions, normal distribution v i ∼ µ(50, 10) and uniform distribution v i ∼ U [0, 100]. We set the parameters such that the average value are same. Nevertheless, our aim is to reveal the general pattern under different distributions and these patterns are independent from these parameters. Score function. We use linear function, σ(θ, o i ) = v i , as the score function. The linear score function is widely used in previous DP auctions, e.g., Talwar, 2007, Xu et al., 2017]. Decreasing sequence. For LAY, we consider different value of a ∈ {1.25, 1.5, 2, 3} in γ = ( a−1 a , a−1 a 2 , . . . ), and evaluate the impact of a on expected social welfare. Benchmark. Since there is no existing DPDM that can be applied in our problem, we design two hypothetical benchmarks. Exponential mechanism without diffusion (EMWD): We apply the exponential mechanism only to the seller's neighbours. The expected social welfare of EMWD can be seen as the lower bound among all DPDMs. Exponential mechanism diffusion (EMD): See the description of EMD in Section 5. We also compare with IDM [Li et al., 2017] (See App. A), which is not DP, to see how much social welfare is sacrificed to achieve DP. Results. Overall, when comparing to IDM, the difference in social welfare of the DPDMs decreases with increases. Then, among DPDMs, EMD performs best in most cases, followed by REC and LAY. Particularly, REC performs very well. The lines of REC even coincide with those of EMD in some cases, e.g., on Facebook & Email-Eu-core in Fig. 2. The deviation of REC from EMD is at most 2.62%. REC performs better than the layered counterpart. EMWD returns the worst expected social welfare. The reason why REC has better expected social welfare than LAY is that in LAY, a probability of 1 − dmax =1 γ is not distributed to any buyer, which means that the seller does not sell the item and the social welfare is 0 with this probability. Next we show the effect of different parameters. (1) Dataset. As shown in each column of Fig. 2, the same pattern can be found for different datasets. (2) Privacy parameter. The expected sw increases with . The less privacy is required, the less noisy is added, and thus the higher probability of returning a result with good social welfare. (3) Valuation. The 1st and the 2nd row of Fig. 2 show the results with normal and uniform distributions, resp.. Under both distributions, REC performs better than LAY. (4) Sequence. Fig. 3 shows the average social welfare is best when a = 1.5, 2 for Hamsterster and when a = 2, 3 for Facebook and Email-Eu-core. When a buyer i with the highest valuation is on a deeper layer, a smaller a leads to a larger probability for the layer where i is and also a larger probability for i. The results verify this argument. In Hamsterster (Facebook, Email-Eu-core), the buyers with the highest valuation are on the 4th (3rd, 2nd) layer. (5) same DP. Fig. 4 shows when the realised privacy is large, the avg. social welfare of REC is greater than that of LAY, while when the realised privacy is small, LAY is better. CONCLUSION AND FUTURE WORK We consider the problem of designing diffusion auction mechanisms that sells a single item on social networks while preserving valuation privacy. We propose two DPDMs, recursive DPDM and layered DPDM. Also, we theoretically show their incentive and privacy properties and empirically show their good performances in social welfare. We could extend this study by considering the following questions: (1) How to design a DPDM for multi-item auctions? (2) How to design a DPDM that preserves both valuation and neighbourhood privacy? and (3) How to design a DPDM that is group IC where no group of buyers can benefit from joint misreporting? 1 α 1 = A θ A θ ≤ Exp θ (T [a w ]) Exp θ (T [a w ]) ≤ k∈T [a w ] exp( σ(θ, o k )) k∈T [a w ] exp( (σ(θ, o k ) − ∆σ)) ≤ exp( ∆σ) (2) When valuation v i ≥ v i , the first ratio is at most 1. We have 1 α 1 = A θ A θ ≤ Exp θ (T (a +1 w )) Exp θ (T (a +1 w )) ≤ k∈T (a +1 w ) exp( (σ(θ, o k ) + ∆σ)) k∈T (a +1 w ) exp( σ(θ, o k )) ≤ exp( ∆σ) In a similar way, we can show that 1 α2 ≤ exp( ∆σ). Therefore we have Proof. Given a global profile θ, for each buyer i with (v i , r i ), we have E LAY [u i (θ)] = (v i − p i (θ))Pr i (θ i ) = vi 0 Pr LAY i ((x, r i ))dx ≥ 0. Therefore, the lemma holds. D PROOF OF LEMMA 5.4 Lemma 5.4. Given a reported global profile θ , layered DPDM LAY is ∆σ-differential private, where is the privacy parameter of LAY. Proof. Given two reported global profiles θ and θ that differ in an arbitrary buyer i's reported valuation such that i reports v i in θ and v i in θ , we consider the probabilities that M (θ) and M (θ ) return a winner w. Without loss of generality, we assume that w is in L , then we have (1) v i < v i . As σ(·) is non-decreasing in v i , the first ratio is at most 1. Then we have (2) v i > v i . In this case, the second ratio is at most 1. Theorem 5.6 Given a global profile θ, the expected social welfare of layered DPDM LAY is at least γ dmax E EMD [sw EMD (θ)]. Proof. Given a global profile θ, the expected social welfare of LAY is E LAY [sw LAY (θ)] = i∈V v i × Pr LAY i (θ i ) = i∈V v i exp( , σ(θ, o i )) j∈L d i 1 γ d i exp( , σ(θ, o j )) ≥ γ dmax i∈N v i exp( , σ(θ, o i )) j∈L d i exp( , σ(θ, o j )) ≥ γ dmax i∈N v i exp( , σ(θ, o i )) j∈V exp( , σ(θ, o j )) = γ dmax E LAY [sw LAY (θ)] (a) Assignment of probabilities to sub-trees. Let T [i] denote the sub-tree rooted by i. T [i] consists of node i and all of i's descendants. Let T (i) denote T [i] with i removed, i.e., T (i) := T [i] \ {i}. Given a sub-tree T [i], REC divides the market in T [i] to |r i | + 1 sub-markets, one for i and each of the other for a sub-tree T [j], where j ∈ r i . Then REC assigns a probability Pr θ i (θ i ) to i with θ i and Pr θ T [j] to each T [j], where j ∈ r i . When the context is clear, we write Pr i and Pr T [j] for Pr θ i (θ i ) and Pr θ T [j] , respectively. We define Pr i later in Step (2).b. For notational convenience, given a set of nodes S ⊆ T , we let Exp(S) be the sum Exp(S) = κ∈S exp( σ(θ , o κ )). Theorem 4.3 ([Archer and Tardos, 2001]). Let Pr i (v i ) be the probability that i wins when she reports v i . A mechanism M = (π, p) is incentive compatible in terms of valuations if and only if, for any i ∈ N , Lemma 4 . 4 . 44Recursive DPDM REC is incentive compatible in terms of both valuations and neighbours. Figure 2 : 2Average social welfare of LAY, REC, EMD, EMWD and IDM with different distributions under fixed sequence with a = 2. Normal distribution is shown in the first row and uniform distribution is shown in the second row. Figure 3 : 3Average social welfare of LAY with different values of a, under normally distributed valuations and linear function. Figure 4 : 4Average social welfare of LAY, REC, EMD, EMWD and IDM under normal distribution, linear function and sequence with a = 2. Horizontal axis represents the value of for EMD, EMWD & LAY, and dmax for REC. Pr[M (θ) = o w ] Pr[M (θ ) = o w ] ≤ exp( ∆σ) × 1≤ <dw exp( ∆σ)≤ exp( d w ∆σ) ≤ exp( d max ∆σ) Layered DPDM LAY is individually rational in terms of both valuations and neighbours. When i is not on layer L , Pr[M (θ)=ow]Pr[M (θ )=ow] = 1 ≤ exp( ∆σ). Otherwise, when i is on layer L , we consider two cases. Pr[M (θ) = o w ] Pr[M (θ ) = o w ] ≤ Exp θ (L ) Exp θ (L ) ≤ j∈L exp( (σ(θ, o j ) + ∆σ)) j∈L exp( σ(θ, o j )) ≤ exp( ∆σ) Then we havePr[M (θ) = o w ] Pr[M (θ ) = o w ] ≤ Exp θ (w) Exp θ (w) ≤ exp( σ(θ, o w )) exp( (σ(θ, o w ) − ∆σ)) ≤ exp( ∆σ) Input: (Sub-)Tree T [i], probabilities Pr T [i]and Pr i Output: Probabilities Pr T [j] and Pr j , j ∈ r i 1: for j ∈ r i doGetPro 2: Calculate Pr T [j] of sub-tree T [j] by Equation (2) 3: Privacy parameter. To verify the performance of our mechanisms, we also vary privacy parameter ∈ {0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3}. Lem. 5.4 and 4.6 show that, under the same input , LAY and REC ensure different privacy levels. To see the performance under the same guaranteed privacy, we set the input as {0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3} for REC and {0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3}d max for the others. https://blogs.gartner.com/avivah-litan/2022/08/05/aimodels-under-attack-conventional-controls-are-not-enough/ https://www.pcipal.com/knowledge-centre/resource/fewerthan-10-of-people-are-confident-about-their-data-security-onsocial-media-according-to-survey-from-pci-pal/ Appendix A IDMHere, we introduce the first diffusion auction for selling single item, IDM[13]. A key concept of IDM is diffusion critical sequence. Given a profile digraph G θ , for any buyers i, j ∈ V θ , i is θ -critical to j, denoted by i θ j, if all paths from s to j in G θ go through i. A diffusion critical sequence of i, denoted by C i , is a sequence of all diffusion critical nodes of i and i itself ordered by θcritical relation. That is, C i = (x 1 , x 2 . . . , x k , i), where x 1 θ x 2 θ . . . θ x k θ i. 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Selling multiple items via social net- works. In Proceedings of the 17th International Con- ference on Autonomous Agents and MultiAgent Systems, pages 68-76, 2018. A survey on differential privacy for unstructured data content. Ying Zhao, Jinjun Chen, ACM Computing Surveys (CSUR). 5410sYing Zhao and Jinjun Chen. A survey on differential privacy for unstructured data content. ACM Computing Surveys (CSUR), 54(10s):1-28, 2022. Differentially private and strategy-proof spectrum auction with approximate revenue maximization. Ruihao Zhu, G Kang, Shin, 2015 IEEE conference on computer communications (INFOCOM). IEEERuihao Zhu and Kang G Shin. Differentially private and strategy-proof spectrum auction with approximate revenue maximization. In 2015 IEEE conference on computer communications (INFOCOM), pages 918-926. IEEE, 2015. Differentially private spectrum auction with approximate revenue maximization. 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[]
[ "Astronomical image time series classification using CONVolutional attENTION (ConvEntion)", "Astronomical image time series classification using CONVolutional attENTION (ConvEntion)" ]
[ "Anass Bairouk [email protected] \nLaboratory of Computer Science\nRobotics and Microelectronics of Montpellier\nUniversity of Montpellier\n34095MontpellierFrance\n", "Marc Chaumont \nLaboratory of Computer Science\nRobotics and Microelectronics of Montpellier\nUniversity of Montpellier\n34095MontpellierFrance\n\nUniversity of Nimes\n30021NîmesFrance\n", "Dominique Fouchez \nCentre of Particle Physics of Marseilles\nAix Marseille Univ\nCNRS/IN2P3\n13009MarseilleFrance\n", "Jerome Paquet \nGroupe AMIS\nPaul Valéry University\nMontpellier 334090MontpellierFrance\n\nINRAE/CIRAD\nLand, Environment, Remote Sensing and Spatial Information -UMR TETIS\nCNRS\n34000MontpellierFrance\n", "Frédéric Comby \nLaboratory of Computer Science\nRobotics and Microelectronics of Montpellier\nUniversity of Montpellier\n34095MontpellierFrance\n", "Julian Bautista \nCentre of Particle Physics of Marseilles\nAix Marseille Univ\nCNRS/IN2P3\n13009MarseilleFrance\n" ]
[ "Laboratory of Computer Science\nRobotics and Microelectronics of Montpellier\nUniversity of Montpellier\n34095MontpellierFrance", "Laboratory of Computer Science\nRobotics and Microelectronics of Montpellier\nUniversity of Montpellier\n34095MontpellierFrance", "University of Nimes\n30021NîmesFrance", "Centre of Particle Physics of Marseilles\nAix Marseille Univ\nCNRS/IN2P3\n13009MarseilleFrance", "Groupe AMIS\nPaul Valéry University\nMontpellier 334090MontpellierFrance", "INRAE/CIRAD\nLand, Environment, Remote Sensing and Spatial Information -UMR TETIS\nCNRS\n34000MontpellierFrance", "Laboratory of Computer Science\nRobotics and Microelectronics of Montpellier\nUniversity of Montpellier\n34095MontpellierFrance", "Centre of Particle Physics of Marseilles\nAix Marseille Univ\nCNRS/IN2P3\n13009MarseilleFrance" ]
[]
Aims. The treatment of astronomical image time series has won increasing attention in recent years. Indeed, numerous surveys following up on transient objects are in progress or under construction, such as the Vera Rubin Observatory Legacy Survey for Space and Time (LSST), which is poised to produce huge amounts of these time series. The associated scientific topics are extensive, ranging from the study of objects in our galaxy to the observation of the most distant supernovae for measuring the expansion of the universe. With such a large amount of data available, the need for robust automatic tools to detect and classify celestial objects is growing steadily.Methods. This study is based on the assumption that astronomical images contain more information than light curves. In this paper, we propose a novel approach based on deep learning for classifying different types of space objects directly using images. We named our approach ConvEntion, which stands for CONVolutional attENTION. It is based on convolutions and transformers, which are new approaches for the treatment of astronomical image time series. Our solution integrates spatio-temporal features and can be applied to various types of image datasets with any number of bands. Results. In this work, we solved various problems the datasets tend to suffer from and we present new results for classifications using astronomical image time series with an increase in accuracy of 13%, compared to state-of-the-art approaches that use image time series, and a 12% increase, compared to approaches that use light curves.
10.1051/0004-6361/202244657
[ "https://export.arxiv.org/pdf/2304.01236v1.pdf" ]
257,751,904
2304.01236
f24052f54fe68df1c40fbe5c027aff39ddcde58c
Astronomical image time series classification using CONVolutional attENTION (ConvEntion) April 5, 2023 April 5, 2023 Anass Bairouk [email protected] Laboratory of Computer Science Robotics and Microelectronics of Montpellier University of Montpellier 34095MontpellierFrance Marc Chaumont Laboratory of Computer Science Robotics and Microelectronics of Montpellier University of Montpellier 34095MontpellierFrance University of Nimes 30021NîmesFrance Dominique Fouchez Centre of Particle Physics of Marseilles Aix Marseille Univ CNRS/IN2P3 13009MarseilleFrance Jerome Paquet Groupe AMIS Paul Valéry University Montpellier 334090MontpellierFrance INRAE/CIRAD Land, Environment, Remote Sensing and Spatial Information -UMR TETIS CNRS 34000MontpellierFrance Frédéric Comby Laboratory of Computer Science Robotics and Microelectronics of Montpellier University of Montpellier 34095MontpellierFrance Julian Bautista Centre of Particle Physics of Marseilles Aix Marseille Univ CNRS/IN2P3 13009MarseilleFrance Astronomical image time series classification using CONVolutional attENTION (ConvEntion) April 5, 2023 April 5, 2023Astronomy & Astrophysics manuscript no. aandaTransformerConvEntionAstronomical Image Time SeriesConvolutional AttentionClassificationSupernovae3D Convolution Network Aims. The treatment of astronomical image time series has won increasing attention in recent years. Indeed, numerous surveys following up on transient objects are in progress or under construction, such as the Vera Rubin Observatory Legacy Survey for Space and Time (LSST), which is poised to produce huge amounts of these time series. The associated scientific topics are extensive, ranging from the study of objects in our galaxy to the observation of the most distant supernovae for measuring the expansion of the universe. With such a large amount of data available, the need for robust automatic tools to detect and classify celestial objects is growing steadily.Methods. This study is based on the assumption that astronomical images contain more information than light curves. In this paper, we propose a novel approach based on deep learning for classifying different types of space objects directly using images. We named our approach ConvEntion, which stands for CONVolutional attENTION. It is based on convolutions and transformers, which are new approaches for the treatment of astronomical image time series. Our solution integrates spatio-temporal features and can be applied to various types of image datasets with any number of bands. Results. In this work, we solved various problems the datasets tend to suffer from and we present new results for classifications using astronomical image time series with an increase in accuracy of 13%, compared to state-of-the-art approaches that use image time series, and a 12% increase, compared to approaches that use light curves. Introduction The astronomical community has been facing a considerable challenge in the last few years as tools for observing the universe continue to improve. Telescopes are becoming more powerful, with the capacity to observe a huge part of the universe and generate a massive amount of data. Processing and analyzing these data are very demanding steps in terms of their computational and human resource requirements. With the promises of The Vera Rubin Observatory Legacy Survey for Space and Time (LSST) (Ivezić et al. 2019), the field will see the discovery of 10 to 100 times more astronomical sources that fluctuate in the night sky. Some of these sources will be entirely new. LSST is prepared to alert the community to 10 million new objects per night, and these objects all need to be classified. There are many types of objects, including active galactic nuclei (AGNs), variables, cepheids, RR Lyrae, and supernovae. The latter stands the most important transient object for cosmology because increasingly large samples of Type Ia supernovae (SNe Ia) are being used to measure luminosity distances as a function of redshift in order to understand the origin of the acceleration of the expansion of the universe. Traditionally, the classification of these objects goes through many processes in a complex pipeline. First, the preprocessing phase known as photometry is conducted on a series of images to extract the flux per band, each band corresponding to a passband-like color filter. The number of bands can vary, depending on the survey, for example SDSS survey (Holtzman et al. 2008;Sako et al. 2014;Frieman et al. 2007) has five bands and the Catalina survey (Drake et al. 2011) has only one band. Then, a time series of brightness changes is generated over time, called the light curves. Afterwards, the light curve is fed to a machine-learning classifier to determine the class of the object. Among all the methods developed to perform such a classification, Möller & de Boissière (2020) introduced a model called SuperNNova: a supernova photometric classification framework that uses a recurrent neural network (RNN) (Rumelhart et al. 1985;Hochreiter & Schmidhuber 1997;Cho et al. 2014) to classify different types of supernovas such as SNIa, SNIb, SNIIP, and others using only light curves. The proposition yields good results because while Bayesian neural networks (BNN) are known to be robust to overfitting and can easily learn from small datasets, they are still significantly more complex than standard neural networks and computationally ex-Article number, page 1 of 11 arXiv:2304.01236v1 [astro-ph.IM] 3 Apr 2023 A&A proofs: manuscript no. aanda pensive. Boone (2019) (winner of the photometric classification challenge PLAsTiCC (PLAsTiCC-team et al. 2018;Hložek et al. 2020)) presented a model based on Gaussian process augmentation of the light curve and then train it on boosted decision tree classifier. Pasquet et al. (2019) created a deep architecture called PELICAN that accepts only light curves and redshifts as input. PELICAN can handle light curves with sparsity and irregular sampling. Some can choose to add more preprocessing before training a model. For instance, Qu et al. (2021) proposed a novel approach where they generated a 2D image heatmap from light curves using 2D Gaussian process regression, which they fed to convolutional neural networks to classify different types of supernovae. The approach yields great results on PLAsTiCC data, with an accuracy of 99.73% on the binary classification of SNIa and non-SNIa. The methods that use light curves for classification still have some limitations. In order to generate a light curve, we should correctly align the two consecutive images and we must lower the quality of one of the two images to subtract them to get the flux, which could lead to a loss of information. Some dedicated algorithms called scene modeling can mitigate such issues on blended objects but are very demanding in terms of computer resources. Most importantly, the scene information, namely, the background of the transient object, is in general not taken into account in the classification. Several recent works have proposed to eliminate the feature extraction and light curve phase and focus on classifying the objects using only images. Carrasco-Davis et al. (2019) and Gómez et al. (2020) used a RNN to classify the sequences after passing the images through a CNN to extract the spatial features. They forwarded the output to the RNN (GRU/LSTM) to extract the temporal characteristics and classify the object, while (Gómez et al. 2020) applied their model to only transient objects and Carrasco-Davis et al. (2019) classified variables and transient. These two papers showed promising results for the astronomical image time series (AITS). Therefore, we followed the same path to improve the classification and also solve some challenges posed by AITS, which have not been tackled before. In particular, image time series (ITS) classification has always been one of the challenging areas of deep learning. In addition to spatial characteristics, you also need to give importance to the temporal aspects, which makes traditional feed-forward networks ineffective. Due to the lack of research carried out on ITS in astronomy, we need to import new technics from other fields of research. Most of the research in ITS classification is done in two major domains: action recognition, where the goal is to classify the type of human action (Shi et al. 2015;Ji et al. 2013), and landscape classification using satellite images (Turkoglu et al. 2021). These two fields have covered many of the essential methods to handle ITS. Then, RNN-based approaches use recurrent neural networks to manage the aspect of time in the classification. These approaches are split into two main categories. The first one handles the spatial features separately from the temporal features. Carrasco-Davis et al. (2019) and Gómez et al. (2020) used precisely this method at the point when the CNN handles the spatial characteristics to pass it later to the RNN, which might be LSTM (Hochreiter & Schmidhuber 1997) or GRU . The second category combines convolution inside the RNN cell, thus maintaining the spatial structure of the input, which leads to extracting spatial-temporal features in the sequence. This method was first introduced by Shi et al. (2015). These authors demonstrated how to create an endto-end trainable model using the convolutional LSTM (ConvL-STM). Experiments indicate that their ConvLSTM network regularly exceeds fully connected LSTM (FC-LSTM) in capturing Spatio-temporal correlations. Using satellite images, Turkoglu et al. (2021) proposed a new type of RNN called ConvSTAR, which has fewer parameters than the LSTM and GRU. Another way of achieving the classification of ITS is by using convolution neural networks. Ji et al. (2013) created a new 3D CNN model for action recognition. This model pulls features from spatial and temporal dimensions, collecting motion information contained in several consecutive frames. Meanwhile, some of the latest developments have abandoned convolutions and RNNs to replace them with only transformers. Liu et al. (2022) andYan et al. (2022) proposed an improved supervised transformer for image classification. On the other hand, Zhou et al. (2022) and Bao et al. (2022) proposed more complex transformers that are self-supervised. In this work, we develop a new deep learning transformerbased architecture to classify AITS. Unlike other works that separate spatial and temporal feature extraction, we combine these two steps by performing a spatio-temporal feature extraction in one step. It improves the capacity of the network to recognize the objects. We also propose a solution for the missing observations problem, which demonstrates a significant improvement in the accuracy of the model. To illustrate the performances of our model, we tested it with actual data from the SDSS survey (Holtzman et al. 2008;Sako et al. 2014;Frieman et al. 2007). In Section 2, we describe the dataset that we used in our work. Section 3 introduces our architecture ConvEntion and describes the role of each component of the model. In Section 4, we present the results of our work with some statistics about the performance and some comparisons with other architectures used for image time series classification. Finally, in Section 5, we present our conclusions and perspectives on this work. Dataset Database description The Sloan Digital Sky Survey (SDSS) (Holtzman et al. 2008;Frieman et al. 2007) is a very ambitious and successful largescale survey program using a dedicated 2.5-meter telescope at Apache Point Observatory, New Mexico, equipped with photometric and spectroscopic instruments that have released images, spectra, and catalog information for several hundred million celestial objects. The dataset used in this paper was collected during the SDSS Supernova Survey (Sako et al. 2014), one of three components (along with the Legacy and SEGUE surveys) of SDSS-II, a three-year extension of the original SDSS that operated from July 2005 to July 2008. The Supernova Survey is a time-domain survey, involving repeat imaging of the same region of the sky every other night, weather permitting. The images are obtained through five wide-band filters (Fukugita et al. 1996) named u', g', r', i', and z', simplified as u, g, r, i, and z in the following, which corresponds to an effective mid-point wavelength of u (365nm), g (475nm), r (658nm), i (806nm), and z (900nm). The survey region observed repeatedly over three years is a 2.5-degree-wide stripe centered on the celestial equator in the Southern Galactic Cap that has been imaged numerous times in the last twenty years, allowing for the construction of a big image database for the discovery of new celestial objects. Most of the sources have included galactic variable stars, active galactic nuclei (AGN), supernovae (SNe), and other astronomical transients, all of which have been processed to generate multi-band (ugriz) light curves. The imaging survey is reinforced by an extensive spectroscopic follow-up program that uses spectroscopic diagnostics to identify SNe and measure their redshifts. Light curves were evaluated during the survey to provide an initial photometric type of the SNe and a selected sample of sources was targeted for spectroscopic observations. In order to investigate the classification from images rather than light curves, we acquired the images from the public SDSS dataset through their platform. Our dataset contains many types of supernovas (see Table 1 and (Sako et al. 2014)). The label of "unknown" mainly represents very sparse or poorly measured transient candidates, "variables" have signals spanning over two seasons, and "AGNs" have a spectral signature. The three other classes are supernovae of type Ia, Ib/c, and II. Among supernovae, the typing is performed from spectroscopy or from the light curve using different machine learning techniques (see Sako et al. 2014). We grouped the non-Ia supernovas because our focus in this study only on the Ia type for their interest in cosmology as standard candles and also because of the small number of non-Ia with spectral signatures. The very small class of three SLSN bright objects has been added to the non-Ia supernovae. Figure 1 shows an example of astronomical image time taken from the SDSS dataset. Object name Count AGN 906 SNIa 499 SNOther 89 Unknown 2009 Variable 3225 SNOther_PT 2041 SNIa_PT 1448 Challenges Most of the astronomical dataset suffers from a number of problems that should be dealt with before feeding it to the classification algorithm. Among difficulties contributing to the challenging nature of AITS, we can mention class imbalance (as shown in Table 1 of our dataset). In particular, we can clearly see that the classes we have are not balanced where the number of samples for variables is much bigger than SNIa. This imbalance significantly impacts machine learning models due to their higher prior probability, which means they tend to overclassify the larger class(es). As a result, instances belonging to the smaller class(es) are more likely to be misclassified than those belonging to the larger class(es). Another problem that impacts the model is missing bands. Indeed, each time an image is acquired in an AITS it is captured through one filter among a set of up to five or more channels. So, an image of a celestial object can be taken in many channels, but not necessarily at the same time. This results in missing bands for a given time of observation (see Figure 3). It is well known that the missing data negatively impacts the performance of the model if it is not dealt with. Gill et al. (2007) stated that an increasingly missing percentage of training data resulted in an increased testing error, which requires a solution to mitigate the impact of missing data. Methods In this section, we propose a neural network based on a combination of convolution and self-attentions. The goal of the model is to handle the challenges that we mentioned previously, such as class imbalance, data sparsity, and missing observations. Figure Then each 3DCNN is fed with a sub-sequence of K inputs of the time series J(∈ R M×H×W×2 for M elements of images of size HxW) to create the new downsized sequence S (∈ R N×H ×W ×D ). S is fed to the positional encoder in order to add the information about the position, which outputs F(∈ R N×H ×W ×D ). Then F is passed to ConvBERT which has L layers. The 3D max-pooling is used to downsize the output of ConvBERT for the classifier 2 represents the general architecture of the ConvEntion model. The model takes as its input the sequence of images that have been rearranged to embed the band information (See Section 3.1 and Figure 4). The sequence first passes through a 3DCNN to downsize its length. It allows for the reduction of the computation complexity of the model and also captures the local characteristics of the objects. The newly constructed sequence by the 3DCNN is fed to a convolutional BERT which then extracts the spatio-temporal features with high-level representation from the input. Finally, we pass the output of the convolutional BERT, which is a projection of our input into a high-level representation subspace, through a 3D max-pooling to downsample it, then it goes on to the final classifier to make the prediction. In the following subsections, we explain each component in depth. Data modeling First, we note that throughout the paper, vectors are given in bold capital letters, sizes in capital letters, and indices in lowercase. To start with the missing data problem, a network dedicated to image time series is usually fed a sequence of images I ∈ R H×W×5 , where H and W are, respectively, the height and width of the image and 5 is the number of channels representing the bands (u, g, r, i, z). However, we know, as explained earlier, some bands are missing in the dataset. To fix this issue, instead of giving the model images with empty channels, thus introducing a bias to the network, we decided to separate the channels as individual images (X ∈ R H×W ) simply by skipping the empty channels. As a consequence, the information about the type of filter, which holds a crucial value for the network to ac-curately discriminate between objects, is also eliminated. In an image with different channels, the order of the channels usually represents the type of filter (see Figure 3). In order to preserve this valuable information, we should add the band type to the new 2D images X. Knowing that the information about the type of the filter is a categorical feature, thus we need to adapt it to the model 2D input representation. To do so, we propose using an embedding layer to encode the channel type before passing the input to the model. For each band (u, g, r, i, z), we assign a unique number id ∈ {1, 2, 3, 4, 5}. Then, an embedding layer BandEmbed converts the band type id, which is a categorical feature, into 2D dense representation E id with E id ∈ R H×W (see Figure 4): E id = BandEmbed(id).(1) The embedding layer is a fully connected layer that is reshaped to a 2D representation. The weights of BandEmbed are J m = Concat(X m , E id ), m ∈ {1, .., M}.(2) The problem of class imbalance is one of the major challenges for any machine learning project. Some tried to solve this problem by adding a new loss function to mitigate the impact of the class imbalance. For example Lin et al. (2017) proposed a loss function called "focal loss" which applies a modulating term to the cross-entropy loss in order to focus the learning on hard misclassified examples. However, this approach tends to produce a vanishing gradient during backpropagation (Hossain et al. 2021). Other solutions propose the use of oversampling such as SMOTE (Chawla et al. 2002). Those authors proposed an approach where they synthesize new samples of the minority class. However, this solution was proposed mainly for tabular data. Knowing that our data are images that contain a much higher number of features than tabular data, it appears obvious that using SMOTE may not be optimal in our case. Dablain et al. (2021) introduced a solution based on SMOTE dedicated to images called DeepSMOTE. It is aimed at generating new images for the minority class. Once again, this approach is unsuitable in our case as our dataset is not composed of images, but of a sequence of images, and it is too expensive to generate a whole new sequence. So, instead of generating a new one, we used data augmentation and weighted random sampling(WRS) (Efraimidis 2015) on our database. We oversampled the dataset, which translates to simply altering the dataset to remove such an imbalance by increasing the number of minority classes and undersampling the data by decreasing the majority classes until we have reached a balanced dataset. In our case, the WRS was applied on a batch level. We generate balanced batches based on the probability of a sample being selected. We weighted each sample according to the inverse frequency of its label's occurrence and then sampled mini-batches from a multinomial distribution based on these weights. This means that samples with high weights are sampled more often for each mini-batch. The same sample can be reused in other mini-batches of the same epoch to increase the minority class, but with a data augmentation applied to it. Different methods of data augmentation were used: for example, a random drop of some steps from the whole sequence to create a new one or a sequence rotation, horizontal and vertical flips, and sequence shifting, where we construct a smaller sequence from the original one which has a bigger length than the input length of ConvEntion. In our implementation, we recall the dataset at every epoch, the transforms operation (augmentation) is executed and then we get different augmented data. Using this oversampling approach has drastically improved the performance of the model. We used the function WeightedRandomS ampler from PyTorch (Paszke et al. 2019) as an implementation of WRS. 3D convolution network: In several deep learning applications, large transformer models have demonstrated fantastic success in obtaining state-ofthe-art results. However, because the original transformer's selfattention mechanism consumes O(M 2 ) time and space with respect to the sequence length, M, training the model for a long sequence is so expensive, it causes the problem called "attention bottleneck" (Wang et al. 2020;Choromanski et al. 2021). The problem is more severe for us because we use convolutions and 3D tensors inside the attention mechanism; for instance, the attention map is of a size H × W, so the complexity of the attention will be O(M 2 × H × W). Thus, our model would then be prohibitively expensive to train. In the last few years, there have been numerous proposals aimed at solving this issue. Wang Article number, page 5 of 11 (2021) presented a novel transformer architecture that uses linear space and time complexity to estimate regular (softmax) full-rank-attention Transformers with proven accuracy. However, all these propositions remain irrelevant in our case because we do not use the standard self-attention mechanism, as the convolutions make it an arduous task. So, the solution we preferred to go with is to reduce the length of the sequence before feeding it to the transformer block. Reducing the sequence must be done without losing relevant information. Thus, we propose using a 3D convolution neural network (3D CNN). A 3D CNN is an improved type version of CNN first proposed by Tran et al. (2014), where it applies a 3D filter to the dataset and the filter moves in three directions to calculate the low-level feature representations. Their output shape is in a 3D volume space. We applied 3DCNN where we input the sequence J to get the reduced new sequence S following the equation: S n = 3DCNN(J (n−1) * K+1 , .., J n * K ), n ∈ {1, .., N}. We let M be the length of the series, J and we fed K inputs of J to the 3DCNN to generate one entry, S , for our transformer. So, in the end, the new sequence, S , will be S ∈ R N×H ×W ×D , where N = M/K, D is the number of channels and H and W are the new height and width. By using the 3DCNN, we reduced the length of the sequence by a factor of K, which also reduced the complexity of the model. The 3DCNN does not just reduce the length of the input sequence, it also captures local spatiotemporal low-level features. The 3DCNN captures these particulate features due to its focus on the local characteristics (space and time) of the sequence, while the transformer focuses on the global characteristics. On the whole, we have reduced the computation without losing essential information that is important for classification. Table 2 summarizes the architecture used inside the 3DCNN. Convolutional BERT After getting the new output S of the 3DCNN, it is time to feed it to what we call convolutional BERT which stands for Convolutional Bidirectional Encoder Representations from Transformers. Transformer and self-attention have become one of the main models that revolutionize deep learning in the last few years, especially in neural language processing (NLP). Self-attention (Bahdanau et al. 2014), also known as intra-attention, is an attention mechanism that connects different positions in a single sequence to compute a representation of the sequence. Here, "attention" refers to the fact that in real life, when viewing a video or listening to a song, we frequently pay more attention to certain details while paying less attention to others, based on the importance of the details. Deep learning uses a similar flow for its attention mechanism, giving particular parts of the data more focus as it is processed. Our intention in using this mechanism is for the model to focus more on the changes happening in the image sequence to better discriminate between astronomical objects. Self-attention layers are the foundation of the transformer block design. Transformers were first introduced by Vaswani et al. (2017), using model-based attention dispensing with recurrence and convolutions entirely. Their work inspired others who used the concept of transformers to achieve even better results. For example, in BERT (Devlin et al. 2019) the authors used only the encoder block by stacking many of them. Even though transformers were widely used in NLP in the last two years, people started implementing these blocks in other domains like image classification. Dosovitskiy et al. (2021) presented a model free from convolutions by using only a transformer to classify images. Garnot et al. (2019) also suggested that they are able to extract temporal characteristics using a custom neural architecture based on self-attention instead of recurrent networks. Their use was not limited to image classification; action recognition was also investigated as in Sharir et al. (2021), where the authors used a transformer-based approach inspired by the work of Dosovitskiy et al. (2021). Liu et al. (2021) did propose a new transformer where they added convolution to the attention mechanisms, making it able to apply convolutions while extracting the temporal features. Positional encoding Because transformers have no recurrence throughout the thumbnail sequence, some information about each thumbnail's relative or absolute position must be injected into the feature map obtained by the 3DCNN to inform the model about the order in the sequence. Similarly to the original transformer paper (Vaswani et al. 2017), we use positional encoding at each layer in the encoder to achieve this. The only difference is that our positional encoding is a 3D tensor, where P ∈ R N×H ×W ×D . Because the positional encoding and the new feature maps have the same dimension, they can be added together. We use sine and cosine functions to encode the position (Vaswani et al. 2017): P (n,2i) = sin(n/10000 2i/D ),(4)P (n,2i+1) = cos(n/10000 2i/D ),(5) where n denotes the position in the sequence of length, N, and i is the channel dimension, while D represent the total number of channel gotten by the 3DCNN. The sinusoidal positional encoding is chosen to make it easy for the model to learn to attend to relative positions. To get the new input for the convolutional BERT, we conducted an element-wise addition between the positional encoding and the feature maps obtained from 3DCNN to obtain the new tensor F ∈ R N×H ×W ×D : F n = S n + P n , n ∈ {1, .., N}. SDSS dataset did not reveal any improvement in the model when using the observation date, as opposed to just using the position. This can be understood because we do the training and the test with the same observation sequence and the network can therefore learn this sequence. On the other hand, not incorporating any information regarding the order of the sequence greatly degraded the performance of the model. As a result, we ultimately chose to use only the position in our model (see Section 4.2 for a discussion). The newly obtained sequence F is fed to a multi-head convolutional attention, which is an improved self-attention that has convolution. Then the multi-head convolutional attention is followed by the second component which is a tiny feed-forward network (FFN) that has convolutions applied to every attention map. Its primary purpose is to transform the attention map into a form acceptable by the next convolutional BERT layer, with the FFN consisting of two convolutional layers with ReLU activation in between. Multi-head convolutional self-attention For this process, we used the model proposed by Liu et al. (2021), with a few modifications where we replaced the last linear layer with a convolution layer. We believe that convolution in self-attention is better than the dot product between the query and the key because the convolution will accurately calculate the similarity, especially when we have 3D feature maps. A query map and a set made up of a pair of key maps and value maps that are encoded to an output using convolutional self-attention. The query map, key maps, value maps, and output are all 3D tensors. Figure 5 represent the general architecture of the multihead ConvAttention. We used a convolution layer to generate the attention model's query, value, and key. The input to the attention model is F ∈ R N×H ×W ×D . We pass each map through a convolution layer to get {Q, K, V} ∈ R N×H ×W ×D , where D = D/T and T represent the number of attention heads. Then we appled a subnetwork, M θ , on the query and the key maps, which consists of an element-wise sum of the query and the key maps fol-lowed by another convolution layer to generate our attention map H (n,m) ∈ R H ×W ×1 : H (n,m) = M θ (Q n , K m ), n, m ∈ {1, .., N}.(7) After getting all the map attentions, H n = {H (n,1) , H (n,2) , ...., H (n,N) }, where H n ∈ R H ×W ×N , we applied a softmax operation along the third dimension of size, N. Then we conducted an element-wise product between the attention map and the value map following the equation: V n = N m=1 S o f tMax(H n ) (n,m) V m .(8) We concatenated the new value representation, V n , obtained from the different attention heads. The multi-head attention is used to attend to input from various representation subspaces jointly: MultiHead(Q, K, V) = Concat(V n 1 , ...., V n T ). Finally, we applied a convolution layer for merging the output of the multi-head and obtaining a high-level representation that groups all the heads. At the end of the network, we pass the encoded sequence to 3D max-pooling and finally to the classifier to make a prediction. Evaluation metrics Accuracy is the probability that an object will be correctly classified. It is defined as the sum of the true positives plus true negatives divided by the total number of individuals tested: Accuracy = T P + T N T P + T N + FP + FN ,(10) where TP, TN, FP, and FN are, respectively, the true positive, true negative, false positive, and false negative. The F1 score is a classification accuracy metric that combines precision and recall. It is a suitable measure of models tested with imbalanced datasets: Precision = T P T P + FP ,(11)Recall = T P T P + FN ,(12)F1 = 2 × Precision × Recall Precision + Recall .(13) Experiments Implementation details The supernovae in our data are not all spectroscopically confirmed, which means that the unconfirmed ones might contain some misclassified objects due to errors from the photometric typing. The model may not generalize due to this data bias. To ensure that our model performs a generalization only on spectroscopically confirmed data, we split up the training process into two steps. We divided the data into two datasets. The first one contains only the photometrically typed data and the second contains spectroscopically confirmed data. We trained the model at first with the photometrically typed data, then we used transfer learning to fine-tune the model on only spectroscopically confirmed data ( Table 5 summarizes the partition of the data). The models are trained using cross-validation of five folds and three ensembles in each fold. All the architectures presented in this paper follow this same process and are implemented using PyTorch (Paszke et al. 2019). We performed an extensive hyperparameter tuning of over 20 models to specify the best hyperparameters for our architecture, which contains 1.3 Million parameters. We conducted a hyperparameter optimization using only a non-confirmed dataset with different parameters, such as sequence length, M, learning rate, lr, 3DCCN sub-sequence length, K, classifier layers' size, number of ConvBERT layers, L, number of Multi-head ConvAttention, T , batch size, and dropout. We used an Adam optimizer (Kingma & Ba 2017), with a value of the learning rate of 10 −3 , and we trained the model with cross-entropy loss and a dropout of 0.3. Hyperparameter tuning involves the number of images K that feed the 3DCNN and the maximum length of the sequence. The best values were K = 3 and M = 99, which means the number of sequences for the convolutional BERT is N = 33. The batch size was 128 sequences which we ran over 100 epochs. We chose the number of convolutional BERT layers to be L = 2 and the number of attention heads T = 4. Also, the images were normalized band-wise, as each band has different characteristics. We used only four classes (AGN, SNIa, Variable, SNOther) to train all the models. The class marked as "unknown" has not been considered in the study. It corresponds to noisy or very sparse data. It can easily be tagged from sparsity or noise in the image metrics and we do not expect any improvement in the classification if such objects are added to the training. We trained all models with 4 GPUs GeForce RTX 2080 Ti, Each model takes about three hours to complete training. The implementation will be released upon publication in our Github page 1 . 1 https://github.com/DaBihy/ConvEntion Results This section provides studies on SDSS comparing the accuracy and F1 score of our proposed solution with other works. Table 3 summarizes the result of different models from different deep learning areas to diversify our benchmark as it contains RNN architectures (SuperNNova, LSTM), CNN-based models such as SCONE, Hybrid models that have CNN and RNN such as (Carrasco-Davis et al. 2019) and (Gómez et al. 2020), and, finally, a transformer-based model. Also, we compared the result using two types of datasets: first, the image dataset and, second, the same dataset object but with the light curves; the goal is to highlight the advantage of using images instead of light curves. Moreover, the different works mentioned in Table 3 were initially proposed for different datasets with different classes and training protocols. Hence, the results do not reflect the quality of these works on other datasets. The goal of the comparison is to give visibility into the performance of our model from a deep learning standpoint and the importance of using image time series from an astronomy perspective. Overall, our model ConvEntion obtains the highest accuracy of 79.83% and F1 score of 70.62%, 13 points higher in accuracy than the best results on images by (Gómez et al. 2020) and 12 points higher in accuracy than the best model using light curves. This confirms the advantage of using images over light curves. This advantage can be explained by the fact that the image contains more information than a single value of flux in a light curve. Hence, a model can learn robustly with the existence of more high-level feature maps. Also, ConvEntion performed better compared to the other image-based models, such as Carrasco-Davis et al. (2019). Additionally, transformers give a remarkable computational advantage because transformers avoid recursion and allow for parallel computation, thus reducing the training time. Our model took only three hours to train, compared to other image-based models which took five hours of training on our GPUs. Our model achieved better results using fewer parameters, compared to the other models trained on image sequences. The main benefit of using a transformer is that it reduces the drop in performance due to long dependencies. Transformers do not rely on past hidden states to capture dependencies with previous features such as RNNs. They instead process a sequence as a whole. Therefore, there is no risk of losing past information. Also, the integration of a spatio-temporal feature extraction helped in getting a better high-level representation of the sequence, in comparison to separating the spatial features from the temporal ones. The two types of features have correlations that may help the model to better discriminate between objects. We can also highlight the importance of separating the band to mitigate the impact of missing observations. Our model performed well, in comparison to that of Gómez et al. (2020) which uses multiple bands, which shows that separating the bands and adding band embedding works better than feeding the network with empty bands. In the study of Carrasco-Davis et al. (2019), the authors trained their model on a dataset that only has a "g" band and they noted that the model can be adapted to classify the image sequence combining information using multiple bands. For the sake of comparison, we trained the image models with all the bands "ugriz" at first and then with only one "g" band. Our model achieved an accuracy of 76.89% and 63.20% in the F1 score using one band ("g") which dropped 7% in comparison to using multiple bands. Meanwhile, Carrasco-Davis et al. (2019) achieved 63% in accuracy and 60% in their F1 score. This shows that our model is more efficient when using multiple bands. This also highlights the impact of band separation to mitigate the impact of the missing observations. Figure 6 illustrates the obtained confusion matrix by Con-vEntion and it shows that the model has well classified the supernovas. Most of the misclassified SNIa are associated with SNOther and vice versa, which is not a serious error. This is even an expected behavior, especially since all types of supernovas share a lot of similarities which may confuse the model. Additionally, with a small dataset like ours, it is normal to have such behavior because the model does not have enough samples to totally discriminate among objects. Meanwhile, variables were the best-classified class in our dataset, with just a bit of confusion with the AGN; this misclassification between AGN and variable can be explained by the class imbalance in our dataset based on the knowledge that the number of variables is higher than in the other classes. Table 4 summarizes the results of different models trained only on three classes (AGN, SN, Variable), where classes SNIa and SNOther are combined into a single class. The goal of this experiment is to see the behavior of our model in discriminating between transient and non-transient objects. We got the best results with an accuracy of 83.90% with an F1 Score of 75.77%. The model was able to classify the SN accurately, with a score of 86% (as shown in Figure 7). The model is able to effectively process a given survey without any loss in performance and without the requirement of pro- Table 5. Count of every object in a dataset of each step in training protocol. Train contains only photometrically typed data, "fine-tune" and "test: contain only spectroscopically confirmed data viding it with the time information for each image. However, when there is a covariate shift, or a mismatch, between the training set and the test set as when using a different dataset with a different observation sequence), incorporating the time information can improve the results. This experimental finding will be further studied and reported in future work using other datasets. Conclusion In this work, we present a method for efficient astronomical image time series classification that is entirely based on the combination of convolutional networks and transformers. Inspired by action recognition and satellite image time series classification, we propose a model ConvEntion that utilizes convolutions and transformers jointly to capture complex spatio-temporal dependencies between distinct steps, leading to accurate predictions based on different observations of an object. The accuracy of our model is better with a high margin of 13%, in comparison to state-of-the-art methods using image data -and even better compared to approaches using light curves. Our model achieves good results on the SDSS dataset, while also being faster thanks to using fewer parameters and parallel computational processes, making it a good candidate for latencysensitive applications such as the real-time thumbnail classifier of astronomical events. Meanwhile, our benchmark stands as clear evidence of the importance of images in the domain of astronomy. Indeed, the images contain more information than the normal light curves, even if they present more difficulties. In the future, we plan to scale up ConvEntion using self-supervised learning to investigate whether the model can generalize even better. With a large amount of unlabeled data in astronomy, we believe that the next step to advance AITS classification is creating self-supervised models. Fig. 1 . 1Sample of some objects present in our dataset. Each image in filter g/i corresponds to a different observation with the same filter. Fig. 2 . 2General architecture of the ConvEntion network. The image time series are first rearranged to embed the band information. Fig. 3 . 3Each image has five filters (u, g, r, i, z), The black channel represents the missing observation Fig. 4 . 4Illustration of the handling of missing information by separating the bands. The empty channels are dropped, then we concatenate each image with a 2D representation of the band used to capture the image. The band embedding contains five band representations. The black channel represents the missing observation learnable. After getting the band embedding, we concatenate it with the new image to get our new input J ∈ R M×H×W×2 that contains the band information, where M is the length of the sequence: Fig. 5 . 5study, we only used information about the position of the image in a sequence. While the observation date could be used as an alternative to the position, this would require adjusting the positional encoding function. Our experiments on the Article number, Convolutional attention (left). Multi-head convolutional attention (right). To obtain the query, key, and value maps, we applied a convolution layer on the feature map obtained from 3DCNN. Fig. 6 . 6Confusion matrix showing the average accuracy and standard deviation of the predictions generated by ConvEntion over crossvalidation of five folds on test data. Fig. 7 . 7Confusion matrix of three classes showing the average accuracy and standard deviation of the predictions generated by ConvEntion over cross-validation of five folds on test data. Table 1 . 1Number of objects per class in the SDSS dataset. PT: Photometrically typed, which means that the SNs are not spectroscopically verified A&A proofs: manuscript no. aandaLayer Layer Parameters Conv3d + BN3d 11 × 11 × 3 × 64, 64 Conv3d + BN3d 5 × 5 × 3 × 128, 128 Conv3d + BN3d 3 × 3 × 3 × 64, 64 Conv3d + BN3d 3 × 3 × 3 × 64, 64 Table 2. 3D CNN architecture where Conv3D is a 3D convolutional element and BN3d is a 3D batch normalization element. et al. (2020) demonstrated that a low-rank matrix could approx- imate the self-attention mechanism. They suggested a new self- attention method that minimizes total self-attention complexity. Choromanski et al. Table 3. Performance comparison in terms of average F1 score and the average of the accuracy of five folds of cross-validation . This table includes only experiments on a dataset with four classes.Table 4. Performance comparison in terms of average F1 score and the average of the accuracy of five folds of cross-validation. This table includes only experiments on a dataset with three classes.Model Bands Type of data Accuracy F1 Score Num params ConvEntion (Ours) ugriz Images 79.83 70.62 1.253M CNN+GRU (Gómez et al. 2020) ugriz Images 66.39 63.22 1.993M ConvEntion (Ours) g Images 76.89 63.20 1.253M CNN+GRU (Gómez et al. 2020) g Images 63.67 61.00 1.992M CNN+LSTM (Carrasco- Davis et al. 2019) ugriz Images 64.08 60.65 2.190M CNN+LSTM (Carrasco- Davis et al. 2019) g Images 63.00 60.00 2.189M SuperNNova (Bayes) (Möller & de Boissière 2020) ugriz Light curves 65.54 55.40 - SITS-BERT (Yuan & Lin 2021) ugriz Light curves 67.43 51.60 0.596M SCONE (CNN) (Qu et al. 2021) ugriz Light curves 62.57 50.43 22.2K SuperNNova (RNN) (Möller & de Boissière 2020) ugriz Light curves 56.30 42.60 - LSTM ugriz Light curves 55.24 40.33 60K Model Bands Accuracy F1 Score ConvEntion (Ours) ugriz 83.90 75.77 ConvEntion (Ours) g 79.47 72.38 CNN+GRU (Gómez et al. 2020) g 74.84 68.95 CNN+LSTM (Carrasco-Davis et al. 2019) g 73.94 67.29 Acknowledgements. 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[ "https://github.com/DaBihy/ConvEntion" ]
[ "A theoretical study of spin filtering and its application to polarizing antiprotons", "A theoretical study of spin filtering and its application to polarizing antiprotons" ]
[ "Donie O&apos;brien \nSchool of Mathematics\nUniversity of Dublin Trinity College\n\n" ]
[ "School of Mathematics\nUniversity of Dublin Trinity College\n" ]
[]
There has been much recent research into possible methods of polarizing an antiproton beam, the most promising being spin filtering, the theoretical understanding of which is currently incomplete. The method of polarization buildup by spin filtering requires many of the beam particles to remain within the beam after repeated interaction with an internal target in a storage ring. Hence small scattering angles, where we show that electromagnetic effects dominate hadronic effects, are important. All spin-averaged and spin-dependent electromagnetic cross-sections and spin observables for elastic spin 1/2 -spin 1/2 scattering, for both point-like particles and non-point-like particles with internal structure defined by electromagnetic form factors, are derived to first order in QED. Particular attention is paid to spin transfer and depolarization cross-sections in antiproton-proton, antiproton-electron and positron-electron scattering, in the low | t | region of momentum transfer. A thorough mathematical treatment of spin filtering is then presented, identifying the key physical processes involved and highlighting the dynamical properties of the physical system. We present and solve sets of differential equations which describe the buildup of polarization by spin filtering in many different scenarios of interest. The advantages of using a lepton target are outlined, and finally a proposal to polarize antiprotons by spin filtering off an opposing polarized electron beam is investigated.iii iv The complete set of spin 0 -spin 1 electromagnetic helicity amplitudes are also presented to first order in QED. These are useful in describing the spin-dependent scattering of deuterons off carbon nuclei for example.A thorough mathematical treatment of spin filtering is then presented, identifying the two key physical processes involved: (a) selective scattering out of the ring and (b) selective spin flip while remaining in the ring. The dynamical properties of the physical system under investigation are highlighted. Sets of differential equations are presented and solved which describe the buildup of polarization by spin filtering in many different scenarios of interest. These scenarios are: 1) spin filtering of a stored beam, 2) spin filtering while the beam is being accumulated, i.e. unpolarized particles are continuously being fed into the beam at a constant rate, 3) unpolarized particles are continuously being fed into the beam at a linearly increasing rate, i.e. the particle input rate is ramped up, 4) the input rate is equal to the rate at which particles are being lost due to scattering beyond the ring acceptance angle, the beam intensity remaining constant, 5) increasing the initial polarization of a stored beam by spin filtering, 6) the input of particles into the beam is stopped after a certain amount of time, but spin filtering continues.
10.33232/bims.0062.26.27
[ "https://arxiv.org/pdf/1003.1604v1.pdf" ]
118,799,723
1003.1604
e26e0a8ead31734bb552b971d162dec0c3d2d782
A theoretical study of spin filtering and its application to polarizing antiprotons 8 Mar 2010 June, 2008 Donie O&apos;brien School of Mathematics University of Dublin Trinity College A theoretical study of spin filtering and its application to polarizing antiprotons 8 Mar 2010 June, 2008 There has been much recent research into possible methods of polarizing an antiproton beam, the most promising being spin filtering, the theoretical understanding of which is currently incomplete. The method of polarization buildup by spin filtering requires many of the beam particles to remain within the beam after repeated interaction with an internal target in a storage ring. Hence small scattering angles, where we show that electromagnetic effects dominate hadronic effects, are important. All spin-averaged and spin-dependent electromagnetic cross-sections and spin observables for elastic spin 1/2 -spin 1/2 scattering, for both point-like particles and non-point-like particles with internal structure defined by electromagnetic form factors, are derived to first order in QED. Particular attention is paid to spin transfer and depolarization cross-sections in antiproton-proton, antiproton-electron and positron-electron scattering, in the low | t | region of momentum transfer. A thorough mathematical treatment of spin filtering is then presented, identifying the key physical processes involved and highlighting the dynamical properties of the physical system. We present and solve sets of differential equations which describe the buildup of polarization by spin filtering in many different scenarios of interest. The advantages of using a lepton target are outlined, and finally a proposal to polarize antiprotons by spin filtering off an opposing polarized electron beam is investigated.iii iv The complete set of spin 0 -spin 1 electromagnetic helicity amplitudes are also presented to first order in QED. These are useful in describing the spin-dependent scattering of deuterons off carbon nuclei for example.A thorough mathematical treatment of spin filtering is then presented, identifying the two key physical processes involved: (a) selective scattering out of the ring and (b) selective spin flip while remaining in the ring. The dynamical properties of the physical system under investigation are highlighted. Sets of differential equations are presented and solved which describe the buildup of polarization by spin filtering in many different scenarios of interest. These scenarios are: 1) spin filtering of a stored beam, 2) spin filtering while the beam is being accumulated, i.e. unpolarized particles are continuously being fed into the beam at a constant rate, 3) unpolarized particles are continuously being fed into the beam at a linearly increasing rate, i.e. the particle input rate is ramped up, 4) the input rate is equal to the rate at which particles are being lost due to scattering beyond the ring acceptance angle, the beam intensity remaining constant, 5) increasing the initial polarization of a stored beam by spin filtering, 6) the input of particles into the beam is stopped after a certain amount of time, but spin filtering continues. Summary Immense efforts, both theoretical and experimental, have been afforded to gaining a better understanding of the spin structure of the nucleon since the startling results from the EMC experiment at CERN in 1988 that the intrinsic valence quarks contribute only a small fraction of the proton's spin. Yet to this day almost nothing is known about the transversity distribution of quarks in the nucleon, the last remaining leading twist piece of the QCD description of the partonic structure of the nucleon in the collinear limit. A high intensity polarized antiproton beam would be required to best analyze the transversity distribution function, via Drell-Yan lepton pair production in the scattering of polarized antiprotons off polarized protons. Unfortunately no high intensity polarized antiproton beam has been achieved to date. Hence there has been much recent research into possible methods of polarizing an antiproton beam, instigated by the recent proposal of the PAX (Polarized Antiproton eXperiments) Collaboration at GSI, Darmstadt. The most promising method under consideration is spin filtering, the theoretical understanding of which is currently incomplete. The method of polarization buildup by spin filtering requires many of the beam particles to remain within the beam after repeated interaction with an internal target in a storage ring. Hence small scattering angles, where we show that electromagnetic effects dominate hadronic effects, are important. The theoretical background to this effort is investigated in this thesis. We derive fully relativistic expressions for all spin-averaged and spin-dependent electromagnetic cross-sections and spin observables for elastic spin 1/2 -spin 1/2 scattering, for both point-like particles and non-point-like particles with internal structure defined by electromagnetic form factors, to first order in QED. Particular attention is paid to spin transfer and depolarization cross-sections in antiprotonproton, antiproton-electron and positron-electron scattering, in the low | t | region of momentum transfer. Of the spin-averaged formula derived we highlight that a generalization of the Rosenbluth formula is presented in a new compact Lorentz invariant form. It is a two-fold generalization in that the masses of both particles are included and both particles are taken to have internal structure determined by electromagnetic form factors. While these results are eventually applied to spin filtering later in the thesis they are not limited to this application. The complete set of spin 1/2 -spin 1/2 helicity amplitudes and spin observables should prove useful to many other areas in particle physics. Acknowledgments Firstly I thank my supervisor Dr. Nigel Buttimore, for his constant support and encouragement. His enthusiasm for the field of research made the process much more enjoyable than it otherwise would have been. I express a special thank you to my parents, for always being there for me, for instilling in me an insatiable thirst for knowledge and for helping me see this thesis through to completion. It is with great pleasure that I dedicate this thesis to them. I wish to thank the International research institutions DESY Zeuthen, ECT* Trento and the University of Crete, where I spent extended periods. I learned much about particle physics, and the process of physical research, at these institutions. The beautiful Alpine surroundings at ECT* Trento provided great inspiration, and is where much of this thesis began to take shape. Thanks to all my team-mates on the Connaught, UCD, TCD and Brookfield tennis teams, for all the laughs and good times on and off the court over the years. Finally I thank the staff of the School of Mathematics, in particular the administrative staff Helen and Karen for doing a fantastic job and making all of our lives much easier. Introduction The great Danish physicist Niels Bohr, it is said, had a good-luck horseshoe hanging in his office. "You don't believe in that nonsense, do you?" a visitor once asked, to which Bohr replied, "Of course not, but they say it brings you good luck whether you believe in it or not." Spin and polarization Spin is a fundamental property of elementary particles. It was introduced theoretically by Wolfgang Pauli in the 1920's to explain how two electrons can exist in the ground state of an atom, while not violating his famous exclusion principle, for which he was awarded the 1945 Nobel Prize. This principle states that two particles satisfying Fermi-Dirac statistics (later called fermions and defined by their half-integer spin) cannot exist in the same state at the same time. Thus the two electrons in the inner shell of an atom must somehow be different, Pauli hypothesized that they have some differentiating characteristic called spin. He theorized one electron to be in a 'spin up' state and the other electron to be in 'spin down' state, thus they do not violate the exclusion principle. Uhlenbeck and Goudschmidt [1] also introduced the concept of spin around the same time as Pauli's work. Pauli's theory of spin was non-relativistic, Paul Dirac developed the relativistic theory of spin in 1928 with the famous Dirac equation of a relativistic electron [2]. Spin was discovered experimentally by the famous Stern-Gerlach experiment in 1922 [3]. Since then it has been an integral part of the Quantum Field Theories that describe particle interactions. A beam of spin 1/2 particles will have the spins of each of the particles in either the 'spin up' or 'spin down' state. For a beam of spin 1/2 particles the polarization is defined as 1.1) where N + and N − are the number of particles in the 'spin up' and 'spin down' states respectively. An unpolarized beam (P = 0) is one where the spin states are randomly distributed, thus for a beam with a large number of spin 1/2 particles, half of the particles will be in the 'spin up' state and half in the 'spin down' state, as seen in Figure 1-2. A polarized beam is one where more of the particles are in one spin state than the other 1 . For example, a 100 % polarized beam (P = 1) has all of the particles in one of the spin states. P = N + − N − N + + N − ,(1. Originally particle physics experiments used unpolarized beams and targets, thus completely overlooking the spins of the particles. This way only total cross-sections instead of spin-dependent cross-sections can be measured. Only a small portion of the reaction can thus be investigated. In the words of the originator of the spin filtering method of polarization buildup, which much of the investigations in this thesis are based on, P. L. Csonka [4] ; "One could, perhaps, say that the physicist who is able to measure only total cross-sections, is like the man in an art gallery who is only told the total weight of each statue, but is kept in ignorance of all other parameters specifying their shapes. Most of us would agree that he is missing something". Nowadays highly polarized beams of certain particles are possible. Polarized electrons and positrons have been used for many decades. Baryons have proved more difficult to polarize. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory, New York is the first accelerator to use a high energy polarized proton beam. A "spin crisis" in the parton model Prior to the European Muon Collaboration (EMC) [5,6] and the Spin Muon Collaboration (SMC) [7] experiments at CERN it was assumed that all the spin of the nucleon was carried by its three constituent valence quarks. The startling results of EMC in 1988 [5] and 1989 [6] showed that the three constituent valence quarks contribute very little to the spin of the nucleon. This caused, what was dubbed "The spin crisis in the parton model" [8,9], prompting a new theoretical investigation into the spin structure of the nucleon, which continues to this day. The phrase "spin crisis" which endures to this day was coined in a beautifully titled paper "A crisis in the parton model: where, oh where is the proton's spin?" by Mauro Anselmino and Elliot Leader [8], and presented at the SPIN 1988 Symposium in Minneapolis, USA. The fact that the two original EMC papers were the most cited experimental papers in the field for three years and have a combined total of over 2500 citations shows the immense effort that has been afforded to solving the "spin crisis". It is now proposed that the spin of the nucleon is made up of the helicity of the constituent quarks ∆q, the helicity of the gluons ∆G, the orbital angular momentum of The diagram on the left shows the naive expectation that the spin of the nucleon is entirely constituted by the three valence quarks. The EMC and SMC results proved that this was not correct. The diagram on the right shows the current more complex view of nucleon spin structure, with contributions to the nucleon spin coming from the valence quarks, sea quarks, gluons and orbital angular momentum. the quarks L q , the orbital angular momentum of the gluons L g and the transversity of the quarks [referred to in the literature as either δq, ∆ T q or h 1 q , we shall use the latter notation], as seen in Figure 1 where the superscript L refers to Longitudinal, and the transverse spin sum rule [10]: S T Nucleon = 1 2 = 1 2 h 1 q + L T q + L T g ,(1. 2.2) where the superscript T refers to Transverse. The current knowledge of these constituents is summarized in Figure 1-4. The contributions L q , L g and ∆q are known from experiment [11,12,13,14] and from Lattice QCD studies [15]. There are currently many theoretical models [16] and experimental programs obtaining information on ∆G, these include HERMES, COM-PASS, JLAB and RHIC. But the last piece of the puzzle, the transversity distribution function h 1 q is to date almost completely unknown. In order to best measure h 1 q , a beam of polarized antiprotons would be required as we explain in the next section. Much of this thesis is devoted to a theoretical investigation of possible methods to polarize an antiproton beam in a storage ring. Antiprotons Antiprotons are the anti-particles of protons, which in turn are the core of the hydrogen atom, the most abundant element in the Universe. The proton was shown to have an internal structure, i.e. not be a point particle, during seminal elastic electron-proton experiments in the Stanford Mark 3 accelerator, from 1954 to 1957 [17]. Robert Hofstadter was awarded the 1961 Nobel prize for this ground-breaking discovery which ushered in a new era of investigation into the structure of the nucleon. A decade later, the much higher energy SLC accelerator was built at Stanford to investigate Deep Inelastic Scattering (DIS) experiments, showing the proton to be made up of point like quarks [18]. Again this achievement warranted the Nobel prize, in 1990 to Jerome Friedman, Henry Kendall and Richard Taylor. The proton consists of uud valence quarks, hence the antiproton consists ofūūd valence quarks. Above, as in the rest of the thesis, we denote antiparticles by an over-bar. In shorthand notation protons are denoted p and hence antiprotons are denotedp. Antiparticles have the opposite electromagnetic charge of their corresponding particle, thus for chargeless particles (e.g. the photon) the antiparticle is the same as the particle. A proton has electromagnetic charge +1 in units where the electron charge is −1, thus an antiproton has electromagnetic charge −1. While the concept of anti-matter often seems mysterious at first glance, it should be remembered that the positron (the anti-particle of the electron) was the third elementary particle discovered [19], after the electron and the photon. Hence anti-matter has been an integral part of physical theories since 1932, the same year the neutron was discovered. Antiprotons were discovered in 1955 by Owen Chamberlain and Emelio Segrè [20], who were awarded the 1959 Nobel prize for this ground-breaking discovery. Professor Chamberlain, who recently passed away, spent the rest of his life contributing great efforts to the investigation of polarization phenomena and spin physics in general [21]. He was the first to investigate a possibility of polarizing antiprotons, and he co-organized the first workshop on polarizing antiprotons at Bodega Bay, California in 1985. The conclusions of this workshop [22], were that a high intensity polarized antiproton beam was not achievable at that time. Another 22 years passed before the International community felt a sequel to this workshop was necessary, during which time interest in polarizing antiprotons grew steadily. There has been much recent interest in producing a high intensity beam of polarized antiprotons, starting in 2004 with a proposal by the Polarized Antiproton eXperiments (PAX ) Collaboration at GSI Darmstadt [23]. Since then many theories have been put forward on how to produce such a beam. So in 2007 a sequel to the Bodega Bay workshop was organized in the newly founded Cockcroft Institute for accelerator research at the Daresbury Laboratory, UK [24]. A thorough investigation of the theoretical aspects of producing a polarized antiproton beam is presented in this thesis. Antiproton-proton colliders have played an important role in the advancement of High Energy Physics. In particular they led to the discovery of the W and Z bosons, and thus to the verification of the Weinberg, Glashow, Salam (1979 Nobel Prize) unified theory of electroweak interactions. This was done by the UA1 and UA2 experiments at the SPS (Super Proton Synchrotron) collider in CERN in 1982 and led to the 1984 Nobel Prize to be awarded to Carlo Rubbia and Simon van der Meer. One hopes that future polarized antiproton-proton colliders will lead to further epoch-making discoveries. Outline of the thesis The major theme of the thesis is a theoretical investigation of the spin filtering method of polarization buildup, and an application of this to producing a high intensity polarized antiproton beam. There is much debate in the International community as to the correct theoretical description of spin filtering. We hope that the thorough analysis of spin filtering presented here will clarify some of this confusion. No high intensity polarized antiproton beam has ever been achieved, and since a high intensity polarized antiproton beam could be used to measure many important quantities in particle physics, it is a main goal of the International community. We first calculate all polarization dependent cross-sections in QED for the processes of interest, then we develop a set of differential equations using these polarization dependent cross-section to describe spin filtering; finally numerical results are obtained from this formalism. In Chapter 2 the motivation for the thesis is outlined. The benefits of a polarized antiproton beam are described, as are all possible methods to produce such a beam. The methods to polarize bunches of other particles and atoms are also presented, such as electrons, positrons, protons, hydrogen and deuterium; and it is explained that none of these can be applied to the elusive case of polarizing antiprotons. It is concluded that spin filtering is the most promising method to produce a high intensity polarized antiproton beam and the chapter concludes with an overview of spin filtering and a description of how it has been verified experimentally. In Chapter 3 all electromagnetic helicity amplitudes and spin observables, accounting for polarization effects in spin 1/2 -spin 1/2 elastic scattering are calculated. Many of these results will be utilized in later chapters when providing a mathematical description of spin filtering, although their use is certainly not limited to this. The spin 1/2 electromagnetic currents are introduced, both for point particles and particles with internal structure determined by electromagnetic form factors. A generic equation is derived that can be used to calculate all polarization phenomena in elastic spin 1/2 -spin 1/2 electromagnetic scattering to first order in QED. We then present results for all electromagnetic helicity amplitudes and spin observables for elastic spin 1/2 -spin 1/2 scattering. The spin-averaged differential cross-section for spin 1/2 -spin 1/2 scattering is also presented in a new compact invariant form. These results are then presented in Chapter 4 for the specific cases of: antiproton-proton, antiproton-electron and positron-electron scattering. Then the cross-sections and spin observables needed for spin filtering are explicitly presented, which will be utilized in the polarization evolution equations developed in Chapter 5. The chapter concludes with a calculation of all spin 0 -spin 1 helicity amplitudes, which describe the scattering of deuterons off a carbon nucleus for example. The theory of spin filtering is developed in Chapter 5. A mathematical description of the related but simpler process of polarization buildup by the Sokolov-Ternov effect is first presented. The ideas presented are utilized in the mathematical descriptions of spin filtering which follow. The rates of change of the number of particles in each spin state are combined into a set of polarization evolution equations which describe the process of polarization buildup by spin filtering. This set of polarization evolution equations is then analyzed and solved, emphasizing the physical implications of the dynamics. The chapter concludes with an investigation of spin filtering of a stored beam. Chapter 6 presents a thorough investigation of spin filtering under various alternate scenarios, which would be of interest to any practical project to produce a high intensity polarized antiproton beam. These scenarios are: 1) spin filtering while the beam is being accumulated, i.e. unpolarized particles are continuously being fed into the beam at a constant rate, 2) unpolarized particles are continuously being fed into the beam at a linearly increasing rate, i.e. the particle input rate is ramped up, 3) the particle input rate is equal to the rate at which particles are being lost due to scattering beyond the ring acceptance angle, the beam intensity remaining constant, 4) increasing the initial polarization of a stored beam by spin filtering, and 5) the input of particles into the beam is stopped after a certain amount of time, but spin filtering continues. As an application of the theoretical work presented throughout the thesis a possible method to produce a high intensity polarized antiproton beam by spin filtering off an opposing polarized electron beam is presented in Chapter 7. It is also outlined how this work can be applied to polarizing antiprotons by spin filtering off a polarized hydrogen target. Firstly a description of the electron cooling technique to refocus the beam after scattering off the target each revolution in order to maintain high beam density is presented. Then the various experimental input parameters, such as revolution frequency, target areal density, target polarization and the effective acceptance angle; needed to obtain realistic numerical estimates from our mathematical formalism are each described. The benefits of using a lepton target are then described, before analyzing the case of spin filtering off an opposing polarized electron beam. Finally spin filtering off a polarized hydrogen target is discussed, in the three cases of hydrogen with only electrons polarized, hydrogen with only protons polarized and finally hydrogen with both electrons and protons polarized. It is shown that electromagnetic effects dominate hadronic effects inp p scattering in the region of low momentum transfer of interest in spin filtering. In Chapter 8 some concluding remarks are presented. Notation and conventions The conventions will mainly follow the book of Peskin and Schroeder [25]. Rationalized units, where = c = 1, will be used throughout the thesis unless otherwise stated. Units in this system are as follows: indices are summed over, is used throughout the thesis, unless otherwise specified. Our conventions for Dirac spinors are presented in Appendix A. The Feynman slash notation / p = γ µ p µ , and the Minkowski metric tensor η µν = diag(+1, −1, −1, −1) are used. The spin four vectors are normalized such that S µ S µ = − 1. We use the shorthand notation A · B for the scalar product in 4- dimensional Minkowski space, where A·B = A µ B µ = η νµ A ν B µ = A 0 B 0 − A·B. The totally antisymmetric tensor ǫ µνρσ , also known as the Levi-Civita symbol, is defined such that ǫ 0123 = + 1 and ǫ 0123 = − 1, as seen in Appendix A. Antiparticles are denoted by an over-bar. In shorthand notation protons are denoted p and hence antiprotons are denotedp. Electrons and positrons are denoted by e − and e + respectively. Time increases from left to right in all Feynman diagrams throughout the thesis. Arrows on particle lines in Feynman diagrams denote the flow of particle number, which is forwards for particles and backwards for antiparticles. We denote the time variable in each of the dynamical systems by τ to avoid confusion with the squared momentum transfer (Mandelstam t variable) used throughout the thesis. The scattering processes investigated in the thesis are always 2 → 2 elastic processes, with the momentum and spin 4-vectors of each particle labeled as: A ( p 1 , S 1 ) + B ( p 2 , S 2 ) −→ A ( p 3 , S 3 ) + B ( p 4 , S 4 ) , with the particles above being the beam (1), target (2), scattered (3) and recoil (4) particles respectively. The four momentum transfer is defined as q = p 3 − p 1 = p 2 − p 4 . The helicity amplitudes are represented by M ( scattered, recoil ; beam, target ) = M ( λ 3 , λ 4 ; λ 1 , λ 2 ) . The arguments are to be read from right to left, as λ 1 and λ 2 correspond to the incoming particles in the reaction and λ 3 and λ 4 correspond to the outgoing particles in the reaction. For spin 1/2 -spin 1/2 scattering the helicities λ i = ± for i ∈ { 1, 2, 3, 4 } are + if the particles spin vector points in the direction of its momentum vector and − if the particles spin vector points in the opposite direction to its momentum vector. The ± in the helicity amplitudes are shorthand for ±1/2 , the helicity of a spin 1/2 particle. The spin observables are represented by K ab for the polarization transfer observable and ( 1 − D ab ) for the depolarization observable, where a, b ∈ { X, Y, Z } for the direction of the particles spin vector where its momentum is along the Z direction. The subscripts are read from right to left, in e.g. K ab where b is the direction of the spin vector of the incoming particle and a is the direction of the spin vector of the outgoing particle. Chapter 2 Motivation "Polarization data has often been the graveyard of fashionable theories. If theorists had their way, they might just ban such measurements altogether out of self-protection." J. D. Bjorken In this chapter the motivation for the present work is discussed. The benefits to the high energy physics community of a high intensity polarized antiproton beam are first presented in section 2.1, by describing the important parameters in particle physics that could be measured and investigated with such a beam. Section 2.2 describes how bunches of other particles and atoms are polarized, such as electrons, positrons, protons, hydrogen and deuterium. Unfortunately none of these tried and tested techniques can be applied to the elusive case of polarizing antiprotons. Section 2.3 describes and compares some possible methods to polarize antiprotons. It is concluded that spin filtering is the most promising technique to produce a polarized antiproton beam as it is the only technique that has been experimentally verified. The chapter concludes with an overview of the theory of spin filtering, and a section showing how spin filtering was verified for polarizing a proton beam by repeated scattering off a polarized hydrogen target in a storage ring by the FILTEX experiment in 1993. Motivation for a polarized antiproton beam A high intensity polarized antiproton beam would provide the unique possibility to measure many very important quantities in particle physics. The most important quantity that could be measured is the transversity distribution of quarks inside protons, which has eluded direct measurement thus far. Two other important investigations, into single spin asymmetries and nucleon electromagnetic form factors, can also be greatly advanced if a high intensity polarized antiproton beam was available. These motivations for producing a high intensity polarized antiproton beam are described in detail below. The transversity distribution function The transversity distribution function is the last leading twist 1 piece of the QCD description of the partonic structure of the nucleon, in the collinear limit 2 , that has not been directly measured. It describes the quark transverse polarization inside a transversely polarized nucleon. In fact, to date, almost nothing is known about the transversity distribution, except for the recent work of Anselmino et al. [26,27]. Unlike the other leading twist distributions [the unpolarized quark distribution q (x, Q 2 ) and the helicity distribution ∆q (x, Q 2 )] which have been measured, the transversity h 1 q (x, Q 2 ) [sometimes referred to in the literature as ∆ T q (x, Q 2 ) or δq (x, Q 2 )] can neither be accessed in deep inelastic scattering of leptons off nucleons, nor can it be reconstructed from the knowledge of q (x, Q 2 ) and ∆q (x, Q 2 ) [23]. In a transversely polarized hadron, h 1 q (x, Q 2 ) is the number density of quarks with momentum fraction x and polarization parallel to that of the hadron, minus the number density of quarks with the same momentum fraction and antiparallel polarization, [28]. One cannot claim to understand the spin structure of the nucleon until all three leading twist structure functions have been measured. i.e. h 1 q (x, Q 2 ) = q ↑ (x, Q 2 ) − q ↓ (x, Q 2 ) In order to best access the transversity distribution function, the double spin asymmetry A TT in the Drell -Yan production of lepton pairs must be measured; thus both initial particles in a reaction must be transversely polarized. It could in future be done for p ↑ p ↑ scattering at RHIC, but this asymmetry is expected to be small from theory [29], as explained below. Also the cross-section for Drell -Yan lepton pair 1 Leading twist means that in the factorization of a physical process the parton distribution function appears in the leading order of 1/Q 2 . 2 The collinear limit is where the intrinsic transverse motion of the quarks is averaged over. production is much higher forp p scattering than for p p scattering (σp p DY ≫ σ p p DY ); because in the former case valance quarks in the proton annihilate with valance antiquarks in the antiproton, as opposed to with sea antiquarks in the second proton in the latter case. The Drell -Yan lepton pair production process [31] is shown in A ij ≡ dσ ↑↑ − dσ ↑↓ dσ ↑↑ + dσ ↑↓ ,(2.A p p TT = d∆σ dσ q e 2 q h p 1 q (x 1 , M 2 ) h p 1q (x 2 , M 2 ) + h p 1q (x 1 , M 2 ) h p 1 q (x 2 , M 2 ) q e 2 q [ q p (x 1 , M 2 )q p (x 2 , M 2 ) +q p (x 1 , M 2 ) q p (x 2 , M 2 ) ] , ≈ d∆σ dσ h p 1 u (x 1 , M 2 ) h p 1ū (x 2 , M 2 ) + h p 1ū (x 1 , M 2 ) h p 1 u (x 2 , M 2 ) u p (x 1 , M 2 )ū p (x 2 , M 2 ) +ū p (x 1 , M 2 ) u p (x 2 , M 2 ) , (2.1.2) where d∆σ and dσ are the polarized and unpolarized cross-sections of the elementary QED process qq → l − l + respectively, M is the invariant mass of the lepton pair, e q is the electromagnetic charge of the quarks and x 1 and x 2 are the fraction of their respective nucleon momentum carried by each of the interacting partons. The leading term in the approximation comes from the fact that the u quark dominates at large x [28]. Whereas for p ↑p ↑ Drell -Yan processes A pp TT = d∆σ dσ q e 2 q h p 1 q (x 1 , M 2 ) hp 1q (x 2 , M 2 ) + h p 1q (x 1 , M 2 ) hp 1 q (x 2 , M 2 ) q e 2 q [ q p (x 1 , M 2 )qp (x 2 , M 2 ) +q p (x 1 , M 2 ) qp (x 2 , M 2 ) ] , ≈ d∆σ dσ h p 1 u (x 1 , M 2 ) hp 1ū (x 2 , M 2 ) u p (x 1 , M 2 )ūp (x 2 , M 2 ) , (2.1.3) the latter of which is much larger since there are more antiquarks in antiprotons mak- [29]. Thus A pp TT , which can only be measured using a polarized antiproton beam, is expected to be much bigger than A p p TT . Note in the literature all quantities in these equations are often written with respect to the proton using the fact that the distribution of antiquarks in a proton is equal to the distribution of quarks in an antiproton etc. but here we want to keep the antiproton distribution functions explicit. Using the above fact, and at x 1 = x 2 , eq. (2.1.3) reduces to ing h p 1 u (x, Q 2 ) = hp 1ū (x, Q 2 ) ≫ h p 1ū (x, Q 2 ) = hp 1 u (x, Q 2 )A pp TT = d∆σ dσ h p 1 u (x 1 , M 2 ) u (x 1 , M 2 ) 2 , (2.1.4) providing a unique direct way to measure a single transversity distribution function [29]. Also q (x, Q 2 ) andq (x, Q 2 ) decrease with increasing x, so to measure A TT large x 1 and x 2 is favoured [29,30]. Interestingly this happens for lower energy scattering again making a low/medium energy facility, such as that proposed by the PAX Collaboration, more suited than RHIC 3 . At RHIC energies, even though A TT could be detected it only measures the transversity of the sea quarks, at the lower PAX energies we could investigate the transversity of the valence quarks [26,30]. Single spin asymmetries Single Spin Asymmetries (SSA), where one of the initial particles in the reaction is polarized in the direction of the arrows below, are defined as A j ≡ dσ ↑ − dσ ↓ dσ ↑ + dσ ↓ , (2.1.5) where j can be either L for longitudinal, or T for transverse. Some data on SSA inp ↑ p Drell-Yan lepton pair production was obtained by the E704 experiment at Fermilab [32,37], but because the collisions were in the energy region of J/ψ production 4 it was difficult to distinguish the Drell-Yan signal from 3 Note there is a proposal to run RHIC at √ s = 50 GeV instead of their usual √ s = 200 GeV which would make it suitable in this regard, but the problem of no antiproton beam would still remain. In Run 6 (2006) RHIC used √ s = 62.4 GeV. 4 The J/ψ particle is the first excited state of charmonium, a meson consisting of one charm quark and one charm antiquark. Two papers by separate experiments announcing its discovery were published on the same day, one group naming it the J particle and the other group naming it the ψ particle. It has come to be known as the J/ψ particle. the large J/ψ production background. The low intensity polarized antiproton beam used in E704 is described in section 2.3.1. Importantly analyzing charm production inp ↑ p scattering will make it possible to disentangle the Sivers [33,34] and the Collins mechanisms [35], of which there is great theoretical interest. In general, both effects contribute to the measured SSA, but in the case of charm production the Collins mechanism drops out. A polarized antiproton beam would allow further analysis of single spin asymmetries inp ↑ p scattering, augmenting the brief Fermilab data on this [32], and adding to the current data on single spin asymmetries which have been observed inp p ↑ and p ↑ p reactions [36,37,38] and the double spin asymmetries observed in p ↑ p ↑ reactions at RHIC [39]. These observed asymmetries are very large, up to 40% [36], prompting Stan Brodsky to call them "the greatest asymmetries ever seen by a human being" constituting "one of the unsolved mysteries of hadron physics". There is much current interest in the theoretical community to try to achieve a satisfactory understanding of these large single spin asymmetries [38,40,41,42,43,44]. Electromagnetic form factors of the proton The fact that nucleons (protons and neutrons) are not point particles and have an internal structure, is parameterized into electromagnetic form factors, as treated later in section 3.1. The Sachs electric and magnetic form factors G E and G M contain information on the finite charge radius of the proton, thus are very important components of a complete understanding of particle physics. They can be measured experimentally but there is not, to date, complete agreement between the experimental results and theoretical models of the form factors [45,46]. There is much current theoretical interest in nucleon time-like form factors [47]. A polarized antiproton beam would enable the first measurement of the moduli and the relative phase of the time-like electric and magnetic form factors G E and G M of the proton. An unexpected Q 2 = − q 2 dependence of the G E (q 2 ) / G M (q 2 ) ratio of the electric and magnetic form factors of the proton, has been observed at the Jefferson laboratory (JLAB), the ratio decreasing monotonically with increasing Q 2 [48,49]. It would be possible to clarify this unexpected Q 2 dependence by a measurement of the relative phases of G E (q 2 ) and G M (q 2 ) in the time-like region, which would constrain and discriminate strongly between the models for the form factors. This phase can be measured for the first time in the reactionsp ↑ p → e + e − andp p ↑ → e + e − [47], the former of which is uniquely possible with a polarized antiproton beam. The JLAB data was obtained by analyzing the polarization transfer reaction p e − ↑ → p ↑ e − , this data could be augmented and checked by analyzing the polarization transfer reaction p e − ↑ →p ↑ e − , which is at the heart of the spin filtering technique discussed in detail throughout this thesis. The relative phase ambiguity can also be addressed by measuring the double spin asymmetry in the reactionp ↑ p ↑ → l + l − , where l is any lepton. This reaction can also be used to analyze the G E − G M separation, thus serving as a check of the Rosenbluth separation in the time-like region [50]. Polarizing bunches of particles or atoms High intensity beams of polarized electrons, positrons and protons, as well as polarized atomic gas targets have been used in high energy physics laboratories throughout the world. We now briefly describe how they are polarized. • Sokolov-Ternov effect ('radiative' or 'self'-polarization): A beam of charged particles circulating in a storage ring is automatically polarized because of a difference in the spin-flip transition rates due to emission of photons by synchrotron radiation induced by bending in the magnetic field of the ring. This method works well for polarizing electrons and positrons, but not for heavier particles such as protons or antiprotons, as explained in section (2.3.3). • Atomic hydrogen and deuterium are polarized by removing atoms in certain hyperfine states, and inducing angular momentum conserving transitions between hyperfine states. • Once hydrogen is polarized the electrons can be stripped off in a magnetic field leaving polarized protons. Unfortunately it is not possible to produce a high intensity polarized antiproton beam using any of these tried and tested methods, as to do so one would need a large supply of antihydrogen atoms. The Sokolov-Ternov effect will be described in section (2.3.3) and we describe how polarized hydrogen is obtained in section (2.2.1). Polarizing hydrogen gas Unpolarized hydrogen atoms in a strong magnetic field equally populate each of four hyperfine states: | ↑ p ↓ e | ↓ p ↓ e | ↓ p ↑ e | ↑ p ↑ e An inhomogeneous magnetic field acts as a Stern-Gerlach apparatus separating the atoms in the states | ↑ p ↑ e and | ↓ p ↑ e from those in the states | ↓ p ↓ e and | ↑ p ↓ e . A sextupole magnet focuses the atoms in one pair of states while defocusing the others. Thus one can extract atoms in the states | ↑ p ↑ e and | ↓ p ↑ e , i.e. hydrogen atoms in which the electrons are totally polarized, but protons unpolarized. If one then requires hydrogen atoms in which the protons are totally polarized but the electrons unpolarized, angular momentum conserving transitions from | ↓ p ↑ e to | ↑ p ↓ e can be induced by a radio frequency field. Hydrogen with both the electrons and protons polarized can be obtained by isolating the | ↑ p ↑ e state, but with only half the intensity of hydrogen with either electrons or protons polarized. In summary there are three types of polarized hydrogen, with all atoms in the hyperfine states as follows where we denote the polarization of the protons in the hydrogen by P p and the polarization of the electrons in the hydrogen by P e . In practice the atoms are not perfectly isolated in certain hyperfine states, thus the electron and proton polarizations in polarized hydrogen are less than one. The HERMES Collaboration at DESY have utilized polarized hydrogen and polarized deuterium targets with P p = 0.9 and/or P e = 0.9 [51], and these targets are now being used by the PAX Collaboration in COSY Jülich for preliminary tests on spin filtering [52,53]. | ↑ p ↑ e + | ↑ p ↓ e =⇒ P Other atoms, such as deuterium, can be polarized analogously. Stripping these atoms of electrons in a magnetic field leaves a polarized ion beam. It is not possible to generate a beam of polarized antiprotons by this means as it is, thus far, not possible to accumulate large numbers of antihydrogen atoms. The RHIC collider in Brookhaven National Laboratory, New York is the world's first high intensity polarized proton accelerator. A high intensity polarized antiproton collider would greatly supplement and add to results obtained at RHIC. Methods to polarize an antiproton beam Now that we have demonstrated the incredible potential of a high intensity polarized antiproton beam, let us investigate the various methods of generating such a beam. Physicists have been trying to produce beams of polarized antiprotons for over 25 years, a great summary of proposed methods is given in Ref. [22]. Atomic beam sources, used in the production of polarized protons and heavy ions will not work because of the annihilation of antiprotons with matter. The E704 experiment at Fermilab has produced polarized antiprotons from the decay of polarized Λ hyperons, but the intensities achieved were too low for current needs. Storing antiprotons in a storage ring would help build up to a high luminosity beam. Spin filtering has been proven to work for protons scattering off a polarized internal hydrogen target in the FILTEX experiment at the TSR ring in Heidelberg in 1992-1993 [54]. Thus spin filtering is the only plausible experimentally tested technique for generating a high intensity polarized antiproton beam. In light of this we devote much of this thesis to the theoretical understanding of spin filtering in general. As an application of our theoretical work we propose a method to polarize antiprotons by spin filtering off an opposing polarized electron beam, and calculate the polarization buildup time and maximum polarization possible in this case. Antihyperon decay Antihyperons are produced when a Multihundred-GeV proton beam strikes a target. The antihyperons decay into antiprotons, which should have the same polarization as the protons from hyperon decay. A polarized antiproton beam of this type was produced at Fermilab's E704 experiment [55]. The low intensity (because it is a tertiary beam) and large phase space made it difficult to store and accelerate these polarized antiprotons; however it was possible to scatter them off a polarized or unpolarized proton target. The polarization of the antiprotons comes from parity violating decays of antilambdas, and the measured polarization was as high as 64 %. The target they used to produce the antilambdas was Beryllium, and their polarized antiproton beam intensities were up to 1.5 × 10 5 s −1 [55]. This method of producing polarized antiprotons is not suitable for high luminosity experiments, such as the high intensity beam a storage ring could provide, which are needed to access transversity and other measurements as outlined in section 2.1. Stern-Gerlach separation A possible method to produce a polarized antiproton beam from an unpolarized antiproton beam is based on the Stern-Gerlach effect. In an inhomogeneous magnetic field the spins of particles, aligned parallel or antiparallel to the field, are deflected in opposite directions and become spatially separated. For this reason this method, proposed by Niinikoski and Rossmanith in 1985 [56], is also called the spin-splitter technique. A major advantage of this method is that the beam can first be accelerated to any desired energy and then polarized, thus avoiding the loss of polarization associated with accelerating polarized beams 5 . In a typical storage ring inhomogeneous magnetic fields are provided by the quadrupoles. It was hoped that the spatial separation of the particles in the two spin states would add up on passing through many quadrupoles, and further over many revolutions in the storage ring; eventually leading to a macroscopic separation of the particles in opposite spin states [56]. One spin state can then be dumped, or flipped, and one is left with a polarized antiproton beam. Unfortunately, after much interest in this technique [57,58], the International Community has doubts as to whether effects in successive quadrupoles will add up coherently [59]. The effects may continuously cancel each other out and one will be left with no net separation of particles in the two spin states. At the very least this method would have to be experimentally verified before being considered a practical method of producing a polarized antiproton beam. Spontaneous synchrotron radiation emission Charged particles emit synchrotron radiation, in the form of photons, when bent in a magnetic field. There is a slight difference in the spin-flip transition cross-sections due to this photon emission: σ e − ↑ → e − ↓ γ = σ e − ↓ → e − ↑ γ , thus over time the beam of charged particles acquires some polarization. The cross-section for a particle in the 'spin up' state to flip to the 'spin down' state on emitting a photon by synchrotron 5 Beams tend to lose some of their polarization at certain depolarization resonance energies during acceleration. This problem can be circumvented by utilizing Siberian Snakes [60], devices which flip the polarization vector of each beam particle by 180 degrees each revolution. Thus any deflections from the polarization axis are canceled out every two revolutions. radiation is different to the cross-section for a particle in the 'spin down' state to flip to the 'spin up' state. The 'spin up' and 'spin down' states are defined as the particle's spin being aligned parallel and antiparallel to the magnetic field of the storage ring respectively. This 'self polarization' is called the Sokolov-Ternov effect after the Russian theorists who discovered it around 1963 [61]. In a perfect storage ring an equilibrium polarization of P ST = 8 / 5 √ 3 ≈ 0.924 is reached [62]. In practice the maximum polarization achievable is slightly less than this ideal value due to imperfections of the magnetic fields in the storage ring. However in less than one hour electron beams at TRISTAN in Japan and HERA in Germany acquired polarizations of about 80 % or more [62]. The Sokolov-Ternov radiative polarization is along the vertical direction perpendicular to the storage ring plane. The effect is much stronger for electrons than for protons as the rate of synchrotron radiation (number of photons emitted per second) is related to the velocity of the particle not its energy. Because (anti)protons are approximately 1800 times more massive than electrons, at a given energy electrons are traveling at a much higher velocity, i.e. much closer to the speed of light (γ e = ( m p / m e ) γ p ≈ 1800 γ p ). If (anti)protons were moving this close to the speed of light they too would become self polarized by the Sokolov-Ternov effect. Thus at a given energy the time taken to polarize electrons by the Sokolov-Ternov effects is much shorter than the time taken to polarize (anti)protons. Even at the 20 TeV of the proposed Superconducting Super Collider (SSC) it would take antiprotons or protons about 10 7 years to acquire a Table 2.1: Properties of some high energy electron storage rings, and proposed proton storage rings. E is the kinetic energy of the beam, γ = 1 − β 2 is the relativistic Lorentz factor, R the mean radius of the storage ring consisting of identical bending magnets of bending radius ρ separated by straight sections combining to give a total circumference 2 πR, N γ is the average number of photons emitted per particle per revolution and τ ST is the time taken to reach the equilibrium Sokolov-Ternov polarization. Parts of this table are reproduced from Refs. [62] and [63]. As one can see this method of polarization buildup takes too long for (anti)proton rings. useful polarization 6 , and much higher kinetic energies would be required to provide a practical method of polarizing antiprotons by the Sokolov-Ternov effect. The time taken to reach the equilibrium polarization is given by [61,62,64]: τ ST = 8 5 √ 3 m ρ 2 R r γ 5 , (2.3.1) where m is the particle's mass, r the classical radius of the particle (electron or proton), = h / 2 π the reduced Planck's constant, R the mean radius of the storage ring consisting of identical bending magnets of bending radius ρ separated by straight sections combining to give a total circumference 2 πR and γ = 1 / 1 − β 2 is the relativistic Lorentz factor, where β = v / c is the ratio of the particles velocity to that of light. Some properties of current and proposed future synchrotrons are presented in Table 2.1. The Sokolov-Ternov effect is similar to systems investigated later in this thesis, and it can be described by systems of differential equations similar to ones we develop to describe spin filtering. Hence to provide a comparison we present and solve a set of differential equations describing the Sokolov-Ternov effect in section 5.1. Synchrotron radiation is the physical principle behind the antenna, emitting photons in the form of radio waves, and some lasers generated by wiggler magnets. The light produced by synchrotron radiation is used by biologists and many storage rings have a second life as intense light sources after high energy physics experiments have ceased. Electrons lose much of their energy in a storage ring due to synchrotron radiation, typically emitting hundreds to thousands of photons per revolution 5 , and as a consequence very high energy electron/positron accelerators, such as the International Linear Collider (ILC), must be linear to avoid this problem. We conclude that this method of polarization buildup would take too long for an antiproton beam to be considered practical at present energies. Polarization of directly produced antiprotons It is well known that the particles produced when a high energy proton beam strikes a target have some polarization at certain production angles. Some of the particles produced will be antiprotons; in fact this is how antiproton beams are obtained [65]. Unfortunately the polarization generally seems to be larger at larger production angles where the cross-sections are smaller [22]. Thus it appears difficult to simultaneously obtain antiprotons with a high polarization and a high beam intensity using this method. The theory of spin filtering The spin filtering method of polarization buildup [4,54,66], described schematically in Figure 2-3, consists of a circulating beam repeatedly interacting with a polarized internal target in a storage ring. Originally proposed by P. L. Csonka in 1968 [4], it is based on the selective removal of particles from the beam, and selective spin-flip while remaining in the beam, due to spin-dependent scattering off a polarized target. Many particles are scattered at small angles but remain in the beam after refocusing each revolution. This introduces a characteristic laboratory frame acceptance angle θ acc , scattering above which causes particles to be lost from the beam. There is also a minimum laboratory frame scattering angle θ min , corresponding to the Bohr radius of the atoms in the target, below which scattering is prevented by Coulomb screening [67], as described in section 7.2.5. The two physical processes that contribute to polarization buildup by spin filtering, as described in Figure 2 along the beam axis and scatter off the target. Particles scattered at angles greater than the acceptance angle θ acc are lost from the beam, while particles scattered at angles less than θ acc pass through a beam focuser and remain in the beam. In this simplistic diagram the beam focuser is represented by a lens, but in reality the beam is focused by electron cooling as explained in section 7.1. Thus particles in one spin state may be scattered out of the beam, or have their spin-flipped while remaining in the beam, at a higher rate than particles in the other spin state. Hence over time one spin state is depleted more than the other leading to a beam polarization. The beam will diverge slightly after many interactions with the target, but can be refocused by beam cooling, as explained in section 7.1. We prove later in the thesis that beam cooling does not depolarize a stored antiproton beam. As the beam polarization increases the beam intensity decreases, when there is scattering out of the beam. So one can obtain beam polarization at the expense of losing beam intensity. Low beam intensity means low event rate, hence low statistics in a measurement, which is never desired. This trade-off between beam polarization and beam intensity is characteristic of spin filtering and must be optimized to produce a sufficient beam polarization while maintaining reasonable beam intensity. An advantage for the spin filtering method is that polarized hydrogen and deuterium jet targets have already been developed for other projects. Highly polarized high density gas jet targets have been used in the HERMES and COMPASS experiments. The HERMES experiment has been decommissioned since the shutdown of the HERA accelerator complex in DESY, and the polarized gas target has been transfered to COSY in Jülich, Germany to be used in spin filtering studies. It is likely that the HERMES polarized gas target will be used by the PAX Collaboration in a future spin filtering Antiproton Polarizer Ring at FAIR, GSI Darmstadt. The following two diagrams provide a schematic representation of the two physical processes, selective scattering out of the ring (left) and selective spinflip (right), that contribute to polarization buildup by spin filtering in a storage ring. Particles in the 'spin up' state are represented by blue squares and particles in the 'spin down' state are represented by yellow squares, while the grey box represents a polarized target. In both cases the beam is initially unpolarized with equal numbers of particles in the 'spin up' and 'spin down' states. Selective scattering out of the ring (left): When interacting with the polarized target at certain energies particles in the 'spin up' state are scattered out of the beam at a higher rate than particles in the 'spin down' state, hence the larger blue arrow than yellow arrow. Thus one is left with a beam that has more particles in the 'spin down' state, i.e. the beam is now polarized, represented by the excess of yellow squares in the final beam. Note that since particles have been scattered out of the ring there are less particles in the beam after interaction than were in the beam initially, this is represented by the smaller final beam. If the target was unpolarized particles in both spin states would be scattered out of the beam at equal rates, thus no polarization buildup would occur via this process. Selective spin-flip (right): On interaction with the polarized target at certain energies, the 'spin up' to 'spin down' spin-flip cross-section is larger than the 'spin down' to 'spin up' spin-flip cross-section. We represent this by different size arrows with colours fading from blue to yellow and from yellow to blue respectively. Thus after interaction with the target the beam will have more particles in the 'spin down' state than in the 'spin up' state, i.e. the beam is now polarized, represented by the excess of yellow squares in the final beam. Note that the beam intensity is the same after interaction with the polarized target in this process since particles are not lost from the beam, they are just flipped from one spin state to the other. If the target was unpolarized particles in both spin states would have their spins flipped at equal rates, thus no polarization buildup would occur via this process. There has been much debate amongst theorists as to what mechanisms are responsible for the polarization buildup in spin filtering. Contributions come from the electromagnetic scattering of beam antiprotons off the electrons in the hydrogen target and from the electromagnetic and hadronic scattering of beam antiprotons off the protons in the hydrogen target. Horowitz and Meyer, in 1994, were the first to highlight the importance to spin filtering of the electrons in the hydrogen target [68,69]. They claim that electrons in a hydrogen target are not massive enough to deflect antiprotons beyond the acceptance angle of any storage ring, a fact which we demonstrate later in the thesis. Thus scattering of the antiprotons off the electrons in a hydrogen target causes no beam losses and any polarization buildup must be due to spin-flip transitions [70]. In 2005 two groups from the Budker Institute for Nuclear Physics, Russia and the Institute for Nuclear Physics, Jülich, Germany claimed that such spin-flip effects are small thus spin filtering off polarized electrons in a hydrogen target will lead to a negligible rate of polarization buildup [71,72,73]. An experiment has been proposed to test this claim [74], by investigating the converse case of whether unpolarized electrons in a helium-4 target depolarize a stored polarized proton beam. The helium-4 target is chosen because its nuclei is spin -0, hence any polarization transfer must come from scattering off its electrons. There are currently two schools of thought regarding spin filtering of antiprotons off a polarized hydrogen target, (1) proposal [66] building on the work of Horowitz and Meyer, which advocates using a hydrogen target with high electron polarization and low proton polarization; and (2) the Budker/Jülich proposal [71,72] to use a hydrogen target with low electron polarization and high proton polarization. As is often the case the matter must be resolved by an experiment [74] to see which method is preferable. There are many advantages of using a lepton target instead of an atomic target, the foremost of which is that antiprotons cannot be absorbed by a lepton target as they are in a atomic target due to annihilation with the protons in the atomic target. This fact has led to two proposals for spin filtering off polarized lepton beams: one off a co-moving polarized positron beam [75] by a group in Mainz, Germany and the other, presented in this thesis, off an opposing polarized electron beam [76]. The momentum of an opposing electron beam causes antiprotons to be scattered beyond acceptance, hence allowing contributions from both of the physical process, selective scattering out of the beam and selective spin-flip, of spin filtering. A thorough treatment of the dynamics of spin filtering has been presented recently by the present author [77,78] and forms much of the later chapters of this thesis. Spin filtering is the only method to produce a polarized antiproton beam in a storage ring that has been successfully tested, by the FILTEX experiment in 1993 [54], as described in section 2.4. As a result, much of this thesis is devoted to a theoretical understanding of the spin filtering process, under various scenarios. Verification of spin filtering Polarization buildup by spin filtering has been proven to work in the FILTEX experiment at the Test Storage Ring (TSR) at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany [54]. We summarize their results below. In the TSR a 23 MeV proton beam was stored and repeatedly made to interact with a polarized internal hydrogen gas target. A polarized hydrogen target with atoms in the hyperfine state | ↑ p ↑ e was used, i.e. where both the protons and electrons are polarized. The target density was 6 × 10 13 polarized hydrogen atoms per cm 2 , and the frequency of revolution was 1.177 MHz [54]. The beam was left to orbit in the ring passing through the target each revolution for times between 30 and 90 minutes; then the polarization was measured. The proton beam was initially unpolarized and over time it gained a small amount of polarization as shown in Figure 2-5. The polarization buildup rate of the proton beam at FILTEX was [54] : d P beam d τ = 0.0124 ± 0.0006 per hour. (2.4.1) After 90 minutes the polarization had increased to 1.86 % and the beam intensity had decreased to 5 % of its original value [54]. But with a better configuration of the experiment, and a dedicated spin filtering polarizer ring, the rate of polarization buildup could be greatly increased. The TSR ring had an acceptance angle measured to be θ acc = 4.4 ± 0.5 mrad, which we could optimize for our needs. The beam lifetime τ * , which we discuss later in the thesis, is the time taken for the number of particles in the beam to decrease by a factor of e = 2.78. The beam lifetime in the TSR during the FILTEX experiment, with the polarized internal target in the ring, was 30 minutes. We show in section 5.3.1 that the polarization achieved after two beam lifetimes is an important measure of a spin filtering scheme. At FILTEX this value was measured to be P beam ( 2 τ * ) = 0.0124. This was just a feasibility test for spin filtering, and while it verifies that the method works, the polarization buildup rate was small. In order to maximize the effect of spin filtering a dedicated spin filtering polarizing ring would need to be built. The PAX Collaboration has recently proposed the construction of such a ring called the Antiproton Polarizer Ring (APR) inside the HESR at FAIR in GSI Darmstadt, Germany. Chapter 3 Generic helicity amplitudes and spin observables "The most incomprehensible thing about the world is that it is comprehensible." Albert Einstein In this chapter all electromagnetic helicity amplitudes and spin observables, accounting for polarization effects in spin 1/2 -spin 1/2 elastic scattering are calculated. Many of these results will be utilized in later chapters when providing a mathematical description of spin filtering, although their use is certainly not limited to this. We begin in section 3.1 by introducing the spin 1/2 electromagnetic currents, both for point particles and particles with internal structure determined by electromagnetic form factors. A generic equation is derived in section 3.2 that can be used to calculate all polarization phenomena in elastic spin 1/2 -spin 1/2 electromagnetic scattering to first order in QED. The spin-averaged differential cross-section for spin 1/2 -spin 1/2 scattering is presented in a new compact invariant form in section 3.3. We then present results for all electromagnetic helicity amplitudes for elastic spin 1/2 -spin 1/2 scattering in section 3.4, and for electromagnetic spin observables for elastic spin 1/2 -spin 1/2 scattering in section 3.5. Spin 1/2 electromagnetic currents In this section the spin 1/2 electromagnetic currents of both point-like particles and non-point-like particles, with internal structure defined by electromagnetic form factors, are introduced. We investigate 2 particle → 2 particle elastic scattering processes in the space-like region, with the mass and the momentum and spin 4vectors of each particle labeled as: A ( M, p 1 , S 1 ) + B ( m, p 2 , S 2 ) −→ A ( M, p 3 , S 3 ) + B ( m, p 4 , S 4 ) , with the particles above being the beam (1), target (2), scattered (3) and recoil (4) particles respectively. The four momentum transfer is defined as q = p 3 − p 1 = p 2 − p 4 . (3.1.1) For 2 particle → 2 particle elastic scattering the spin-averaged differential cross section is related to the helicity amplitudes M(λ 3 λ 4 ; λ 1 λ 2 ) by s d σ d Ω = 1 ( 8 π ) 2 λ 1 λ 2 λ 3 λ 4 1 ( 2 s A + 1 ) ( 2 s B + 1 ) | M( λ 3 , λ 4 ; λ 1 , λ 2 ) | 2 , (3.1.2) where λ 1 , λ 2 and λ 3 , λ 4 are the helicities of the initial and final particles respectively, s A and s B are the spins of the two particles in the elastic process, and the s and t are Mandelstam variables [79] defined in Appendix B. We label the mass of particle A, taken to be an antiproton, as M and the mass of particle B, taken to be an electron or a proton, as m. Define electromagnetic form factors F 1 (q 2 ) and F 2 (q 2 ), with normalization F 1 (0) = 1 and F 2 (0) = κ p = µ p − 1, the anomalous magnetic moment of the proton, where q 2 = t in the t-channel case that we are solely interested in. Form factors are empirical quantities, obtained from experiment, which describe the fact that protons are not point-like particles and have an internal structure. They include all effects of the strong nuclear interaction inside the proton, hence are very difficult to calculate theoretically. The Sachs electric G E (t) = F 1 (t) + the electron current for particle B is j µ B = − i eū ( p 4 , λ 4 ) γ µ u ( p 2 , λ 2 ) , (3.1.3) where e is the electron charge and u ( p 2 , λ 2 ) andū ( p 4 , λ 4 ) are the spinors of the incoming and outgoing electron respectively. Generalizing this to a non-point-like particle, such as the proton, with a finite extent defined by the Pauli and Dirac electromagnetic form factors F 1 (q 2 ) and F 2 (q 2 ), one obtains the most general (anti)proton 1 electromagnetic current for particle A: J µ A = ± i e pū (p 3 , λ 3 ) F 1 q 2 γ µ + F 2 (q 2 ) 2 M i σ µ ν ( p 3 − p 1 ) ν u(p 1 , λ 1 ) , (3.1.4) where e p = − ep = − e is the charge on the proton and consequently the upper sign 1 We are only interested in t-channel elastic scattering, i.e. not s-channel antiproton-proton annihilation which is suppressed in comparison to the t-channel amplitude in the low | t | region of interest in a storage ring, as explained in Figure 4-1. In this case an antiproton can be treated as a negatively charged proton, with internal electromagnetic structure described by the same form factors as the proton. Hence, as is customary in the phenomenology literature [68,81], u andū spinors can be used for the antiproton current in the t-channel instead of the v andv anti-spinors required in the treatment of annihilation. The treatment presented here is in exact agreement with a treatment where the proton current involves spinors and the antiproton current involves anti-spinors. The minus signs introduced into eq. (3.2.11) because of the anti-spinor completeness relations of eqs. (A.8 and A.10) only contribute to terms that vanish when the traces are evaluated using the trace theorems of eq. (A.14). Hence all cross-sections and spin observables for t-channel elastic antiproton-antiproton, antiproton-proton and proton-proton scattering are equal to first order in QED. The anti-spinor formalism is less general then the one presented here as one is forced, due to the Gordon decomposition identity for anti-spinors derived in Appendix C, to specify that the particle and antiparticle have opposite anomalous magnetic moments. Hence the results could not be applied to antiproton-neutron scattering for example, as they can be in the present formalism. is for the antiproton current and the lower sign is for the proton current, and where σ µ ν ≡ i 2 [ γ µ , γ ν ] = i 2 ( γ µ γ ν − γ ν γ µ ) . (3.1.5) Using the Gordon decomposition identity [82,83] : u(p ′ ) γ µ u(p) =ū(p ′ ) ( p + p ′ ) µ 2 M + i σ µ ν ( p ′ − p ) ν 2 M u(p) , (3.1.6) which is derived in Appendix C, the (anti)proton electromagnetic current can be written as J µ A = ± i e pū (p 3 , λ 3 ) G M γ µ − F 2 p µ 1 + p µ 3 2 M u(p 1 , λ 1 ) , (3.1.7) which has a simpler gamma matrix structure than eq. (3.1.4), hence will allow for easier computations. It is important to note that the structureless limit (point-like particles) is obtained when F 1 (q 2 ) = 1 and 2 F 2 (q 2 ) = 0, hence G M (q 2 ) = G E (q 2 ) = 1. Applying this condition to the proton current in eq. (3.1.7) one obtains the electron current presented in eq. (3.1.3). Because we are interested in both antiproton-proton and antiproton-electron scattering, we shall calculate the generic case of the elastic scattering of two structured spin 1/2 particles. One can then take the one particle pointlike limit to account for antiproton-electron scattering, or the two particle point-like limit to account for positron-electron scattering. Hence we generalize the electron current to account for non-point-like particles by using J µ B = − i eū(p 4 , λ 4 ) g M γ µ − f 2 p µ 2 + p µ 4 2 m u(p 2 , λ 2 ) , (3.1.8) where we label the Dirac, Pauli and Sachs electromagnetic form factors of particle B by lowercase f 1 (t), f 2 (t), g M (t) = f 1 (t) + f 2 (t) and g E (t) = f 1 (t) + f 2 (t) t / (4 m 2 ) respectively, with the usual normalizations f 1 (0) = 1 and f 2 (0) = κ = µ − 1 therefore g M (0) = µ and g E (0) = 1. The above electromagnetic currents, and hence all results derived using them, can be applied to electrically neutral particles which nevertheless have an anomalous magnetic moment, such as the neutron, by encompassing the charge e into the definitions of the form factors and using the new normalizations F n 1 (0) = 0 and F n 2 (0) = κ n = −1.913 in units of nuclear magnetons. Generic elastic spin 1/2 -spin 1/2 calculation The helicity amplitudes can be related to the currents, to first order in QED, by the relation i M(λ 3 λ 4 ; λ 1 λ 2 ) = J µ A ( M, λ 3 , λ 1 ) − i η µν q 2 J ν B ( m, λ 4 , λ 2 ) . (3.2.1) Therefore the generic electromagnetic amplitude for a structured spin 1/2 particle of mass M scattering elastically off a structured spin 1/2 particle of mass m, via single t-channel photon exchange, is i M(λ 3 λ 4 ; λ 1 λ 2 ) = ± (− i) 3 e 2ū (p 3 , λ 3 ) G M γ µ − F 2 2 M R µ u(p 1 , λ 1 ) (3.2.2) × η µν q 2 ū(p 4 , λ 4 ) g M γ ν − f 2 2 m r ν u(p 2 , λ 2 ) , = ± i e 2 q 2ū (p 3 , λ 3 ) G M γ µ − F 2 2 M R µ u(p 1 , λ 1 ) Aū (p 4 , λ 4 ) g M γ µ − f 2 2 m r µ u(p 2 , λ 2 ) B where the upper sign is for p p orpp scattering and the lower sign is forp p scattering, and where we have defined R µ ≡ p µ 1 + p µ 3 and r ν ≡ p ν 2 + p ν 4 ,(3. 2.3) and used the fact that ep = −e p = e. Note that in the t-channel the hermiticity of the electromagnetic currents implies that the form factors are real functions of [84]. Now one must obtain q 2 = t (i.e. F 1 = F * 1 , f 1 = f * 1 , F 2 = F * 2 , f 2 = f * 2 and hence G M = G * M and g M = g * M )|M| 2 = M M * = e 4 q 4 (A B) (A B) * = e 4 q 4 A B B * A * ,(3.p 1 , λ 1 p 2 , λ 2 p 3 , λ 3 p 4 , λ 4 γ µ e p p e 00 00 00 00 11 11 11 11 antiproton-electron and positron-electron scattering respectively. As elsewhere in the thesis the time axis increases from left to right. The antiproton-proton case is generic and encompasses the other two cases. In this section we calculate all spin-dependent phenomena for electromagnetic spin 1/2 -spin 1/2 elastic scattering, of particles with internal structure described by form factors, to first order in QED. p 1 , λ 1 p 2 , λ 2 p 3 , λ 3 p 4 , λ 4 γ µ + e + e e e p 1 , λ 1 p 2 , λ 2 p 3 , λ 3 p 4 , λ 4 γ µ where, using relations in Appendix A, the complex conjugates of A and B, defined in eq. (3.2.2), are A * =ū(p 1 , λ 1 ) G M γ ν − F 2 2 M R ν u(p 3 , λ 3 ) , (3.2.5) B * =ū(p 2 , λ 2 ) g M γ ν − f 2 2 m r ν u(p 4 , λ 4 ) . (3.2.6) Substituting the above into eq. (3.2.4), requires using the completeness relations: u(p i , λ i )ū(p i , λ i ) = 1 2 / p i + m i 1 + γ 5 / S i , (3.2.7) where λ i and S i are the helicity and spin four vector of the particle with momentum p i where i ∈ {1, 2, 3, 4}, normalized such that S µ i S iµ = − 1 and constrained by the orthogonality condition p µ i S iµ = 0. The mass of the particle with momentum p i is denoted by m i where we have that m 1 = m 3 = M and m 2 = m 4 = m. The result is a generic equation for all polarization phenomena in elastic spin 1/2 -spin 1/2 electromagnetic scattering to first order in QED: 16 q e 4 | M | 2 = (3.2.8) Tr / p 4 + m 1 + γ 5 / S 4 g M γ ν − f 2 r ν 2 m / p 2 + m 1 + γ 5 / S 2 g M γ µ − f 2 r µ 2 m × Tr / p 1 + M 1 + γ 5 / S 1 G M γ µ − F 2 R µ 2 M / p 3 + M 1 + γ 5 / S 3 G M γ ν − F 2 R ν 2 M This generic equation can thus be used to calculate all helicity amplitudes and spin observables by substituting specific values for the spin (S i ) and momenta (p i ) four vectors, and can describe equal particle scattering in the case f 1 → F 1 , f 2 → F 2 and m → M. This also applies to antiproton-electron scattering by setting one particle to be point-like using f 1 → 1 and f 2 → 0 and hence g M → 1, in which case the first trace simplifies to the familiar electron trace: Tr / p 4 + m 1 + γ 5 / S 4 γ ν / p 2 + m 1 + γ 5 / S 2 γ µ , (3.2.9) which when polarization effects are averaged over gives the familiar spin-averaged electron trace Tr / p 2 + m γ µ / p 4 + m γ ν = 4 p 2µ p 4ν + p 2ν p 4µ + t 2 η µν , (3.2.10) which has been evaluated using the trace theorems presented in Appendix A. Equation (3.2.8) can be generalized to directly evaluate the squares of all electromagnetic helicity amplitudes. We introduce the constants ǫ i multiplying each S i , where i ∈ {1, 2, 3, 4}. 16 q e 4 | M | 2 = (3.2.11) Tr / p 4 + m 1 + ǫ 4 γ 5 / S 4 g M γ ν − f 2 r ν 2 m / p 2 + m 1 + ǫ 2 γ 5 / S 2 g M γ µ − f 2 r µ 2 m × Tr / p 1 +M 1 + ǫ 1 γ 5 / S 1 G M γ µ − F 2 R µ 2 M / p 3 +M 1 + ǫ 3 γ 5 / S 3 G M γ ν − F 2 R ν 2 M Later we will set ǫ i = ±1 to account for different helicity states. Equation Table 3.1. Substituting combinations of these four vectors into eq. (3.2.11), and computing the traces, will provide expressions for all spin-dependent cross-sections for elastic spin 1/2 -spin 1/2 scattering to first order in QED. The traces were computed using the computer algebraic program Mathematica and its add on package Tracer [87]. The Mathematica code for a typical calculation is presented in Appendix E. Centre-of-Mass Momenta vectors p 1 = ( E A , 0, 0, k ) p 3 = ( E A , k sin θ, 0, k cos θ ) p 2 = ( E B , 0, 0, − k ) p 4 = ( E B , − k sin θ, 0, − k cos θ ) Centre-of-Mass Normal spin vectors S N 1 = ( 0, 0, 1, 0 ) S N 3 = ( 0, 0, 1, 0 ) S N 2 = ( 0, 0, 1, 0 ) S N 4 = ( 0, 0, 1, 0 ) Centre-of-Mass Transverse spin vectors S T 1 = ( 0, 1, 0, 0 ) S T 3 = ( 0, cos θ, 0, − sin θ ) S T 2 = ( 0, 1, 0, 0 ) S T 4 = ( 0, − cos θ, 0, sin θ ) Centre-of-Mass Longitudinal spin vectors The spin 4-vectors of Table 3.1 satisfy the general expression: S L 1 = 1 M ( k, 0, 0, E A ) S L 3 = 1 M ( k, E A sin θ, 0, E A cos θ ) S L 2 = 1 m (− k, 0, 0, E B ) S L 4 = 1 m (− k, E B sin θ, 0, E B cos θ )S µ = 1 M p ·ŝ , Mŝ + p ·ŝ E + M p , (3.2.12) where p, E and M are the momentum 3-vector, the energy and the mass of the particle in question andŝ is a unit 3-vector identifying a generic spatial direction [88,89]. Whenŝ is parallel (or antiparallel) to p then the particle is longitudinally polarized while ifŝ is perpendicular to p then the particle is transversely polarized. From this equation one can easily verify that S µ S µ = −1 and p µ S µ = 0, where p µ = (E , p), as is seen to be satisfied by all of the vectors in Table 3.1. Spin-averaged cross-section The spin-averaged differential cross-section for t-channel elastic spin 1/2 -spin 1/2 scattering can be obtained by setting each S i = 0 in eq. (3.2.11), and multiplying by 2 4 = 16 to counter the four factors of 1/2 from the spin-dependent completeness relations that are absent in the spin-averaged completeness relations. One obtains | M | 2 = e q 4 Tr / p 4 + m g M γ ν − f 2 r ν 2 m / p 2 + m g M γ µ − f 2 r µ 2 m × Tr / p 1 + M G M γ µ − F 2 R µ 2 M / p 3 + M G M γ ν − F 2 R ν 2 M (3.3.1) These traces can be evaluated using the trace theorems of Appendix A to obtain Tr / p 1 + M G M γ µ − F 2 R µ 2 M / p 3 + M G M γ ν − F 2 R ν 2 M (3.3.2) = 4 G 2 M p 1µ p 3ν + p 1ν p 3µ + t 2 η µν − 4 G M F 2 R µ R ν + 2 F 2 2 R µ R ν 1 − t 4 M 2 The result follows from eq. (3.3.1), after including a factor of 1/4 from averaging over initial spin states and summing over final spin states, as presented in our recent paper [90] : s α 2 d σ d Ω = 4 m 2 g 2 E − t g 2 M 4 m 2 − t 4 M 2 G 2 E − t G 2 M 4 M 2 − t ( M 2 − m 2 ) 2 − s u t 2 + 2 m Mg E G E t 2 + 1 2 g 2 M G 2 M , (3.3.3) where we have used the electromagnetic coupling constant (fine structure constant) α = e 2 /4 π and the Sachs electric form factor of each particle: G E q 2 = F 1 q 2 + t 4 M 2 F 2 q 2 and g E q 2 = f 1 q 2 + t 4 m 2 f 2 q 2 . (3.3.4) Equation (3.3.3) , describing the spin-averaged elastic scattering of any two spin 1/2 particles or antiparticles, is a generalization of the famous Rosenbluth formula for elastic electron-proton scattering [91]. It corresponds to a twofold generalization of the Rosenbluth formula in that the mass of neither particle has been neglected and the internal structure of both particles is included, and is expressed here in a new invariant form. Helicity amplitudes Helicity is defined as the projection of the particles spin 3-vector in the direction of its momentum 3-vector. Helicity is a discretized quantity, having values of either ± / 2 for a spin 1/2 particle, because the spin of a particle with respect to an axis is quantized. Helicity states are always longitudinally polarized, i.e. either along direction of motion, which we denote by + for + 1/2, or opposite direction of motion, which we denote by − for − 1/2 in spin 1/2 -spin 1/2 scattering 4 . The helicity amplitudes for 2 particle → 2 particle scattering processes were introduced by Jacob and Wick [92], and are represented by M ( scattered, recoil ; beam, target ) = M ( λ 3 , λ 4 ; λ 1 , λ 2 ) . (3.4.1) The arguments are to be read from right to left, as λ 1 and λ 2 correspond to the incoming particles in the reaction and λ 3 and λ 4 correspond to the outgoing particles in the reaction. For spin 1/2 -spin 1/2 scattering the helicities λ i = ± for i ∈ { 1, 2, 3, 4 } are + if the particles spin vector points in the direction of its momentum vector and − if the particles spin vector points in the opposite direction to its momentum vector. The ± in the helicity amplitudes are shorthand for ±1/2 , the helicity of a spin 1/2 particle. For 2 particle → 2 particle spin 1/2 -spin 1/2 Parity Invariance and Time-reversal Invariance, which are strictly satisfied in both electromagnetic and hadronic reactions, provide relations between these 16 helicity amplitudes. Parity changes the direction of motion but does not change the spin, hence it flips the helicity of a particle. Therefore Parity Invariance acts on the helicity amplitudes as follows 5 [92,93]: Time-reversal Invariance means that the amplitude for a reaction is unchanged if the direction of time is reversed. It acts on the helicity amplitudes by interchanging incoming and outgoing particles as follows [92,93]: Hence there are six independent helicity amplitudes, φ 1 , φ 2 , φ 3 , φ 4 , φ 5 and φ 6 , for 2 particle → 2 particle spin 1/2 -spin 1/2 scattering, as follows: i.e. their helicities reversed, and by single-spin-flip only one particle in the reaction has its spin flipped, while non-spin-flip means none of the particles in the reaction have their spins flipped. The non-spin-flip amplitudes, φ 1 and φ 3 , are also known as spin-elastic amplitudes. Note that for identical particle scattering one also has that φ 6 = − φ 5 and hence there are only five independent helicity amplitudes in this case, as will be shown. M ( −λ 3 , −λ 4 ; −λ 1 , −λ 2 ) = (−1) λ−µ M ( λ 3 , λ 4 ; λ 1 , λ 2 ) , (3.4.3) where λ ≡ λ 1 − λ 2 and µ ≡ λ 3 − λ 4 ,M ( λ 1 , λ 2 ; λ 3 , λ 4 ) = (−1) λ−µ M ( λ 3 , λ 4 ; λ 1 , λ 2 ) .φ 1 ≡ M(+, + ; +, +) = M(−, − ; −, −) φ 2 ≡ M(+, + ; −, −) = M(−, − ; +, +) φ 3 ≡ M(+, − ; +, −) = M(−, + ; −, +) The multiplicity of an independent helicity amplitude is defined as the number of times it appears in the set of total helicity amplitudes, as seen in eq. (3.4.7). Hence φ 1 , φ 2 , φ 3 and φ 4 each have multiplicity 2 while φ 5 and φ 6 have multiplicity 4. Thus one can write the spin-averaged differential cross-section in terms of the independent helicity amplitudes as s d σ d Ω = 1 (8 π) 2 λ 1 λ 2 λ 3 λ 4 1 4 | M( λ 3 , λ 4 ; λ 1 , λ 2 ) | 2 , = 1 2 (8 π) 2 |φ 1 | 2 + |φ 2 | 2 + |φ 3 | 2 + |φ 4 | 2 + 2 |φ 5 | 2 + 2 |φ 6 | 2 . (3.4.8) We now calculate the modulus of all six independent helicity amplitudes for elastic scattering of spin 1/2 -spin 1/2 particles or antiparticles to first order in QED, by substituting the four longitudinal spin vectors (four pure helicity states) from Table 3.1 into eq. (3.2.11) and setting specific values (±1) for the ǫ's to obtain φ 1 ±α = s − m 2 − M 2 t 1+ t 4 k 2 f 1 F 1 − f 1 F 1 − f 2 F 1 − f 1 F 2 − 1 2 f 2 F 2 1 − t 4 k 2 φ 2 ±α = 1 2 m k f 1 − k m f 2 M k F 1 − k M F 2 + s − m 2 − M 2 − 2 k 2 4 m M 1 + t 4 k 2 f 2 F 2 φ 3 ±α = s − m 2 − M 2 t f 1 F 1 + f 2 F 2 2 1 + t 4 k 2 (3.4.9) φ 4 = −φ 2 φ 5 ±α = s (4 k 2 + t) −t f 1 F 1 m 4 k 2 1− m 2 − M 2 s − F 1 f 2 2 m + t f 2 F 2 16 m k 2 1+ m 2 − M 2 s φ 6 ±α = s (4 k 2 + t) −t f 1 F 1 M 4 k 2 M 2 − m 2 s − 1 + F 2 f 1 2 M − t f 2 F 2 16 M k 2 1+ M 2 − m 2 s in agreement with those found by other methods [94,95,96]. The linear combinations φ + ≡ φ 1 + φ 3 and φ − ≡ φ 1 − φ 3 appear often in the observables φ 1 + φ 3 ±α = − g M G M + 1 + t 4 k 2 2 f 1 F 1 s − m 2 − M 2 t + f 2 F 2 , (3.4.10) φ 1 − φ 3 ±α = − g M G M ,(3. 4.11) along with the simpler combinations φ 2 + φ 4 = 0 and φ 2 − φ 4 = 2 φ 2 . On comparison to previously published proton-proton helicity amplitudes [94,95,96], and using the fact that ep = −e p , one finds that the ± α factors above are plus for like-charged particles (e.g. proton-proton or antiproton-antiproton scattering) and minus for unlike-charged particles (e.g. antiproton-proton scattering). The above t-channel helicity amplitudes can be transformed into the corresponding s-and u-channel helicity amplitudes using crossing symmetry [50,97]. Spin observables The spin observables for a 2 particle → 2 particle elastic reaction, as introduced in [94,96], are described in Table 3.2. In this section we present expressions for them to first order in QED for t-channel electromagnetic scattering. In particular the spin transfer and depolarization observables will play an important role in the remainder of the thesis. Polarization transfer observables Setting S 1 = S 4 = 0 in eq. (3.2.11) and subtracting the spin-averaged differential cross-section gives a generic equation for spin transfer from initial particle B to final antiparticle A : d σ d Ω K ij = α 2 g M G M 8 m M s t 2 F 2 t q · S 3 4 m 2 f 1 p 3 · S 2 + f 2 2 m 2 + M 2 − s q · S 2 + p 3 · S 2 t − q · S 2 t ) ] − 16 m 2 M 2 F 1 g E p 1 · S 2 q · S 3 + 4 M 2 G E 4 m 2 f 1 ( p 3 · S 2 q · S 3 − S 2 · S 3 t ) + t f 2 ( p 3 · S 2 q · S 3 + q · S 2 ( p 2 + p 4 ) · S 3 − S 2 · S 3 t ) ] } , (3.5.1) into which specific vectors S 2 and S 3 will be inserted to give the various polarization transfer observables. Scattering is in the XZ plane, so the coordinates are X (Transverse), Y (Normal) and Z (Longitudinal), where in the above equation Table 3.2: The 16 spin observables of a 2 particle → 2 particle elastic reaction. An upward pointing triangle represents that the beam is polarized, while the absence of a triangle represents an unpolarized beam. The right hand column shows the symbols for the spin observables that will be used throughout the thesis, as defined in Ref. [94]. The subscripts i, j, k, l ∈ { X, Y, Z } correspond to the direction of the polarization of each particle. Time increases from left to right as elsewhere in the thesis. i, j ∈ {X, Y, The spin observables K XX , K YY and K ZZ The spin observable K XX is obtained by inserting the transverse polarized spin vectors Table 3.1 into the generic spin transfer equation (3.5.1): S 2 = S T 2 and S 3 = S T 3 fromd σ d Ω K XX = α 2 g M G M 8 k 2 m M s 4 m 2 f 1 M 2 F 1 − k 2 F 2 + f 2 − 4 k 2 M 2 F 1 + 4 k 4 + 4 k 2 + t √ k 2 + m 2 √ k 2 + M 2 F 2 . (3.5.2) Inserting the normal polarized spin vectors S 2 = S N 2 and S 3 = S N 3 from Table 3.1 into the generic spin transfer equation (3.5.1) gives d σ d Ω K YY = 2 α 2 s t m M g E g M G E G M . (3.5.3) Inserting the longitudinally polarized spin vectors S 2 = S L 2 and S 3 = S L 3 from Table 3.1 into the generic spin transfer equation (3.5.1) gives d σ d Ω K ZZ = α 2 g M G M 8 k 2 s 2 t s 2 − M 2 − m 2 2 4 k 2 + t f 1 F 1 + s 4 k 2 f 1 − t f 2 4 k 2 F 1 − t F 2 . (3.5.4) The spin observables K XZ and K ZX When the spin four vectors S 2 = S L 2 and S 3 = S T 3 from Table 3.1 are substituted into the generic spin transfer equation (3.5.1) we obtain d σ d Ω K XZ = α 2 g M G M 16 M s 3/2 t − t ( 4 k 2 + t ) k 4 s + M 2 − m 2 t f 2 F 2 + 4 f 1 M 2 s + m 2 − M 2 F 1 − 2 k 2 s F 2 . (3.5.5) Inserting the spin four vectors S 2 = S T 2 and S 3 = S L 3 from Table 3.1 we obtain d σ d Ω K ZX = α 2 g M G M 16 m s 3/2 t − t ( 4 k 2 + t ) k 4 s + m 2 − M 2 t f 2 F 2 + 4 F 1 m 2 s + M 2 − m 2 f 1 − 2 k 2 s f 2 . (3.5.6) As expected by parity and time-reversal invariance we confirm that K XY = K YX = K YZ = K ZY = 0 . (3.5.7) Depolarization spin observables The observable D ij is the polarization remaining after interaction with the target, while of interest here is the loss of polarization after interaction with the target, i.e. (1 − D ij ). We present results to leading order in small | t |. Here setting S 2 = S 4 = 0 in the generic eq. (3.2.11) and subtracting the spin-averaged differential crosssection, then subtracting the result from the spin-averaged equation gives a generic depolarization equation for antiparticle A, into which the various vectors S 1 and S 3 can be substituted. The Mathematica code to derive this generic depolarization equation is included in Appendix E, but the equation itself is too long to be practical to include here. The ≈ sign means to first order in small | t |. The spin observables ( 1 − D XX ), ( 1 − D YY ) and ( 1 − D ZZ ) Substituting the transverse spin vectors S 1 = S T 1 and S 3 = S T 3 from Table 3.1 into the generic depolarization equation gives d σ d Ω ( 1 − D XX ) ≈ − 2 α 2 F 2 1 k 2 m 2 t s m 4 k 2 + M 2 f 2 1 + s k 4 f 2 2 (3.5.8) − k 2 m 2 s + M 2 − m 2 f 1 f 2 . Inserting the normal polarized spin vectors S 1 = S N 1 and S 3 = S N 3 from Table 3.1 into the generic depolarization equation gives d σ d Ω ( 1 − D YY ) = α 2 2 s g 2 M G 2 M , complete to all orders in t. (3.5.9) Inserting the longitudinal spin vectors S 1 = S L 1 and S 3 = S L 3 from Table 3.1 into the generic depolarization equation gives d σ d Ω ( 1 − D ZZ ) ≈ − 2 α 2 F 2 1 k 2 M 2 t s M 4 k 2 + m 2 f 2 1 + s k 4 f 2 2 (3.5.10) − k 2 M 2 s + m 2 − M 2 f 1 f 2 . The spin observables ( 1 − D XZ ) and ( 1 − D ZX ) Substituting the spin vectors S 1 = S L 1 and S 3 = S T 3 from Table 3.1 into the generic depolarization equation gives d σ d Ω ( 1 − D XZ ) ≈ 4 α 2 f 2 1 F 2 1 s t 2 2 k 4 + m 2 M 2 + k 2 s − 2 k 2 , (3.5.11) and d σ d Ω ( 1 − D ZX ) , found by inserting into the generic depolarization equation the spin vectors S 1 = S T 1 and S 3 = S L 3 from Table 3.1, is the same as above to leading order in t. Note this is just the leading t part of the spin-averaged case. As expected by parity and time-reversal invariance we confirm that D XY = D YX = D YZ = D ZY = 0 . (3.5.12) Spin asymmetries For electromagnetic scattering, to first order in QED, all single and triple spin asymmetries are zero, 5.13) and the double spin asymmetries equal the polarization transfer spin observables: A i = C ijk = 0 where i, j, k ∈ { X, Y, Z } ,(3.A ij = K ij where i, j ∈ { X, Y, Z } . (3.5.14) as a consequence of the tree-level electromagnetic helicity amplitudes all being real, and the fact that φ 2 = − φ 4 to first order in QED 6 . There are also 'four spin measurement' spin observables C ijkl as described in Table 3.2 where the polarization of the beam, target, scattered and recoil particles are measured. These 'four spin measurement' spin observables are not needed for spin filtering as the polarization of the recoil particle will not be measured, hence they will not be treated in this thesis. For a treatment of this see Refs. [96,98,99]. Radiative corrections are not treated in this thesis. In elastic e − p scattering radiative corrections are estimated to give corrections of 1% -3%, and are only considered in very high precision experiments. This is treated in Refs. [100,101,102,103]. Chapter 4 Specific helicity amplitudes and spin observables "Mathematics, rightly viewed, posses not only truth, but a supreme beauty -a beauty cold and austere, like that of a sculpture." Bertrand Russell The expressions for the generic spin 1/2 -spin 1/2 electromagnetic helicity amplitudes and spin observables derived in Chapter 3 are presented in this chapter for the specific cases of: antiproton-proton scattering in section 4.2, antiproton-electron scattering in section 4.3 and positron-electron scattering in section 4.4. Then the specific cross-sections and spin observables needed for spin filtering are explicitly presented in section 4.5. These results are of importance to many areas of particle physics, and will be utilized throughout the remainder of this thesis. The chapter concludes with a calculation of all spin 0 -spin 1 helicity amplitudes in section 4.6, which describe the scattering of deuterons off a carbon nucleus for example. Introduction In this chapter we calculate all spin-averaged and spin-dependent electromagnetic cross-sections for elastic antiproton-proton, antiproton-electron and positron-electron scattering, to first order in QED. These are required in the descriptions of spin filtering that follow later in the thesis. In the region of low momentum transfer of importance in storage rings electromagnetic effects dominate over hadronic effects. So we calculate the electromagnetic contribution to these cross-sections, and focus on the low | t | behaviour. As explained in section 7.2.5 there is a minimum momentum transfer q min (and correspondingly a minimum scattering angle θ min ), below which scattering is prevented by Coulomb screening, corresponding to an impact parameter of the Bohr radius a B = 52900 fm of the atom [67]. Antiprotons flying past an atom at impact parameters greater than a B see the atom as neutral and do not interact with the atom [72]. For momentum transfers of q > q min = 1/a B the energy transfered is greater than the binding energy of the hydrogen atom, hence to a good approximation the antiproton scatters from free protons and electrons in the hydrogen atom [68,69] , i.e. d σ d Ω p H ≈ d σ d Ω p p + d σ d Ω p e − (4.1.1) where H represents a hydrogen atom. Since the objective is to polarize the antiproton beam the transfer of polarization from initial target particles to the final antiproton beam is of principal importance. Hence a thorough investigation of the elastic polarization transfer reactions p + p ↑ −→p ↑ + p (4.1.2) p + e − ↑ −→p ↑ + e − (4.1.3) is presented. The effects of depolarization on the antiproton beam are also calculated. We are also interested in the interactions of an antiproton beam with an opposing polarized electron beam, which eq. (4.1.3) equally describes. Coulomb screening also introduces a minimum scattering angle, θ min , in this case; due to the impact parameter being limited by half of the average separation of electrons in the beam, as explained in section 7.2.5. 4.2p p helicity amplitudes and spin observables In this section we present the expressions for the electromagnetic helicity amplitudes and spin observables in the specific case of antiproton-proton scattering. We have only derived expressions for t-channel scattering, while for antiproton-proton scattering there is also a contribution from s-channel (annihilation) scattering, as shown in Figure 4 t c = − 8 π α β lab σp p tot 1 + ρ 2 ≈ − 0.001 (GeV/c) 2 , (4.2.1) where the electromagnetic interaction dominates the hadronic interaction, as derived in section 7.4.1. Here the laboratory velocity is β lab = s (s − 4 M 2 ) / (s − 2 M 2 ) and ρ = Re{φ h + } / Im{φ h + } the ratio of real to imaginary parts of the hadronic non-flip amplitude 1 . The electromagnetic helicity amplitudes and spin observables forp p scattering can be obtained by setting equal masses and form factors ( f 1 = F 1 , f 2 = F 2 , g E = G E , g M = G Mφ 1 − α = s + 4 k 2 2 t + M 2 2 k 2 F 2 1 − 2 F 1 F 2 + t − 4 k 2 8 k 2 F 2 2 , φ 2 − α = − φ 4 − α = M 2 2 k 2 F 2 1 − F 1 F 2 + s ( t + 8 k 2 ) 32 M 2 k 2 − 1 2 F 2 2 , φ 3 − α = s − 2 M 2 t F 2 1 + F 2 2 2 1 + t 4 k 2 , (4.2.2) φ 5 − α = 1 2 M s (4 k 2 + t) −t M 2 2 k 2 F 2 1 − F 1 F 2 + t 8 k 2 F 2 2 , 1 The totalp p cross-section behaves like σp p tot ≈ 75.5/β lab mb for energies up to 1 GeV [66,134], and ρ 2 ≈ 0 at all energies measured thus far (see Figure 4-12 of Ref. [140]). Therefore t c is energy independent, at least up to energies of 1 GeV. where φ 6 = − φ 5 in this case, as expected. The generic spin transfer equation for this case is d σ d Ω K ij = α 2 G 2 M 8 M 2 s t 2 q · S 2 q · S 3 4 M 2 − 2 s − t t F 2 2 (4.2.3) + 4 [ 2 p 3 · S 2 q · S 3 + q · S 2 ( p 2 + p 4 ) · S 3 ] M 2 t F 2 G E + 16 M 4 G E ( q · S 2 q · S 3 − S 2 · S 3 t G E ) , and the spin transfer observables now become d σ d Ω K XX = α 2 G 2 M 8 s k 2 M 2 4 M 4 F 2 1 − 8 k 2 M 2 F 1 F 2 + 4 k 4 + k 2 + t 4 s F 2 2 , d σ d Ω K YY = 2 α 2 s t M 2 G 2 E G 2 M , (4.2.4) d σ d Ω K ZZ = α 2 G 2 M 8 k 2 s t s 4 k 2 + t F 2 1 + 4 k 2 F 1 − t F 2 2 . Forp p elastic scattering K XZ = K ZX , and we obtain d σ d Ω K XZ = d σ d Ω K ZX , (4.2.5) = α 2 G 2 M 2 M t √ s − t ( 4 k 2 + t ) k 4 M 2 F 2 1 2 − k 2 F 1 F 2 + t F 2 2 8 . The depolarization spin observables to leading order in small | t | become d σ d Ω ( 1 − D XX ) ≈ − 2 α 2 F 2 1 k 2 M 2 s t k 2 + M 2 M 2 F 1 − 2 k 2 F 2 2 , d σ d Ω ( 1 − D YY ) = α 2 2 s G 4 M complete to all orders in t , d σ d Ω ( 1 − D ZZ ) ≈ − 2 α 2 F 2 1 k 2 M 2 s t k 2 + M 2 M 2 F 1 − 2 k 2 F 2 2 , (4.2.6) d σ d Ω ( 1 − D XZ ) ≈ d σ d Ω ( 1 − D ZX ) ≈ d σ d Ω ≈ 4 α 2 F 4 1 s t 2 2 k 2 + M 2 2 . Note that to first order in small | t | the antiproton and proton form factors can be approximated as F 1 ≈ 1, F 2 ≈ µ p − 1 (i.e. G M ≈ µ p and G E ≈ 1), where µ p is the magnetic moment of the proton. The t-channel spin-averaged differential cross-section for this case simplifies to s α 2 d σ d Ω = 4 M 2 G 2 E − t G 2 M 4 M 2 − t 2 −s u t 2 + 2 M 2 G 2 E t 2 + 1 2 G 4 M . (4.2.7) The above helicity amplitudes and spin observables are seen to be correct by a comparison to the relations between the helicity amplitudes and spin observables presented in Ref. [104] and in Table A.10.5 of Ref. [98]. In addition to antiproton-proton scattering the results of this section equally apply to any spin 1/2 (anti)baryon-(anti)baryon elastic electromagnetic scattering in the t-channel. Also, given that quarks are spin 1/2 particles, the above expressions could be applied to t-channel (anti)quark-(anti)quark electromagnetic scattering if in future quarks are found to have an internal structure. 4.3p e − helicity amplitudes and spin observables The spin observables forp e scattering can be obtained by setting f 1 = 1 and f 2 = 0 (i.e. g E = g M = 1) in the expressions from sections (3.4 and 3.5), to account for the elastic scattering of a structured particle off a point-like particle. These are required by the PAX Collaboration to analyze the buildup of polarization of an antiproton beam by interactions with the electrons in a hydrogen target, or interactions with a polarized electron beam as treated in Chapter 7. The helicity amplitudes in section 3.4 for antiproton-electron elastic collisions become φ 1 α = s − m 2 − M 2 1 + t 4 k 2 F 1 t − F 1 − F 2 , φ 2 α = − φ 4 α = m M F 1 2 k 2 − m F 2 2 M , φ 3 α = s − m 2 − M 2 1 + t 4 k 2 F 1 t , (4.3.1) φ 5 α = s ( 4 k 2 + t ) − t m ( s − m 2 + M 2 ) F 1 4 k 2 s , φ 6 α = − s ( 4 k 2 + t ) − t M ( s + m 2 − M 2 ) F 1 4 k 2 s − F 2 2 M . The generic spin transfer equation for this case is d σ d Ω K ij = − 2 α 2 m M G M s t G E S 2 · S 3 + F 2 2 M 2 p 3 · S 2 − F 1 t q · S 2 q · S 3 , (4.3.2) which is a generalization of equation (3) of Ref. [68], which was at the root of the initial interpretation of the FILTEX results on spin filtering in 1994. We do not neglect the 1/M 2 term, or make the non-relativistic approximations of eqs. (4.3.6), as done in Ref. [68]. The spin transfer observables for antiproton-electron elastic scattering are d σ d Ω K XX = α 2 m G M 2 k 2 M s M 2 F 1 − k 2 F 2 , d σ d Ω K YY = 2 α 2 m M G M G E s t , d σ d Ω K ZZ = α 2 G M 8 k 2 s 2 t s 2 − M 2 − m 2 2 4 k 2 + t F 1 + 4 k 2 s 4 k 2 F 1 − t F 2 , (4.3.3) d σ d Ω K XZ = α 2 G M 4 M s 3/2 t − t (4 k 2 + t ) k 4 M 2 s + m 2 − M 2 F 1 − 2 k 2 s F 2 , d σ d Ω K ZX = α 2 m F 1 G M 4 s 3/2 t − t ( 4 k 2 + t ) k 4 s − m 2 + M 2 . The depolarization spin observables to leading order in small | t | for this case are d σ d Ω ( 1 − D XX ) ≈ − m 2 α 2 F 2 1 2 k 2 s 2 t s − m 2 + M 2 2 , d σ d Ω ( 1 − D YY ) = α 2 2 s G 2 M complete to all orders in t , d σ d Ω ( 1 − D ZZ ) ≈ −M 2 α 2 F 2 1 2 k 2 s 2 t s + m 2 − M 2 2 , (4.3.4) d σ d Ω ( 1 − D XZ ) ≈ d σ d Ω ( 1 − D ZX ) ≈ d σ d Ω ≈ 4 α 2 F 2 1 s t 2 s k 2 + m 2 M 2 . The spin-averaged differential cross-section from section 3.3 for this case simplifies to s α 2 d σ d Ω = 4 M 2 G 2 E − t G 2 M 4 M 2 − t ( M 2 − m 2 ) 2 − s u t 2 + 2 m M G E t 2 + 1 2 G 2 M , (4.3.5) the familiar Rosenbluth formula [91], where we have not neglected the mass of the electron. The low energy (non-relativistic) limit of the above equations can be obtained using s = ( E cm 1 + E cm 2 ) 2 ≈ ( m + M ) 2 ≈ M 2 , G E ≈ 1 , (4.3.6) G M ≈ µ p = 1 + κ p . Of particular importance is the non-relativistic limit of (d σ/d Ω) K YY in eq. (4.3.3) for polarization transfer inp e − ↑ →p ↑ e − scattering, which using t = − 4 k 2 sin 2 θ 2 where θ is the Centre-of-Mass scattering angle gives d σ d Ω K YY ≈ − α 2 m ( 1 + κ p ) 2 k 2 M sin 2 θ 2 . (4.3.7) This is precisely equation (4) of Ref. [68], used extensively throughout their paper and early PAX calculations. The equations presented in this thesis generalize this work to relativistic energies. Non-relativistic expressions for the spin transfer observables have also been presented recently in Ref. [88,107], in the context of a proposal to polarize antiprotons by repeated interaction with a co-moving polarized positron beam at very low relative velocities [75]. In addition to antiproton-electron elastic scattering the results of this section apply also to any (anti)baryon-(anti)lepton elastic scattering. In particular they apply to antiproton-muon elastic scattering which may be of use in spin filtering, where the fact that muons have approximately 200 times the mass of electrons will greatly enhance the polarization transfer cross-sections K XX and K YY in eq. e + e − helicity amplitudes and spin observables The electromagnetic helicity amplitudes and spin observables for t-channel elastic positron-electron scattering can be obtained by making the transformations f 1 = F 1 → 1, f 2 = F 2 → 0, g E = G E → 1, g M = G M → 1φ 1 − α = s + 4 k 2 2 t + m 2 2 k 2 , φ 2 − α = − φ 4 − α = m 2 2 k 2 , φ 3 − α = s − 2 m 2 t 1 + t 4 k 2 , (4.4.1) φ 5 − α = m 4 k 2 s (4 k 2 + t) −t , where again φ 6 = − φ 5 in this case, as was found forp p scattering. The generic spin transfer equation for this case is d σ d Ω K ij = − 2 α 2 m 2 s t S 2 · S 3 − q · S 2 q · S 3 t , (4.4.2) and the spin transfer observables for e + e − elastic scattering are d σ d Ω K XX = α 2 m 2 2 s k 2 , d σ d Ω K YY = 2 α 2 m 2 s t , (4.4.3) d σ d Ω K ZZ = α 2 8 k 2 s t s 4 k 2 + t + 16 k 4 . For e + e − elastic scattering K XZ = K ZX hence one obtains d σ d Ω K XZ = d σ d Ω K ZX = α 2 m 4 t √ s − t ( 4 k 2 + t ) k 4 . The depolarization spin observables to leading order in small | t | for this case are d σ d Ω ( 1 − D XX ) ≈ − 2 α 2 m 2 k 2 s t k 2 + m 2 , d σ d Ω ( 1 − D YY ) = α 2 2 s complete to all orders in t , d σ d Ω ( 1 − D ZZ ) ≈ − 2 α 2 m 2 k 2 s t k 2 + m 2 , (4.4.4) d σ d Ω ( 1 − D XZ ) ≈ d σ d Ω ( 1 − D ZX ) ≈ d σ d Ω ≈ 4 α 2 s t 2 2 k 2 + m 2 2 . The t-channel spin-averaged differential cross-section for this case simplifies to s α 2 d σ d Ω = −s u t 2 + 2 m 2 t 2 + 1 2 , (4.4.5) which can be written in the more familiar form [108] d σ d Ω = α 2 s t 2 2 m 2 − s 2 + s t + t 2 2 . (4.4.6) In elastic electron-positron scattering, also known as Bhabha scattering [109], for very low momentum transfer the t-channel part of the spin-averaged differential cross-section dominates over the s-channel part. But the complete spin-averaged Bhabha differential cross-section must include both t-channel and s-channel Feynman diagrams in the amplitude, as shown in Figure 4-1, to give s d σ d Ω = t-channel part α 2 t 2 2 m 2 − s 2 + s t + t 2 2 (4.4.7) + α 2 s 2 2 m 2 − t 2 + s t + s 2 2 s-channel part + α 2 s t ( s + t ) 2 − 4 m 4 cross-term part . where one sees that the t-channel part transforms into the s-channel part, and viceversa, by the well known crossing symmetry obtained by interchanging s and t. The results of this section also apply to any combination of t-channel elastic (anti)lepton-(anti)lepton, (anti)lepton-(anti)quark and (anti)quark-(anti)quark electromagnetic scattering. This is particularly useful since the masses of each particle have been retained in the equations. Observables needed for spin filtering In this section we present the leading t approximation of all spin observables needed for spin filtering. These expressions will be integrated over a range of t later in the thesis. All expressions are written in terms of invariants using eqs. (4.5.2 and 4.5.4) above. The low | t | approximation of the form factors 2 , F 1 (t) ≈ 1 and F 2 (t) ≈ µ p − 1 therefore G E (t) ≈ 1 and G M (t) ≈ µ p , are used, as is seen to be valid in the dipole model for the Sachs form factors with Λ 2 = 0.71 (GeV/c) 2 obtained from a best fit to experimental data [110,111]. Spin filtering is an azimuthally symmetrical process, as will be explained in Chapters 5 and 6, hence the transverse contributions will be averaged in what follows. G E (t) = G M (t) µ p = 1 ( 1 − t/Λ 2 ) 2 ,(4. The Centre-of-Mass expressions presented throughout this chapter and the previous chapter can be transformed into invariants, or into the Laboratory reference frame using the following relations [112]: λ ≡ 4 k 2 s = s − ( m + M ) 2 s − ( m − M ) 2 , (4.5.2) = s s − 4 M 2 when m = M , (4.5.3) d σ d t = π k 2 d σ d Ω = 4 π s λ d σ d Ω , (4.5.4) k = M √ s p lab , (4.5.5) t = − 2 k 2 ( 1 − cos θ ) = − 4 k 2 sin 2 θ 2 , (4.5.6) where θ is the Centre-of-Mass scattering angle and λ is a Lorentz invariant. Of interest in eq. (4.5.6) are the particular cases that t = 0 when θ = 0 and t = − 4 k 2 when θ = π, corresponding to total backward scattering. One sees that for elastic scattering t < 0 for all Centre-of-Mass scattering angles θ = 0. Antiproton -proton scattering In this section, where the masses of the two particles are equal, λ is the invariant defined in eq. (4.5.3). The ≈ sign refers to the first term in the expansion in t. To leading order in small | t | the relevant spin observables for single photon exchange antiproton-proton scattering are: K XX + K YY 2 d σ d t ≈ 4 π α 2 M 2 µ 2 p λ t , (1 − D XX ) + (1 − D YY ) 2 d σ d t ≈ − π α 2 (λ + 4 M 2 s) λ 2 M 2 s 2 t 2 M 2 s + λ (1 − µ p ) 2 , K ZZ d σ d t ≈ 4 π α 2 µ 2 p s t λ λ + 2 M 2 s , (4.5.7) (1 − D ZZ ) d σ d t ≈ − 2 π α 2 (λ + 4 M 2 s) λ 2 M 2 s 2 t 2 M 2 s + λ (1 − µ p ) 2 . The leading t approximation of the spin-averaged cross-section for this case is: d σ d t ≈ 4 π α 2 λ ( s − 2 M 2 ) 2 t 2 . (4.5.8) Antiproton -electron scattering In this section, where the masses of the two particles are not equal, λ is the invariant defined in eq. (4.5.2). The leading t terms of the relevant observables for antiprotonelectron scattering are: K XX + K YY 2 d σ d t ≈ 4 π α 2 m M µ p λ t , (1 − D XX ) + (1 − D YY ) 2 d σ d t ≈ − 4 π m 2 α 2 ( s − m 2 + M 2 ) 2 λ 2 t , K ZZ d σ d t ≈ 4 π α 2 µ p λ t s − m 2 − M 2 , (4.5.9) (1 − D ZZ ) d σ d t ≈ − 8 π M 2 α 2 ( s + m 2 − M 2 ) 2 λ 2 t . The leading t approximation of the spin-averaged cross-section for this case is: d σ d t ≈ 4 π α 2 λ ( s − m 2 − M 2 ) 2 t 2 . (4.5.10) Spin 0 -spin 1 helicity amplitudes In this section we calculate the electromagnetic helicity amplitudes for spin 0 -spin 1 scattering 3 , generalizing the spin 0 -spin 1 helicity amplitudes presented to leading order in the low | t | approximation in Ref. [113]. While not being directly used later in the thesis this calculation uses the formalism developed earlier, and would be applicable to the scattering of deuterons 4 (which are spin 1) off a carbon nucleus (spin 0). A spin 1 particle has three possible spin states, represented by −1, 0 and +1, while a spin 0 particle has only one spin state. We represent the helicity amplitudes for spin 0 -spin 1 scattering as M( λ b ; λ a ) where λ a and λ b are the helicities of the incoming and outgoing spin 1 particle respectively, hence λ a , λ b ∈ {−, 0, +}. Hence have multiplicity 2, H 2 has multiplicity 4 and H 4 has multiplicity 1. Thus one can write the spin-averaged differential cross-section in terms of the independent helicity amplitudes as s d σ d Ω = 1 (8 π) 2 λa λ b 1 ( 2 s A + 1 ) ( 2 s B + 1 ) | M( λ b ; λ a ) | 2 , = 1 (8 π) 2 λa λ b 1 3 | M( λ b ; λ a ) | 2 , = 1 3 (8 π) 2 2 |H 1 | 2 + 4 |H 2 | 2 + 2 |H 3 | 2 + |H 4 | 2 , (4.6.5) where s A = 0 and s B = 1 are the spins of the two particles in the elastic scattering process. The helicity amplitudes for elastic spin 0 -spin 1 electromagnetic scattering can be found by calculating M( λ b ; λ a ) = j µ J µ ( λ b ; λ a ) q ν q ν , (4.6.6) where j µ and J µ are the spin 0 and spin 1 electromagnetic currents respectively, as defined below. The spin 0 current is very simple as it has no helicity structure. The electromagnetic current for a spin 0 particle of charge Z e and form factor F 0 (q 2 ) is simply [84,114] j µ = Z e F 0 q 2 ( p + p ′ ) µ . (4.6.7) Since the deuteron is a spin 1 object its electromagnetic structure is described by three form factors, charge monopole G C , charge quadrupole G Q and magnetic dipole G M , assuming parity and time-reversal invariance. The most general form of the deuteron electromagnetic current, assuming Parity and Time-reversal invariance is [115,116,117]: J µ ( λ b ; λ a ) = e ǫ * ρ ( p b , λ b ) F d 1 q 2 R µ η ρ σ − F d 2 (q 2 ) 2 M 2 d R µ q ρ q σ − G d 1 q 2 ( η µ ρ η ν σ − η µ σ η ν ρ ) q ν ǫ σ ( p a , λ a ) , (4.6.8) where M d is the mass of the deuteron, R µ = p µ a + p µ b and ǫ σ ( p a , λ a ) is a polarization 4-vector restricted to three independent components by the condition F d 1 (0) = 1, F d 2 (0) = Q + µ d − 1, Q d 1 (0) = µ d ,(4.= ( E d , 0, 0, k ) where E d = k 2 + M 2 d is the energy of the deuteron, are as follows [118]: Initial                    ǫ µ ( p a , +) = 1 √ 2 ( 0, −1, − i, 0 ) ǫ µ ( p a , 0 ) = 1 M d ( k, 0, 0, E d ) ǫ µ ( p a , −) = 1 √ 2 ( 0, 1, − i, 0 ) (4.6.10) Final                    ǫ * µ ( p b , +) = 1 √ 2 ( 0, − cos θ, i, sin θ ) ǫ * µ ( p b , 0 ) = 1 M d ( k, E d sin θ, 0, E d cos θ ) ǫ * µ ( p b , −) = 1 √ 2 ( 0, cos θ, i, − sin θ ) (4.6.11) where θ is the CM scattering angle and i = √ −1. The momentum transfer is q µ = η µ ν q ν = ( 0, − k sin θ, 0, k − k cos θ ) . (4.6.12) Combining all of the above into eq. (4.6.6) gives the helicity amplitudes for a deuteron of mass M d colliding with a spin zero nucleus of charge Z e and electromagnetic form factor F 0 (t), as follows: H 1 Z e 2 F 0 = (s − u) −F d 1 t + F d 2 4 M 2 d + G d 1 1 + t 4k 2 ,(4.H 2 Z e 2 F 0 = 1 M 2 d + 1 k 2 (s − u) −F d 1 + F d 2 t 4 M 2 d + t G d 1 + 2 k √ s M d G d 1 2 −t − 1 2 k 2 , (4.6.14) H 3 Z e 2 F 0 = F d 1 (s − u) 4 k 2 − F d 2 (s − u) 4 M 2 d + G d 1 1 + t 4 k 2 , (4.6.15) H 4 Z e 2 F 0 = − F d 1 (s − u) 2 2 t + 1 M 2 d + 1 k 2 + F d 2 t (s − u) 8 M 2 d 1 M 2 d + 1 k 2 + 2 G d 1 (s − u) 4 M 2 d + t 4 k 2 + 1 . The Sokolov-Ternov effect The fact that an electron beam acquires a 'self-polarization' due to the emission of synchrotron radiation in a storage ring is called the Sokolov-Ternov effect [61], and has been described in section 2.3.3 of this thesis. It turns out that one can describe the Sokolov-Ternov effect by a system of polarization evolutions equations very similar to a scenario of spin filtering when there is no scattering out of the ring. The physical principles behind both systems are identical, the polarization buildup in both being induced by a discrepancy in the spin-flip transition rates. The spin-flip in the Sokolov-Ternov effect is induced by synchrotron radiation as a result of the charged particle being bent in a magnetic field, whereas in spin filtering the spin-flip is induced by interactions with a polarized internal target. In order to introduce the mathematics of systems of polarization evolution equations we now present and solve a set of polarization evolution equations that describe the Sokolov-Ternov effect. This will provide a stepping-stone to the mathematical description of spin filtering which follows later in the chapter. The intermediate details of the calculations will be presented here, but omitted in later sections. Please note as described in section 2.3.3 that this effect is much stronger for electrons than (anti)protons and the Sokolov-Ternov 'self-polarization' is not a practical method to produce a polarized antiproton beam at present energies. We denote the transition rates in this section by W ab instead of σ ab , where a, b ∈ {+, −}, to distinguish from the spin filtering spin-flip case. It is important to note that there is no target in the Sokolov-Ternov effect, which makes it physically different to spin filtering, although it can be described similarly. The number of particles in the 'spin up' and 'spin down' states can change by two mechanisms while emitting synchrotron radiation: (1) 'spin up' particles can be flipped to 'spin down' particles, the cross-section for which we label as W +− ; and (2) 'spin down' particles can be flipped to 'spin up' particles, the cross-section for which we label as W −+ . Mechanism Therefore the Sokolov-Ternov effect can be described by the polarization evolution equations d d τ   N + N −   =   − W +− W −+ W +− − W −+     N + N −   . (5.1.1) where τ is the time variable 1 . As long as the two spin-flip transition rates W +− and W −+ are not equal, i.e. W +− = W −+ , there will be a buildup of beam polarization over time. It has been found that there is a slight difference between these rates, and this is the basis of the original Sokolov-Ternov idea [61]. This system is identical to the system presented later in eq. (5.2.8), which describes spin filtering when there is no scattering out of the ring, except for the values of the matrix entries. Solving this system gives eigenvalues λ 1 = 0 and λ 2 = − ( W −+ + W +− ), leading to eigenvectors v 1 =   W −+ W +−   and v 2 =   1 − 1   . And the solution to the system is   N + (τ ) N − (τ )   = c 1 v 1 e λ 1 τ + c 2 v 2 e λ 2 τ , = c 1   W −+ W +−   e 0 + c 2   1 − 1   e − ( W −+ + W +− ) τ , where c 1 and c 2 are constants to be determined from the initial conditions, hence N + (τ ) = c 1 W −+ + c 2 e − ( W −+ + W +− ) τ , N − (τ ) = c 1 W +− − c 2 e − ( W −+ + W +− ) τ ,(5.c 1 = N 0 W −+ + W +− and c 2 = N 0 2 W +− − W −+ W −+ + W +− , and thus the complete solution P(τ ) = N + − N − N + + N − = W −+ − W +− W −+ + W +− 1 − e − ( W −+ + W +− ) τ . (5.1.3) The spin-flip transition rates W +− and W −+ are defined from the theory of synchrotron radiation as [62,61] W +− = W 0 2 1 + 8 5 √ 3 and W −+ = W 0 2 1 − 8 5 √ 3 ,(5.W −+ + W +− = W 0 , W −+ − W +− = − 8 5 √ 3 W 0 . (5.1.6) One can now present the complete solutions for the number of particles in both the 'spin up' and 'spin down' states as a function of time τ : N + (τ ) = N 0 2 + 8 N 0 10 √ 3 e − W 0 τ − 1 , (5.1.7) N − (τ ) = N 0 2 − 8 N 0 10 √ 3 e − W 0 τ − 1 , (5.1.8) which can trivially be seen to satisfy N(τ ) ≡ N + (τ ) + N − (τ ) = N 0 = constant. The steady state polarization (Sokolov-Ternov polarization) P ST is reached when 1.9) and one has the complete solution τ > 1 / ( W −+ + W +− ) = W −1 0 ≡ τ ST where τ ST is presented in eq. (2.3.1) 2 P ST = W −+ − W +− W −+ + W +− = − 8 5 √ 3 ≈ − 0.924 ,(5.P(τ ) = P ST 1 − e − W 0 τ = − 8 5 √ 3 1 − e − τ / τ ST . (5.1.10) In a perfect ring one obtains an equilibrium polarization of 92.4% after time τ ST . In practice the maximum polarization achieved is slightly less than this due to imperfections in the magnetic fields of real synchrotrons. Some parameters, including τ ST , of current and proposed future synchrotrons are presented in Table 2 Here we label the cross-section for a particle in the 'spin up' state to be scattered out of the beam as σ out + , the cross-section for a particle in the 'spin down' state to be scattered out of the beam as σ out − , the cross-section for a particle in the 'spin up' state to be flipped to the 'spin down' state while remaining in the beam as σ +− and the cross-section for a particle in the 'spin down' state to be flipped to the 'spin up' state while remaining in the beam as σ −+ . In order for each of these processes to contribute to polarization buildup of the beam we must have σ out + = σ out − and σ +− = σ −+ respectively. These cross-sections will be used below in the mathematical evolution equations to describe the rate of buildup of polarization by spin filtering. Polarization evolution equations In this section we develop sets of differential equations that describe the buildup of polarization of an antiproton beam by spin filtering. Consistency checks are then performed on the systems of equations, which provides a chance to highlight the underlying physical phenomena under investigation. The method of polarization buildup by spin filtering has been outlined in section 2.3.5, which the reader may wish to read again before continuing here. Since the cross-sections for an interaction between a beam particle and a particle in the target are low we neglect the effects of multiple scattering, which has a cross-section orders of magnitude smaller than single scattering. Hence we consider two possibilities each time the beam passes through the target: (1) a beam particle can pass through the target without interaction, or (2) a beam particle can scatter off at most one of the target particles. Interactions with residual gas due to a non-perfect vacuum in the storage ring can also be neglected as the density of the internal target is much higher than the density of the residual gas. The high density of the target, and the fact that it is a constantly replenished gas jet, ensures that there is no significant target depolarization. Recall the two physical processes that contribute to spin filtering shown in Fig d d τ   N + N −   = − n ν   σ out + + σ +− − σ −+ − σ +− σ out − + σ −+     N + N −   , (5.2.1) where τ is the time variable, n is the areal density of the target, ν is the revolution frequency of the beam and N + (τ ) and N − (τ ) are the number of beam particles in the 'spin up' and 'spin down' states at time τ respectively. For a beam that is initially unpolarized one imposes the following initial condi- tions N + (0) = N − (0) = N 0 2 , (5.2.2) where N 0 is the total number of particles in the beam initially. We define the beam intensity N(τ ) ≡ N + (τ ) + N − (τ ) as the total number of particles in the beam at time τ , and the beam total spin at time τ as J(τ ) ≡ N + (τ ) − N − (τ ) so that the polarization of the beam at time τ is simply given by P(τ ) = N + (τ ) − N − (τ ) N + (τ ) + N − (τ ) = J(τ ) N(τ ) . (5.2.3) The change in beam polarization as the number of particles in each of the spin states changes is very important throughout the thesis, so we plot this dependence and highlight a few points in Figure 5-3. A problem with spin filtering where particles are scattered out of the beam is that while the beam polarization increases the beam intensity decreases. We propose possible solutions to this problem in Chapter 6. The behaviour of the number of particles in the 'spin up' and 'spin down' states, along with the polarization, as time increases is shown in Figure 5-4. Note some treatments of spin filtering investigate a scenario where no particles are scattered out of the beam, i.e. the maximum scattering angle for the process is less than the ring acceptance angle, which is the case for antiprotons scattering off electrons in an atomic target [68,69,71,72,76] and for antiprotons scattering off a co-moving beam of electrons or positrons [75]. In these scenarios only selective spinflip can contribute to polarization buildup, and one avoids the problem of decreasing beam intensity. The low density of the targets currently available causes the rate of polarization buildup using these methods to be slow, but the enhanced cross-sections at low energies suggested in Refs. [75,88,107] may counteract this difficulty. We analyze such systems later in the thesis. Before solving this system of polarization evolution equations we shall prove a number of short lemma's providing a consistency check that the equations accurately describe the physical phenomena we wish to model. This also provides a chance to highlight the dynamical properties of the physical system, as this plays a major role in the rest of the thesis. Lemma 1 If σ out + = σ out − and σ +− = σ −+ there will be no buildup of beam polarization, but there will still be loss of beam intensity N(τ ). Proof: When σ out + = σ out + + σ +− − σ +− − σ +− σ out + + σ +−     N + N −   , (5.2.4) i.e. d N + d τ = − n ν σ out + + σ +− N + − σ +− N − , d N − d τ = − n ν − σ +− N + + σ out + + σ +− N − , (5.2.5) which can be added and subtracted to give and we see that N(τ ) will decrease exponentially, provided that σ out + = 0, and J(τ ) which is zero initially will always remain zero (i.e. if J(0) = 0 then J(τ ) = 0 for all τ ). So there will be no polarization buildup. Also in the case when the beam is initially polarized (J(0) = 0) its polarization will decrease exponentially to zero, remembering the cross-sections are positive quantities. d N(τ ) d τ = − n ν σ out + N(τ ) , d J(τ ) d τ = − n ν σ out + + 2 σ +− J(τ ) ,(5. We later show that when the internal target is not polarized σ out + = σ out − and σ +− = σ −+ . Thus there will be no polarization buildup by spin filtering if the internal target is unpolarized. Lemma 2 If σ out + = σ out − = 0 there will be no loss of beam intensity N(τ ) = Constant = N 0 , but there may still be polarization buildup. Proof: When σ out + = σ out − = 0 (this happens when the maximum scattering angle for the process is less than the ring acceptance angle) the polarization evolution equations reduce to d d τ   N + N −   = − n ν   σ +− − σ −+ − σ +− σ −+     N + N −   , (5.2.8) i.e. d N + d τ = − n ν ( σ +− N + − σ −+ N − ) , d N − d τ = − n ν ( − σ +− N + + σ −+ N − ) ,(5. 2.9) and adding these gives d N d τ = d d τ ( N + + N − ) = d N + d τ + d N − d τ , = − n ν ( σ +− N + − σ −+ N − − σ +− N + + σ −+ N − ) = 0 .d J d τ = d d τ ( N + − N − ) = d N + d τ − d N − d τ , = − 2 n ν ( σ +− N + − σ −+ N − ) , which leads to a non-zero J(τ ) (i.e. non-zero polarization) provided that σ +− = σ −+ . The system without scattering out of the ring described in eq. Corollary 1 When there is no scattering out of the beam, i.e. σ out + = σ out − = 0, the condition d N + d τ = − d N − d τ ,(5. 2.10) must be satisfied. Proof: We have shown that when there is no scattering out of the ring the equations reduce to eqs. (5.2.9) which can immediately be seen to satisfy eq. (5.2.10). Lemma 3 When there is no spin-flip, i.e. σ +− = σ −+ = 0, the change in one spin state should not depend on the number of particles in the other spin state, i.e. the equations should decouple. Proof: When there is no spin-flip σ +− = σ −+ = 0 and thus the polarization evolution equations reduce to [71,72]. As we show in Chapter 7, the maximum scattering angle of antiprotons scattering off atomic electrons is 0.54 mrad, below the acceptance angle of a typical storage ring, thus there is no scattering out of the beam. Since there is no scattering out of the beam, and spin-flip transitions are negligible the Budker-Jülich groups conclude that polarized electrons in an atomic target are not effective in transferring polarization to an antiproton beam by spin filtering [71,72]. To force some antiprotons to be scattered out of the beam, and to avoid the problem of loss of beam intensity due to antiprotons annihilating with the protons in an atomic target, we suggest to use an opposing polarized electron beam of sufficient energy to scatter some antiprotons beyond the ring acceptance angle [76]. Such a system is treated in Chapter 7. The rate of polarization buildup using this method is slow due to the low densities of polarized electron beams currently available [76], but the enhanced cross-sections at low energies suggested in Refs. [75,88,107] may counteract this difficulty. d N + d τ = − n ν σ out + N + , d N − d τ = − n ν σ out − N − ,(5. Differentiating eq. (5.2.3) and rearranging gives d P d τ = 1 − P 2 2 1 N + d N + d τ − 1 N − d N − d τ ,(5.d P d τ = − n ν 2 1 − P 2 σ out + − σ out − , (5.2.14) which is exactly the important equation (3) from Ref. [69], the first theoretical description of the FILTEX results. Integrating the above equation, and using a result from the next section eq. (5.2.21), leads to P(τ ) = tanh − n ν 2 σ out + − σ out − τ , = tanh ( − n ν P T A out τ ) . (5.2.15) Which was proposed initially as a model of the rate of polarization buildup in spin filtering [54,69]. Since then the importance of scattering within the beam has been highlighted and a more complex treatment of spin filtering is required, involving the polarization transfer to and depolarization of particles scattering within ring acceptance. Therefore it is seen that when spin-flip effects are neglected the theoretical treatment of spin filtering presented in this thesis reduces to the initial naive treatments where polarization transfer and depolarization effects of scattering within ring acceptance were not included. σ out + , σ out − , σ +− , σ −+ and the spin observables The spin observables of a spin 1/2 -spin 1/2 scattering process are defined in section 3.5. In spin filtering where the polarization of the recoiled target particle is not important one is interested in the polarization transfer, depolarization and double spin asymmetry spin observables. These have been calculated for electromagnetic antiproton-proton and antiproton-electron elastic scattering in Chapter 4. The spin transfer observable has been calculated for low energy antiproton-positron scattering in Ref. [88]. A large increase of this spin transfer cross-section at very low energies is the basis for the proposal to polarize antiprotons by interaction with a co-moving polarized positron beam presented in Ref. [75]. It is claimed in Ref. [107] that the polarization transfer cross-section for e p or e +p (like charges) scattering is enhanced at very low relative velocities, but by much less than that claimed in Refs. [75,88]. An experiment has been proposed to test, and distinguish between, these claims [119]. The cross-sections σ out + , σ out − , σ +− and σ −+ can be related to the spin observables that have been calculated in Chapters (3 and 4) by the following relations [72]: σ out + ≡ I out + P T A out , (5.2.16) σ out − ≡ I out − P T A out , (5.2.17) σ +− ≡ L in + P T 2 ( A in − K in ) , (5.2.18) σ −+ ≡ L in − P T 2 ( A in − K in ) , (5.2.19) where P T is the polarization of the target, and L in = ( I in − D in ) / 2 is a loss of polarization quantity. These relations involve integration of the spin observables presented in Chapter 4 over the following angular ranges, as seen in Table 5.1. The "in" subscript refers to particles that are scattered at small angles ≤ θ acc remaining in the beam, and the "out" subscript refers to particles that are scattered out of the beam. Thus the integrals over scattering angle θ are labeled "in" where the range of integration is θ min ≤ θ ≤ θ acc , "out" where the range of integration is 3 θ acc < θ ≤ θ max , where θ max is the maximum scattering angle for the process in the given reference frame 4 , and "all" = "in" + "out" where the range of integration is θ min ≤ θ ≤ θ max ; as seen in Table 5.1. I = dσ / dΩ is the spin-averaged differential cross-section and A, K and D are the double spin asymmetry, polarization transfer and depolarization spin observables respectively as calculated in Chapter 4. 3 While not occurring in the case of antiprotons which we focus on here, an additional effect must be accounted for in the case of polarization buildup of a proton beam by spin filtering off a hydrogen target [120], as in the FILTEX experiment. Because the final state particles are identical and hence indistinguishable, u-channel p p scattering can contribute; i.e. protons from the hydrogen target can be back scattered into the circulating proton beam. This happens when the beam protons are back scattered into the angular range (π − θ acc ) ≤ θ ≤ π in the CM frame. This effect can be accounted for in the above formalism by changing the angular ranges in the CM frame to θ min ≤ θ ≤ θ acc plus (π − θ acc ) ≤ θ ≤ π for the "in" integrations and θ acc < θ < (π − θ acc ) for the "out" integrations. The physical result of this effect is to lessen the rate of decrease of beam intensity. 4 In the CM frame θ cm max = π corresponding to total backward scattering, but in other frames, for example the LAB frame, this extreme value is not reached for some reactions and θ lab max < π. All cross-sections and spin observables contributing to spin filtering are azimuthally averaged, due to the geometry of the scattering, where the scattering plane can be at any azimuthal angle. Hence single spin observables, for example the analyzing power, do not contribute to the polarization evolution equations because they vanish when azimuthally averaged. The cylindrical symmetry of the system also implies that the ring acceptance angle has no azimuthal dependence. Note the following linear combinations of the cross-sections I out = σ out + + σ out − 2 , (5.2.20) P T A out = σ out + − σ out − 2 , (5.2.21) L in = σ +− + σ −+ 2 , (5.2.22) P T ( A in − K in ) = σ +− − σ −+ . (5.2.23) Again to ensure consistency and to highlight the physical properties we are trying to describe mathematically we prove a number of short Lemma's on the above relations between the cross-sections and the spin observables. Lemma 4 If the target is unpolarized (P T = 0) then one has that σ out + = σ out − and σ +− = σ −+ so no polarization buildup will occur. Proof: Setting P T = 0 into the eqs. ( + = I out = σ out − and σ +− = L in = σ −+ . Once this is satisfied it is proved in Lemma 1 that no polarization buildup will occur in this case. Lemma 5 The spin-flip cross-sections should depend only on spin observables relating to particles scattering within the ring, i.e. only to "in" spin observables which are integrated from θ min to θ acc . Proof: This is immediately satisfied by the relations in eqs. (5.2.18 and 5.2.19). Lemma 6 The cross-section differences σ out where we have also transformed from the cross-sections to the spin observables which have already been calculated. The parameters n and ν are the target areal density and the beam revolution frequency respectively. The systems presented in eqs. (5.2.1 and 5.2.24) are identical and from now on we concentrate on the latter as its solution is more illustrative of the underlying physical phenomena, and the dependence on the target polarization is explicit. In particular one immediately sees that when the target is unpolarized no beam polarization buildup occurs, as when P T = 0 the system reduces to two uncoupled separable first order ODE's as in eq. (5.2.6) with solutions as presented in eq. (5.2. 7) showing P(τ ) = 0 for all τ if P T = 0. The parameters in the matrix of coefficients of eq. (5.2.24) depend on the state of the target polarization, i.e. longitudinal or transverse, as seen in Table 5.1. Transverse polarization requires Longitudinal polarization requires Table 5.1: The entries in the system of equations for polarization buildup involve angular integration over the spin observables presented in Chapter 4. X, Y and Z are the coordinate axes where the beam is moving in the positive Z direction. The minimum value for θ (θ min ) relates to the average transverse electron separation for a pure electron target and to the Bohr radius for an atomic gas target, θ acc is the ring acceptance angle and θ max is the maximum scattering angle for the process. I out = 2 π θmax θacc d σ d Ω sin θ dθ I out = 2 π θmax θacc d σ d Ω sin θ dθ A out = 2 π θmax θacc A XX + A YY 2 d σ d Ω sin θ dθ A out = 2 π θmax θacc A ZZ d σ d Ω sin θ dθ A all = 2 π θmax θ min A XX + A YY 2 d σ d Ω sin θ dθ A all = 2 π θmax θ min A ZZ d σ d Ω sin θ dθ K in = 2 π θacc θ min K XX + K YY 2 d σ d Ω sin θ dθ K in = 2 π θacc θ min K ZZ d σ d Ω sin θ dθ D in = 2 π θacc θ min D XX + D YY 2 d σ d Ω sin θ dθ D in = 2 π θacc θ min D ZZ d σ d Ω sin θ dθ Solving the polarization evolution equations We have shown in the previous section that when circulating at frequency ν, for a time τ , in a ring with a polarized internal target of areal density n and polarization P T oriented normal to the ring plane, (or longitudinally with rotators) [72]. In this section we solve this system of polarization evolution equations. The eigenvalues of the matrix of coefficients are found to be d d τ   N J   = − n ν   I out P T A out P T ( A all − K in ) I all − D in     N J   ,(5.λ 1 = − n ν ( I out + L in + L d ) and λ 2 = − n ν ( I out + L in − L d ) , (5.3.2) where the discriminant L d of the quadratic equation for the eigenvalues is L d = P 2 T A out (A all − K in ) + L 2 in .N(τ ) = e λ 1 τ ( L d − L in ) + e λ 2 τ (L d + L in ) N 0 2 L d , (5.3.4) J(τ ) = e λ 1 τ − e λ 2 τ (A all − K in ) N 0 P T 2 L d . (5.3.5) The time (τ ) dependence of the polarization of the beam is given by P (τ ) = J(τ ) N(τ ) = − P T ( A all − K in ) L in + L d coth (L d n ν τ ) . (5.3.6) The expression for P (τ ) is proportional to P T which confirms that if the target polarization is zero there will be no polarization buildup in the beam. The approximate rate of change of polarization for sufficiently short times, and the limit of the polarization for large times are respectively: d P d τ ≈ − n ν P T (A all − K in ) , (5.3.7) P max = lim τ → ∞ P (τ ) = − P T A all − K in L in + L d . (5.3.8) In order to compare to earlier treatments of spin filtering notice that in the absence of scattering within the ring, when all "in" spin observables are zero, one has that L d = P T A out and eq. (5.3.6) reduces to eq. (5.2.15) which was the initial treatment of spin filtering proposed in 1993 where scattering within ring acceptance was not included. Having said that, the general behaviour of equations 5.2.15 and 5.3.6 are similar. For pure electromagnetic scattering the double spin asymmetries equal the polarization transfer spin observables [90], thus one can simplify the above equations using A in = K in , A out = K out and A all = K all ; hence A all − K in = K out . At this point one may wish to find expressions for N + (τ ) and N − (τ ) which can easily be obtained from eqs. N + (τ ) = N(τ ) + J(τ ) 2 , = N 0 4 L d e λ 1 τ [ L d − L in + ( A all − K in ) P T ] + e λ 2 τ [ L d + L in − ( A all − K in ) P T ] , (5.3.9) N − (τ ) = N(τ ) − J(τ ) 2 , = N 0 4 L d e λ 1 τ [ L d − L in − ( A all − K in ) P T ] + e λ 2 τ [ L d + L in + ( A all − K in ) P T ] , (5.3.10) which can easily be seen to satisfy N + (0) = N − (0) = N 0 / 2, the correct initial conditions. If the target is unpolarized (P T = 0) one sees that N + (τ ) = N − (τ ) therefore no polarization buildup will occur, as was required by Lemma 4. Beam lifetime and figure of merit The beam lifetime τ * , the time taken for the beam intensity to decrease by a factor of e ≈ 2.718, i.e. N ( τ * ) = N 0 /e, can be obtained from eq. (5.3.4). One finds beam, and is given by τ * ≈ 1 n ν I out ,(5.FOM(τ ) = P 2 (τ ) N(τ ) = J 2 (τ ) N(τ ) . (5.3.12) The figure of merit for the above case is FOM(τ ) = (A all − K in ) 2 N 0 P 2 T 2 L d e λ 1 τ − e λ 2 τ 2 e λ 1 τ ( L d − L in ) + e λ 2 τ (L d + L in ) . approximately twice the beam lifetime. Thus the optimum time for polarization buildup is twice the lifetime of the beam, as also found in Ref. [66]. The behaviours of the beam intensity, beam polarization and the figure of merit as time increases are shown in Figure 5-6. Note the characteristic trade-off of spin filtering: as the beam polarization increases the beam intensity greatly decreases. In Chapter 6 we address possible ways of circumventing this drawback, for instance by continuously inputting particles into the beam. Pure electromagnetic scattering For pure electromagnetic scattering A ij = K ij where i, j ∈ { X, Y, Z }, so the system simplifies to d d τ   N J   = − n ν   I out P T K out P T K out I all − D in     N J   , (5.3.15) the coefficient matrix of which is symmetric, and the solutions become P(τ ) = J(τ ) N(τ ) = − K out P T L in + L d coth (L d n ν τ ) , (5.3.16) L d = P 2 T K 2 out + L 2 in . (5.3.17) The approximate rate of change of polarization for sufficiently short times, and the limit of the polarization for large times simplify to respectively: d P dτ ≈ − n ν P T K out P max = lim τ →∞ P(τ ) = − K out P T L in + L d . (5.3.18) We now show that the maximum polarization achieved cannot exceed one, i.e. |P max | ≤ 1. From the definition of L d for the case of pure electromagnetic scattering we have L 2 d = P 2 T K 2 out + L 2 in ⇒ L d ≥ |P T K out | which can be used to obtain |P max | ≤ L d L in + L d ≤ L d L d = 1 , (5.3.19) since L in is a non-negative quantity. Note the upper bound |P max | = 1 only happens for L in = 0, i.e. no polarization is lost. Fraction of antiprotons lost per revolution We now calculate the fraction of the antiproton beam that is lost per revolution. Define ∆τ to be the time taken for one revolution, thus ν ∆τ = 1, and ∆N to be the change in beam intensity during this time. Manipulating the d N / d τ equation gives us ∆N ∆τ = − n ν I out N − n ν P T A out J , ∆N N = − n ν I out ∆τ − n ν P T A out J N ∆τ , = − n I out − n P T A out J N , = − n ( I out + Pp P T A out ) . (5.3.20) Since ∆N is the change in beam intensity during time ∆τ , and the beam intensity decreases with time, the quantity ∆N is negative. But the fraction of the antiprotons lost per revolution is n ( I out + Pp P T A out ) which is positive, and thus the fraction of antiprotons remaining in the beam per revolution is 1 − n ( I out + Pp P T A out ). The fraction of antiprotons lost per revolution is not constant in time, it depends on the polarization of the antiproton beam Pp , which is zero initially. The fraction of antiprotons lost per revolution decreases as the polarization of the antiproton beam increases. This makes physical sense, since as the antiproton beam polarization increases there will be fewer particles in the spin state that is scattered out more often. This can also be used to obtain the beam lifetime. Since the frequency ν is the number of revolutions per second, the fraction of particles lost from the beam per second is n ν ( I out + Pp P T A out ), hence the fraction of particles lost from the beam in τ seconds is n ντ ( I out + Pp P T A out ) . The beam lifetime is the time taken for the beam intensity to decrease by a factor of 1 / e, i.e. the time taken for a fraction 1 − 1 / e = (e − 1) / e of the beam particles to be lost due to scattering out of the ring. Thus we can calculate the beam lifetime (τ * ) by solving n ν τ * ( I out + Pp P T A out ) = e − 1 e , (5.3.21) leading to the beam lifetime τ * = e − 1 n ν e ( I out + Pp P T A out ) , = 1 n ν ( I out + Pp P T A out ) − 1 n ν e ( I out + Pp P T A out ) , (5.3.22) the first term of which is e ≈ 2.718 time larger than the second. Note that since I out > Pp P T A out , this expression for the beam lifetime limits to the one calculated by the other method above are equal to leading order. Of importance here is the fact that the beam lifetime is not constant, it increases as the beam polarization increases; which makes physical sense as there will be fewer particles in the spin state that is scattered out more often. Chapter 6 Various scenarios of spin filtering "Let no one ignorant of Mathematics enter here." Plato, inscription over the entrance to the Academy. A major problem with spin filtering is that as the beam polarization increases the beam intensity decreases, since particles are being continuously scattered out of the beam. While one may obtain a polarized antiproton beam its intensity may be too low to be of use in any experiment. In this chapter we investigate the possibility of continuously inputting unpolarized particles into the beam to counteract this loss of beam intensity. We present a thorough investigation of spin filtering under various alternate scenarios of interest to any practical project to produce a high intensity polarized antiproton beam. These scenarios are: 1) spin filtering while the beam is being accumulated, i.e. unpolarized particles are continuously being fed into the beam at a constant rate, 2) unpolarized particles are continuously being fed into the beam at a linearly increasing rate, i.e. the particle input rate is ramped up, 3) the particle input rate is equal to the rate at which particles are being lost due to scattering beyond the ring acceptance angle, the beam intensity remaining constant, 4) increasing the initial polarization of a stored beam by spin filtering, and finally 5) the input of particles into the beam is stopped after a certain amount of time, but spin filtering continues. The five sections of this chapter each treat one of the scenarios of spin filtering listed above, in that order. Accumulation of antiprotons in the ring In the discussion so far we have only considered polarizing an antiproton beam when the beam is already accumulated in the storage ring. The PAX Collaboration plans to obtain their antiproton beam by collecting the produced antiprotons from high energy interactions of protons on targets of light nuclei, such as Beryllium. The antiprotons will be continuously fed into the storage ring at a fixed rate and accumulated, hence increasing the beam intensity, allowing for a greater luminosity in an experiment. The PAX Collaboration estimates the production rate of antiprotons as being 10 7 per second [23]. Since 10 11 antiprotons are required in the storage ring, antiprotons will be fed into the storage ring at a rate of 10 7 per second for 10 4 seconds [23]. We now consider a system where spin filtering occurs as the antiprotons are being fed into the ring. The original system of equations must be amended to account for this constant accumulation. The effect will be to add a term β to the d N(τ ) / d τ equation, where β is the constant rate at which antiprotons are fed into the ring; while the d J(τ ) / d τ equation remains unchanged. The initial conditions are N(0) = N 0 , which will be set to zero in section 6.1.1, and J(0) = 0. The new system of differential equations is d N(τ ) d τ = − n ν [ I out N(τ ) + P T A out J(τ ) ] + β , (6.1.1) d J(τ ) d τ = − n ν [ P T ( A all − K in ) N(τ ) + ( I all − D in ) J(τ ) ] . (6.1.2) By differentiating eq. (6.1.2) with respect to τ and substituting in eq. (6.1.1) one obtains an inhomogeneous second order linear differential equation with constant coefficients for J(τ ): d 2 J(τ ) d τ 2 − ( λ 1 + λ 2 ) d J(τ ) d τ + λ 1 λ 2 J(τ ) = − n ν P T ( A all − K in ) β , (6.1.3) the solution of which is J(τ ) = P T ( A all − K in ) 2 L d λ 1 λ 2 λ 2 ( λ 1 N 0 + β ) e λ 1 τ − λ 1 ( λ 2 N 0 + β ) e λ 2 τ + β ( λ 1 − λ 2 ) ] .N(τ ) = 1 2 L d λ 1 λ 2 λ 2 ( λ 1 N 0 + β ) ( L d − L in ) e λ 1 τ (6.1.5) + λ 1 ( λ 2 N 0 + β ) ( L in + L d ) e λ 2 τ + β ( I all − D in ) ( λ 2 − λ 1 ) . As a consistency check one can see that these solutions for J(τ ) and N(τ ) satisfy the initial conditions J(0) = 0 and N(0) = N 0 , and in the particular case when β = 0 the above expressions reduce to the solution of the homogeneous system presented in section 5.3. Dividing J(τ ) by N(τ ) we obtain an expression for the polarization P(τ ) as a function of time, P(τ ) = − ( A all − K in ) P T L in + L d    2 1 − λ 2 [ e λ 1 τ ( λ 1 N 0 +β ) − β ] λ 1 [ e λ 2 τ ( λ 2 N 0 +β ) − β ] − 1    . (6.1.6) When the particle input rate is zero (i.e. β = 0) the above equation simplifies to P(τ ) = − ( A all − K in ) P T L in + L d 2 1 − e ( λ 1 −λ 2 ) τ − 1 = − ( A all − K in ) P T L in + L d coth (L d n ν τ ) , (6.1.7) which is the solution of the homogeneous case presented in eq. (5.3.6). Using a Taylor Series expansion we find that the approximate initial rate of polarization buildup for each of these cases (N 0 = 0 with β = 0 and N 0 = 0 with β = 0) is the same as in the homogeneous case (N 0 = 0 with β = 0): d P dτ ≈ − n ν P T (A all − K in ) . (6.1.8) The maximum polarization achievable is the limit as time approaches infinity: P max = lim τ → ∞ P(τ ) = − P T ( A all − K in ) I all − D in , (6.1.9) which is independent of both N 0 and β, however note that in taking this limit we used the fact that β = 0. If β was equal to zero then the maximum polarization achievable would equal that from the homogeneous case; as can be easily seen from eq. (6.1.6) remembering that λ 1 ≤ λ 2 ≤ 0. Thus for the complete case there are just two values of the maximum polarization, one for β = 0 and one for all β = 0. The figure of merit for this inhomogeneous case is: FOM(τ ) = P 2 (τ ) N(τ ) = J 2 (τ ) N(τ ) = ( A all − K in ) 2 P 2 T 2 L d λ 1 λ 2 × c 1 e λ 1 τ − c 2 e λ 2 τ + β ( λ 1 − λ 2 ) 2 c 1 ( L d − L in ) e λ 1 τ + c 2 ( L in + L d ) e λ 2 τ + β ( I all − D in ) ( λ 2 − λ 1 ) , (6.1.10) where for convenience we have defined the two constants c 1 = λ 2 ( λ 1 N 0 + β ) and c 2 = λ 1 ( λ 2 N 0 + β ). Note the FOM will not have a maximum in finite time if the accumulation rate β is high enough to make the beam intensity a constant or increase with time. If this happens the FOM will increase monotonically. No initial beam Of interest is the particular case when N 0 = 0, i.e. there are no particles in the beam initially. In this case the above solutions simplify to J(τ ) = β P T ( A all − K in ) 2 L d λ 1 λ 2 λ 2 e λ 1 τ − 1 + λ 1 1 − e λ 2 τ , (6.1.11) N(τ ) = β 2 L d λ 1 λ 2 λ 2 ( L d − L in ) e λ 1 τ + λ 1 ( L in + L d ) e λ 2 τ (6.1.12) + ( I all − D in ) ( λ 2 − λ 1 ) ] , P(τ ) = − ( A all − K in ) P T L in + L d    2 1 − ( 1−e λ 1 τ ) λ 2 ( 1−e λ 2 τ ) λ 1 − 1    , (6.1.13) FOM(τ ) = ( A all − K in ) 2 P 2 T β 2 L d λ 1 λ 2 × (6.1.14) λ 2 e λ 1 τ − 1 + λ 1 1 − e λ 2 τ 2 λ 2 ( L d − L in ) e λ 1 τ + λ 1 ( L in + L d ) e λ 2 τ + ( I all − D in ) ( λ 2 − λ 1 ) . Interestingly the β dependence of P(τ ) vanishes in this case, i.e. the polarization buildup rate is independent of the rate at which antiprotons are fed into the ring, if there are no particles in the beam initially. But we have used the fact that β = 0 to obtain the above result. We should note the obvious physical fact that if N 0 = 0 and β = 0 i.e. there are no particles in the beam initially and no particles are fed into the beam, then there will never be any particles in the beam; so measuring the beam polarization is meaningless. Notice that in this case where N 0 = 0, the figure of merit is proportional to the particle input rate β. Summary The results obtained thus far for the polarization buildup in various scenarios can be summarized as P(τ ) =                                                              0 for β = 0 & N 0 = 0 − ( A all − K in ) P T L in + L d coth (L d n ν τ ) for β = 0 & N 0 = 0 − ( A all − K in ) P T L in + L d    2 1 − ( 1−e λ 1 τ ) λ 2 ( 1−e λ 2 τ ) λ 1 − 1    for β = 0 & N 0 = 0 − ( A all − K in ) P T L in + L d    2 1 − λ 2 [ e λ 1 τ ( λ 1 N 0 +β ) − β ] λ 1 [ e λ 2 τ ( λ 2 N 0 +β ) − β ] − 1    for β = 0 & N 0 = 0 where, as usual, β is the constant rate at which particles are fed into the beam and N 0 is the number of particles in the beam initially. Constant beam intensity In this case the accumulation rate is set specifically so that extra particles are fed into the beam at such a rate so that the beam intensity is kept constant, i.e. fed in at such a rate to balance the rate at which particles are scattered out of the beam. The system of equations is much simpler in this case. Here N(τ ) = N 0 is a constant, hence d N(τ )/d τ = 0, and the J(τ ) equation becomes a first order linear ODE with constant coefficients d J(τ ) d τ + n ν ( I all − D in ) J(τ ) = − n ν ( A all − K in ) P T N 0 , (6.2.1) and imposing the initial conditions N(0) = N 0 and J(0) = 0 one obtains the solution J(τ ) = − ( A all − K in ) P T N 0 ( I all − D in ) 1 − e − n ν ( I all − D in ) τ . (6.2.2) Now the polarization as a function of time can be presented P(τ ) = J(τ ) N(τ ) = J(τ ) N 0 = − ( A all − K in ) P T ( I all − D in ) 1 − e − n ν ( I all − D in ) τ . (6.2.3) To find the maximum polarization achievable, i.e. the limit as time tends to infinity, we note that I all > D in thus − n ν ( I all − D in ) < 0 and hence one obtains P max = lim τ → ∞ P(τ ) = − ( A all − K in ) P T I all − D in , (6.2.4) which is the same as in the inhomogeneous case of section 6.1 when β = 0. The initial rate of polarization buildup can be obtained by expanding P(τ ) as a Taylor expansion in n ν τ . Assuming n ν τ is small we neglect terms of second or higher order giving d P d τ ≈ − n ν ( A all − K in ) P T , (6.2.5) as it was in the homogeneous case presented in eq. (5.3.7). The figure of merit in this case is easily obtained is constant in this case, and beam polarization P as time (scaled by the beam lifetime τ * ) increases. The figure of merit is also shown on the graph, with the same scales as the other functions. The graph just shows general trends and is not in exact numerical correspondence to the equations, which will be presented in Chapter 7. The beam lifetime is defined as earlier, in the absence of particle input to the beam. FOM(τ ) = ( A all − K in ) 2 P 2 T N 0 ( I all − D in ) 2 1 − e − n ν ( I all − D in ) τ 2 ,(6. Approximating the critical input rate The accumulation rate needed to keep the beam intensity constant is important, as this critical rate divides the solution of the system into two physically distinct cases. Smaller accumulation rates than this critical value cause the beam intensity to decrease, hence the FOM will have a maximum in finite time. Larger values than the critical value cause the beam intensity to increase continuously, hence the FOM will increase monotonically. We can see from eq. (5.3.4) that N(τ ) does not decrease linearly with time τ . So the accumulation rate needed to keep the beam intensity constant, say f (τ ), will not be linear in τ . We now derive the function f (τ ) and obtain a linear approximation to it, which can be used in the inhomogeneous case treated in section 6.1. We must solve N(τ ) = N hom (τ ) + f (τ ) = N 0 , i.e. N(τ ) = e λ 1 τ ( L d − L in ) + e λ 2 τ (L d + L in ) N 0 2 L d + f (τ ) = N 0 , (6.2.7) which leads to f (τ ) = N 0 2 L d 2 L d − e λ 1 τ ( L d − L in ) − e λ 2 τ (L d + L in ) . (6.2.8) A linear approximation f L (τ ) to f (τ ) can be found by Taylor expanding the exponentials to order τ (which is valid since n ν τ is small), to obtain f L (τ ) = n ν N 0 I out τ , (6.2.9) which is in the linear form + β c τ where β c = n ν N 0 I out is the critical value of β which when added to the d N(τ ) / d τ differential equation in section 6.1 approximately makes the beam intensity constant. The input rate is ramped up In this section we investigate a scenario where unpolarized particles are input into the beam at a linearly increasing rate, i.e. the input rate is ramped up. This is accounted for by the following system of polarization evolution equations d N(τ ) d τ = − n ν [ I out N(τ ) + P T A out J(τ ) ] + β τ , (6.3.1) d J(τ ) d τ = − n ν [ P T ( A all − K in ) N(τ ) + ( I all − D in ) J(τ ) ] , (6.3.2) where β τ is the rate at which particles are fed in, the input ramped up at a rate proportional to the time elapsed. The initial conditions are N(0) = N 0 which we may later set to zero, and J(0) = 0. By differentiating eq. (6.3.2) with respect to τ and substituting in eq. (6.3.1) one obtains a second order linear inhomogeneous differential equation for J(τ ): d 2 J(τ ) d τ 2 − ( λ 1 + λ 2 ) d J(τ ) d τ + λ 1 λ 2 J(τ ) = − n ν P T ( A all − K in ) β τ (6.3.3) the solution of which is J(τ ) = F λ 2 λ 1 e λ 1 τ + F λ 1 λ 2 e λ 2 τ + β ( A 1 τ + A 2 ) . (6.3.4) Where for convenience we have defined the constants A 1 ≡ − n ν P T ( A all − K in ) λ 1 λ 2 , (6.3.5) A 2 ≡ 2 n 2 ν 2 P T ( A all − K in ) ( L in + I out ) λ 2 1 λ 2 2 , (6.3.6) F λ 2 λ 1 ≡ n ν ( A all − K in ) N 0 P T + β ( A 1 − λ 2 A 2 ) λ 2 − λ 1 , (6.3.7) obtained by imposing the initial conditions J(0) = 0 and N(0) = N 0 thus d J(0)/d τ = − n ν P T ( A all − K in ) N 0 . The function F λ 1 λ 2 is F λ 2 λ 1 with λ 1 and λ 2 interchanged. Differentiating eq. (6.3.4) with respect to τ and substituting into eq. (6.3.2) gives an expression for N(τ ): N(τ ) = −1 ( A all − K in ) P T F λ 2 λ 1 e λ 1 τ ( L in − L d ) + F λ 1 λ 2 e λ 2 τ ( L in + L d ) + β A 1 n ν + ( I out + 2 L in ) ( A 1 τ + A 2 ) . (6.3.8) As a consistency check it can be seen that the inhomogeneous solutions for J(τ ) and N(τ ) satisfy the initial conditions, and that when β = 0 they reduce to the solutions of the homogeneous system eq. (5.2.24) presented in section 5.3. Dividing J(τ ) by N(τ ) we obtain an expression for the polarization as a function of time (τ ), P(τ ) = − P T ( A all − K in ) L in + L d   2 1 − e λ 1 τ F λ 2 λ 1 ( λ 2 −λ 1 ) − β [ A 1 ( 1−λ 2 τ )−λ 2 A 2 ] e λ 2 τ F λ 1 λ 2 ( λ 1 −λ 2 ) − β [ A 1 ( 1−λ 1 τ )−λ 1 A 2 ] − 1   . (6.3.9) When β = 0 the above equation simplifies to P(τ ) = − P T ( A all − K in ) L in + L d coth (L d n ν τ ) , (6.3.10) which is the solution of the homogeneous case eq. (5.2.24) presented in section 5.3. Of interest is the case when N(0) = N 0 = 0, i.e. there are no particles in the beam initially. To obtain this result we set N 0 = 0 in the above equation to obtain P(τ ) = − P T ( A all − K in ) L in + L d    2 1 − ( e λ 1 τ −1 ) λ 2 A 2 −A 1 ( e λ 1 τ +λ 2 τ −1 ) ( e λ 2 τ −1 ) λ 1 A 2 −A 1 ( e λ 2 τ +λ 1 τ −1 ) − 1    , (6.3.11) where for β = 0 the β dependence vanishes. Using a Taylor Series expansion we obtain the approximate initial rate of polarization buildup 3.12) identical to that of the homogeneous case presented in section 5.3. The maximum polarization achievable is the limit as time approaches infinity: d P dτ ≈ − n ν P T (A all − K in ) ,(6.P max = lim τ →∞ P(τ ) = − P T ( A all − K in ) I all − D in = − P T ( A all − K in ) I out + 2 L in . (6.3.13) The above expression is only valid for β = 0, the β = 0 expression is presented in section 5.3. For this inhomogeneous case the figure of merit is: FOM(τ ) = P 2 (τ ) N(τ ) = J 2 (τ ) N(τ ) = (6.3.14) − P T ( A all − K in ) F λ 2 λ 1 e λ 1 τ + F λ 1 λ 2 e λ 2 τ + β ( A 1 τ + A 2 ) 2 F λ 2 λ 1 e λ 1 τ (L in + L d ) − F λ 1 λ 2 e λ 2 τ (L in + L d ) + β A 1 n ν + (I all − D in ) (A 1 τ + A 2 ) . If the particle accumulation rate β τ is high enough to make the beam intensity constant or increase with time the figure of merit will be a monotonically increasing function of time, i.e. it will not have a maximum in finite time. Stored beam with initial polarization We now solve the homogeneous system where the initial polarization is not zero. This will be used if two methods of polarizing antiprotons are combined, i.e. if antiprotons were produced with a small polarization by some other method and one wanted to increase that polarization by spin filtering in a storage ring, where the luminosity could also be increased. In this section the beam has been stored and there is no further input of particles into the beam. The system of differential equations, the eigenvalues and eigenvectors are the same as section 5.3, but one of the initial conditions is different. The new initial conditions are N(0) = N 0 > 0 the total number of particles in the beam initially, and J(0) = J 0 = 0 ⇒ N + (0) = N − (0) i.e. initially the beam is polarized. Note that since the number of particles in the beam in one particular spin state must not be greater than the total number of particles in the beam the bound |J 0 | ≤ N 0 is respected. A negative value for J 0 simply implies that the antiproton beam is initially polarized in the opposite direction to the polarization direction of the target. Enforcing these initial conditions leads to the solutions N(τ ) = ( J 0 P T A out − N 0 L in ) e λ 1 τ − e λ 2 τ + N 0 L d e λ 1 τ + e λ 2 τ 2 L d , (6.4.1) J(τ ) = [ N 0 P T ( A all − K in ) + J 0 L in ] e λ 1 τ − e λ 2 τ + J 0 L d e λ 1 τ + e λ 2 τ 2 L d ,(6.P(τ ) = L d J 0 − tanh (L d n ν τ ) [ L in J 0 + N 0 P T ( A all − K in ) ] L d N 0 + tanh (L d n ν τ ) [ L in N 0 − J 0 P T A out ] . (6.4.3) Denoting the initial polarization P(0) = J 0 / N 0 ≡ P 0 the above can be written as P(τ ) = L d P 0 − tanh (L d n ν τ ) [ L in P 0 + P T ( A all − K in ) ] L d + tanh (L d n ν τ ) [ L in − P 0 P T A out ] . (6.4.4) The approximate rate of change of polarization for sufficiently short times is found by Taylor expanding to first order in τ d P dτ ≈ n ν A out P 2 0 − ( A all − K in ) P T − 2 P 0 L in . (6.4.5) The limit as time goes to infinity of P(τ ) in eq. (6.4.4) is lim τ → ∞ P(τ ) = P 0 ( L d − L in ) − P T ( A all − K in ) ( L in + L d ) − A out P 0 P T , (6.4.6) which of course agrees with the earlier maximum polarization if J 0 = 0 (i.e. P 0 = 0). The figure of merit for this case is: FOM(τ ) = P 2 (τ ) N(τ ) = J 2 (τ ) N(τ ) = (6.4.7) N 0 [ P T ( A all − K in ) + P 0 L in ] e λ 1 τ − e λ 2 τ + P 0 L d e λ 1 τ + e λ 2 τ 2 Note that in terms of the number of particles in each spin state the initial conditions for an initially polarized beam, N(0) = N 0 and J(0) = J 0 , are N + (0) = N 0 2 ( 1 + P 0 ) and N − (0) = N 0 2 ( 1 − P 0 ) . (6.4.8) An unpolarized target A special case of this system deserves comment. Given that the beam is initially polarized what happens if the target is unpolarized? One would imagine that the beam polarization should decrease and eventually reach zero. We now analyze the equations of section 6.4 when the target is unpolarized (i.e. P T = 0) and use the fact that L d = P 2 T A out (A all − K in ) + L 2 in = L in when P T = 0 to obtain the beam polarization as a function of time P(τ ) = P 0 e ( λ 1 − λ 2 ) τ , (6.4.9) which is an exponentially decreasing function of τ for λ 1 − λ 2 < 0. The beam polarization will not decrease in the special case of λ 1 − λ 2 = 0, but this only happens when L in = 0, i.e. when there is no depolarization. The special case of λ 1 − λ 2 = 0, which does not lead to polarization buildup as seen from eq. (6.4.2), would be avoided by any experimental effort, thus is omitted from the rest of the discussion. The limit of beam polarization for large times when P T = 0 is lim τ → ∞ P(τ ) = 0 . (6.4.10) Thus, as expected, if the beam is initially polarized and the target unpolarized then the beam polarization will decrease with time and eventually the beam polarization will reduce to zero. Thus a beam cannot gain polarization from an unpolarized target by spin filtering. The figure of merit in this case simplifies to FOM(τ ) = N 0 P 2 0 e ( 2 λ 1 −λ 2 ) τ which is a monotonically decreasing function of τ . One can derive a polarization half-life in this case, the time taken for the polarization to decrease by a factor of 2, by solving P(τ ) = P 0 / 2 to obtain τ 1 2 = ln 2 λ 2 − λ 1 = ln 2 2 n ν L d . (6.4.11) This scenario occurs in an electron cooler, a device used to focus the beam in many storage rings 1 . The beam passes through a co-moving beam of unpolarized electrons with low transverse momentum, in order to dampen the transverse momentum of the antiprotons in the stored beam. But the low electron areal densities in cooler beams, where typically n ≈ 10 −18 fm −2 = 10 −19 mb −1 , causes the polarization half-life to be very large. Thus our work shows that this depolarization effect is negligible, in agreement with Ref. [106]. Since electron cooling is a necessary part of the spin filtering process of polarization buildup it is very important that the depolarization caused by electron cooling is negligible. It has recently been suggested that the positron-antiproton polarization transfer observable is very much enhanced at low energies [88,107]. This enhancement is the basis of the recent proposal by Th. Walcher et al. to polarize an antiproton beam by repeated interaction with a co-moving polarized positron beam in a storage ring [75]. All of the antiprotons remain within the beam in this scenario and one avoids the problem of the antiprotons annihilating with protons in an atomic gas target. This large enhancement of the polarization transfer observable at low energies is due to the unlike charges of the positron and antiproton, and does not occur for the like charges case of antiproton-electron scattering. Hence this does not affect the conclusion that depolarization of an antiproton beam in an electron cooler is negligible. A critical value for the target polarization The beam polarization will also decrease for low values of the target polarization. In fact there is a critical value of the target polarization P T which keeps the beam polarization constant. If the target polarization is above this critical value the polarization of the beam will increase, and if the target polarization is below this critical value the beam polarization will decrease. The critical value is obtained by solving P(τ ) = L d P 0 − tanh (L d n ν τ ) [ L in P 0 + P T ( A all − K in ) ] L d + tanh (L d n ν τ ) [ L in − P 0 P T A out ] = P 0 , (6.4.12) for P T , where the time dependence will cancel leading to P critical T = 2 P 0 L in P 2 0 A out − ( A all − K in ) . (6.4.13) For target polarizations below this critical value the maximum beam polarization occurs at time τ = 0, and for target polarizations above this critical value the maximum beam polarization occurs at large times τ → ∞. Particles fed in for a limited time The Heaviside step function could be used in the system of equations to explain the case of particles input into a beam for a certain amount of time after which the input is turned off and no more particles are fed into the beam, but spin filtering continues 2 . This scenario is under consideration by the PAX Collaboration [23], but there has been no theoretical treatment of it to date. The Heaviside function is a piecewise continuous function which is zero in one region and one everywhere else, it is used in many mathematical modeling problems to describe an external effect turned on or off after a certain duration of time. In our case this external effect is the input of particles into the beam. The Heaviside function is defined as H ( τ − τ c ) =    0 if τ < τ c 1 if τ ≥ τ c (6.5.1) and is used to describe an external effect turned on at time τ c , but we require an external effect on initially and turned off at time τ c , thus we need H (τ ) − H ( τ − τ c ) =    1 if 0 ≤ τ < τ c 0 if τ ≥ τ c (6.5.2) Note that in the special case when τ c = 0, [ H (τ ) − H (τ ) ] = 0, as this is in the second region. This describes a physical situation where particles are being fed in for zero seconds, which is the same as saying no particles are fed in, so the second order ODE should be homogeneous in this case, which it is. The extra term to add to the d N / d τ equation to account for particles being fed in at a constant rate β per second for τ c seconds after which the input is switched off is β [ H (τ ) − H ( τ − τ c ) ]. The equations are now broken into two pieces, i.e. discontinuous, but piecewise continuous. The solutions will be broken into two regions 0 ≤ τ < τ c and τ ≥ τ c , where the solutions in the region 0 ≤ τ < τ c should equal those in the inhomogeneous case presented in section 6.1. The initial conditions will be N(0) = 0 and J(0) = 0, thus J ′ (0) = 0. The Heaviside function above is included in the second order ODE for J(τ ) to obtain: d 2 J(τ ) d τ 2 − ( λ 1 + λ 2 ) d J(τ ) d τ + λ 1 λ 2 J(τ ) (6.5.3) = − n ν P T ( A all − K in ) β [ H (τ ) − H ( τ − τ c ) ] , which can be solved by Laplace Transform methods, as described in Appendix F, to obtain J(τ ) C 1 =        λ 1 1 − e λ 2 τ + λ 2 e λ 1 τ − 1 if 0 ≤ τ < τ c λ 1 e λ 2 ( τ − τc ) − e λ 2 τ + λ 2 e λ 1 τ − e λ 1 ( τ − τc ) if τ ≥ τ c (6.5.4) where for convenience we have defined the constant factor C 1 ≡ β P T ( A all − K in ) 2 L d λ 1 λ 2 . (6.5.5) One sees from eq. (6.5.4) that J(τ ) = 0 for all τ when τ c = 0, which is physically reasonable as there are never any particles in the beam if τ c = 0. Also the P T factor in C 1 indicates that J(τ ) will always be zero if P T = 0 (i.e. if the target is unpolarized). It can also be seen that the complete solution in the region τ ≥ τ c is the combination of the solution in the region 0 ≤ τ < τ c and an additional part dependent on τ c ; which is C 1 λ 1 e λ 2 ( τ − τc ) − 1 + λ 2 1 − e λ 1 ( τ − τc ) , and immediately one sees that when τ = τ c this additional part vanishes. So when τ = τ c , i.e. at the boundary between the two regions, the two solutions match. Therefore the solution for J(τ ) is continuous as expected. The expression for J(τ ) in the first region 0 ≤ τ < τ c of eq. (6.5.4) is equal to the solution of the inhomogeneous system presented in eq. (6.1.11). A similar analysis as that done for J(τ ) reveals the second order ODE for N(τ ) d 2 N(τ ) d τ 2 − ( λ 1 + λ 2 ) d N(τ ) d τ + λ 1 λ 2 N(τ ) = n ν ( I all − D in ) β [ H (τ ) − H ( τ − τ c ) ] .N(τ ) C 2 =                                                        λ 1 1 − L in + L d I all − D in e λ 2 τ + λ 2 L in − L d I all − D in e λ 1 τ − 1 if 0 ≤ τ < τ c λ 1 e λ 2 ( τ − τc ) − L in + L d I all − D in e λ 2 τ + λ 2 L in − L d I all − D in e λ 1 τ − e λ 1 ( τ − τc ) if τ ≥ τ c > 0 0 if τ c = 0 (6.5.7) where again for convenience we have defined a constant factor C 2 ≡ − β ( I all − D in ) 2 L d λ 1 λ 2 . (6.5.8) Again one sees that the complete solution in the region τ ≥ τ c > 0 is the combination of the solution in the region 0 ≤ τ < τ c plus an additional part dependent on τ c ; which is C 2 λ 1 e λ 2 ( τ − τc ) − 1 + λ 2 1 − e λ 1 ( τ − τc ) . Immediately we see that when τ = τ c this additional part vanishes, thus the solution for N(τ ) is continuous. The solution in the first region 0 ≤ τ < τ c is equal to the solution from our inhomogeneous case presented in eq. (6.1.12), and it satisfies the initial condition N(0) = 0. We now present results for the polarization P(τ ) = J(τ ) / N(τ ) as a function of time, in both regions. The polarization is undefined when there are no particles in the beam, thus we need not treat the case τ c = 0. As expected in the 0 ≤ τ < τ c region P(τ ) equals the solution of our inhomogeneous case presented in eq. (6.1.13), and in the region τ ≥ τ c > 0 one finds P(τ ) = (6.5.9) P T ( A all − K in ) λ 1 e λ 2 τ e − λ 2 τc − 1 + λ 2 e λ 1 τ 1 − e − λ 1 τc λ 1 e λ 2 τ [ ( I all − D in ) e −λ 2 τc − ( L in + L d ) ] + λ 2 e λ 1 τ [ ( L in − L d ) − ( I all − D in ) e −λ 1 τc ] . The approximate initial rate of polarization buildup and the maximum polarization achievable will both reside in the 0 ≤ τ < τ c region, and thus will be identical to those presented in section 6.1. This is because the maximum polarization achievable occurs when the input rate is never switched off, i.e. in the 0 ≤ τ < τ c region. Chapter 7 Numerical results "The whole point of physics is to work out a number, with decimal points etc.! Otherwise you haven't done anything." Richard Feynman As an application of the theoretical work presented throughout the thesis we now investigate a possible method to produce a high intensity polarized antiproton beam by spin filtering off an opposing polarized electron beam. It is also outlined how this work can be applied to polarizing antiprotons by spin filtering off a polarized hydrogen target. Firstly a description of the electron cooling technique to refocus the beam after scattering off the target each revolution in order to maintain high beam density is presented in section 7.1. Then the various experimental input parameters, such as revolution frequency, target areal density, target polarization and the effective acceptance angle, needed to obtain realistic numerical estimates from our mathematical formalism are each described in section 7.2. The benefits of using a lepton target are described in section 7.3, before analyzing the case of spin filtering off an opposing polarized electron beam. Finally spin filtering off a polarized hydrogen target is discussed in section 7.4, in the three cases of hydrogen with only electrons polarized, hydrogen with only protons polarized and finally hydrogen with both electrons and protons polarized. In section 7.4.1 it is shown that electromagnetic effects dominate hadronic effects inp p scattering in the region of low momentum transfer of interest in spin filtering. Beam cooling After interaction with the internal target in the storage ring many of the beam particles do not move exactly along the beam axis, i.e. they have acquired a small deflection angle. This causes the beam to spread transversely and this process is called beam "heating", in analogy to the random motion of atoms in a hot thermodynamic gas. In order to maintain a well ordered beam, where all particles move as collinearly as possible and also to increase the beam transverse density, a method to counteract this beam spread is required. Fortunately such a method exists and has been utilized successfully in many experiments world wide over the past three decades. The method is called Electron Cooling and was invented in 1966 by G. I. Budker at the Institute for Nuclear Physics (INP) laboratory (later renamed the Budker Institute of Nuclear Physics (BINP) in his honour) in Novosibirsk [122], as a way to increase the luminosity of p p andp p colliders. Electron cooling was first tested in 1974 with 68 MeV protons at the NAP-M storage ring of INP Novosibirsk [123], and is currently operational at over a dozen storage rings world wide. The terminology of "cooling" is analogous to thermodynamic cooling, as the random transverse motion of the (anti)protons is dampened by cooling the beam. A "hot" beam has many particles with large transverse motion, whereas a "cool" (or "cold") beam has low transverse motion, i.e. all particles move collinear to the beam axis. An electron cooler is a device inserted into the storage ring, where the antiproton beam passes through a co-moving cold electron beam, and on multiple Coulomb scattering with the electrons the transverse motion of the antiprotons is reduced, i.e. the antiproton beam phase-space density is increased. One immediately thinks of an analogy with the temperatures of mixed gases: Gas A with a high temperature is mixed with gas B having a low temperature, after some time the combined mixture tends to a uniform temperature which is midway between the initial temperatures of the individual gases. If the gases could be separated afterwards, one could say that gas B has reduced the temperature of gas A. Fortunately the electrons can be injected into, and extracted out of, an antiproton beam easily by magnets which deflect charged particles at different angles depending on the mass of the charged particles. Given that the mass of the antiprotons is approximately 1800 times that of the electrons, the electrons can easily be completely removed from the mixture. The velocity of the electrons in a cooler is carefully set to equal the average velocity of the antiprotons, to maximize the interaction time. The antiprotons undergo Coulomb scattering in the electron "gas" and lose transverse energy, which is transfered to the co-moving electrons until some thermal equilibrium is attained. The electrons get "heated up" but are discarded after each pass and new cold electrons are injected continuously. Electron cooling is conventionally used on low to medium energy (anti)protons. Many laboratories are now investigating high energy electron cooling [124] and it has been shown to work on 8.9 GeV antiprotons in Fermilab [125]. GSI Darmstadt are investigating electron cooling the antiproton beam in the HESR at up to about 8 GeV [28]. It is expected that in the near future high energy electron cooling will be commonplace in many laboratories. In section 6.4 of this thesis we have proven that the depolarization of an antiproton beam due to electron cooling is negligible because of the low areal density of electrons in a cooler beam. Therefore the beam can be refocused after interaction with the target each revolution, without losing significant beam polarization. Input parameters As shown in the system of polarization evolution equations of Chapter 5, to give the highest possible antiproton polarization after a given filter time, the maximum antiproton revolution frequency, maximum target areal density and maximum target polarization are required. We now investigate each of these in turn before computing the numerical quantities. Obviously one can maximize the revolution frequency by using a very high energy beam in an extremely small circumference storage ring, but this is limited by the power of the magnets to bend high energy antiprotons around such a small circum-ference. A plausible example is treated in [75] with GeV/c will be stored in a 574 metre circumference ring, giving a velocity of β = 0.998 and a revolution frequency of 521628 Hz [23]. Maximum revolution frequency Having more than one target in the ring, or having the electron beam overlap with the antiproton beam at more than one point in the ring, has exactly the same effect as increasing the revolution frequency, i.e. using two targets has the same effect as doubling the revolution frequency, using three targets has the same effect as tripling the revolution frequency etc. This is obvious since we are using the revolution frequency as a measure of the number of times the beam passes through the target. Having R targets in the ring will increase the rate of polarization buildup by a factor of R. The ring is limited by space, and such targets would have to be purposefully built, so realistically R = 1 is most likely for PAX , anything above R = 5 would be very challenging. The effect of multiple targets in the storage ring could be included in our system of polarization evolution equations simply by multiplying the entire coefficient matrix by a parameter R, where R is the number of targets in the ring. This would carry directly into each of the solutions presented in this thesis under the substitution n ν → n ν R. Note one must set R = 1 to compare to other work in the field which assume only one target in the ring. One could choose ν = 5 MHz as a best case scenario available in the near future. If necessary two interaction regions of the opposing electron beam with the antiproton beam could be used to achieve this effective revolution frequency. Maximum target areal density As shown in Chapter 5 the rate of polarization buildup is highly dependent on the target areal density. One requires as high as possible a target areal density to achieve the highest rate of polarization buildup. In this section we review the maximum areal densities of different types of target currently available. The FILTEX polarized hydrogen target, developed in the early 1990's, had an areal density of 6 × 10 13 atoms per cm 2 [54]. Since then some advances have been made in polarized atomic gas targets, in particular by the HERMES Collaboration. The HERMES Collaboration has produced and used polarized hydrogen and deuterium targets with densities of up to 10 14 atoms per cm 2 [51]. It is expected that this maximum areal density of polarized atomic gas targets could be increased by a factor of 100 in the near future. For a recent review of polarized gas targets see Ref. [129]. The PANDA Collaboration, also at the Facility for Antiproton and Ion Research (FAIR) at GSI Darmstadt, aim to produce a hydrogen pellet target of areal density up to 10 16 atoms per cm 2 [126], similar to the target in operation at the WASA detector in COSY Jülich. The maximum areal densities of polarized electron beams is many orders of magnitude lower than that of atomic targets, because of the electromagnetic repulsion felt by the like-charge electrons in the beam. This effect is absent for atoms which are electrically neutral. Typical areal densities of polarized electron beams produced thus far is 10 8 electrons per cm 2 . Given the enormous research and development effort that is currently been afforded to the International Linear Collider (ILC) project, which will use electron and positron beams, the maximum electron beam areal densities can be expected to be increased in the near future [130,131]. Perhaps polarized electron beams with areal densities of up to 10 9 electrons per cm 2 , or even 10 10 electrons per cm 2 may be available in the next decade. It has been claimed that electrons stored in a Penning trap may soon reach areal densities of 10 12 electrons per cm 2 [66]. For a realistic maximum value of the polarized electron beam areal density available in the near future we use 10 12 electrons per cm 2 in our numerical calculations. Maximum target polarization Beams of electrons and positrons with polarizations of up to P T = 0.9 have been produced and utilized in many laboratories [127,128]. Polarized internal targets of atomic hydrogen and deuterium with polarizations of nuclei, electrons or both of up to P T = 0.9 have also been constructed, in particular the HERMES polarized hydrogen and deuterium targets which have been operated both longitudinally and transversely polarized [51,129]. There is an uncertainty of about 3% to 5% in the measurement of these polarizations, thus polarizations of above 0.95 are impractical to produce. As a target polarization that should be state of the art in the coming decade before PAX will be realized we pick a target polarization of P T = 0.9 in the following numerical calculations. Effective acceptance angle The acceptance angle θ acc introduced earlier, is an idealistic simplification. It assumes all beam particles are moving exactly along the infinitely narrow beam axis, hence neglecting the following two effects: (a) In reality the beam has a finite extent, i.e. a r.m.s. radius of about 8 mm at the target [23]. So some particles are moving collinearly with the beam but at a distance of a few millimetres from the beam axis. (b) Not all particles move collinearly, many particles have a slight angle of motion with respect to the beam axis due to scattering off the target. Some of these particles that are scattered at or even slightly less than the idealistic acceptance angle will be lost. Hence there is an effective ring acceptance angle θ effective acc which is always less than the idealistic ring acceptance angle θ naive acc , i.e. θ effective acc < θ naive acc . The effect of this is to lessen the region of integration for the "in" spin observables thus reducing them, and increase the region of integration for the "out" spin observables thus increasing them. This effect also reduces the beam lifetime. These effects have been investigated by the PAX Collaboration for the polarized proton beam scattering off an internal hydrogen target at COSY Jülich by comparing the theoretically calculated idealistic loss cross-section to the experimentally measured beam lifetime [132]; the results were as follows. They calculate the ratio of the effective spin-averaged loss cross-section σ out effective to the naive spin-averaged loss cross-section σ out naive , the former of which is proportional to (θ effective acc ) −2 and the latter to (θ naive acc ) − In the numerical calculations that follow we shall take these results and use an effective acceptance angle that is 0.94 times the idealistic acceptance angle of the storage ring. Minimum scattering angle From quantum mechanics one has the relation, |q| b = , between the modulus of the three-momentum transfer (|q|) in an interaction and the impact parameter (b). Since is set to one throughout this thesis one can relate the modulus of the minimum three-momentum transfer of an interaction to the maximum impact parameter by: |q min | = 1 b max , (7.2.5) and using − t = |q| 2 one has that the minimum squared momentum transfer is − t min = 1 b 2 max . (7.2.6) This minimum value for | t | ensures that there is no singularity from the 1/t dependence of many of the spin observables presented in Chapter 4. There are two cases of interest to us: where α is the fine structure constant and m the mass of the electron, and the relation a B = (α m) −1 has been used. One can convert this to the minimum laboratory frame scattering angle θ min using eq. (7.4.10) to obtain [67] θ min = 1 p lab a B , (7.2.8) where p lab is the laboratory frame antiproton momentum. Using the relations p = T ( T + 2 M ) and a B = (α m) −1 , where T is the Laboratory frame kinetic energy of the antiprotons, one can rewrite this as [72] θ min = α m T (T + 2 M ) , (7.2.9) where M is the mass of the (anti)proton. For FILTEX kinetic energies of T = 23 MeV one finds that θ min ≈ 0.02 mrad, far below both the acceptance angle of any storage ring and the maximum angle antiprotons are scattered off stationary electrons, hence verifying eq. (7.4.13). For scattering angle less than this, corresponding to impact parameters greater than the Bohr radius, the Coulomb fields of the atomic electron and proton of the hydrogen atom screen each other, hence antiprotons do not interact with the hydrogen atom. 1) A polarized hydrogen target 2) A polarized electron beam For an electron beam of areal density n particles per femtometre squared the average distance between electrons is 1/ √ n fm. If an antiproton passes exactly equidistant from two electrons in the beam it will feel no force as the Coulomb fields of the two electrons will cancel each other. The maximum impact parameter is one-half of the average electron separation, i.e. b max = 1/(2 √ n ). Therefore − t min = 1 b 2 max = 4 n , (7.2.10) where one must convert the areal density n into units of (MeV/c) 2 using the conversion factor 1 fm −2 = 38937.9323 (MeV/c) 2 [134]. By Taylor expanding the left hand side of eq. (7.3.2) with respect to the ring frame 1 scattering angle θ r one obtains the approximate relation, valid for small θ r and small | t | θ r ≈ √ − t p r p , (7.2.11) where p r p is the antiproton momentum in the ring frame. Therefore the minimum ring frame scattering angle in this case is θ r min ≈ √ − t min p r p = √ 4 n p 1 . (7.2.12) Results for the minimum squared momentum transfer and minimum ring frame scattering angle for various values of the electron beam areal density are presented in There are many advantages of using a polarized lepton target (or beam) over a polarized internal atomic target (hydrogen or deuterium) for spin filtering: 1. There is no loss of beam intensity due to annihilation of the antiprotons with protons as there is in the nuclear targets. 2. The polarization observables for antiproton -electron scattering are calculable in perturbative QED (as presented in this thesis), whereas for an atomic target currently less known hadronic polarization observables contribute. 3. Residual gas does not build up over time in the storage ring as would happen if an atomic target was used. The first of these is by far the most important, and has caused many groups to investigate methods to polarize antiprotons by spin filtering off lepton beams and targets. We now investigate various scenarios of spin filtering off pure lepton beams and targets. Antiprotons scattering off stationary electrons The maximum laboratory frame scattering angle for antiprotons scattering off stationary electrons is m / M = 0.54 mrad as shown in eq. (7.3.5) and Figure. 7-1 (a). This is below the acceptance angle of any storage ring [23,66] so that all scattering off atomic electrons will be within the ring. Therefore stationary electrons can only contribute to the polarization buildup of the antiproton beam by selective spin-flip. The Budker and Jülich groups claim that spin-flip effects while scattering within the ring are small forp e − →p e − scattering; hence stationary electrons, and very low energy electrons such as in an atomic target, are not effective in transferring polarization to an antiproton beam [71,72]. We propose the use of an opposing polarized electron beam of sufficient energy to increase the scattering angles of the antiprotons beyond acceptance as seen in Figure 7-1 (b)-(f). This is the subject of the remainder of the chapter. Note another solution to this problem would be to use a polarized muon target. Stationary muons, having much more mass than electrons (m µ ≈ 200 m e ), would provide a maximum laboratory frame antiproton scattering angle of θ µ max ≈ m µ / M = 113 mrad. This is far beyond the ring acceptance angles under consideration at PAX , hence allows selective scattering out of the beam to contribute to polarization buildup. Another positive aspect of muons is that they can be produced automatically highly polarized, through the decays of charged pions. Charged pions decay into muons and (anti)neutrinos: π + → µ + ν µ and π − → µ −ν µ . Since neutrino's have only one possible polarization state, angular momentum conservation forces the produced muons to be polarized. Polarized muon beams have been used in many experiments, among them the seminal EMC and SMC experiments which ushered in a new era of interest in spin physics and in particular in the spin structure of nucleons. But it is feared that the density of such polarized muon beams will be too low with today's technologies to provide a reliable method of polarizing antiprotons [135]. At higher energies the plots of the relationship between θ r and t become skewed towards higher | t | as shown in An opposing electron beam A crucial variable for spin filtering is the acceptance angle, as it defines which particles are scattered out of the beam and which are scattered at small angles remaining in the beam. This angle is with respect to the beam axis, so for a stationary target in a storage ring the acceptance angle is the scattering angle in the LAB frame. We are now investigating the use of a colliding electron beam instead of a stationary target. Here the acceptance angle does not correspond to either the scattering angle in the Centre-of-Mass frame or the LAB frame. We want to use a frame in which the antiproton scattering angle with respect to the beam axis still corresponds to the angle featuring in our equations. Note the scattering angle of the electrons is irrelevant here, and its use will be avoided by conservation of four-momentum. A ring frame for elastic antiproton electron scatterinḡ p ( P 1 , M ) + e − ( P 2 , m) →p ( P 3 , M ) + e − ( P 4 , m) is defined as follows 2 : P 1 = ( E 1 , 0, 0, p 1 ) P 3 = ( E 3 , p 3 sin θ r , 0, p 3 cos θ r ) P 2 = (E 2 , 0, 0, −p 2 ) P 4 = (E 1 − E 3 + E 2 , −p 3 sin θ r , 0, p 1 − p 2 − p 3 cos θ r ) Looking at E 2 = p 2 + m 2 for the final state electron gives ( E 1 + E 2 − E 3 ) 2 = p 2 3 sin 2 θ r + ( p 1 − p 2 − p 3 cos θ r ) 2 + m 2 . After substituting out the unknowns E 3 and p 3 this gives 3 an equation for the ring frame antiproton scattering angle θ r in terms of the Mandelstam variable t = ( p 3 − p 1 ) 2 , cos θ r = t − 2 M 2 + E 1( t − 2 E 2 1 ) p 2 − E 1 ( 2 E 1 E 2 + t ) p 1 p 1 E 2 + p 2 E 1 2 p 1 ( 2 E 1 E 2 + t ) p 1 + ( 2 E 2 1 − t ) p 2 2 ( p 1 E 2 + p 2 E 1 ) 2 − M 2 . (7.3.1) A Taylor expansion to O(t) of the above provides some clarity of the behaviour at small | t |: cos θ r ≈ 1 + t 2 p 2 1 . (7.3.2) Equation (7.3.1), the ring frame analogy of the much simpler Centre-of-Mass frame relation eq. (4.5.6), is used to graph θ r versus t in Figure 7-1. There is no backward antiproton scattering, i.e. θ r max ≤ π/2, for opposing electron beam momentum of p 2 < M p 1 E 1 + p 1 + M , (7.3.3) using the p 2 ≈ E 2 approximation to make the inequality strict. In this region the maximum ring frame antiproton scattering angle is given, via eq. (7.3.1), by [71] sin θ r max = p 1 E 2 + p 2 E 1 M ( p 1 − p 2 ) , (7.3.4) which, using sin θ ≈ θ for small angles, limits to the correct formula θ r max ≈ sin θ r max = m M = 0.54 mrad , (7.3.5) for stationary electrons (i.e. p 2 = 0 and hence E 2 = m). Notice this maximum scattering angle for stationary electrons is independent of the antiproton momentum. One can raise θ r max , and hence scatter more antiprotons out of the beam, simply by increasing the electron momentum p 2 . Raising it sufficiently beyond the acceptance angle will increase K out and the rate of buildup of polarization. Equation (7.3.4) can be used to derive a relation for the electron momentum needed to scatter antiprotons beyond the ring acceptance angle θ r acc , hence allowing selective scattering out of the beam to contribute to polarization buildup. This happens for electron momentum p 2 > p out where on assuming θ r acc is small and p 2 ≈ E 2 , i.e. that the electron mass is small compared to its momentum, one obtains Note that typical storage rings have acceptance angles in the range of 1 mrad to 50 mrad. The Lorentz invariant λ in the ring frame is p out ≈ M p 1 θ r acc E 1 + p 1 .(7.λ = 4 k 2 cm s = 4 ( p 1 E 2 + p 2 E 1 ) 2 ,(7.s = M 2 + m 2 + 2 E 1 E 2 + 2 p 1 p 2 . (7.3.8) The maximum squared momentum transfer | t |, corresponding to total backward scattering, is given in the ring frame by . The acceptance angle θ acc is plotted as a horizontal dashed line showing the region where particles are scattered "out" of the ring (t 1 to t 3 ) and the regions where particles are scattered at small angles remaining "in" the ring (t min to t 1 and t 3 to t 4 ). Note t min corresponds to the minimum scattering angle θ min , scattering below which is prevented by Coulomb screening. The squared momentum transfer t 2 corresponds to the maximum scattering angle θ max . Table 7.3: The entries in the system of polarization evolution equations, for both longitudinal and transverse polarization, integrated with respect to squared momentum transfer t. D ZZ d σ d t d tln a b = ln b a −1 = − ln b a . To present numerical results for the case of section 6.2 where particles are fed into the beam at such a rate that the beam intensity remains constant, we also need I out : I out = 8 π 2 α 2 λ s − m 2 − M 2 2 t 1 t 3 1 t 2 d t , = 8 π 2 α 2 λ s − m 2 − M 2 2 1 t 3 − 1 t 1 . (7.3.12) Since all antiproton-electron scattering is purely electromagnetic we now have all expressions needed to present the polarization buildup as a function of time from eq. (5.3.16). One can obtain the values of t 1 and t 3 for various acceptance angles by solving eq. (7.3.1). Table 7.4 presents results for values of p 1 = 15 GeV/c and p 2 = 50 MeV/c, where one sees from eq. (7.3.4) that θ r max = 107.248 mrad. The maximum electron beam momentum for this value of antiproton momentum, in order for the antiproton scattering angle to be less than π/2, is found from eq. (7.3.3) to be p 2 = 454.478 MeV/c. Raising the opposing electron beam momentum has the same physical effect, that more antiprotons are scattered out of the beam, as lowering the acceptance angle. The acceptance angle is a fixed parameter of a storage ring whereas the momentum of the opposing electron beam can easily be altered. Therefore it makes sense to pick a typical ring acceptance angle and investigate what effect changing the opposing electron beam momentum has on the polarization buildup time. This is done in Table 7.5 which follows. As correctly emphasized in Ref. [75] the important practical parameters for a method to polarize antiprotons are: -the polarization buildup time, -the degree of polarization achieved after this time, -the number of antiprotons available after this time, and -the phase space of the polarized antiprotons. The latter is a measure of how focused the final polarized antiproton beam is, and MeV/c, where one sees from eq. (7.3.4) that θ r max = 107.248 mrad and from eq. (7.3.9) that t 4 = − 2.32223 × 10 6 (MeV/c) 2 . Here we take the most optimistic t min = − 1.5575 × 10 −9 (MeV/c) 2 corresponding to an electron beam areal density of 10 12 cm 2 as discussed in Table 7.1. θ r acc [mrad] t 1 [(MeV/c) 2 ] t 3 [(MeV/c) 2 ] ln t 3 t 1 ln t 1 t 4 t min t 3 1 − 224 given that the beam can be focused by electron cooling we do not worry about this parameter here. In section 5.3.1 it has been shown that the figure of merit FOM (τ ) = P 2 (τ ) N (τ ) has a maximum at twice the beam lifetime. Of interest then is the polarization achieved after two beam lifetimes, i.e. the polarization achieved when the beam intensity has decreased by a factor of e 2 ≈ 7.389. The later provides a combined measure of the second and third points listed above. Equations (5.3.14 and 5.3.18) can be used to provide an estimate of the polarization achieved after two beam lifetimes: P (τ optimum ) = P ( 2 τ * ) ≈ − 2 P T K out I out . (7.3.13) It is interesting to notice that this optimum polarization is not dependent on the electron beam areal density n or the antiproton beam revolution frequency ν, although the time taken to achieve this polarization is strongly dependent on both of these parameters, as shown in eq. (5.3.14). In fact to this approximation, which is valid at times of twice the beam lifetime, the optimum polarization achieved is dependent only on energy and the ring acceptance angle, i.e. θ acc or t 1 and t 3 . Using the expressions for K out and I out presented in eqs. (7.3.10 and 7.3.12) one can write P (τ optimum ) = P ( 2 τ * ) ≈ 2 µ p P T s − m 2 − M 2 ln t 3 t 1 1 t 3 − 1 t 1 ,(7. 3.14) which we now maximize with respect to acceptance angle and energy. One finds that P (τ optimum ) is maximal for large t 1 ≈ t 3 , i.e. for large θ acc ≈ θ max , as shown in Figure 7-4, and also for high energies. Note these equalities cannot be strict otherwise P (τ optimum ) is undefined, but this is physically reasonable as if θ acc = θ max there is no scattering out of the ring and hence the beam lifetime τ * is undefined. Since P ( τ optimum ) is maximal at high energies we perform the following numerical calculations at the HESR energies. Thus the antiproton beam will have ring frame momentum 15 GeV/c. Stored antiprotons, having already been accelerated to these energies, could be polarized directly in the HESR ring. This eliminates the need for a purpose built low energy Antiproton Polarizing Ring, providing another advantage of this method over spin filtering off a polarized atomic gas target. This will also avoid the problem of having to accelerate a polarized antiproton beam past depolarizing resonances in the storage ring. Numerical results: Initial treatment Taking the discussion of section 7.2.4 into account we fix the effective acceptance angle at θ r acc = 50 mrad, the highest acceptance angle under consideration by the PAX Collaboration [23]. We shall investigate spin filtering with HESR parameters of antiproton momentum of 15 GeV and revolution frequency of 521628 Hz. For this ring acceptance angle and antiproton momentum one finds from eqs. (7.3.3 and 7.3.6) that the minimum and maximum opposing electron beam momentum, in order to provide scattering out of the beam but no backward scattering, are 23.4339 MeV/c and 454.478 MeV/c respectively. p 2 [MeV/c] t 1 [(MeV/c) 2 ] t 3 [(MeV/c) 2 ] t 4 [(MeV/c) 2 ] ln t 3 t 1 ln t 1 t 4 t min t 3 50 − Table 7.5: Values of t 1 , t 3 and t 4 , in units of (MeV/c) 2 , and the natural logarithms involving them appearing in the integrated spin observables, for various opposing electron beam momenta, in units of MeV/c, obtained by solving eq. (7.3.1). Results are for fixed values of θ r acc = 50 mrad and p 1 = 15 GeV/c, where one sees from eq. (7.3.6) that p out = 23.4339 MeV/c. Again we take the most optimistic t min = − 1.5575 × 10 −9 (MeV/c) 2 corresponding to an electron beam areal density of 10 12 cm 2 as discussed in Table 7.1. Using the values 5 of ln(t 3 /t 1 ) and ln(t 1 t 4 /t min t 3 ) presented in Table 7.5 one can obtain numerical values of K out , L in , L d and I out using eqs. (7.3.10 to 7.3.12). Hence using the results of section 5.3 one can obtain numerical values of the maximum polarization achievable for various opposing electron beam momenta, as presented in Table 7.7. The beam lifetime, τ * , is the time taken for the beam intensity to decrease by a factor of e = 2.718. The figure of merit has a maximum at twice the beam lifetime, as described in section 5.3.1, hence we are principally interested in the polarization achieved after this time. Table 7.7: The maximum polarization achievable, the beam lifetime and the optimum polarization achieved after two beam lifetimes for a stored antiproton beam, for various opposing electron beam momenta, in units of MeV/c. Results are for fixed values of θ r acc = 50 mrad, p 1 = 15 GeV/c, electron beam polarization P T = 0.9, electron beam areal density n = 10 12 cm −2 = 10 −15 mb −1 and antiproton revolution frequency 521628 Hz. The values for the integrated spin observables are taken from Table 7.6, and inserted into eq. (5.3.18). The positive values for the induced antiproton polarization indicate that it is orientated in the same direction as the polarization of the electron beam. This is a result of the negative sign of K out , the positive signs of the other integrated spin observables presented in Table 7.6 and the overall minus sign of eqs. The maximum polarization achievable, satisfying the relation of eq. (5.3.18), increases with increasing opposing electron beam momentum as a consequence of L in and L d decreasing faster than K out with increasing opposing electron beam momen-tum. While the polarization achieved after two beam lifetimes in the above investigations are very high one must note that the time taken to reach these polarizations is impractically long. In practice one requires the beam lifetime to be of the order of a few hours, as opposed to the 10 11 seconds ≈ 3171 years in the above cases! One notes that the closer t 1 is to t 3 , i.e. the closer θ acc is to θ max , the lower the rate that particles are scattered out of the beam and hence the longer the beam lifetime. Since spin filtering must continue for two beam lifetimes to achieve the optimum polarization the beam lifetime in practice must not be greater than a few hours. Therefore one should maximize P ( 2 τ * ) in eq. (7.3.14) subject to the constraint that τ * < 5 hours, which will allow for the optimum polarization to be achieved in less than 10 hours. This constraint can be presented in terms of t 1 and t 3 as τ * = 2 n ν I out ≤ 5 hours = 18000 seconds ⇒ I out ≥ 1 9000 n ν , ∴ 1 t 3 − 1 t 1 ≥ λ 72000 π 2 α 2 n ν ( s − m 2 − M 2 ) 2 > 0 . (7.3.15) The minimum value for this difference is essentially energy independent, but highly dependent on the electron beam areal density n and the antiproton beam revolution frequency ν. We shall present the numerics for two cases: (1) for n = 10 12 cm −2 the best electron beam areal density that will be available now or in the near future, and (2) for an ideal case of n = 10 20 cm −2 assuming great advances in electron beam areal densities. The latter simply provides a verification that the method of antiproton polarization buildup by spin filtering off an opposing polarized electron beam works in principle. One finds the minimum value for the difference of reciprocals of t, as presented in eq. (7.3.15), to be 5.28527 × 10 6 (MeV/c) −2 for case (1) and 0.0528527 (MeV/c) −2 for case (2). Unfortunately with today's electron beam areal densities and antiproton beam revolution frequencies the rate at which antiprotons are scattered out of the ring will be very low, even with the lowest possible ring acceptance angles of about 1 mrad. An extreme case provides a lower bound on the antiproton beam lifetime with today's technologies: 15 GeV/c antiprotons scattering off a 454 MeV/c opposing electron beam in a ring with lowest possible acceptance angle of 1 mrad and antiproton beam revolution frequency 521628 Hz, which still gives a beam lifetime of 2.63468 × 10 8 seconds ≈ 8.35 years. Hence with today's electron beam areal densities and antiproton beam revolution frequencies the beam lifetime will always be of the order of years instead of the required hours. Thus the constraint presented in eq. (7.3.15) is too strict with current parameters, and one cannot maximize the polarization achieved after two beam lifetimes subject to it. We now investigate the scenario where unpolarized antiprotons are continuously fed into the beam at such a rate to cancel the rate that antiprotons are being scattered out of the beam, the beam intensity remaining constant, as presented in section 6.2. The time taken to reach 8% polarization without any loss of beam intensity is presented for various opposing electron beam momenta. Inserting the integrated spin observables presented in Table 7.6 into eq. (6.2.3), where A all − K in = K out in the pure electromagnetic case of interest here and I all − D in = I out + 2 L in , gives the results presented in Table 7 to reach 8% polarization in the scenario where unpolarized particles are fed into the beam at the same rate particles are being scattered out of the beam, the beam intensity remaining constant N (τ ) = N 0 , for various opposing electron beam momenta, in units of MeV/c. Results are for fixed values of θ r acc = 50 mrad, p 1 = 15 GeV/c, electron beam polarization P T = 0.9, electron beam areal density n = 10 12 cm −2 = 10 −15 mb −1 and antiproton revolution frequency 521628 Hz. The values for the integrated spin observables are taken from Table 7.6, and inserted into eq. (6.2.3). Due to the long times taken to reach significant polarization it is apparent that neither of these methods of obtaining a high intensity polarized antiproton beam are practical at present. The key parameters limiting the rate of polarization buildup are the areal density of the opposing polarized electron beam, and the revolution frequency of the antiproton beam in the storage ring. The antiproton beam revolution frequency and the areal density of polarized electron beams would have to increase by many orders of magnitude in order for these methods to provide significant polarization in a few hours, which would be required. However, considering the immense research and development that will take place in the near future on electron and positron beams at the International Linear Collider (ILC), advances in electron beam areal densities can be expected in the coming years [130,131]. Numerical results: An ideal case To conclude this treatment let us now investigate an ideal case, assuming fanciful values for the key parameters that are not currently achievable, but may be achievable in the future, to show that in principle the method works. Let us assume a ring frame acceptance angle of 50 mrad and a very high electron beam areal density of 10 20 cm −2 which gives t min = − 1.5575 × 10 −1 (MeV/c) 2 . We shall investigate spin filtering with HESR parameters of antiproton momentum of 15 GeV and revolution frequency of 521628 Hz. For this ring acceptance angle and antiproton momentum one finds from eqs. (7.3.3 and 7.3.6) that the minimum and maximum opposing electron beam momentum, in order to provide scattering out of the beam but no backward scattering, are 23.4339 MeV/c and 454.478 MeV/c respectively. Since t min is much larger in this case compared to the previous treatment one expects L in to be much smaller in this treatment. As a consequence in this case both the maximum polarization achievable and the optimum polarization achieved after two beam lifetime should be larger, and the beam lifetime should be much shorter. A similar analysis to that presented above gives the results presented in the following tables. p 2 [MeV/c] t 1 [(MeV/c) 2 ] t 3 [(MeV/c) 2 ] t 4 [(MeV/c) 2 ] ln t 3 t 1 ln t 1 t 4 t min t 3 50 − Table 7.11: The maximum polarization achievable, the beam lifetime τ * and the polarization achieved after spin filtering for two beam lifetimes for various opposing electron beam momenta, in units of MeV/c. Results are for idealistic fixed values of θ r acc = 50 mrad, p 1 = 15 GeV/c, electron beam polarization P T = 0.9, electron beam areal density n = 10 20 cm −2 = 10 −7 mb −1 and antiproton revolution frequency 521628 Hz. The values for the integrated spin observables are taken from Table 7.10, and inserted into eq. (5.3.16). One emphasizes the best case presented above as achieving an antiproton beam polarization of 51.8% after just 1.84 hours when the beam intensity has decreased by a factor of e 2 = 7.389, by spin filtering off an opposing electron beam of momentum 200 MeV/c. Since the beam intensity has decreased significantly by the time the polarization reaches a high percentage, we now redo the analysis for the scenario where unpolarized particles are fed into the beam at the same rate particles are being scattered out of the beam, the beam intensity remaining constant N(τ ) = N 0 , as presented in section 6.2. The time taken to reach 15% without any loss of beam intensity is presented for various opposing electron beam momenta. Inserting the integrated spin observables presented in Table 7.10 into eq. (6.2.3), where A all − K in = K out in the pure electromagnetic case of interest here and I all − D in = I out + 2 L in , gives the results presented in Table 7 to reach 15% polarization in the scenario where unpolarized particles are fed into the beam at the same rate particles are being scattered out of the beam, the beam intensity remaining constant N (τ ) = N 0 , for various opposing electron beam momenta, in units of MeV/c. Results are for fixed values of θ r acc = 50 mrad, p 1 = 15 GeV/c, electron beam polarization P T = 0.9, electron beam areal density n = 10 20 cm −2 = 10 −7 mb −1 and antiproton revolution frequency 521628 Hz. The values for the integrated spin observables are taken from Table 7.10, and inserted into eq. (6.2.3). Notice that in the best case presented in Table 7.12, for an opposing electron beam of momentum 100 MeV/c, the antiproton beam polarization builds up to 15% in only 2515 seconds ≈ 42 minutes, while maintaining constant beam intensity. While this idealistic treatment might be far from today's technologies, requiring an increase of eight orders of magnitude in the product n ν, it highlights that the method of polarizing an antiproton beam by spin filtering off an opposing polarized electron beam works very well in principle. A co-moving lepton beam It has recently been proposed by Th. Walcher et al. [75,136] that a polarized positron beam, with very low relative momentum to an antiproton beam, could transfer polarization to the antiproton beam. This method of spin filtering is entirely based on selective spin flip while scattering within the ring, as a co-moving positron beam cannot scatter antiprotons out of the ring. Coulomb forces dominate the spin transfer observables at such low energies, and the antiproton-positron interaction is chosen because of the attraction of unlike charges. A positron beam can easily be polarized by the Sokolov-Ternov effect as described in sections (2.3.3 and 5.1). The basis for this proposal is a dramatic enhancement of the polarization transfer cross-section in the reactionp e + ↑ →p ↑ e + as calculated in Ref. [88]. Such a dramatic enhancement, of over nine orders of magnitude, has been called into question by many groups, who point out that, if true, multiple scattering effects should not be neglected. In particular Ref. [107] claims that there is an enhancement at low energies but by many orders of magnitude less than that claimed in Ref. [88], and that the polarization transfer cross-section is still far too low to make the Walcher et al. proposal practical at present. It also remains to be seen if the depolarization observable also gets enhanced greatly at such low relative velocities. An experiment has been proposed to test these claims [119]. As encountered earlier the relatively low areal densities of polarized positron beams is a crucial disadvantage of this method. Any advances in the technology of obtaining high intensity positron beams would greatly benefit this proposal. An interesting application of a great increase in the areal densities of polarized electron beams is that an electron cooler could utilize a high intensity polarized electron beam. This would allow for an antiproton beam (or a beam of any other particles for that matter) to be both cooled and polarized after repeated interaction with the electron cooler. This would represent a major advance in accelerator physics and hadron storage rings. Spin filtering off a polarized hydrogen target 7.4.1 Electromagnetic and hadronic scattering The critical squared momentum transfer t c , below which electromagnetic effects dominate and above which hadronic effects dominate inp p scattering, is now derived. For both hadronic and electromagnetic scattering, at low | t |, the non-spin-flip amplitudes φ 1 + φ 3 ≡ φ + dominate the spin-averaged differential cross-section. The where θ e = 0.54 mrad is the maximum angle antiprotons are scattered by stationary (or atomic) electrons as shown in eq. (7.3.5) and Figure 7-1 (a). Since spin filtering utilizes angles below 152 mrad, electromagnetic (QED) effects provide a good approximation to the totalp p interaction in this region. In conventional particle physics experiments particles must be scattered out of the beam pipe into detectors for measurements to be made, hence no direct experimental observations can be made for scattering at angles below the ring acceptance angle. Thus there is very little experimental data on this region of very low angle scattering, which is of interest only in storage rings. In particular hadronic antiproton-proton amplitudes are completely unknown in this kinematical region. The LHC very forward detector TOTEM hopes to obtain some data on low angle proton-proton scattering in the near future [142]. One can derive an expression for the antiproton momentum in the laboratory frame that makes the squared momentum transfer for total backward scattering equal to the critical squared momentum transfer below which electromagnetic effects dominate hadronic effects. Solving the equation t c = − 4 k 2 cm = − 4 M 2 p 2 lab /s one obtains p lab ≈ 31.6 MeV/c, below which all scattering is electromagnetically dominated. Polarization states of a hydrogen target Unpolarized hydrogen atoms in a strong magnetic field equally populate each of four hyperfine states: | ↑ p ↓ e | ↓ p ↓ e | ↓ p ↑ e | ↑ p ↑ e It is explained in section 2.2.1 how these hyperfine states and pairs of hyperfine states can be isolated to give polarized hydrogen. We are particularly interested in three types of polarized hydrogen target, with all atoms in the hyperfine states as follows | ↑ p ↑ e + | ↑ p ↓ e =⇒ P p = 1 and P e = 0 (7.4.14) | ↑ p ↑ e + | ↓ p ↑ e =⇒ P p = 0 and P e = 1 (7.4.15) | ↑ p ↑ e =⇒ P p = 1 and P e = 1 (7.4.16) where we denote the polarization of the electrons in the hydrogen by P e and the polarization of the protons in the hydrogen by P p . In practice the atoms are not perfectly isolated in certain hyperfine states, thus the electron and proton polarizations in polarized hydrogen are less than one. The HERMES Collaboration have utilized polarized hydrogen targets with P e = 0.9 and/or P p = 0.9 [51]. Spin filtering off a polarized hydrogen target in each of the polarization states presented in eqs. (7.4.14-7.4.16) can be treated similarly to the treatment presented in section 7.3 using the spin observables for antiproton-proton and antiproton-electron scattering presented in sections (4.5.1 and 4.5.2) respectively. In the second case where the electrons in the hydrogen target are polarized but the protons are unpolarized one has that σ out + = σ out − , i.e. particles in both spin states are scattered out of the beam at equal rates, since only the protons are massive enough to scatter the antiprotons beyond the ring acceptance angle. Thus while there is scattering out of the ring, it is not spin-dependent and does not lead to a buildup of beam polarization. Therefore only selective spin-flip in electromagnetic antiprotonelectron elastic scattering can contribute to polarization buildup in this case, yet one also has the negative effects of antiprotons being scattered out of the beam and annihilating with the protons in the hydrogen target; decreasing the beam intensity. The only advantage this case has over using a pure lepton target, considering it has the disadvantages listed above, is that the areal densities of electrons in atomic targets is greater than those achievable in pure lepton targets to date. Some experimental tests must now be carried out to decide which of the three possible states of a polarized atomic target would be most effective in polarizing an antiproton beam by spin filtering [74]. The Antiproton Decelerator (AD) storage ring at CERN is the only source of antiprotons in the required energy range. Consequently spin filtering studies of antiprotons scattering off a polarized hydrogen target at the AD ring are planned in the near future [53,143,144]. In particular these experiments should determine which of the antiproton-proton spin-dependent cross-sections proposed Antiproton Polarizing Ring (APR) to produce a high intensity polarized antiproton beam at the FAIR facility at GSI Darmstadt. It is well known that there is a significant spin-dependence of the proton-proton total cross-section. This was the basis of the FILTEX experiment, and the re-sulting polarization buildup of the proton beam was a consequence of this spindependence. At FILTEX laboratory frame kinetic energies of 23 MeV one has that 1 2 σ p p ↑↓ − σ p p ↑↑ = 122 mb [68]. Any measured polarization buildup of the AD antiproton beam during the proposed spin filtering experiments will provide direct evidence of, and the polarization buildup rate will provide a measure of, a spindependence in the antiproton-proton total cross-section. Measurements of the spin-dependent antiproton-proton cross-sections at the AD ring will also provide the first experimental results to test and distinguish between the current models [73,140,145,146,147] of the hadronic antiproton-proton interaction in the non-perturbative regime. We have pointed out that a pure lepton target has many advantages over an atomic target, but has the disadvantage of lower target densities because of the electromagnetic repulsion of the leptons in a pure lepton target which is less severe in the electrically neutral atomic targets. Consequently it is now appropriate to outline some possible, albeit far fetched, solutions to this problem: 1) Positronium, an electrically neutral electron-positron bound state could in future be used as a high density polarized pure lepton target. Being electrically neutral it should allow for similar densities as atomic targets. 2) Similarly muonium, an electrically neutral electron-antimuon bound state could possibly be used as above. 3) A polarized muon target/beam would allow for spin-dependent scattering out of the ring, as discussed in section 7.3.2. Because muons have approximately 200 times the mass of electrons using a muon target will enhance the polarization transfer cross-sections K XX and K YY as seen in eq. (4.3.3). 4) A polarized tau lepton target or beam would be even more favourable to our needs, because of the very large mass of the tau lepton (m τ ≈ 3477 m e ≈ 17 m µ ). We stress that these ideas are far fetched, and very far from today's technologies, in particular because of the very low mean lifetimes of the particles discussed. However they may be practical in future, and in the mean time they might prompt other solutions to the problem of low target areal densities. Chapter 8 Conclusions "I was born not knowing and have had only a little time to change that here and there." Richard Feynman There has been much recent research into possible methods of polarizing an antiproton beam, the most promising being spin filtering, the theoretical understanding of which is currently incomplete. The method of polarization buildup by spin filtering requires many of the beam particles to remain within the beam after repeated interaction with an internal target in a storage ring. Hence small scattering angles, where it is shown that electromagnetic effects dominate hadronic effects, are important. All spin-averaged and spin-dependent cross-sections and spin observables for elastic spin 1/2 -spin 1/2 scattering, for both point-like particles and non-point-like particles with internal structure defined by electromagnetic form factors, have been presented to first order in QED. Particular attention is paid to spin transfer and depolarization cross-sections in antiproton-proton, antiproton-electron and positronelectron scattering, in the low | t | region of momentum transfer. Of the spin-averaged formula derived we highlight that a generalization of the Rosenbluth formula has been presented in a new compact Lorentz invariant form. It is a two-fold generalization in that the masses of both particles are included and both particles are taken to have internal structure determined by electromagnetic form factors. While these results are eventually applied to spin filtering later in the thesis they are not limited to this application. The complete set of spin 1/2 -spin 1/2 helicity amplitudes and spin observables should prove useful to many other areas in particle physics. The complete set of spin 0 -spin 1 electromagnetic helicity amplitudes have also been presented to first order in QED. These are useful in describing the spindependent scattering of deuterons off carbon nuclei for example. A thorough mathematical treatment of spin filtering has also been presented, identifying the two key physical processes involved: (a) selective scattering out of the ring and (b) selective spin flip while remaining in the ring. The dynamical properties of the physical system have also been highlighted and analyzed. Sets of differential equations which describe the buildup of polarization by spin filtering have been presented and solved in many different scenarios of interest. These scenarios are: 1) spin filtering of a stored beam, 2) spin filtering while the beam is being accumulated, i.e. unpolarized particles are continuously being fed into the beam at a constant rate, 3) unpolarized particles are continuously being fed into the beam at a linearly increasing rate, i.e. the particle input rate is ramped up, 4) the input rate is equal to the rate at which particles are being lost due to scattering beyond the ring acceptance angle, the beam intensity remaining constant, 5) increasing the initial polarization of a stored beam by spin filtering, 6) the input of particles into the beam is stopped after a certain amount of time, but spin filtering continues. The depolarization of a polarized beam on interaction with an unpolarized target or beam, as in the important case of electron cooling, has also been investigated and shown to be negligible. There are advantages of using a lepton target instead of an atomic gas target for spin filtering, principal amongst them that antiprotons will not annihilate with the target as they do with the protons in the atomic targets, leading to a loss of beam intensity. Since electrons in an atomic target are not massive enough to scatter antiprotons beyond the acceptance angle of any storage ring we have proposed using an opposing polarized electron beam, of momentum large enough to provide scattering of antiprotons beyond ring acceptance, as a possible method to polarize antiprotons by spin filtering. This is presented as a practical application of the theoretical work presented throughout the thesis. The areal density of the polarized electron beam is identified as the key parameter limiting the rate of antiproton polarization buildup in this proposal. After analyzing this proposal it is concluded that the areal densities of electron beams currently available would have to increase significantly in order for this method of polarizing an antiproton beam to be practical. While this thesis is devoted to investigating spin-dependent antiproton interactions and the theoretical background to spin filtering in light of a possible method to polarize antiprotons, we emphasize that much of the work presented is applicable to many other areas of particle physics. Possible extensions to this work would include redoing the analysis for the case of antiprotons repeatedly interacting with a polarized deuterium target. Polarized deuterium targets have successfully been utilized by the HERMES experiment, and would be available for spin filtering studies in future. In the kinematical regime of interest in spin filtering antiproton-deuterium scattering consists of antiproton-electron and antiproton-deuteron scattering. The helicity amplitudes and spin observables for electromagnetic antiproton-electron scattering have been presented in this thesis, and those for antiproton (spin 1/2) -deuteron (spin 1) scattering could be derived in analogy to those for spin 1/2 -spin 1/2 and spin 0 -spin 1 scattering presented here. We hope that the treatment of spin filtering presented in this thesis will clarify some of the confusion in the theoretical literature, and perhaps play some part in the eventual achievement of a high intensity polarized antiproton beam. Measurements obtained using such a beam should lead to a better understanding of the spin structure of the protons and neutrons. Given that the proton is the nucleus of the hydrogen atom, the most abundant element in the Universe, one cannot overstate the importance of a better understanding of its internal structure. The hermitian conjugates of spinors u = u( p, λ ) are as follows u † =ū γ 0 γ 0 † = γ 0 γ † 5 = γ 5 u † = γ 0 † u = γ 0 u γ † µ = γ 0 γ µ γ 0 (A.6) γ 0 = γ 0 (γ 0 ) 2 = γ 0 γ 0 = I 4×4 . The completeness relations for the sum of spinors are where the polarization four-vector S µ (p, λ) of a particle is orthogonal to its momentum four vector, i.e. S µ (p, λ) p µ = 0, and is normalized such that S µ S µ = −1. Tr [ γ µ γ ν γ ρ γ σ ] = 4 ( η µ ν η ρ σ − η µ ρ η ν σ + η µ σ η ν ρ ) , Tr γ 5 = Tr γ 5 γ µ = Tr γ µ γ ν γ 5 = 0 , (A.14) Tr γ µ γ ν γ ρ γ σ γ 5 = − 4 i ǫ µ ν ρ σ . The totally antisymmetric permutation tensor, also known as the Levi-Civita symbol, is defined as and it satisfies the following contraction identities: ǫ µ ν ρ σ ǫ µ ω λ τ = δ ν τ δ ρ λ δ σ ω + δ ν ω δ ρ τ δ σ λ + δ ν λ δ ρ ω δ σ τ − δ ν ω δ ρ λ δ σ τ − δ ν τ δ ρ ω δ σ λ − δ ν λ δ ρ τ δ σ ω , ǫ µ ν ρ σ ǫ µ ν λ τ = − 2 ( δ ρ λ δ σ τ − δ ρ τ δ σ λ ) , ǫ µ ν ρ σ ǫ µ ν ρ τ = − 6 δ σ τ , (A. 16) ǫ µ ν ρ σ ǫ µ ν ρ σ = − 24 , where the Kronecker delta is defined, not using the Einstein summation convention, as δ µ ν =    1 if µ = ν 0 if µ = ν (A.17) and when using the Einstein summation convention is used to sum over repeated indices µ ∈ { 0, 1, 2, 3 } one has δ µ µ = δ 0 0 + δ 1 1 + δ 2 2 + δ 3 3 = 4. Appendix B Relations between Mandelstam variables Initial and final state particles, which in elastic scattering are the same, are on-shell thus p 1 · p 1 = p 3 · p 3 = M 2 and p 2 · p 2 = p 4 · p 4 = m 2 , where M and m are the masses of the two particles in the elastic process. One can square the above relations to obtain s = m 2 + M 2 + 2 p 1 · p 2 = m 2 + M 2 + 2 p 3 · p 4 , t = 2 m 2 − 2 p 2 · p 4 = 2 M 2 − 2 p 1 · p 3 , (B.3) u = m 2 + M 2 − 2 p 2 · p 3 = m 2 + M 2 − 2 p 1 · p 4 , hence one sees that p 1 · p 2 = p 3 · p 4 and p 1 · p 4 = p 2 · p 3 , (B.4) and adding, using conservation of four momentum, gives the defining relation for Mandelstam variables s + t + u = 2 m 2 + 2 M 2 . (B.5) The above relation is the special case for elastic scattering of the general relation s + t + u = 4 i=1 m 2 i . (B.6) Rearranging the t equation gives p 1 · p 3 = M 2 − t 2 and p 2 · p 4 = m 2 − t 2 . (B.7) Similarly for s and u p 1 · p 4 = p 2 · p 3 = 1 2 m 2 + M 2 − u , (B.8) p 1 · p 2 = p 3 · p 4 = − 1 2 m 2 + M 2 − s . (B.9) For convenience define R µ = p µ 1 + p µ 3 and r ν = p ν 2 + p ν 4 , thus we have R 2 = R µ R µ = p 2 1 + 2 p 1 · p 3 + p 2 3 = 2 M 2 + 2 M 2 − t = 4 M 2 − t , r 2 = r ν r ν = · · ·· = 4 m 2 − t . (B.10) Now R · r = R µ r µ = ( p µ 1 + p µ 3 ) p 2 µ + p 4 µ = p 1 · p 2 + p 1 · p 4 + p 2 · p 3 + p 3 · p 4 , = 2 − 1 2 m 2 + M 2 − s + 2 1 2 m 2 + M 2 − u = s − u , (B.11) similarly r · p 1 = r · p 3 = R · p 2 = R · p 4 = 1 2 ( s − u ) . (B.12) Appendix C Derivation of the Gordon decomposition identities The Dirac gamma matrix structure of the most general proton electromagnetic current can greatly be simplified using the Gordon Decomposition Identity [82,83], which we now derive. The commutation and anticommutation relations where [ γ µ , γ ν ] = γ µ γ ν − γ ν γ µ and { γ µ , γ ν } = γ µ γ ν + γ ν γ µ , (C.2) and η µν = diag(+1, −1, −1, −1) the Minkowski metric tensor, can be used to write [ γ µ , γ ν ] = γ µ γ ν − γ ν γ µ = γ µ γ ν − (2 η µ ν − γ µ γ ν ) = 2 γ µ γ ν − 2 η µ ν . Hence i σ µ ν = − ( γ µ γ ν − η µ ν ) = η µ ν − γ µ γ ν , but equivalently i σ µ ν = η µ ν − (2 η µ ν − γ ν γ µ ) = γ ν γ µ − η µ ν . on the above equation to obtain v(p ′ ) i σ µ ν ( p ′ ν − p ν ) v(p) =v(p ′ ) − M γ µ − ( p + p ′ ) µ + γ µ ( − M ) v(p) , =v(p ′ ) − 2 M γ µ − ( p + p ′ ) µ v(p) . Now rearranging gives the Gordon Decomposition identity for anti-spinors: The Laplace transforms used in solving the differential equations in section 6.5 are: v(p ′ ) γ µ v(p) =v(p ′ ) − ( p + p ′ ) µ 2 M − i σ µ ν ( p ′ − p ) ν 2 M v(p) .L { f ′ (τ ) } = s L { f (τ ) } − f (0) , (F.2) L { f ′′ (τ ) } = s 2 L { f (τ ) } − s f (0) − f ′ (0) , (F.3) L { H ( τ − τ c ) } = e − τc s s , (F.4) L { H ( τ ) } = 1 s = L { 1 } , (F.5) where f (τ ) is an arbitrary function, ′ denotes first derivative and ′′ denotes second derivative. One obtains f (0) and f ′ (0) from the initial conditions of the differential equation. The Heaviside step function, H ( τ − τ c ), is defined as H ( τ − τ c ) =    0 if τ < τ c 1 if τ ≥ τ c (F.6) The Inverse Laplace Transforms go from right to left in the above list, for example Figure 1 - 1 : 11The great physicists Wolfgang Pauli (left) and Niels Bohr musing over the spin of a spinning top toy, trying to gain insight into the nature of spin in particle and nuclear physics. Figure 1 - 2 : 12The upper beam is unpolarized, with equal number of particles in the 'spin up' and 'spin down' states, which in reality are randomly distributed. The lower beam is 100% polarized, with all particles in the 'spin up' state. In practice the maximum beam polarizations achievable are about 90%. Figure 1 - 3 : 13These diagrams describe the internal structure of protons and neutrons, according to the theory of Quantum Chromo-Dynamics (QCD). - 3 . 3Gluons, being massless spin-1 bosons, cannot be transversely polarized, hence there is no gluon transversity. This gives us the longitudinal spin sum rule as follows: Figure 1 - 4 : 14The contributions to the spin of the nucleon: the helicity of the constituent quarks ∆q, the orbital angular momentum of the quarks L q , the orbital angular momentum of the gluons L g all of which are known; the partly known helicity of the gluons ∆G and the unknown transversity of the quarks h 1 q . Figure 2 - 1 : 21The Drell -Yan lepton pair production process p p →l l X, via a virtual photon γ * . Figure 2 - 1 . 21The double spin asymmetry, an experimentally measurable quantity, is defined as Figure 2 - 2 : 22The two spin-flip Feynman diagrams that contribute to the Sokolov-Ternov effect. In (a) an electron in the 'spin up' state gets flipped to the 'spin down' state, while in (b) an electron in the 'spin down' state gets flipped to the 'spin up' state; after emitting a photon due to synchrotron radiation induced by bending in a magnetic field. The crosssections for these two processes are not equal and as such there will be a gradual buildup of polarization in the beam, known as the Sokolov-Ternov effect. - 4 , 4are: (a) spin selective scattering out of the beam, and (b) selective spin-flip while remaining in the beam Figure 2 - 3 : 23This diagram describes the spin filtering technique. Beam particles travel Figure 2 - 4 : 24Figure 2-4: The following two diagrams provide a schematic representation of the two physical processes, selective scattering out of the ring (left) and selective spinflip (right), that contribute to polarization buildup by spin filtering in a storage ring. Particles in the 'spin up' state are represented by blue squares and particles in the 'spin down' state are represented by yellow squares, while the grey box represents a polarized target. In both cases the beam is initially unpolarized with equal numbers of particles in the 'spin up' and 'spin down' states. Figure 2 - 5 : 25The results of the FILTEX experiment, showing polarization buildup over time. The solid lines show the best fit to the data with a rate of polarization buildup of 1.24 × 10 −2 h −1 . The dashed lines are based on the expected buildup rate from the model presented in Ref.[54], from where this plot has been reproduced with permission from the authors. F 2 Figure 3 - 1 : 231(t) t/ (4 M 2 ) and magnetic G M (t) = F 1 (t) + F 2 (t) form factors are used. In the t-channel, also known as the space-like region, the form factors are real functions of t. Although not treated in this thesis it is worth mentioning that this is not true in the s-channel, also known as the time-like region, where the form factors are complex functions of s. For a treatment of polarization observables in the time-like region see Ref.[80].FromFigure 3-1, and using the Feynman rules for QED presented in Appendix D, The electron, proton and antiproton electromagnetic currents; j µ , J µ p and J μ p respectively. Time increases from left to right. These Feynman diagrams are converted into the mathematical expressions for the currents presented in this section using the Feynman rules of Appendix D. The shaded circle in the proton and antiproton currents describe that these are not point particles, and have an internal structure described by form factors. Figure 3 - 2 : 32The Feynman diagrams for single photon exchange in antiproton-proton, (3.2.11) is used to derive the helicity amplitudes and spin observables throughout the remainder of this chapter.The spin four vectors are now normalized so that all ǫ i = +1 corresponds to the helicity amplitude φ 1 = M( +, + ; +, + ), and the ±1 in the helicity amplitudes now relate to the signs of the ǫ i . The momenta and longitudinal, transverse 3 and normal spin four vectors in the Centre-of-Mass frame are presented in 3. 1 : 1Momenta and spin 4-vectors in the Centre-of-Mass frame. The Centre-of-Mass energies of particles A and B are E A = √ k 2 + M 2 and E B = √ k 2 + m 2 respectively, where k is the modulus of the Centre-of-Mass 3-momentum. The Centre-of-Mass scattering angle is denoted by θ. keeping in mind that each λ i is ±1/2 . Thus this reduces the 16 helicity amplitudes to eight independent ones: M(+, + ; +, +) = (−1) 0−0 M(−, − ; −, −) = M(−, − ; −, −) M(+, + ; +, −) = (−1) −1−0 M(−, − ; −, +) = −M(−, − ; −, +) M(+, + ; −, +) = (−1) 1−0 M(−, − ; +, −) = −M(−, − ; +, −) M(+, − ; +, +) = (−1) 0−(−1) M(−, + ; −, −) = −M(−, + ; −, −) M(−, + ; +, +) = (−1) 0−1 M(+, − ; −, −) = −M(+, − ; −, −) M(+, + ; −, −) = (−1) 0−0 M(−, − ; +, +) = M(−, − ; +, +) M(+, − ; −, +) = (−1) −1−1 M(−, + ; +, −) = M(−, + ; +, −) M(+, − ; +, −) = (−1) −1−(−1) M(−, + ; −, +) = M(−, + ; −, the eight remaining helicity amplitudes to six independent ones, by the relations, M(+, − ; +, +) = (−1) 1−0 M(+, + ; +, −) = −M(+, + ; +, −) M(−, + ; +, +) = (−1) −1−0 M(+, + ; −, +) = −M(+, + ; −, +, + ; +, −) = − M(−, − ; −, +) = − M(+, − ; +, +) = M(−, + ; −, −) φ 6 ≡ M(+, + ; −, +) = − M(−, − ; +, −) = − M(−, + ; +, +) = M(+, − ; −, −) As can be seen from eq. (3.4.7), φ 1 and φ 3 are non-spin-flip amplitudes, φ 2 and φ 4 are double-spin-flip amplitudes and φ 5 and φ 6 are single-spin-flip amplitudes. By double-spin-flip we mean that both particles in the reaction have their spins flipped, - 1 . 1However the t-channel results are dominant in the low | t | region of interest in a storage ring, as explained inFigure 4-1, and electromagnetic effects also dominate over the hadronic effects in this low | t | region. The results in this section are important in the region where | t | < | t c | for antiproton-proton collisions with total cross section σp p tot , defined by[95,105] and m = M) in the expressions provided in sections (3.4 and 3.5). These are required by the PAX Collaboration to analyze the buildup of polarization of an antiproton beam by interactions with the protons in a hydrogen target. The helicity amplitudes in section 3.4 now become and M → m in the expressions provided in sections (3.4 and 3.5). The helicity amplitudes from section 3.4 now become Figure 4 - 1 : 41The two tree-level Feynman diagrams for elastic positron-electron scattering, also known as Bhabha scattering. The antiproton-proton case is analogous. Contributions to the full amplitude M = M t + M s come from the t-channel (left diagram), being proportional to 1/ t, and s-channel (right diagram), being proportional to 1/s, as shown. For low momentum transfer (small | t |), and also at high energies (large s), the t-channel contributions dominate. As always time increases from left to right. from section 3.4 are equally valid here and one obtainsλ − µ = ( λ 0 − λ a ) − ( λ 0 − λ b ) = λ b − λ a .Hence Parity Invariance defined in eq. (3.4.3) gives the following relations between the helicity amplitudes M(− ; −) = (−1) − ; +) = (−1) 1−(−1) M(+ ; −) = M(+ ; −) the nine helicity amplitudes to five independent ones. Applying Timereversal Invariance, defined in eq. (3.4.5), further reduces to four independent helicity amplitudes by the following relations: M(+ ; 0) = (−1) 1−0 M(0 ; +) = − M(0 ; +) M(− ; 0) = (−1) −1−0 M(0 ; −) = − M(0 ; −) can present the four independent helicity amplitudes for spin 0 -spin 1 scattering:H 1 ≡ M(+ ; +) = M(− ; −) H 2 ≡ M(+ ; 0) = M(0 ; −) = − M(0 ; +) = − M(− ; 0) H 3 ≡ M(+ ; −) = M(− ; +) (4.6.4) H 4 ≡ M(0 ; 0) ,where it is seen that H 1 and H 4 are the non-spin-flip amplitudes, while H 2 and H 3 are the spin-flip amplitudes. The non-spin-flip amplitudes, H 1 and H 4 , are also known as spin-elastic amplitudes. It can be seen from eq. (4.6.4) that H 1 and H 3 p µ a ǫ µ ( p a , λ a ) = 0. The quantities F d 1 , F d 2 and G d 1 are the electromagnetic form factors of the deuteron with normalizations is the quadrupole moment of the deuteron in units of e/M 2 d and µ d is the magnetic dipole moment of the deuteron in units of e/2 M d . The complete set of initial and final deuteron polarization vectors, in the Centreof-Mass (CM) frame, where the initial momentum of the deuteron is p a above expressions the dependence on the mass of the spin-0 nucleus is in the ( s − u ) terms. Polarization buildup by spin filtering "If, as I have reason to believe, I have disintegrated the nucleus of the atom, this is of greater significance than the war." Ernest Rutherford, apologizing for absence from a meeting of the International Antisubmarine Warfare Committee. The theory of spin filtering is developed in this chapter. A mathematical description of the related but simpler process of polarization buildup by the Sokolov-Ternov effect is first presented in section 5.1. The ideas presented are utilized in the mathematical descriptions of spin filtering which follow. In section 5.2 the rates of change of the number of particles in each spin state are combined into a set of polarization evolution equations which describe the process of polarization buildup by spin filtering. This set of polarization evolution equations is then analyzed and solved in section 5.3, emphasizing the physical implications of the dynamics. ( 1 ) 1constitutes a decrease in the number of 'spin up' particles (N + ) and an increase in the number of 'spin down' particles (N − ), while mechanism (2) constitutes a decrease in the number of 'spin down' particles and an increase in the number of 'spin up' particles. This explains the signs of the coefficients in eq. (5.1.1) which follows. Figure 5 - 1 : 51sees that N + (τ ) + N − (τ ) = c 1 ( W −+ + W +− ) = constant ≡ N 0 A schematic graph of the Sokolov-Ternov effect, showing that the number of particles in the 'spin up' state (N + ) decreases with time while the number of particles in the 'spin down' state (N − ) increases with time. Hence |P| = |(N + − N − )/(N + + N − )|, the absolute value of the beam polarization, increases with time. One sees that the relations dN + /dτ = − dN − /dτ and N + + N − = N 0 are satisfied throughout. The graph just shows general trends, therefore we do not specify the units of the time axis. also that d N + / d τ = − d N − / d τ as it must be if there is no scattering out of the ring. Imposing the initial conditions N + (0) = N − (0) = N 0 / 2, corresponding to a beam that is initially unpolarized, gives the constants . 1 . 1The general behaviour of the number of particles in the 'spin up' and 'spin down' states, along with the polarization buildup, due to the Sokolov-Ternov effect are plotted in Figure 5-1. Figure 5 - 2 : 52These diagrams describing the two physical processes, selective scattering out of the beam (left) and selective spin-flip while remaining in the beam (right), that contribute to polarization buildup by spin filtering in a storage ring have been explained in Figure 2-4. number of particles in the 'spin up' state can change by three means: (1) 'spin up' particles being scattered out of the beam, the cross-section for which we label as σ out + , (2) 'spin up' particles being flipped to 'spin down' particles while remaining in the beam, the cross-section for which we label as σ +− , and (3) 'spin down' particles being flipped to 'spin up' particles while remaining in the beam, the cross-section for which we label as σ −+ . Mechanisms(1)and(2)constitute a decrease in the number of 'spin up' particles and (3) constitutes an increase in the number of 'spin up' particles. This explains the signs of the coefficients in eq. (5.2.1) which follows. Correspondingly the number of particles in the 'spin down' state can also change by three means: (1) 'spin down' particles being scattered out of the beam, the cross-section for which we label as σ out − , (2) 'spin down' particles being flipped to 'spin up' particles while remaining in the beam (σ −+ ) and (3) 'spin up' particles being flipped to 'spin down' particles while remaining in the beam (σ +− ). All of this can be expressed in the following set of polarization evolution equations: NFigure 5 - 3 : 53+ / N − ratio Polarization versus N + / N − ratio N + = N − 2 N − 3 N − 10 N − 19 N − 100 N −This graph, from equation 5.2.3, shows how the polarization (P) changes as the ratio (N + / N − ) of the number of particles in the 'spin up' state to the number of particles in the 'spin down' state changes. The horizontal axis is a log scale. The change in polarization as the number of particles in each of the spin states changes is very important throughout the thesis, so we highlight a few points in the table. The second last entry in the table is particularly relevant as the target used in our numerical calculations is 90% polarized. Note that 0 ≤ |P| ≤ 1 and when N − > N + the polarization is defined to be negative, i.e. −1 ≤ P < 0. Figure 5 - 4 : 54A schematic graph showing that the number of particles in the 'spin up' (N + ) and 'spin down' (N − ) states each decrease with time, but at different rates. Hence the beam polarization P = (N + − N − )/(N + + N − ) increases with time. The time axis is scaled by the beam lifetime τ * , as described in section 5.3.1. Figure 5 - 5 : 55A schematic graph of the system treated in Lemma 2 and Corollary 1, showing that the number of particles in the 'spin up' state (N + ) increases with time while the number of particles in the 'spin down' state (N − ) decreases with time. Hence the beam polarization P = (N + − N − )/(N + + N − ) increases with time. One sees that the relations dN + /dτ = − dN − /dτ and N = N + + N − = N 0 are satisfied throughout. The graph just shows general trends, therefore we do not define the units of the time axis. So we have d / d τ [ N + (τ ) + N − (τ ) ] = 0 which implies N + (τ ) + N − (τ ) = constant = N(0). Thus there will be no loss of particles, as expected. Subtracting eqs. (5.2.9) from each other gives the system which describes the Sokolov-Ternov effect presented in section 5.1. The systems describing these two physical processes should be similar as they are both governed solely by spin-flip transitions. In spin filtering the spinflip transitions are induced by scattering off the polarized internal target while in the Sokolov-Ternov effect the spin-flip transitions are induced by spontaneous synchrotron radiation emission of photons while the charged particles of the beam are being bent in the magnetic field of the ring. In fact these systems are identical except for the interpretations of the matrix entries, and that because there is no target in the Sokolov-Ternov effect, the system of equations describing it does not depend on a target areal density n. The solution of the system presented in eq. (5.2.8) is identical to the solution of the Sokolov-Ternov system presented in detail in section 5.1. an uncoupled system of equations as required. The solutions of the above equations, for an initially unpolarized beam, that if in addition σ out + = σ out − in eqs. (5.2.12) then no polarization buildup occurs, as shown in Lemma 1. It is claimed by the Budker-Jülich groups that the spinflip transition rates are negligible for antiprotons scattering off polarized electrons in a hydrogen target P σ +− − σ −+ should both be proportional to the target polarization P T . Proof: This is immediately satisfied by the relations in eqs. (5.2.21 and 5.2.23). While the system of polarization evolution equations involving the variables N + (τ ) and N − (τ ) presented in eq. (5.2.1) is very transparent, one is more interested in the variables N(τ ) = N + (τ ) + N − (τ ) and J(τ ) = N + (τ ) − N − (τ ) which immediately lead to P(τ ) = J(τ ) / N(τ ). We can transform the system of two first order ODE's in variables N + (τ ) and N − (τ ) presented in eq. (5.2.1) to the following system of two first order ODE's in variables N(τ ) and J(τ ) [T ( A all − K in ) I all − D in rate of change of the number of beam particles N(τ ) = N + (τ ) + N − (τ ) and their total spin J(τ ) = N + (τ ) − N − (τ ) I out , L in and L d are all non-negative. As a consequence the eigenvalues are non-positive and λ 1 ≤ λ 2 ≤ 0. When there is no scattering out of the ring all of the "out" integrations are zero, and one finds that L d = L in and hence λ 1 = − 2 n ν L d and λ 2 = 0. Now enforcing the initial conditions N(0) = N 0 the total number of particles in the beam initially, and J(0) = 0 ⇒ N + (0) = N − (0) = N 0 / 2 i.e. initially the beam is unpolarized, one obtains the solutions: Figure 5 - 6 : 56accurate form of which is derived in section 5.3.3. The Figure Of Merit (FOM) provides a measure of the quality of the polarized A schematic graph showing the behaviour of the beam intensity N and beam polarization P as time (scaled by the beam lifetime τ * ) increases. The behaviour of the figure of merit FOM is also shown on the graph, being blown up to clearly show it has a maximum at twice the beam lifetime. figure of merit gives the optimum polarization buildup time, taking into account the trade-off between decreasing beam intensity and increasing beam polarization. Solving d FOM / d τ = 0 yields τ optimum ≈ 2 n ν I out ≈ 2 τ * , (5.3.14) . (6.1.4) with respect to τ and substituting into eq. (6.1.2) gives an expression for N(τ ) : Figure 6 - 1 : 61monotonically with time. The behaviour of the beam intensity, beam polarization and figure of merit as functions of time are plotted in Figure 6-1. One notes that in this case the figure of merit increases monotonically and hence does not have a maximum in finite time. Therefore in this scenario it would be best to continue the spin filtering process for as long as possible before extracting the polarized antiproton beam. A schematic graph showing the behaviour of the beam intensity N , which to the solutions of the original homogeneous system presented in eqs. (5.3.4 and 5.3.5) when J 0 → 0. The beam lifetime is the same to leading approximation as in the homogeneous case when J(0) = 0. Dividing J(τ ) by N(τ ) provides the time dependence of the polarization of the beam conditions are N(0) = 0 and N ′ (0) = β [ 1 − H ( −τ c ) ], the latter of which deserves comment. The rate N ′ (0) should be β when τ c > 0 and 0 when τ c = 0, corresponding to the case of particles being fed in for zero seconds. Note that H ( −τ c ) = 1 when τ c = 0 and H ( −τ c ) = 0 when τ c > 0, and note physically that τ c , the duration for which particles are fed into the beam, cannot be negative. The Heaviside function in this initial condition forces the solution for N(τ ) to be split into three regions. On solving by Laplace Transform methods one obtains upper bound on the velocity of the antiprotons is the speed of light c = 3 × 10 8 m s −1 . The revolution frequency (ν) is simply the reciprocal of the time taken for one revolution, which in turn is the circumference of the storage ring (L) divided by the velocity of the antiprotons (v = β c), i.e. an antiproton polarizer ring of circumference L = 75 m and a beam velocity of β = v / c = 0.5 giving a revolution frequency of 2 MHz. The PAX Collaboration proposes an Antiproton Polarizer Ring (APR) of circumference L = 86.5 m with antiprotons of kinetic energy 250 MeV which, using the relation p = T ( T + 2 M ), corresponds to a momentum of 729.13 MeV/c and a velocity of β = 0.6136, thus they propose a revolution frequency of ν = 2.12662 MHz [23]. The High Energy Storage Ring (HESR) of the proposed Facility for Antiproton and Ion Research (FAIR) at GSI Darmstadt provides a high energy example. Antiprotons with momentum 15 For the case of a polarized hydrogen target the maximum impact parameter is given by the Bohr radius a B . For an antiproton scattering off a hydrogen atom at impact parameters greater a B the Coulomb fields of the atomic electron and proton screen each other, hence the antiproton sees the atom as electrically neutral and does not interact with it. Hence we have the relation− t min = 1 a 2 B = α 2 m 2 = 0.000013912 (MeV/c) 2 , (7.2.7) 1.5575 × 10 −12 8.3 × 10 −11 10 10 10 −16 − 1.5575 × 10 −11 2.6 × 10 −10 10 11 10 −15 − 1.5575 × 10 −10 8.3 × 10 −10 10 12 10 −14 − 1.5575 × 10 −9 2.6 × 10 −9 Table 7.1: Results for the minimum squared momentum transfer t min and minimum ring frame scattering angle θ r min for various values of the electron beam areal density n. The ring frame antiproton momentum is fixed at p 1 = 15 GeV/c. The conversion factor 1 cm −2 = 10 −26 fm −2 has been used. 7.3 Spin filtering off a polarized electron beam 7.3.1 The advantages of using a lepton target Plot of scattering angle θ r verses t for 20 MeV/c electrons Figure 7 - 1 : 71(a) Ring frame antiproton scattering angle θ r versus squared momentum transfer t for antiprotons of momentum 729 MeV/c scattering off electrons in an atomic target, where one confirms that the maximum antiproton scattering angle is 0.54 mrad. (b)-(f ) With a colliding electron beam the maximum ring frame antiproton scattering angle increases as the electron beam momentum increases in a direction opposite to that of the antiproton beam. The plots show the relationship between θ r and t given in eq. (7.3.1).Plot of scattering angle θ r verses t for 300 MeV/c electrons Figure 7 - 2 : 72Ring frame antiproton scattering angle θ r versus squared momentum transfer t for antiprotons of momentum 15 GeV/c scattering off an opposing electron beam of increasing energy. The plots become more skewed towards higher | t | for higher opposing electron beam momentum. The plots show the relationship between θ r and t given in eq. (7.3.1). Figure 7 - 72 above for HESR energies, where the antiprotons have ring frame momentum 15 GeV/c. Figure 7 - 3 : 73Plot of scattering angle θ versus squared momentum transfer t, in the region of opposing electron beam momentum defined by eq. (7.3.3) Figure 7 - 4 : 74A surface plot of the dependence on t 1 and t 3 in P ( τ optimum ). The units of each axis are (MeV/c) 2 . = σp p ↑↓ − σp p ↑↑ ,(7.4.17) for longitudinal and transverse polarizations respectively is largest, and at what laboratory frame antiproton kinetic energies in the AD range50 − 200 MeV they are maximal. This should provide essential information on the optimal parameters of a 2 × 2 Pauli spin matrices, with i = √ −1, and the 2 × 2 identity and zero matrices respectively. Feynman slash notation / p = γ µ p µ , and the usual conven-tion of Greek characters representing four dimensional space-time indices {0, 1, 2, 3} and Latin characters representing three dimensional space indices {1, 2, 3} are used throughout the thesis. The Dirac equation is ( i γ µ ∂ µ − m ) ψ(x) = 0 , (A.3) where the plane wave solutions are ψ(x) = u(p) e −ip·x and ψ(x) = v(k) e +ik·x , (A.4) for particles and antiparticles respectively, where u are the spinors for particles and v are the anti-spinors for antiparticles. Applying the operator − ( iγ µ ∂ µ + m ) to the left of both sides of the Dirac equation yields the Klein-Gordon equation: ∂ µ ∂ µ + m 2 ψ(x) = 0 . (A.5) k − m) 1 + γ 5 / S(k, λ ′ ) , (A.10) The spinors satisfy the Dirac equation as follows:/ p − m u(p) = 0 , ( / k + m ) v(k) = 0 , (A.11) calculations can be simplified using:γ µ γ µ = 4 , γ µ γ ν γ µ = −2 γ ν , (A.13) γ µ γ ν γ ρ γ µ = 4 η νρ , γ µ γ ν γ ρ γ σ γ µ = − 2 γ σ γ ρ γ ν .Traces of products of gamma matrices are as followsTr [ odd # of γ matrices ] = 0 ,Tr [ γ µ γ ν ] = 4 η µ ν , Figure B- 1 : 1The Mandelstam variables are defined from the general two particle to two particles scattering process, where p 1 and p 2 are the momenta of the incoming particles and p 3 and p 4 are the momenta of the outgoing particles.The Mandelstam variables[79] are defined as followss = ( p 1 + p 2 ) 2 = ( p 3 + p 4 ) 2 , t = ( p 4 − p 2 ) 2 = ( p 1 − p 3 ) 2 , (B.1) u = ( p 3 − p 2 ) 2 = ( p 1 − p 4 ) 2 ,where we have used the law of conservation of four momentump 1 + p 2 = p 3 + p 4 . (B.2) µ , γ ν ] = σ µν and { γ µ , γ ν } = 2 η µν , (C.1) e methods are used to solve the differential equations involving the Heaviside step function in section 6.5. All Laplace transform results needed are presented here.The Laplace transform L { f (τ ) } = F (s) of a function f (τ ) is defined as L { f (τ ) } = F (s) = ∞ 0 e −s τ f (τ ) d τ . (F.1)It is a linear operator, which can be proven as follows:L { c 1 f 1 (τ ) + c 2 f 2 (τ ) } = ∞ 0 e −s τ [ c 1 f 1 (τ ) + c 2 f 2 (τ ) ] d τ , −s τ f 2 (τ ) d τ , = c 1 L { f 1 (τ ) } + c 2 L { f 2 (τ ) } ,where c 1 and c 2 are constants and f 1 (τ ) and f 2 (τ ) are arbitrary functions. Differential equations involving the Heaviside step function can be solved by taking the Laplace transform of the entire equation, applying the relations which follow, solving for L{ f (τ ) } and then taking the inverse Laplace transform of what is left to obtain the solution f (τ ) to the differential equation. L −1 e − τc s s = H ( τ − τ c ) . (F.7) The Laplace inverse is defined, as always, such that L L −1 {f (τ )} = f (τ ) = L −1 {L {f (τ )}} . 0.5ex]$N_+\,/\,N_-$ Polarization versus $N_+\,/\,N_-$ ratio To my brother Dennis, sister Edel and Laura Gibson for constantly lifting my spirits and for always being around when I need them. I thank my office mates in G29 and WR 2.3 for all the good times, and fellow Ph.D. students David Henry, Sinead Keegan, Derek Kitson, Stephan Meissner and Richie Morrin for mutual encouragement to keep us all motivated. I would also like to thank everyone who offered to proof-read this thesis, perhaps they did not know what they were getting themselves into! I thank the Irish Research Council for Science, Engineering and Technology (IRC-SET) for a Postgraduate Scholarship, without which I would never have been able to embark on this research. The funding provided offered me the opportunity to attend many International conferences and interact with some of the leading scientists in this field, which proved invaluable in progressing my research. In particular advice from Professor Mauro Anselmino, Professor Elliot Leader, Professor Hans-Otto Meyer, Dr. Frank Rathmann and Dr. Werner Vogelsang proved very useful. I am very grateful to Professor Peter Hogan for all of his encouragement and advice during my Undergraduate studies. In particular for the many conversations we had, and the insight he provided, when I was deciding on the field of research for my Ph.D. 1.1) where i and j can be either L for longitudinal, or T for transverse. For p ↑ p ↑ Drell -Yan processes the transverse double spin asymmetry is[29] Table Z}.Unpolarized: 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 d σ d Ω 1 particle polarized: Single spin asymmetries 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 A i 2 particles polarized: Polarization transfer: 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 K ij Depolarization: 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 D ij Double spin asymmetries: 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 A ij 3 particles polarized: 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 C ijk 4 particles polarized: 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 C ijkl Table 7 .1. 7n [cm −2 ] n [fm −2 ] t min [(MeV/c) 2 ] θ r min [mrad] 10 8 10 −18 − 1.5575 × 10 −13 2.6 × 10 −11 Table 5 . 51 shows similar expressions but integrated with respect to scattering angle θ.Now L d follows directly from the above expressions via eq. (5.3.17). The natural logarithms are expressed so that they are all positive, i.e. their arguments are greater than one, by using Table 7.4: Values of t 1 and t 3 , in units of (MeV/c) 2 , and the natural logarithms involving them appearing in the integrated spin observables, for various acceptance angles, in units of mrad, obtained by solving eq. (7.3.1). Results are for values of p 1 = 15 GeV/c and p 2 = 50.988 − 2.32222 × 10 6 9.24199 25.6962 2 − 899.809 − 2.32219 × 10 6 7.85584 27.0824 5 − 5617.55 − 2.32195 × 10 6 6.02427 28.9140 10 − 22381.3 − 2.32111 × 10 6 4.64158 30.2966 20 − 88135.2 − 2.31770 × 10 6 3.26946 31.6688 50 − 498035 − 2.29123 × 10 6 1.52617 33.4120 498035 − 2.291 × 10 6 − 2.322 × 10 6 1.52617 33.4120 100 − 520975 − 5.230 × 10 6 − 5.238 × 10 6 2.30641 33.4452 200 − 539229 − 1.119 × 10 7 − 1.119 × 10 7 3.03262 33.4782 300 − 546436 − 1.718 × 10 7 − 1.718 × 10 7 3.44796 33.4914 400 − 550267 − 2.317 × 10 7 − 2.317 × 10 7 3.74039 33.4983454 − 551672 − 2.641 × 10 7 − 2.641 × 10 7 3.86869 33.5009 p 2 [ 2MeV/c] Table 7.6: The integrated spin observables, in units of millibarns, for various opposing electron beam momenta, in units of MeV/c. Results are for fixed values of θ r acc = 50 mrad, p 1 = 15 GeV/c and electron beam polarization P T = 0.9.I out [mb] K out [mb] L in [mb] L d [mb] 50 0.00257272 − 0.00232374 0.01068 0.0108828 100 0.00282946 − 0.0017559 0.00267275 0.00310499 200 0.00288982 − 0.00115439 0.000668854 0.00123563 300 0.00290078 − 0.000874996 0.000297386 0.000841777 400 0.00290458 − 0.000711905 0.000167314 0.0006622 454 0.00290566 − 0.000648743 0.00012989 0.000598142 p 2 [MeV/c] P max τ * [seconds] P ( 2 τ * ) 50 0.096990 7.45156 × 10 11 0.096990 100 0.273516 6.77541 × 10 11 0.269873 200 0.545529 6.63390 × 10 11 0.424074 300 0.691294 6.60883 × 10 11 0.412933 400 0.772397 6.60019 × 10 11 0.372750 454 0.801983 6.59772 × 10 11 0.350905 .8 below.Table 7.8: The maximum polarization achievable and the time taken, in units of seconds,p 2 [MeV/c] P max τ 8% [seconds] 50 0.087385 1.97920 × 10 11 100 0.193311 1.25265 × 10 11 200 0.245759 1.78586 × 10 11 300 0.225285 2.40580 × 10 11 400 0.197800 3.06730 × 10 11 454 0.184451 3.44399 × 10 11 498035 − 2.291 × 10 6 − 2.322 × 10 6 1.52617 14.9914 Table 7.9: Values of t 1 , t 3 and t 4 , in units of (MeV/c) 2 , and the natural logarithms involving them appearing in the integrated spin observables, for various opposing electron beam momenta, in units of MeV/c, obtained by solving eq. (7.3.1). Results are for idealistic fixed values of θ r acc = 50 mrad and p 1 = 15 GeV/c. Here we take the very far fetched t min = − 1.5575 × 10 −1 (MeV/c) 2 corresponding to an electron beam areal density of 10 20 cm 2 .p 2 [MeV/c] Table 7.10: The integrated spin observables, in units of millibarns, for various opposing electron beam momenta, in units of MeV/c. Results are for fixed values of θ r acc = 50 mrad, p 1 = 15 GeV/c and electron beam polarization P T = 0.9.p 2 [MeV/c] P max τ * [seconds] P ( 2 τ * )100 − 520975 − 5.230 × 10 6 − 5.238 × 10 6 2.30641 15.0246 200 − 539229 − 1.119 × 10 7 − 1.119 × 10 7 3.03262 15.0575 300 − 546436 − 1.718 × 10 7 − 1.718 × 10 7 3.44796 15.0707 400 − 550267 − 2.317 × 10 7 − 2.317 × 10 7 3.74039 15.0777 454 − 551672 − 2.641 × 10 7 − 2.641 × 10 7 3.86869 15.0802 I out [mb] K out [mb] L in [mb] L d [mb] 50 0.00257272 − 0.00232374 0.00479191 0.00522841 100 0.00282946 − 0.0017559 0.00120068 0.00198469 200 0.00288982 − 0.00115439 0.000300831 0.00108163 300 0.00290078 − 0.000874996 0.00013382 0.000798785 400 0.00290458 − 0.000711905 0.0000753084 0.000645125 454 0.00290566 − 0.000648743 0.000058469 0.000586789 50 0.208713 7452 0.208648 100 0.496115 6775 0.459284 200 0.751524 6634 0.517906 300 0.844405 6609 0.455736 400 0.889346 6600 0.395042 454 0.904861 6598 0.367339 .12.Table 7.12: The maximum polarization achievable and the time taken, in units of seconds,p 2 [MeV/c] P max τ 15% [seconds] 50 0.172036 3241 100 0.302115 2515 200 0.297568 3851 300 0.248546 5597 400 0.209713 7882 454 0.193168 9504 See Table 2.1. More correctly for a point-like electron F 2 (0) = α/(2 π) + O(α 2 )[84]. This is a result of the anomalous magnetic moment of the electron and was first calculated by Julian Schwinger shortly after the famous Shelter Island Conference in 1947[85]. The anomalous magnetic moment of the electron has now been calculated to order α 4 in QED and agrees with the experimentally measured value to 10 decimal places[86], making QED one of the most accurately verified theories in the history of physics. Transverse to the direction of motion but still in the scattering plane, sometimes called Sideways and denoted S. Remembering we set = 1 throughout the thesis. In the general case there would be a quantity η = ηC ηD ηA ηB (− 1) sA+sB −sC −sD , where s and η correspond to the spin and intrinsic parity of each particle in the reaction, as a factor on the right hand side of eq. (3.4.3). But for elastic scattering, which we are solely interested in, particles A and C are the same and particles B and D are the same; therefore η = (−1) 0 = 1. See expressions for the spin observables in terms of helicity amplitudes presented Ref.[104] and inTable A.10.5 of Ref.[98]. Writing F 1 (t) and F 2 (t) in terms of G E (t) and G M (t) and then using a Taylor expansion of eq. (4.5.1) one obtains the Taylor expansions F 1 (t) = 1 + 2.30777 t + 4.37246 t 2 + O t 3 and F 2 (t) = 1.79285 + 5.5594 t + 12.2483 t 2 + O t 3 . This is the only section of the thesis where the particles involved are not spin 1/2.4 The nucleus of a deuterium atom, a bound state consisting of a proton and a neutron, is called a deuteron. We denote the time variable in each of the dynamical systems by τ to avoid confusion with the squared momentum transfer (Mandelstam t variable) used throughout the thesis. For times well above τ = τ ST , 1 − e − τ / τ ST ≈ 1. The time τ ST depends strongly on the Lorentz γ factor and the mean radius R of the storage ring but is typically of the order of minutes or hours for electron storage rings, seeTable 2.1. L d { ( P 0 P T A out − L in ) ( e λ 1 τ − e λ 2 τ ) + L d ( e λ 1 τ + e λ 2 τ ) } . Electron cooling is described in section 7.1. This section makes extensive use of Laplace transform methods in solving differential equations. All results needed are presented in Appendix F. The ring frame is described in detail in section 7.3.3. If one of the particles is initially at rest the ring frame equals the laboratory frame.3 Many of the expressions in this chapter can be re-written in terms of the relativistic velocity β i = p i /E i of each particle, where i ∈ {1, 2}. In particular p 1 E 2 + p 2 E 1 = E 1 E 2 (β 1 + β 2 ) andE 1 E 2 − p 1 p 2 = E 1 E 2 (1 − β 1 β 2 ). Since for antiproton-electron scattering the longitudinal spin-transfer observable from section 4.5.2 is greater than the transverse spin-transfer observable we do the numerical calculation for the longitudinal case. It has been shown at RHIC that one can rotate the polarization of the beam from longitudinal to transverse, or vice versa, without any loss of polarization. Since the stable spin direction in a storage ring is transverse, it is likely that the beam will circulate in the ring with transverse polarization but be rotated to longitudinal directly before the target and back to transverse directly after the target. This is how RHIC operates[133]. The antiproton beam will eventually be utilized in a transversely polarized state in order to measure A T T and hence obtain information on the transversity distribution of quarks in the nucleon. At these high energies t 3 ≈ t 4 , due to the skewness of the relationship between θ and t in the ring frame, as seen inFigure 7-2. Hence the major contribution to ln(t 1 t 4 /t min t 3 ) comes from ln(t 1 /t min ). using the ring frame expressions of eqs. (7.3.7 and 7.3.8).The spin observables needed for spin filtering presented in section 4.5 can be converted into the ring frame by using eq. (7.3.7). Figure 7-3 shows how the regions of scattering angle defined by the acceptance angle θ r acc can be converted into regions of squared momentum transfer t. Hence the angular regions of integration of the spin observables, presented inTable 5.1 can be presented as regions of integration over squared momentum transfer t inTable 7.3. Using the expressions of section 4.5.2 andTable 7.3, one has that for longitudinal 4 polarization:Transverse polarization requires Longitudinal polarization requirest 4 leading t imaginary part of the hadronic amplitude is given by the Optical Theorem:and the leading t part of thep p electromagnetic amplitude is given bywhere F 1 ≈ 1 for small | t | and e δ i ≈ 1 as δ is small[137]. The Coulomb phaseshift, e δ i , accounts for the small correction to the single-photon exchange amplitude coming from multi-photon exchange[138]. The hadronic amplitude iswhere ρ = ρ(s, t) = Re{φ h + } / Im{φ h + } the ratio of real to imaginary parts of the hadronic non-flip amplitude. For small | t | one has that e b t ≈ 1[95,105]. Thus to leading order in small | t | the spin-averagedp p cross-section isThe electromagnetic and hadronic amplitudes are of equal size wheni.e. whenwhere the relations k cm √ s = p lab M and E lab = (s − 2 M 2 ) /(2M) have been used.Finally using the relativistic laboratory velocity β lab = p lab / E lab gives[95,105]the critical squared momentum transfer below which electromagnetic effects dominate and above which hadronic effects dominate .At high energies β lab ≈ 1 and ρ 2 ≈ 0[139,140], giving the often used resultThis region of momentum transfer is referred to in the literature as the Coulomb-Nuclear-Interference (CNI) region.In analogy to t c above, we now derive an expression for the critical antiproton laboratory frame scattering angle below which the electromagnetic interaction dominates the hadronic interaction inp p scattering. It is then shown that the scattering angles of importance in spin filtering are below this critical angle, and hence the electromagneticp p cross-sections calculated in Chapters 3 and 4 provide a good approximation to the totalp p interaction in this region.Using eqs. (4.5.5 and 4.5.6) and the relation between the scattering angles in the Centre-of-Mass and LAB frames sin θ cm sin θ lab = p lab k cm ,(7.4.9)and Taylor expanding sin 2 θ lab for small θ lab gives − t ≈ p 2 lab θ 2 lab . (7.4.10)So the critical scattering angle below which the electromagnetic interaction dominates the hadronic interaction is(7.4.11)and inserting the expression for t c in eq. (4.2.1) one obtains that electromagnetic effects dominate hadronic effects for laboratory scattering angleswhere M and T are the antiproton mass and kinetic energy in the Laboratory frame respectively[72], and the numerical result is for FILTEX kinetic energies of T = 23 MeV where σp p tot ≈ 325 mb[134,140]and ρ ≈ 0.1[140,141]. This gives the strong inequality[72]θ min ≪ θ e ≪ θ acc ≪ θ c lab , (7.4.13)Using the above relations one can computēwhere the Feynman slash notation / p = γ µ p µ has been used. Now simplify by using the Dirac equation and its conjugatewhere M is the mass of the particle, to obtainRearranging gives the general form of the Gordon Decomposition identity:Another Gordon decomposition identity, which we now derive, can be used to simplify the gamma matrix structure of the antiproton current using the v anti-spinors.Equation (C.3) for anti-spinors givesBut now we must use the Dirac equation for antiparticles and its conjugateAppendix DFeynman rules for QEDThe Lagrangian for Quantum Electrodynamics (QED) isMomentum is conserved at each vertex, fermion loops receive an additional factor of (−1) and undetermined loop momenta are integrated over by:We work in the Feynman gauge where the gauge parameter is set to ξ = 1. The charge factor Q = −1 for an electron or antiproton, and Q = +1 for a positron or proton. Time increases from left to right in all Feynman diagrams throughout the thesis.Appendix E Sample Mathematica codeThe Mathematica code to derive the generic depolarization equation from eq. (3.2.11)is included in this appendix. The Mathematica add on package for High Energy Physics Tracer.m[87]is used, which defines G5 ≡ γ 5 and uses Spur[] for Trace. All other QED calculations in the thesis are done analogously. Only input commands are included. All parameters are as defined and used throughout the thesis. . M(+ , + ; + , + , M(+, + ; +, +) ≡ φ 1 . M(− , − ; + , + , M(−, − ; +, +) . M(+ , + ; − , − , φ 2 10. M(−, + ; −, +M(+, + ; −, −) ≡ φ 2 10. M(−, + ; −, +) . M(+ , − ; + , − , φ 3 11. M(−, + ; +, −M(+, − ; +, −) ≡ φ 3 11. M(−, + ; +, −) . M(+ , − ; − , + , φ 4 12. M(+, − ; −, −M(+, − ; −, +) ≡ φ 4 12. M(+, − ; −, −) . M(+ , + ; + , − , φ 5 13. M(−, − ; +, −M(+, + ; +, −) ≡ φ 5 13. M(−, − ; +, −) . M(+ , + ; − , + , φ 6 14. M(−, − ; −, +M(+, + ; −, +) ≡ φ 6 14. M(−, − ; −, +) . M(+ , − ; + , + , φ 7 15. M(−, + ; −, −M(+, − ; +, +) ≡ φ 7 15. M(−, + ; −, −) . M(− , + ; + , + , φ 8 16. M(−, − ; −, −M(−, + ; +, +) ≡ φ 8 16. M(−, − ; −, −) . << Tracer/Tracer.m. << Tracer/Tracer.m; . = G[le, U] + G , le, G5]**G[le, S1= G[le, U] + G[le, G5]**G[le, S1]; . U Sp3 = G[le, ] + G[le, G5]**g[le, SP3 = G[le, U] + G[le, G5]**G[le, S3]; Spin trace *). (* Spin trace *) . = Spintrace, G[le, P1 + M U]**sp1**(h G[le, p1.{nu} + p3.{nu}))**G[le, p3 + M U]**SP3**(H G[le, {mu}SpinTrace = G[le, p1 + M U]**SP1**(H G[le, {nu}] + F (p1.{nu} + p3.{nu}))**G[le, p3 + M U]**SP3**(H G[le, {mu}] . + , p1.{mu} + p3.{mu}+ F (p1.{mu} + p3.{mu})); . = Spintrace2, Fullsimplify, SpinTrace/.le -> 1SpinTrace2 = FullSimplify[SpinTrace/.le -> 1]; Spinless trace *). (* Spinless trace *) . = Spinlesstrace, le2, p4 + m U]**(h G[le2, {mu}] + f (p2.{mu} + p4.{mu}))**G[le2, p2 + m U]**(h G[le2,{nu}] + f (p2.{nu} + p4.{nu})SpinlessTrace = G[le2, p4 + m U]**(h G[le2, {mu}] + f (p2.{mu} + p4.{mu}))**G[le2, p2 + m U]**(h G[le2,{nu}] + f (p2.{nu} + p4.{nu})); . Nospur, 1NoSpur[1]; . = Spinlesstrace2, Fullsimplify, SpinlessTrace/.le2 -> 2SpinlessTrace2 = FullSimplify[SpinlessTrace/.le2 -> 2]; Contract all Lorentz indices *). (* Contract all Lorentz indices *) . Simplify, ContractEpsGamma[SpinTrace2 SpinlessTrace2msq1 = Simplify[ContractEpsGamma[SpinTrace2 SpinlessTrace2]]; . Fullsimplify, msq1msq2 = FullSimplify[msq1]; Take the depolarization observable away from the spin-averaged *). (* Take the depolarization observable away from the spin-averaged *) . = Spinaveraged, Simplify, msq2/.{S1 -> 0, S3 -> 0}SpinAveraged = Simplify[msq2/.{S1 -> 0, S3 -> 0}]; SpinTerm1 = SpinAveraged -msq2. SpinTerm1 = SpinAveraged -msq2; Now make algebraic simplifications *). (* Now make algebraic simplifications *) . = Spinterm2, Simplify, SpinTerm1/.{p2.p4 -> m^2 -t/2, p1.p3 -> M^2 -t/2, p4.p4 -> m^2, p3.p3 -> M^2, p2.p2 -> m^2, p1.p1 -> M^2, p3.p4 -> s/2 -M^2/2 -m^2/2, p1.p2 -> s/2 -M^2/2 -m^2/2, p3.p2 -> M^2/2 + m^2/2 -u/2, p1.p4 -> M^2/2 + m^2/2 -u/2, S1.p1 -> 0, S3.p3 -> 0}SpinTerm2 = Simplify[SpinTerm1/.{p2.p4 -> m^2 -t/2, p1.p3 -> M^2 -t/2, p4.p4 -> m^2, p3.p3 -> M^2, p2.p2 -> m^2, p1.p1 -> M^2, p3.p4 -> s/2 -M^2/2 -m^2/2, p1.p2 -> s/2 -M^2/2 -m^2/2, p3.p2 -> M^2/2 + m^2/2 -u/2, p1.p4 -> M^2/2 + m^2/2 -u/2, S1.p1 -> 0, S3.p3 -> 0}]; . = Spinterm3, Simplify, SpinTerm2/.{p3 -p1 -> q, p2 -p4 -> q, u -> 2m^2 + 2M^2 -s -t}SpinTerm3 = Simplify[SpinTerm2/.{p3 -p1 -> q, p2 -p4 -> q, u -> 2m^2 + 2M^2 -s -t}]; . = Spinterm4, Simplify, SpinTerm3/.{p3.S1 -> q.S1, p1.S3 -> -q.S3}SpinTerm4 = Simplify[SpinTerm3/.{p3.S1 -> q.S1, p1.S3 -> -q.S3}]; . = Spinterm5, Fullsimplify, SpinTerm4 /.{F -> -F 2/2M, f -> -f 2/2m}SpinTerm5 = FullSimplify[SpinTerm4 /.{F -> -F 2/2M, f -> -f 2/2m}]; . = Genericdepolarisationequation, Fullsimplify, SpinTerm5 /.{H -> F 1GenericDepolarisationEquation = FullSimplify[SpinTerm5 /.{H -> F 1 h -> f 1 + f 2, p1.S3 -p3.S3 -> -q.S3, -p1.S3 + p3.S3 -> q.S3, p2.S1 -p4. S1 -> q.S1}+ F 2, h -> f 1 + f 2, p1.S3 -p3.S3 -> -q.S3, -p1.S3 + p3.S3 -> q.S3, p2.S1 -p4.S1 -> q.S1}] The output is now the generic depolarization equation, for the electromagnetic scattering of two non-identical spin 1/2 particles with internal structure defined by form factors, used in the thesis. All helicity amplitudes and spin observables for elastic spin 1/2 -spin 1/2 scattering to first order in QED can be obtained from eq. 3.2.11) in a similar fashionThe output is now the generic depolarization equation, for the electromagnetic scat- tering of two non-identical spin 1/2 particles with internal structure defined by form factors, used in the thesis. All helicity amplitudes and spin observables for elastic spin 1/2 -spin 1/2 scattering to first order in QED can be obtained from eq. (3.2.11) in a similar fashion. . S A Goudschmidt, G H Uhlenbeck, Nature. 117264Bibliography [1] S. A. Goudschmidt and G. H. Uhlenbeck, Nature 117, 264 (1926). . P A M Dirac, Proc. Roy. Soc. Lond. 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[ "Autocomplete Repetitive Stroking with Image Guidance", "Autocomplete Repetitive Stroking with Image Guidance" ]
[ "Yilan Chen \nCity University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong\n", "Kin Chung Kwan \nCity University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong\n", "Li-Yi Wei \nCity University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong\n", "Adobe Research \nCity University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong\n", "Hongbo Fu \nCity University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong\n" ]
[ "City University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong", "City University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong", "City University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong", "City University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong", "City University of Hong Kong\nUniversity of Konstanz\nCity University of Hong\nKong" ]
[]
a) user input (b) suggestion (c) accept (d) type visualization (e) result Figure 1: Example of our system workflow. A user stipples over a leaf region of a reference image (a) while our system predicts what she might draw next (b) (blue strokes: inferred exemplars; pale red region: inferred target region; semi-transparent strokes: system suggestions), which is then accepted by the user (c) (green strokes: user inputs or accepted suggestions in this scene). (d) visualizes all the manually drawn content in black (261 strokes) and autocompleted content in red (3510 strokes). (e) shows the final result with different repetitive stroke patterns over different regions. Our autocomplete system can reduce tedious repetitive inputs, while being fully under user control.ABSTRACT Image-guided drawing can compensate for the lack of skills but often requires a significant number of repetitive strokes to create textures. Existing automatic stroke synthesis methods are usually limited to predefined styles or require indirect manipulation that may break the spontaneous flow of drawing. We present a method to autocomplete repetitive short strokes during users' normal drawing process. Users can draw over a reference image as usual. At the same time, our system silently analyzes the input strokes and the reference to infer strokes that follow users' input style when certain repetition is detected. Users can accept, modify, or ignore the system predictions and continue drawing, thus maintaining the fluid control of drawing. Our key idea is to jointly analyze image regions and operation history for detecting and predicting repetitions. The proposed system can effectively reduce users' workload in drawing repetitive short strokes and facilitates users in creating results with rich patterns.
10.1145/3478512.3488595
[ "https://arxiv.org/pdf/2108.07115v1.pdf" ]
237,091,866
2108.07115
fa7b29d6b54ab433eef5d78791fa7140a7a6734f
Autocomplete Repetitive Stroking with Image Guidance Yilan Chen City University of Hong Kong University of Konstanz City University of Hong Kong Kin Chung Kwan City University of Hong Kong University of Konstanz City University of Hong Kong Li-Yi Wei City University of Hong Kong University of Konstanz City University of Hong Kong Adobe Research City University of Hong Kong University of Konstanz City University of Hong Kong Hongbo Fu City University of Hong Kong University of Konstanz City University of Hong Kong Autocomplete Repetitive Stroking with Image Guidance CCS CONCEPTS • Computing methodologies → Computer graphics• Human- centered computing → User interface design KEYWORDS Interactive SystemAutocompletionDigital DrawingPredictionTexture Synthesis a) user input (b) suggestion (c) accept (d) type visualization (e) result Figure 1: Example of our system workflow. A user stipples over a leaf region of a reference image (a) while our system predicts what she might draw next (b) (blue strokes: inferred exemplars; pale red region: inferred target region; semi-transparent strokes: system suggestions), which is then accepted by the user (c) (green strokes: user inputs or accepted suggestions in this scene). (d) visualizes all the manually drawn content in black (261 strokes) and autocompleted content in red (3510 strokes). (e) shows the final result with different repetitive stroke patterns over different regions. Our autocomplete system can reduce tedious repetitive inputs, while being fully under user control.ABSTRACT Image-guided drawing can compensate for the lack of skills but often requires a significant number of repetitive strokes to create textures. Existing automatic stroke synthesis methods are usually limited to predefined styles or require indirect manipulation that may break the spontaneous flow of drawing. We present a method to autocomplete repetitive short strokes during users' normal drawing process. Users can draw over a reference image as usual. At the same time, our system silently analyzes the input strokes and the reference to infer strokes that follow users' input style when certain repetition is detected. Users can accept, modify, or ignore the system predictions and continue drawing, thus maintaining the fluid control of drawing. Our key idea is to jointly analyze image regions and operation history for detecting and predicting repetitions. The proposed system can effectively reduce users' workload in drawing repetitive short strokes and facilitates users in creating results with rich patterns. INTRODUCTION Drawing is a common form of artistic expression. By varying the stroke, texture, and shading, artists can create drawings with various styles [5]. Yet, it remains a largely manual process that may require significant artistic expertise and repetitive manual labor. Various methods have been proposed to synthesize user-initiated repetitive strokes [22,48] to reduce the manual labor. However, such methods still require sufficient artistic expertise or experience for high-level picture composition. One common way to overcome this skill barrier is to use a reference photo as a scaffold for drawing, i.e., tracing a reference photo physically via transparent papers or digitally via layers in digital drawing applications. With a given reference, many methods exist to automate the synthesis of details, such as contours, textures, or strokes [3, 4, 7, 13, 25-27, 37, 43, 46], with the effects tunable via input parameters or exemplars. However, since these algorithms largely predefine the behaviors, their results may look canned ( Figure 3) and cannot give users a sense of ownership. Furthermore, tweaking parameters or providing exemplars can break the spontaneous flow of direct drawing manipulation, (a) our result (b) produced with [1] (c) produced with [30] Figure 3: Our work is designed to reduce the workload of completing repetitive patterns during the manual drawing process. The full control of the drawing process leads to more dynamic results than (b) Photoshop's Art History Brush Tool [1] and (c) StippleShop [30]. which is important to creative decision making [18] and essential to a user's enjoyment and exploration [40]. Manual drawing provides sufficient freedom for individual expressing even when scaffolded with a reference image [47], and its typical interface (e.g., brush, eraser) is familiar to general users. Thus, we aim to enhance the manual drawing process and the typical UI design, by automating tedious repetitions. Our idea is to bridge the two extremes: manual drawing, which allows full control but can be tedious; and image-based algorithmic synthesis, which saves efforts but provides limited user control and interactivity. As the first attempt towards this goal, our approach focuses on autocompleting repetitive short strokes, which are very common in pen-and-ink drawing (Figure 2), under the guidance of a reference image. Like typical digital drawing applications, users can draw freely on a reference image with our system. Meanwhile, our system analyzes the relationships between user inputs and the reference image, detects potential repetitions, and suggests what users might want to draw next. Users can accept, reject, or ignore the suggestions and continue drawing, thus maintaining the fluid control of drawing. See Figure 1 for an example scenario. The challenge of autocompletion is to predict suggestions that respect both users' inputs and the reference image. Our method is inspired by image analogy [13] and operation history analysis and synthesis [48] while leveraging two key insights. First, since the act of drawing repetitive strokes usually indicates specific intentions (e.g., filling an object or hatching a shading region), we use the common image features among the coherent repetitive strokes to infer the intended regions. Second, the drawing usually relates to the underlying reference image (e.g., the density of strokes with respect to the image lightness). Therefore, we analyze the properties of both the drawing and the reference image to infer possible relationships as contextual constraints for stroke prediction. We implemented a prototype and conducted a pilot study with participants in different backgrounds to evaluate its utility and usability. The quantitative analysis and qualitative feedback, as well as various drawing results created by users, suggest that our system effectively reduces users' workload in drawing repetitive short strokes and facilitates users in creating results with rich patterns. RELATED WORK 2.1 Image-assisted Drawing Many drawing support tools adopt reference images and provide intelligent assistance to novices, e.g., beautifying users' sketches with extracted image features [20,25,41,47], or providing educational guidance to novice users [16,31,45]. We share a similar goal to [3,10,43] so as to reduce the user workload. However, these works use predefined algorithms to generate strokes along cursor movement and only take users' input as an indicator of where to render, thus greatly limiting users' artistic freedom. In contrast, we aim to provide more flexibility between automatic synthesis and manual artistic control by autocompleting tedious repetitions during users' normal drawing processes. Image-based Artistic Rendering Our work is related to image-based artistic rendering (IB-AR) [23], especially stroke-based methods and example-based methods. Stroke-based methods create artistic results from images by strategically generating brushstrokes whose properties (e.g., position, density, orientation, color, size) are related to the image properties (e.g., gradient, edge, color, salience) [12]. Among those methods, the closest to ours are the early image-based pen-and-ink rendering methods [14,38], which allow users to input sample elements for distribution. However, users have to prepare the sample elements separately (usually as a standalone file) and then tweak parameters to view the rendered output. In contrast, our system lets users directly specify exemplars on a reference image while silently inferring the distribution properties. Example-based methods aim to model the visual features of example images for transferring. There are two major modeling approaches: the parametric approach [8,9,19] that is based on the summary statistics of stroke characteristics and thus preserves the global textures better, and the non-parametric approach [7,13,21] that is based on patch-wise mapping and thus captures the local structures better. We combine both methods for generating strokes: the parametric approach to infer statistical relationships between stroke properties and image features, and the patch-wise matching method to preserve the local arrangements of strokes. Stylit [7] allows users to stylize a rendered ball and simultaneously propagates the style to arbitrary 3D shapes. Our method shares a similar idea in interactive style propagation but with two main differences. First, instead of propagating a style globally, we propagate a style to its perceptually similar local areas so that users can conveniently define different styles in different areas. Second, we represent drawings as discrete stroke operations instead of raster textures for better preserving their structures and enabling procedural editing [39], such as changing the color or size of the drawn strokes. Operation History-assisted Authoring Operation histories [33] have been leveraged in different authoring tasks, such as sketching [48], animation [34,49], modeling [35,42], beautification of freehand drawings [6], and handwritings [53]. Our work is most closely related to that by Xing et al. 's [48], which autocompletes repetitive sketching by analyzing the dynamic operations recorded during authoring. Our method extends their work to consider additional information from a reference image and thus enables the propagation of strokes to regions with similar image attributes such as color or semantic meaning. In our use scenario, an operation is an input stroke, so our work is also related to stroke pattern analysis and synthesis [2,4,15,17,22]. These works disregard the temporal relationship among past strokes and do not use image guidances and thus are different from ours. To sum up, we list our major differences from the discussed closely related works in Table 1. Table 1: The differences between our tool and closely related works. "batch" means the generation is performed in a batch, based on predefined attributes; "dynamic" means the generation is performed based on dynamic operation history. "direct" means users can specify a style by directly operating on the output. "Y" and "N" represent yes and no, respectively, for using image references. Our prototype follows a standard digital drawing interface, with the added autocomplete feature, as shown in Figure 4. A user draws on top of a reference image displayed semi-transparently on the main canvas, while our system analyzes the input strokes and the reference image in the background. Autocomplete In the autocomplete mode, our system automatically analyzes whenever the user finishes a new stroke. When a potential repetition is detected, our system highlights the currently repetitive strokes and an inferred propagation region, updates the inferred parameters in the filling property panel, and generates autocompletion suggestions. Users can accept or reject all the suggestions via hotkeys, accept part of them via lasso selection, or ignore them and continue to draw ( Figure 5). The suggestions will keep updating according to user inputs. Our system provides a set of tools to refine the autocompleted results. Propagation region editing. Users can create/add/subtract a new region using the intelligent scissors tool [32] or expand an existing region by a fixed width ( Figure 4e) for stroke autocompletion. Figure 6 shows an example of creating a new region for stroke regeneration. Density editing. Users can tweak three parameters to adjust the density of the generated strokes: the average spacing, the lightness coefficient and the gradient coefficient. The latter two define the relationships between density and image lightness/gradient, respectively. Our system automatically updates these parameters upon prediction, and the updated parameters provide a starting point for users to manipulate. Figure 7 shows an example. Orientation editing. Our system automatically predicts whether the input exemplar correlates with the image flow, which can also be tweaked by users manually. Users can also modify the image flow field via the gesture brush, and the touched strokes will be rotated to align with the gesture direction. See Figure 8 for an example. Interactive Editing (a) (8, 0, 0) (b) (15, 0, 0) (c) (8, 0.2, 0) (d) (8, Auxiliary Functions Our prototype also includes the auxiliary functions below. These are not unique to our system but can facilitate the usual drawing processes. Post-edit stroke properties. Users can select the existing strokes and edit their properties, such as size and color. Auto-color. This function, when toggled on, can automatically colorize strokes with color from the reference image. Switch view. Users can press the space key to switch between the canvas view, reference view, and pure drawing view. OUR APPROACH To support the autocomplete functionalities described in Section 3, our system involves two key algorithm steps: (1) inferring the input exemplar, the output region, and the contextual constraints from the stroke history and the reference image; (2) synthesizing suggestive strokes accordingly. This section first describes how p v neighborhood radius local orientation Figure 10: Illustration of our synthesis algorithm. We synthesize the predicted strokes (in green) from previously drawn strokes (in gray) by matching their neighborhoods. to synthesize (Section 4.1) strokes, assuming all the information is available, and then explains how to infer (Section 4.2) the necessary information for synthesis. Stroke Synthesis Problem statement. The inputs to our stroke synthesis method include an exemplar consisting of repetitive strokes, the reference image , a target region mask , an orientation map , and a radius map . Pixel values of denote the extents of stroke spacing: a smaller value leads to a denser distribution. Our goal is to compute an output set of strokes over the output region , such that is similar to with respect to . We describe how to infer , , , and from user interactions with in Section 4.2. Key idea. We extend the discrete element texture synthesis method [29,48], which represents strokes as point samples and iteratively improves the sample distribution by minimizing the neighborhood difference between the exemplar and the output, with an additional reference image. First, we combine sample neighborhoods [29] with image features [13] for measuring neighborhood difference. Second, the range and orientation of sample neighborhoods are determined by the radius and orientation maps inferred from the reference image. Figure 10 shows our key idea. Stroke representation. As shown in Figure 9a, a stroke is an ordered list of sample points, each with a timestamp and appearance attributes such as thickness and color. Here we focus on autocompleting short strokes, so we represent each stroke by its centroid and the average direction for efficiency during synthesis, without considering any other information of the original stroke. To take the drawing order into consideration, we obtain the dominant direction by averaging the vectors from the start point to each subsequent point. After synthesis, we reconstruct all the sample points according to the updated centroid and direction. Initialization. We pre-process the target region mask by removing the area occupied by existing strokes in the same layer to avoid cluttering, and then initialize the output by generating sample positions with Poisson-disk sampling based on the radius map . For each sampled position, we copy the input stroke with the smallest image feature distance , which will be explained in Equation (2). We then optimize the output for a few objectives, as detailed below. Neighborhood term. We define the neighborhood of a stroke as both its neighboring strokes as well as an ( ) × ( ) image patch around its centroid, where ( ) is the radius value at . Prior methods (e.g. [29]) determine the neighboring strokes by spatial distances. Thus, the neighborhood radius should be large enough in order to capture an underlying pattern. However, this might include redundant strokes and thus decrease the performance. Therefore, we adopt Zhao et al.'s method [52] to automatically find a minimum representative neighborhood, considering not only the spatial distance between strokes but also their locations. As depicted in Figure 9b, we set the neighborhood radius of the center stroke to 2 ( ). We then divide all the strokes within the neighborhood radius into four quadrants with respect to the local frame defined by the orientation at ( ), and collect the nearest strokes from each quadrant as the representative neighborhood, denoted as N( ). In our implementation, we set = 4 for the input exemplar and = 1 for the output strokes to ensure that each output neighborhood can be maximally matched. For a stroke and a neighboring stroke ′ ∈ N( ), we compute their difference in position and direction as: ( ′ , ) = 1 ( ) ( ) −1 ( ′ ) − ( ) , ( ) −1 ( ′ ) − ( ) ,(1) which is computed in the local frame defined by the radius map and orientation map . Therefore, the neighborhood distance between an output stroke and an input stroke is: ℎ ( , ) = ∑︁ ′ ∈N( ) ˆ( ′ , ) −ˆ( ′ , ) 2 + | ( ) − ( )| 2 ,(2) where ′ is the matched input sample for ′ via the Hungarian algorithm [28,29], the second term measures the image feature distance , and (= 0.1 in our implementation) controls the relative weighting. We use the mean * color of an × patch at the stroke centroid as the image feature vector. The overall neighborhood term to minimize is: ℎ ( , ) = ∑︁ ∈ min ∈ ℎ ( , ).(3) Correction term. Since the neighborhood term is a one-way matching from the output neighborhoods to the input neighborhoods, sometimes the optimization would tend to leave out some void regions. Besides, the neighborhood term does not preserve strokes' alignment to the image (e.g., Figure 11e). To address these issues, we apply a correction term. We compute a weighted centroidal Voronoi diagram from all the strokes' center points, using 1 as weight, and denote the computed region centroids as {¯}. Thus we can minimize the distance between each output stroke centroid and the region centroid, defined as follows: ( ) = ∑︁ ∈ | ( ) −¯( )| 2 .(4) Solver. The energy function we aim to minimize is defined as: ( , ) = (1 − ) ℎ + .(5) We iteratively minimize the energy function following the EM methodology in [29]. In each iteration, for each output stroke , we search for the most matched input stroke to minimize ℎ , compute the Voronoi diagram centroid¯to minimize , and solve a least-squares system combining both terms. Let be the total number of iterations. For the −th iteration, we set = ( / ) 2 , which means that more weight is given to ℎ in the beginning of iterations, so that we can optimize the neighborhood distribution first before doing corrections, which leads to better results. Figures 11b to 11d show the iterative optimization process of both the objectives. In comparison, Figure 11e shows the result without the correction term and Figure 11f shows the result without using the image neighborhood in both initialization and optimization. Inference In this section, we describe how to infer , , , and used for our synthesis method in Section 4.1 from user interactions with . Input exemplar . This step aims to detect whether stroke repetitions exist and obtain the repetitive group as an exemplar for the synthesis process. Since people usually draw strokes in a coherent manner [48] and they usually have specific intentions when drawing repetitive strokes, we assume the example strokes to be temporally consecutive and have certain similar properties. We start from the last stroke input by the user and search backward in the stroke sequence to incrementally find strokes that have similar shape and image features to the last stroke. Specifically, the stroke shape similarity is measured with the Fréchet distance, and the image features include * color (weighted by 0.12, 0.44, and 0.44 to suppress the impact of lightness) and precomputed semantic segmentation [51] at a stroke's center. We compare the standard deviation of a feature in the traversed strokes against a threshold (15/255 for the color feature, 1 for the segmentation feature) for similarity measurement. The back-traversal stops when the next stroke does not contain any similar feature or > 50. These strokes serve as the input exemplar for the synthesis process. See Figure 12 for an example of the incremental searching process. Output region . The shared features of the obtained stroke exemplar also indicate the intended region. For instance, if all of the exemplar strokes are inside the same object segmentation region, it is very likely that the user intends to fill that region. Therefore, we . The cumulative number is determined when both cost curves exceed the threshold. Note that the third region prediction result is only for demonstration: since the exemplar only contains one stroke (i.e., = 1), it is not considered a valid exemplar and will not be used for synthesis. use the shared features obtained in the exemplar grouping process to find a similar region for output. Since there are only two features in our implementation, we simply obtain the region by GrabCut [36] if the * color feature is shared among the exemplar strokes, directly take the corresponding segmentation if the semantic feature is shared, and take the intersection if both features are shared. See Figure 12 for an example. When there are multiple disconnected regions, we retain the nearest region to the user's last stroke and discard the rest, because it is less natural to propagate to distant regions. Contextual constraints. Since the drawing usually relates to the underlying reference image, we analyze the properties of both the drawn strokes and the reference image to infer possible relationships that control the global distribution of strokes. Orientation . Artists usually adjust the stroke directions to convey curvatures, but they may sometimes randomize or fix the stroke orientation regardless of the depicted objects to create different visual effects. Therefore, the problem is to decide which case the input exemplar implies. We first compute the edge tangent field (ETF) [24] for the reference image and then calculate the angles between the exemplar strokes and the ETF directions at their centroids. If the standard deviation of the angles is small (less than 15 degrees), we consider the stroke orientations to be related to the ETF and take the ETF as the orientation field; otherwise, we set a default global coordinate frame to each point of the orientation field. Radius . Since density is inversely proportional to the spacing between strokes, we reframe the problem as predicting a radius map that controls the extent of stroke neighborhoods. First, we compute the distance from each exemplar stroke to its nearest neighbor. We assume a linear relationship between these minimum distances and the image features, including image lightness and gradient strength at a stroke's centroid, represented as: = 1 · t,(6) where t denotes the coefficients to solve. With the fitted linear model, if the squared correlation value is lower than 0.5 (the closer to 1, the better explanation), we use the model to compute a radius map. Otherwise, we consider the density as uniform and create a constant radius map with the average spatial distance of the exemplar. We then update the UI with the computed coefficients. EVALUATION We conducted a pilot study to evaluate the utility and usability of our approach. We compared three modes through quantitative analysis and qualitative feedback. Autocomplete Users have full access to our prototype, including autocomplete and interactive editing. Interactive batch filling (aka batch mode) Users are required to create a texture example first and then manually specify the properties for batch filling. It simulates the sequential procedure in many IB-AR methods (e.g., [38]), although they rarely allow users to directly define examples on target images. This mode is performed on our system with the autocomplete function off. Fully manual drawing (aka manual mode) Users have to manually draw each stroke without any automatic synthesis. We also tested the expressiveness of our system through an open creation session and obtained comments for future improvements. Target Session The goal of this session is to compare the three interaction modes in utility and usability. Since we aim to facilitate image-scaffolded drawing, we hope to include general users from different background while focusing more on less skillful users, who are more likely to use reference images. We thus recruited 12 participants, including nine novices with little drawing experiences, two amateurs with some experiences (P3, P4), and a student majored in illustration (P5). Most of the studies were conducted on a Lenovo Miix 520 tablet with stylus in a lab environment, except two studies conducted remotely with mouse due to the pandemic. The study procedure consisted of the following parts and took each participant about two hours in total. Tutorial. Each participant was first given a brief introduction to our system and then asked to fill the apple in Figure 4 with short hatches as a warm-up task. They were encouraged to vary the density and orientation of input strokes and get familiar with the features of our system. Target tasks. We used a within-subjects design, where each participant was asked to reproduce two target drawings ( Figure 13) in all the three modes: autocomplete, interactive batch filling, and fully manual drawing. The target drawings include an object and a landscape, which are common illustration topics (e.g., Figure 2). The assigned order of modes was counter-balanced among all the participants. Since we focus on region filling, we asked the participants to draw the outlines of both images in advance, so that they could focus on drawing the textures during the study. We encouraged the participants to finish each drawing as soon as possible, preferably in a dozen of minutes, but without any hard time limit. After completing the two drawings in each mode, each participant filled in a NASA-TLX questionnaire [11]. At the end, we asked the participants about their preferred mode, usage experience and other comments. Open session The goal of this session is to observe users' interaction with our system and learn about users' subjective experience. We invited seven participants (one professional artist, two amateurs and four novices) for this session. They were asked to create a drawing freely from the same reference image (Figure 15a) with our system. The reference image was a portrait photo, which is also common in illustrations. The only requirement was that the drawings should contain some repetitive content. We again gave a tutorial in the beginning and conducted the task on a Lenovo Miix 520 tablet with stylus. The participants were encouraged to think aloud and describe their thought process and interactions during this session. After this task, participants could optionally create more drawings with any images they want. Since our prototype does not contain all common functions in commercial drawing tools, we allow the participants to retouch the result drawings without adding more strokes in Photoshop. Results and Observations Workload. Figure 14a shows the perceived workload scores from the target session. Generally, the autocomplete mode received the lowest (i.e., best) scores for almost all the factors. One-way ANOVA showed the three modes have significant difference in physical demand (F=10.69, p < 0.001) while no significant difference in other factors. Regarding the physical demand, post-hoc pairwise tests showed that the autocomplete mode and batch mode were both rated significantly lower than manual mode, while had no significant difference from each other. This matches our expectation, since automatic synthesis should only reduce physical load and not cause extra pressure than manual work. Efficiency. We calculate the average completion time (Figure 14b) and stroke count (Figure 14c) in each mode and each task. Generally, the system synthesized about 82% strokes in the autocomplete mode and about 92% strokes in the batch mode. Although the manual mode took the shortest time for the participants to complete, it also resulted in the fewest total number of strokes. We thus calculated the strokes per minute for each mode: autocomplete (111.03, SD=38.76), batch (101.98, SD=45.13), manual (115.95, SD=46.73). It turns out automatic generation did not improve the efficiency, probably because the users spent extra time adjusting and experimenting with the generated effects instead of just drawing strokes. It should be noted that such directed tasks omit the time for exploring alternative patterns, which, however, might be high in a fully manual case. Quality. We asked 30 external volunteers to evaluate the quality of participants' drawings, as shown in Figure 19. We randomized all the drawings created by the participants, showed each output drawing alongside the target drawing, and asked volunteers to rate the resemblance of the output drawing to the target drawing, on a scale from 1 (very dissimilar) to 5 (very similar). The volunteers were instructed to focus more on the overall stroke distributions and flows instead of individual stroke thickness and detailed shapes. We calculated the average scores for each mode: autocomplete (3.10, SD=1.24), batch (3.09, SD=1.21), manual (2.98,SD=1.20). The quality of the drawings created with automatic synthesis is slightly better than the fully manual drawings, but without significant difference. From the participants' perspective, three novices commented the automated strokes were better than their manual strokes, because they tend to lose patience when manually drawing all strokes, which results in worse quality. Each case is marked with the # of manual and autocompleted strokes. Preferred Mode. Seven participants preferred the autocomplete mode while the rest five participants preferred the batch mode. Generally, the autocomplete mode is considered more convenient, yet less precise; the batch mode is considered more precise, but requires too many interactions. P12 commented, "the autocomplete mode is more straightforward, because you can see the filled effects instantly without doing a lot of manipulation beforehand; while in the batch mode, you have to remember the meaning of parameters and tweak them in order to create strokes." P10 also said, "Compared with batch filling, the autocomplete mode provides a quick guess of filled regions and allows me to get the results more quickly with less work." However, the autocomplete mode is "less accurate at some vague and detailed regions, such as the shadows of the boat, where it tends to include some unwanted regions, so I have to manually subtract those regions, which is a bit tedious", according to P3. The professional, P5, also preferred the batch mode for being able to precisely select the regions. Therefore, we consider the autocomplete function and the interactive editing function are complementary in usability. Creation Results and Experience. Figure 15 shows the outcomes from the open session. Although from the same reference image and widely using repetitive short strokes, the study participants were able to create different results by varying the stroke shapes and arrangement. Figures 16 and 17 demonstrate some sample results. Regarding the creation experience, one user said "it is playful, the final result is also good"; two users described it as "encouraging", because the system allows beginners to quickly create stylistic drawings; one user commented that she "felt creative when drawing with this system", because she could test out patterns over image regions conveniently and she was more comfortable with drawing from a reference image than from scratch. The professional suggested that the tool itself was somewhat limited to pointillism and hatching styles, but can be helpful in adding interesting textures into color paintings (e.g., Figure 16i). Two users commented that the reduction of workload is useful, but they also complained about some inaccurate inference of autocompletion. We will discuss about this problem in Section 7. CONCLUSION We have presented a method to help users autocomplete repetitive short strokes with guidance from reference images while maintaining the flexible control of manual drawing. By extending operation history analysis and synthesis with image analysis, our method is able to generate results that adapt to reference images and users' prior inputs. We conducted a pilot study to validate the usefulness of our approach and show various drawing results from the users. LIMITATIONS AND FUTURE WORK From our observation and users' feedback, we identified several improvement opportunities. Improve accuracy of autocompletion. We rely on simple * color and semantic segmentation for region inference. While color feature is sufficient for most cases, regions with similar colors but different semantics will require sufficient segmentation accuracy for region inference (Figures 13c and 13g). Since our segmentation map is precomputed, taking users' input as additional cues might help improve the segmentation accuracy (e.g., using interactive semantic segmentation methods like [50]). Resolve visual blocking. Since the drawing and the system suggestions are overlaid on the reference image, it might be difficult for users to refer to the image when selecting parts of the suggestions (e.g., Figure 18) or adding a new layer of strokes. Although users can switch the views via a hotkey, it might be helpful to provide some reference information, like image darkness or boundaries, through additional visual hints [45,47]. Consider relationships with higherlevel image features. We only consider the relationships between strokes and low-level image features, like colors and flows, over regions. By considering higherlevel image features, such as elements and edges, it is possible to extend the scope of autocompletion, such as autocomplting the sparse flowers in the foreground of Figure 16i through the correspondences between strokes and elements. Support more stroke types. Our method only supports short strokes, while artists also use long repetitive strokes frequently [5]. It is worth investigating the possibility of incorporating continuous strokes [44] in our analysis and synthesis framework and extending the support for different input strokes. Figure 16: Sample results. In each example, the left column shows the reference images, the middle column visualizes the manual strokes (black) and autocompleted results (red) of the final drawings on the right column. In the last example, the strokes are created with our system first and then imported into Photoshop for background coloring. Figure 2 : 2Inspiring manual drawings by artists. Figure 4 : 4User interface, consisting of a central drawing canvas (a), a toolbar for drawing and selection (b), a toggleswitch of the autocomplete mode (c), a brush property toolbar (d), a filling property toolbar (e), and a layers panel (f). Figure 5 : 5An example of autocompletion. The user selects part of the suggestions via the lasso selection tool (a) with the result in (b), continues to draw leading to the updated suggestions (c), and accepts all the suggestions via a hotkey (d). The blue strokes in (a) and (c) indicate inferred exemplars from user-input strokes. Figure 6 : 6Region editing example. The initial prediction (a) contains only the brown region. The user-specified region (b) contains the entire apple, with the corresponding synthesis result in (c). Figure 7 :Figure 8 : 78Density editing example with different values of spacing, lightness and gradient parameters. Larger spacing parameters lead to sparser strokes, while larger lightness and gradient parameters lead to larger stroke density variations. Orientation editing example. (a) User gesture. (b) Orientation field updated based on the user gesture and the original image flow field. (c) Updated result. (d) A result without any orientation field. Figure 9 : 9(a) A stroke, with centroid and dominant direction . (b) The neighborhood of the black stroke includes the ( = 1 in this example) closest strokes (in green) from each quadrant and the middle image patch (blue pixel grid). Figure 11 : 11Iteration process in (b) to (d) and ablation studies in (e) and (f). Without the correction term the predicted strokes tend to clutter together as in (e). Without the image term the predicted strokes might not follow the reference sufficiently as in (f). Figure 12 : 12An example of predicting the input exemplar and output region. The left column shows the input stroke sequence visualized in black dots (only a few indices are shown for clarity) on the reference image (top) and the image features (bottom). The right columns show the threshold lines and the image feature cost curves for 10 , 11 , 12 respectively (top), and the corresponding predicted output regions (bottom) Figure 13 : 13Target session tasks. Reference photos in (a) and (e), and the corresponding sample outputs in (b) and (f). Figure 14 :Figure 15 : 1415Target sessions results. (a) Average NASA-TLX scores from 12 participants. The lower the better. (b) Average completion time. (c) Average stroke counts. The number of system-generated strokes is labeled in each column. Example drawing results from the open session. Figure 18 : 18Example of visual blocking. Left: reference image. Right: canvas view. Figure 17 : 17Additional results forFigure 16. 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[ "Published as a conference paper at ICLR 2021 BENCHMARKS FOR DEEP OFF-POLICY EVALUATION", "Published as a conference paper at ICLR 2021 BENCHMARKS FOR DEEP OFF-POLICY EVALUATION" ]
[ "Justin Fu [email protected] \nUC Berkeley\n\n", "Mohammad Norouzi [email protected] \nGoogle Brain 3 DeepMind\n\n", "Ofir Nachum [email protected] \nGoogle Brain 3 DeepMind\n\n", "George Tucker \nGoogle Brain 3 DeepMind\n\n", "Ziyu Wang \nGoogle Brain 3 DeepMind\n\n", "Alexander Novikov ", "Mengjiao Yang \nGoogle Brain 3 DeepMind\n\n", "Michael R Zhang \nGoogle Brain 3 DeepMind\n\n", "Yutian Chen ", "Aviral Kumar \nUC Berkeley\n\n", "Cosmin Paduraru ", "Sergey Levine \nUC Berkeley\n\n", "Tom Le Paine " ]
[ "UC Berkeley\n", "Google Brain 3 DeepMind\n", "Google Brain 3 DeepMind\n", "Google Brain 3 DeepMind\n", "Google Brain 3 DeepMind\n", "Google Brain 3 DeepMind\n", "Google Brain 3 DeepMind\n", "UC Berkeley\n", "UC Berkeley\n" ]
[]
Off-policy evaluation (OPE) holds the promise of being able to leverage large, offline datasets for both evaluating and selecting complex policies for decision making. The ability to learn offline is particularly important in many real-world domains, such as in healthcare, recommender systems, or robotics, where online data collection is an expensive and potentially dangerous process. Being able to accurately evaluate and select high-performing policies without requiring online interaction could yield significant benefits in safety, time, and cost for these applications. While many OPE methods have been proposed in recent years, comparing results between papers is difficult because currently there is a lack of a comprehensive and unified benchmark, and measuring algorithmic progress has been challenging due to the lack of difficult evaluation tasks. In order to address this gap, we present a collection of policies that in conjunction with existing offline datasets can be used for benchmarking off-policy evaluation. Our tasks include a range of challenging high-dimensional continuous control problems, with wide selections of datasets and policies for performing policy selection. The goal of our benchmark is to provide a standardized measure of progress that is motivated from a set of principles designed to challenge and test the limits of existing OPE methods. We perform an evaluation of state-of-the-art algorithms and provide open-source access to our data and code to foster future research in this area † .Published as a conference paper at ICLR 2021 commonly explored by modern deep reinforcement learning algorithms(Bellemare et al., 2013;Brockman et al., 2016) with standardized evaluation protocols and metrics. Our goal is to provide a set of tasks with a range of difficulty, excercise a variety of design properties, and provide policies with different behavioral patterns in order to establish a standardized framework for comparing OPE algorithms. We put particular emphasis on large datasets, long-horizon tasks, and task complexity to facilitate the development of scalable algorithms that can solve high-dimensional problems.Our primary contribution is the Deep Off-Policy Evaluation (DOPE) benchmark. DOPE is designed to measure the performance of OPE methods by 1) evaluating on challenging control tasks with properties known to be difficult for OPE methods, but which occur in real-world scenarios, 2) evaluating across a range of policies with different values, to directly measure performance on policy evaluation, ranking and selection, and 3) evaluating in ideal and adversarial settings in terms of dataset coverage and support. These factors are independent of task difficulty, but are known to have a large impact on OPE performance. To achieve 1, we selected tasks on a set of design principles outlined in Section 3.1. To achieve 2, for each task we include 10 to 96 policies for evaluation and devise an evaluation protocol that measures policy evaluation, ranking, and selection as outlined in Section 3.2. To achieve 3, we provide two domains with differing dataset coverage and support properties described in Section 4. Finally, to enable an easy-to-use research platform, we provide the datasets, target policies, evaluation API, as well as the recorded results of state-of-the-art algorithms (presented in Section 5) as open-source. . Counterfactual reasoning and learning systems: The example of computational advertising. . Deep reinforcement learning in a handful of trials using probabilistic dynamics models. , et al. Reinforcement learning benchmarks and bake-offs ii.
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[ "https://arxiv.org/pdf/2103.16596v1.pdf" ]
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2103.16596
d86dfdbb8eab91cf23b81c541b4f741f88b7d756
Published as a conference paper at ICLR 2021 BENCHMARKS FOR DEEP OFF-POLICY EVALUATION Justin Fu [email protected] UC Berkeley Mohammad Norouzi [email protected] Google Brain 3 DeepMind Ofir Nachum [email protected] Google Brain 3 DeepMind George Tucker Google Brain 3 DeepMind Ziyu Wang Google Brain 3 DeepMind Alexander Novikov Mengjiao Yang Google Brain 3 DeepMind Michael R Zhang Google Brain 3 DeepMind Yutian Chen Aviral Kumar UC Berkeley Cosmin Paduraru Sergey Levine UC Berkeley Tom Le Paine Published as a conference paper at ICLR 2021 BENCHMARKS FOR DEEP OFF-POLICY EVALUATION Off-policy evaluation (OPE) holds the promise of being able to leverage large, offline datasets for both evaluating and selecting complex policies for decision making. The ability to learn offline is particularly important in many real-world domains, such as in healthcare, recommender systems, or robotics, where online data collection is an expensive and potentially dangerous process. Being able to accurately evaluate and select high-performing policies without requiring online interaction could yield significant benefits in safety, time, and cost for these applications. While many OPE methods have been proposed in recent years, comparing results between papers is difficult because currently there is a lack of a comprehensive and unified benchmark, and measuring algorithmic progress has been challenging due to the lack of difficult evaluation tasks. In order to address this gap, we present a collection of policies that in conjunction with existing offline datasets can be used for benchmarking off-policy evaluation. Our tasks include a range of challenging high-dimensional continuous control problems, with wide selections of datasets and policies for performing policy selection. The goal of our benchmark is to provide a standardized measure of progress that is motivated from a set of principles designed to challenge and test the limits of existing OPE methods. We perform an evaluation of state-of-the-art algorithms and provide open-source access to our data and code to foster future research in this area † .Published as a conference paper at ICLR 2021 commonly explored by modern deep reinforcement learning algorithms(Bellemare et al., 2013;Brockman et al., 2016) with standardized evaluation protocols and metrics. Our goal is to provide a set of tasks with a range of difficulty, excercise a variety of design properties, and provide policies with different behavioral patterns in order to establish a standardized framework for comparing OPE algorithms. We put particular emphasis on large datasets, long-horizon tasks, and task complexity to facilitate the development of scalable algorithms that can solve high-dimensional problems.Our primary contribution is the Deep Off-Policy Evaluation (DOPE) benchmark. DOPE is designed to measure the performance of OPE methods by 1) evaluating on challenging control tasks with properties known to be difficult for OPE methods, but which occur in real-world scenarios, 2) evaluating across a range of policies with different values, to directly measure performance on policy evaluation, ranking and selection, and 3) evaluating in ideal and adversarial settings in terms of dataset coverage and support. These factors are independent of task difficulty, but are known to have a large impact on OPE performance. To achieve 1, we selected tasks on a set of design principles outlined in Section 3.1. To achieve 2, for each task we include 10 to 96 policies for evaluation and devise an evaluation protocol that measures policy evaluation, ranking, and selection as outlined in Section 3.2. To achieve 3, we provide two domains with differing dataset coverage and support properties described in Section 4. Finally, to enable an easy-to-use research platform, we provide the datasets, target policies, evaluation API, as well as the recorded results of state-of-the-art algorithms (presented in Section 5) as open-source. . Counterfactual reasoning and learning systems: The example of computational advertising. . Deep reinforcement learning in a handful of trials using probabilistic dynamics models. , et al. Reinforcement learning benchmarks and bake-offs ii. INTRODUCTION Reinforcement learning algorithms can acquire effective policies for a wide range of problems through active online interaction, such as in robotics (Kober et al., 2013), board games and video games (Tesauro, 1995;Mnih et al., 2013;Vinyals et al., 2019), and recommender systems (Aggarwal et al., 2016). However, this sort of active online interaction is often impractical for real-world problems, where active data collection can be costly (Li et al., 2010), dangerous (Hauskrecht & Fraser, 2000;Kendall et al., 2019), or time consuming (Gu et al., 2017). Batch (or offline) reinforcement learning, has been studied extensively in domains such as healthcare (Thapa et al., 2005;Raghu et al., 2018), recommender systems (Dudík et al., 2014;Theocharous et al., 2015;Swaminathan et al., 2017), education (Mandel et al., 2014), and robotics (Kalashnikov et al., 2018). A major challenge with such methods is the off-policy evaluation (OPE) problem, where one must evaluate the expected performance of policies solely from offline data. This is critical for several reasons, including providing high-confidence guarantees prior to deployment (Thomas et al., 2015), and performing policy improvement and model selection (Bottou et al., 2013;Doroudi et al., 2017). The goal of this paper is to provide a standardized benchmark for evaluating OPE methods. Although considerable theoretical (Thomas & Brunskill, 2016;Swaminathan & Joachims, 2015;Jiang & Li, 2015;Wang et al., 2017;Yang et al., 2020) and practical progress (Gilotte et al., 2018;Nie et al., 2019;Kalashnikov et al., 2018) on OPE algorithms has been made in a range of different domains, there are few broadly accepted evaluation tasks that combine complex, high-dimensional problems 2 BACKGROUND Figure 1: In Off-Policy Evaluation (top) the goal is to estimate the value of a single policy given only data. Offline Policy Selection (bottom) is a closely related problem: given a set of N policies, attempt to pick the best given only data. We briefly review the off-policy evaluation (OPE) problem setting. We consider Markov decision processes (MDPs), defined by a tuple (S, A, T , R, ρ 0 , γ), with state space S, action space A, transition distribution T (s |s, a), initial state distribution ρ 0 (s), reward function R(s, a) and discount factor γ ∈ (0, 1]. In reinforcement learning, we are typically concerned with optimizing or estimating the performance of a policy π(a|s). The performance of a policy is commonly measured by the policy value V π , defined as the expected sum of discounted rewards: V π := E s0∼ρ0,s1:∞,a0:∞∼π ∞ t=0 γ t R(s t , a t ) . (1) If we have access to state and action samples collected from a policy π, then we can use the sample mean of observed returns to estimate the value function above. However, in off-policy evaluation we are typically interested in estimating the value of a policy when the data is collected from a separate behavior policy π B (a|s). This setting can arise, for example, when data is being generated online from another process, or in the purely offline case when we have a historical dataset. In this work we consider the latter, purely offline setting. The typical setup for this problem formulation is that we are provided with a discount γ, a dataset of trajectories collected from a behavior policy D = {(s 0 , a 0 , r 0 , s 1 , . . .)}, and optionally the action probabilities for the behavior policy π B (a t |s t ). In many practical applications, logging action propensities is not possible, for example, when the behavior policy is a mix of ML and hard-coded business logic. For this reason, we focus on the setting without propensities to encourage future work on behavior-agnostic OPE methods. For the methods that require propensities, we estimate the propensities with behavior cloning. The objective can take multiple flavors, as shown in Fig. 1. A common task in OPE is to estimate the performance, or value, of a policy π (which may not be the same as π B ) so that the estimated value is as close as possible to V π under a metric such as MSE or absolute error. A second task is to perform policy selection, where the goal is to select the best policy or set of policies out of a group of candidates. This setup corresponds to how OPE is commonly used in practice, which is to find the best performing strategy out of a pool when online evaluation is too expensive to be feasible. DOPE: DEEP OFF-POLICY EVALUATION The goal of the Deep Off-Policy Evaluation (DOPE) benchmark is to provide tasks that are challenging and effective measures of progress for OPE methods, yet is easy to use in order to better facilitate research. Therefore, we design our benchmark around a set of properties which are known to be difficult for existing OPE methods in order to gauge their shortcomings, and keep all tasks amenable to simulation in order for the benchmark to be accessible and easy to evaluate. TASK PROPERTIES We describe our motivating properties for selecting tasks for the benchmark as follows: High Dimensional Spaces (H) High-dimensionality is a key-feature in many real-world domains where it is difficult to perform feature engineering, such as in robotics, autonomous driving, and more. In these problems, it becomes challenging to accurately estimate quantities such as the value function without the use of high-capacity models such a neural networks and large datasets with wide state coverage. Our benchmark contains complex continuous-space tasks which exercise these challenges. Long Time-Horizon (L) Long time horizon tasks are known to present difficult challenges for OPE algorithms. Some algorithms have difficulty doing credit assignment for these tasks. This can be made worse as the state dimension or action dimension increases. Sparse Rewards (R) Sparse reward tasks increase the difficulty of credit assignment and add exploration challenges, which may interact with data coverage in the offline setting. We include a range robotics and navigation tasks which are difficult to solve due to reward sparsity. Temporally extended control (T) The ability to make decisions hierarchically is major challenge in many reinforcement learning applications. We include two navigation tasks which require high-level planning in addition to low-level control in order to simulate the difficulty in such problems. EVALUATION PROTOCOL Figure 2: Error is a natural measure for off-policy evaluation. However for policy selection, it is sufficient to (i) rank the policies as measured by rank correlation, or (ii) select a policy with the lowest regret. The goal of DOPE to provide metrics for policy ranking, evaluation and selection. Many existing OPE methods have only been evaluated on point estimates of value such as MSE, but policy selection is an important, practical use-case of OPE. In order to explicitly measure the quality of using OPE for policy selection, we provide a set of policies with varying value, and devise two metrics that measure how well OPE methods can rank policies. For each task we include a dataset of logged experiences D, and a set of policies {π 1 , π 2 , ..., π N } with varying values. For each policy, OPE algorithms must use D to produce an estimate of the policy's value. For evaluation of these estimates, we provide "ground truth values" {V π1 , V π2 , ..., V π N } that are computed by running the policy for M ≥ 1000 episodes, where the exact value of M is given by the number of episodes needed to lower the error bar on the ground truth values to 0.666. The estimated values are then compared to these ground truth values using three different metrics encompassing both policy evaluation and selection (illustrated in Figure 2; see Appendix A.1 for mathematical definitions). Absolute Error This metric measures estimate accuracy instead of its usefulness for ranking. Error is the most commonly used metric to assess performance of OPE algorithms. We opted to use absolute error instead of MSE to be robust to outliers. Regret@k This metric measures how much worse the best policies identified by the estimates are than the best policy in the entire set. It is computed by identifying the top-k policies according to the estimated returns. Regret@k is the difference between the actual expected return of the best policy in the entire set, and the actual value of the best policy in the top-k set. Rank correlation This metric directly measures how well estimated values rank policies, by computing the correlation between ordinal rankings according by the OPE estimates and ordinal rankings according to the ground truth values. DOMAINS DOPE contains two domains designed to provide a more comprehensive picture of how well OPE methods perform in different settings. These two domains are constructed using two benchmarks previously proposed for offline reinforcement learning: RL Unplugged (Gulcehre et al., 2020) and D4RL (Fu et al., 2020), and reflect the challenges found within them. The DOPE RL Unplugged domain is constrained in two important ways: 1) the data is always generated using online RL training, ensuring there is adequate coverage of the state-action space, and 2) the policies are generated by applying offline RL algorithms to the same dataset we use for evaluation, ensuring that the behavior policy and evaluation policies induce similar state-action distributions. Using it, we hope to understand how OPE methods work as task complexity increases from simple Cartpole tasks to controlling a Humanoid body while controlling for ideal data. On the other hand, the DOPE D4RL domain has: 1) data from various sources (including random exploration, human teleoperation, and RL-trained policies with limited exploration), which results in varying levels of coverage of the state-action space, and 2) policies that are generated using online RL algorithms, making it less likely that the behavior and evaluation policies share similar induced state-action distributions. Both of these result in distribution shift which is known to be challenging for OPE methods, even in simple tasks. So, using it we hope to measure how well OPE methods work in more practical data settings. DOPE RL UNPLUGGED DeepMind Control Suite (Tassa et al., 2018) is a set of control tasks implemented in MuJoCo (Todorov et al., 2012). We consider the subset included in RL Unplugged. This subset includes tasks that cover a range of difficulties. From Cartpole swingup, a simple task with a single degree of freedom, to Humanoid run which involves control of a complex bodies with 21 degrees of freedom. All tasks use the default feature representation of the system state, including proprioceptive information such as joint positions and velocity, and additional sensor information and target position where appropriate. The observation dimension ranges from 5 to 67. Datasets and policies We train four offline RL algorithms (D4PG (Barth-Maron et al., 2018), ABM (Siegel et al., 2020), CRR (Wang et al., 2020) and behavior cloning), varying their hyperparameters. For each algorithm-task-hyperparameter combination, we train an agent with 3 random seeds on the DM Control Suite dataset from RL Unplugged and record policy snapshots at exponentially increasing intervals (after 25k learner steps, 50k, 100K, 200K, etc). Following Gulcehre et al. (2020), we consider a deterministic policy for D4PG and stochastic policies for BC, ABM and CRR. The datasets are taken from the RL Unplugged benchmark, where they were created by training multiple (online) RL agents and collecting both successful and unsuccessful episodes throughout training. All offline RL algorithms are implemented using the Acme framework (Hoffman et al., 2020). DOPE D4RL Gym-MuJoCo tasks. Gym-MuJoCo consists of several continuous control tasks implemented within the MuJoCo simulator (Todorov et al., 2012) and provided in the OpenAI Gym (Brockman et al., 2016) benchmark for online RL. We include the HalfCheetah, Hopper, Walker2D, and Ant tasks. We include this domain primarily for comparison with past works, as a vast array of popular RL Gym-MuJoCo datasets and policies. For each task, in order to explore the effect of varying distributions, we include 5 datasets originally proposed by Fu et al. (2020). 3 correspond to different performance levels of the agent -"random", "medium", and "expert". We additionally include a mixture of medium and expert dataset, labeled "medium-expert", and data collected from a replay buffer until the policy reaches the medium level of performance, labeled "medium-replay". For policies, we selected 11 policies collected from evenly-spaced snapshots of training a Soft Actor-Critic agent (Haarnoja et al., 2018), which covers a range of performance between random and expert. Maze2D and AntMaze tasks. Maze2D and AntMaze are two maze navigation tasks originally proposed in D4RL (Fu et al., 2020). The domain consists of 3 mazes ranging from easy to hard ("umaze", "medium", "large"), and two morphologies: a 2D ball in Maze2D and the "Ant" robot of the Gym benchmark in AntMaze. For Maze2D, we provide a less challenging reward computed base on distance to a fixed goal. For the AntMaze environment reward is given only upon reaching the fixed goal. Maze2D and AntMaze datasets and policies. Datasets for both morphologies consists of undirect data navigating randomly to different goal locations. The datasets for Maze2D are collected by using a high-level planner to command waypoints to a low-level PID controller in order to reach randomly selected goals. The dataset in AntMaze is generated using the same high-level planner, but the low-level planner is replaced with a goal-conditioned policy trained to reach arbitrary waypoints. Both of these datasets are generated from non-Markovian policies, as the high-level controller maintains a history of waypoints reached in order to construct a plan to the goal. We provide policies for all environments except "antmaze-large" by taking training snapshots obtained while running the DAPG algorithm (Rajeswaran et al., 2017). Because obtaining high-performing policies for "antmaze-large" was challenging, we instead used imitation learning on a large amount of expert data to generate evaluation policies. This expert data is obtained by collecting additional trajectories that reach the goal using a high-level waypoint planner in conjunction with a low-level goal-conditioned policy (this is the same method as was used to generate the dataset, Sec. 5 (Fu et al., 2020)). Adroit tasks. The Adroit domain is a realistic simulation based on the Shadow Hand robot, first proposed by Rajeswaran et al. (2017). There are 4 tasks in this domain: opening a door ("door"), pen twirling ("pen"), moving a ball to a target location ("relocate"), and hitting a nail with a hammer ("hammer"). These tasks all contain sparse rewards and are difficult to learn without demonstrations. Adroit datasets and policies. We include 3 datasets for each task. The "human" dataset consists of a small amount of human demonstrations performing the task. The "expert" dataset consists of data collected from an expert trained via DAPG (Rajeswaran et al., 2017). Finally, the "cloned" dataset contains a mixture of human demonstrations and data collected from an imitation learning algorithm trained on the demonstrations. For policies, we include 11 policies collected from snapshots while running the DAPG algorithm, which range from random performance to expert performance. BASELINES AND RESULTS The goal of our evaluation is two-fold. First, we wish to measure the performance of a variety of existing algorithms to provide baselines and reference numbers for future research. Second, we wish to identify shortcomings in these approaches to reveal promising directions for future research. BASELINES We selected six methods to evaluate, which cover a variety of approaches that have been explored for the OPE problem. Fitted Q-Evaluation (FQE) As in Le et al. (2019), we train a neural network to estimate the value of the evaluation policy π by bootstrapping from Q(s , π(s )). We tried two different implementations, one from Kostrikov & Nachum (2020) Model-Based (MB) Similar to Paduraru (2007), we train dynamics and reward models on transitions from the offline dataset D. Our models are deep neural networks trained to maximize the log likelihood of the next state and reward given the current state and action, similar to models from successful model-based RL algorithms (Chua et al., 2018;Janner et al., 2019). We follow the setup detailed in Zhang et al. (2021). We include both the feed-forward and auto-regressive models labeled MB-FF and MB-AR respectively. To evaluate a policy, we compute the return using simulated trajectories generated by the policy under the learned dynamics model. Importance Sampling (IS) We perform importance sampling with a learned behavior policy. We use the implementation from Kostrikov & Nachum (2020) 3 , which uses self-normalized (also known as weighted) step-wise importance sampling (Precup, 2000). Since the behavior policy is not known explicitly, we learn an estimate of it via a max-likelihood objective over the dataset Variational Power Method (VPM) This method runs a variational power iteration algorithm to estimate the importance weights d π (s, a)/d π B (s, a) without the knowledge of the behavior policy. It then estimates the target policy value using weighted average of rewards similar to the DICE method. Our implementation is based on the same network and hyperparameters for OPE setting as in Wen et al. (2020). We further tune the hyper-parameters including the regularization parameter λ, learning rates α θ and α v , and number of iterations on the Cartpole swingup task using ground-truth policy value, and then fix them for all other tasks. RESULTS To facilitate aggregate metrics and comparisons between tasks and between DOPE RL Unplugged and DOPE D4RL, we normalize the returns and estimated returns to range between 0 and 1. For each set of policies we compute the worst value V worst = min{V π1 , V π2 , ..., V π N } and best value V best = max{V π1 , V π2 , ..., V π N } and normalize the returns and estimated returns according to x = (x − V worst )/(V best − V worst ). We present results averaged across DOPE RL Unplugged in Fig. 4, and results for DOPE D4RL in Fig. 5. Overall, no evaluated algorithm attains near-oracle performance under any metric (absolute error, regret, or rank correlation). Because the dataset is finite, we do not expect that achieving oracle performance is possible. Nevertheless, based on recent progress on this benchmark (e.g., Zhang et al. (2021)), we hypothesize that the benchmark has room for improvement, making it suitable for driving further improvements on OPE methods and facilitating the development of OPE algorithms that can provide reliable estimates on the types of high-dimensional problems that we consider. While all algorithms achieve sub-optimal performance, some perform better than others. We find that on the DOPE RL Unplugged tasks model based (MB-AR, MB-FF) and direct value based methods (FQE-D, FQE-L2) significantly outperform importance sampling methods (VPM, DICE, IS) across all metrics. This is somewhat surprising as DICE and VPM have shown promising results in other settings. We hypothesize that this is due to the relationship between the behavior data and evaluation policies, which is different from standard OPE settings. Recall that in DOPE RL Unplugged the behavior data is collected from an online RL algorithm and the evaluation policies are learned via offline RL from the behavior data. In our experience all methods work better when the behavior policy is a noisy/perturbed version of the evaluation policy. Moreover, MB and FQE-based methods may implicitly benefit from the architectural and optimization advancements made in policy optimization settings, which focus on similar environments and where these methods are more popular than importance sampling approaches. In Fig. A.2 we show the rank correlation for each task in DOPE RL Unplugged. Most tasks follow the overall trends, but we will highlight a few exceptions. 1) Importance sampling is among the best methods for the humanoid run task, significantly outperforming direct value-based methods. We present more detailed results, separated by task, in Appendix A.2. Note in particular how in Table A.2.2, which shows the regret@1 metric for different D4RL tasks, the particular choice of dataset for the Gym-MuJoCo, Adroit, and AntMaze domains causes a significant difference in the performance of OPE methods. This indicates the importance of evaluating multiple distinct datasets, with different data distribution properties (e.g., more narrow datasets, such as expert data, vs. broader datasets, such as random data), as no tested method is reliably robust to the effects of dataset variation. High-dimensional tasks requiring temporally extended control were also challenging, as highlighted by the performance on the AntMaze domain. No algorithm was able to achieve a good absolute error value on such tasks, and importance sampling was the only method able to achieve a correlation consistently above zero, suggesting that these more complex tasks are a particularly important area for future methods to focus on. RELATED WORK Off-policy evaluation (OPE) has been studied extensively across a range of different domains, from healthcare (Thapa et al ) are performed using the MSE metric, and they do not provide standardized datasets. In contrast, we provide a variety of policies for each problem which enables one to evaluate metrics such as ranking for policy selection, and a wide range of standardized datasets for reproducbility. CONCLUSION We have presented the Deep Off-Policy Evaluation (DOPE) benchmark, which aims to provide a platform for studying policy evaluation and selection across a wide range of challenging tasks and datasets. In contrast to prior benchmarks, DOPE provides multiple datasets and policies, allowing researchers to study how data distributions affect performance and to evaluate a wide variety of metrics, including those that are relevant for offline policy selection. In comparing existing OPE methods, we find that no existing algorithms consistently perform well across all of the tasks, which further reinforces the importance of standardized and challenging OPE benchmarks. Moreover, algorithms that perform poorly under one metric, such as absolute error, may perform better on other metrics, such as correlation, which provides insight into what algorithms to use depending on the use case (e.g., policy evaluation vs. policy selection). We believe that OPE is an exciting area for future research, as it allows RL agents to learn from large and abundant datasets in domains where online RL methods are otherwise infeasible. We hope that our benchmark will enable further progress in this field, though important evaluation challenges remain. As the key benefit of OPE is the ability to utilize real-world datasets, a promising direction for future evaluation efforts is to devise effective ways to use such data, where a key challenge is to develop evaluation protocols that are both reproducible and accessible. This could help pave the way towards developing intelligent decision making agents that can leverage vast banks of logged information to solve important real-world problems. A APPENDIX A.1 METRICS The metrics we use in our paper are defined as follows: Absolute Error We evaluate policies using absolute error in order to be robust to outliers. The absolute error is defined as the difference between the value and estimated value of a policy: AbsErr = |V π −V π |(2) Where V π is the true value of the policy, andV π is the estimated value of the policy. Regret@k Regret@k is the difference between the value of the best policy in the entire set, and the value of the best policy in the top-k set (where the top-k set is chosen by estimated values). It can be defined as: Regret @ k = max i∈1:N V π i − max j∈topk(1:N ) V π j(3) Where topk(1 : N ) denotes the indices of the top K policies as measured by estimated valuesV π . Rank correlation Rank correlation (also Spearman's ρ) measures the correlation between the ordinal rankings of the value estimates and the true values. It can be written as: RankCorr = Cov(V π 1:N ,V π 1:N ) σ(V π 1:N )σ(V π 1:N )(4) A.2 DETAILED RESULTS Detailed results figures and tables are presented here. We show results by task in both tabular and chart form, as well as scatter plots which compare the estimated returns against the ground truth returns for every policy. A.2.1 CHART RESULTS First we show the normalized results for each algorithm and task. Figure 3 : 3Online evaluation of policy checkpoints for 4 Offline RL algorithms with 3 random seeds. We observe a large degree of variability between the behavior of algorithms on different tasks. Without online evaluation, tuning the hyperparameters (e.g., choice of Offline RL algorithm and policy checkpoint) is challenging. This highlights the practical importance of Offline policy selection when online evaluation is not feasible. SeeFigure A.7 for additional tasks. methods have been evaluated and developed on these tasks(Schulman et al., 2015; Lillicrap et al., 2015; Schulman et al., 2017; Fujimoto et al., 2018; Haarnoja et al., 2018). 3 and another from Paine et al. (2020) labeled FQE-L2 and FQE-D respectively to reflect different choices in loss function and parameterization. Figure 4 : 4D, as advocated byXie et al. (2018); Hanna et al. (2019). In order to be able to compute log-probabilities when the target policy is deterministic, we add artificial Gaussian noise with standard deviation 0.01 for all deterministic target policies. DOPE RL Unplugged Mean overall performance of baselines. Figure 5 : 5DOPE D4RL Mean overall performance of baselines. Figure 6 :Figure 7 : 67Rank correlation for each baseline algorithm for each RL Unplugged task considered. Scatter plots of estimate vs ground truth return for MB-AR and FQE-D on selected tasks. 2) while MB-AR and FQE-D are similar overall, there are a few tasks where the difference is large, for example FQE-D outperfroms MB-AR on finger turn hard, and manipulator insert ball, where as MB-AR outperforms FQE-D on cartpole swingup, fish swim, humanoid run, and manipulator insert peg. We show the scatter plots for MB-AR and FQE-D on these tasks in Fig 7 which highlights different failure modes: when MB-AR performs worse, it assigns similar values for all policies; when FQE-D performs worse, it severely over-estimates the values of poor policies. ., 2005 ; 2005Raghu et al., 2018; Nie et al., 2019), to recommender systems(Li et al., 2010; Dudík et al., 2014; Theocharous et al., 2015), and robotics(Kalashnikov et al., 2018). While a full survey of OPE methods is outside the scope of this article, broadly speaking we can categories OPE methods into groups based the use of importance sampling(Precup, 2000), value functions(Sutton et al., 2009; Migliavacca et al., 2010; Sutton et al., 2016; Yang et al., 2020), and learned transition models(Paduraru, 2007), though a number of methods combine two or more of these components(Jiang & Li, 2015; Thomas & Brunskill, 2016; Munos et al., 2016). A significant body of work in OPE is also concerned with providing statistical guarantees(Thomas et al., 2015). Our focus instead is on empirical evaluation -while theoretical analysis is likely to be a critical part of future OPE research, combining such analysis with empirical demonstration on broadly accepted and standardized benchmarks is likely to facilitate progress toward practically useful algorithms.Current evaluation of OPE methods is based around several metrics, including error in predicting the true return of the evaluated policy(Voloshin et al., 2019), correlation between the evaluation output and actual returns(Irpan et al., 2019), and ranking and model selection metrics(Doroudi et al., 2017). As there is no single accepted metric used by the entire community, we provide a set of candidate metrics along with our benchmark, with a detailed justification in Section 5. Our work is closely related to (Paine et al., 2020) which studies OPE in a similar setting, however in our work we present a benchmark for the community and compare a range of OPE methods. Outside of OPE, standardized benchmark suites have led to considerable standardization and progress in RL(Stone & Sutton, 2001; Dutech et al., 2005; Riedmiller et al., 2007). The Arcade Learning Environment (ALE) (Bellemare et al., 2013) and OpenAI Gym (Brockman et al., 2016) have been widely used to compare online RL algorithms to good effect. More recently, Gulcehre et al. (2020); Fu et al. (2020) proposed benchmark tasks for offline RL. Our benchmark is based on the tasks and environments described in these two benchmarks, which we augment with a set of standardized policies for evaluation, results for a number of existing OPE methods, and standardized evaluation metrics and protocols. Voloshin et al. (2019) have recently proposed benchmarking for OPE methods on a variety of tasks ranging from tabular problems to image-based tasks in Atari. Our work differs in several key aspects. Voloshin et al. (2019) is composed entirely of discrete action tasks, whereas out benchmark focuses on continuous action tasks. Voloshin et al. (2019) assumes full support for the evaluation policy under the behavior policy data, whereas we designed our datasets and policies to ensure that different cases of dataset and policy distributions could be studied. Finally, all evaluations in Voloshin et al. (2019 Figure A. 1 :Figure A. 2 :Figure A. 3 :Figure A. 4 :Figure A. 5 :Figure A. 6 : 123456Absolute error for each baseline algorithm for each RL Unplugged task considered. Rank correlation for each baseline algorithm for each RL Unplugged task considered. Regret@1 for each baseline algorithm for each RL Unplugged task considered. Absolute error for each baseline algorithm for each D4RL task domain considered. Rank correlation for each baseline algorithm for each D4RL task domain considered. Regret@1 for each baseline algorithm for each D4RL task domain considered. Figure A. 8 : 8Scatter plots of estimate vs ground truth return for each baseline on each task in DOPE RL Unplugged. Figure A. 9 :Figure 9Scatter plots of estimate vs ground truth return for each baseline on each task in A.10: Scatter plots of estimate vs ground truth return for each baseline on each task in DOPE D4RL (part 2). FigureFigure A.11: Scatter plots of estimate vs ground truth return for each baseline on each task in A.12: Scatter plots of estimate vs ground truth return for each baseline on each task in DOPE D4RL (part 4). Figure A.13: Scatter plots of estimate vs ground truth return for each baseline on each task in DOPE D4RL (part 5). Figure A.14: Scatter plots of estimate vs ground truth return for each baseline on each task in DOPE D4RL (part 6). Table 1 : 1Task statistics for RLUnplugged tasks (top) and D4RL tasks (bottom). Dataset size is the number of (s, a, r, s ) tuples. For each dataset, we note the properties it possesses: high dimensional spaces (H), long time-horizon (L), sparse rewards (R), temporally extended control (T). 1 2 3 4 5 6 7 8 Checkpoint id 0 20 40 60 80 100 Retrun (d=0.995) D4PG BC ABM CRR cartpole swingup 1 2 3 4 5 6 7 8 Checkpoint id 20 30 40 50 60 Retrun (d=0.995) D4PG BC ABM CRR fish swim 1 2 3 4 5 6 7 8 Checkpoint id 20 40 60 80 100 120 140 160 Retrun (d=0.995) D4PG ABM CRR BC walker walk 1 2 3 4 5 6 7 8 Checkpoint id 10 20 30 40 50 Retrun (d=0.995) D4PG ABM CRR BC manipulator insert ball Doubly-Robust (DR) We perform weighted doubly-robust policy evaluationThomas & Brunskill (2016) using the implementation of Kostrikov & Nachum (2020) 3 . Specifically, this method combines the IS technique above with a value estimator for variance reduction. The value estimator is learned using deep FQE with an L2 loss function. More advanced approaches that trade variance for bias exist (e.g., MAGIC (Thomas & Brunskill, 2016)), but we leave implementing them to future work.DICE This method uses a saddle-point objective to estimate marginalized importance weights d π (s, a)/d π B (s, a); these weights are then used to compute a weighted average of reward over the offline dataset, and this serves as an estimate of the policy's value in the MDP. We use the implementation fromYang et al. (2020) corresponding to the algorithm BestDICE. 4 Note that within the MB and FQE methods, design details can create a significant difference in performance. For example model architecture (MB-AR vs MB-FF) and implementation differences (FQE-D vs FQE-L2) show differing performance on certain tasks.On DOPE D4RL, direct value based methods still do well, with FQE-L2 performing best on the Absolute Error and Regret@1 metrics. However, there are cases where other methods outperform FQE. 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Michael R Zhang, Thomas Paine, Ofir Nachum, Cosmin Paduraru, George Tucker, ziyu wang, and Mohammad Norouzi. Autoregressive dynamics models for offline policy evaluation and optimization. In International Conference on Learning Representations, 2021. URL https: //openreview.net/forum?id=kmqjgSNXby. Table A . AVariational power method −0.35 ±0.10 −0.10 ±0.11Table A.2: Spearman's rank correlation (ρ) coefficient (bootstrap mean ± standard deviation) between different OPE metrics and ground truth values at a discount factor of 0.995. In each column, rank correlation coefficients that are not significantly different from the best (p > 0.05) are bold faced. Methods are ordered by median. Also seeTable A.3 and Table A.1 for Normalized Regret@5 and Average Absolute Error results. Variational power method 0.50 ±0.13 0.37 ±0.04 0.45 ±0.13 0.02 ±0.02 0.56 ±0.08 Doubly Robust (IS, FQE) 0.28 ±0.05 0.09 ±0.05 0.56 ±0.12 Variational power method 0.41 ±0.02 0.39 ±0.02 0.52 ±0.20 0.32 ±0.02 0.41 Doubly Robust (IS, FQE) 0.02 ±0.01 0.05 ±0.07 0.30 ±0.10 Table A.3: Normalized Regret@5 (bootstrap mean ± standard deviation) for OPE methods vs. ground truth values at a discount factor of 0.995. In each column, normalized regret values that are not significantly different from the best (p > 0.05) are bold faced. Methods are ordered by median.Finally, we present scatter plots plotting the true returns of each policy against the estimated returns. Each point on the plot represents one evaluated policy.1: Average absolute error between OPE metrics and ground truth values at a discount factor of 0.995 In each column, absolute error values that are not significantly different from the best (p > 0.05) are bold faced. Methods are ordered by median. Cartpole Cheetah Finger Fish Humanoid swingup run turn hard swim run Rank Correlation btw. OPE and ground truth Importance Sampling −0.23 ±0.11 −0.01 ±0.12 −0.45 ±0.08 −0.17 ±0.11 0.91 ±0.02 Best DICE −0.16 ±0.11 0.07 ±0.11 −0.22 ±0.11 0.44 ±0.09 −0.10 ±0.10 Variational power method 0.01 ±0.11 0.01 ±0.12 −0.25 ±0.11 0.56 ±0.08 0.36 ±0.09 Doubly Robust (IS, FQE) 0.55 ±0.09 0.56 ±0.08 0.67 ±0.05 0.11 ±0.12 −0.03 ±0.12 Model based -FF 0.83 ±0.05 0.64 ±0.08 0.08 ±0.11 0.95 ±0.02 0.35 ±0.10 FQE (distributional) 0.69 ±0.07 0.67 ±0.06 0.94 ±0.01 0.59 ±0.10 0.74 ±0.06 FQE (L2) 0.70 ±0.07 0.56 ±0.08 0.83 ±0.04 0.10 ±0.12 −0.02 ±0.12 Model based -AR 0.91 ±0.02 0.74 ±0.07 0.57 ±0.09 0.96 ±0.01 0.90 ±0.02 Walker Walker Manipulator Manipulator Median ↑ stand walk insert ball insert peg Rank Correlation btw. OPE and ground truth Importance Sampling 0.59 ±0.08 0.38 ±0.10 −0.72 ±0.05 −0.25 ±0.08 −0.17 Best DICE −0.11 ±0.12 −0.58 ±0.08 0.19 ±0.11 −0.35 ±0.10 −0.11 0.61 ±0.08 0.41 ±0.09 0.01 Doubly Robust (IS, FQE) 0.88 ±0.03 0.85 ±0.04 0.42 ±0.10 −0.47 ±0.09 0.55 Model based -FF 0.82 ±0.04 0.80 ±0.05 0.06 ±0.10 −0.56 ±0.08 0.64 FQE (distributional) 0.87 ±0.02 0.89 ±0.03 0.63 ±0.08 −0.23 ±0.10 0.69 FQE (L2) 0.96 ±0.01 0.94 ±0.02 0.70 ±0.07 −0.48 ±0.08 0.70 Model Based -AR 0.96 ±0.01 0.98 ±0.00 −0.33 ±0.09 0.47 ±0.09 0.90 Cartpole Cheetah Finger Fish Humanoid swingup run turn hard swim run Regret@5 for OPE vs. ground truth Importance Sampling 0.73 ±0.16 0.40 ±0.21 0.64 ±0.05 0.12 ±0.05 0.31 ±0.09 Best DICE 0.68 ±0.41 0.27 ±0.05 0.44 ±0.04 0.35 ±0.24 0.84 ±0.22 0.61 ±0.12 0.99 ±0.00 FQE (L2) 0.06 ±0.04 0.17 ±0.05 0.30 ±0.11 0.50 ±0.03 0.99 ±0.00 Model based -FF 0.02 ±0.02 0.24 ±0.12 0.43 ±0.04 0.00 ±0.00 0.44 ±0.02 FQE (distributional) 0.03 ±0.09 0.11 ±0.09 0.10 ±0.12 0.49 ±0.06 0.24 ±0.15 Model based -AR 0.00 ±0.02 0.01 ±0.02 0.63 ±0.11 0.03 ±0.02 0.32 ±0.06 Walker Walker Manipulator Manipulator Median ↓ stand walk insert ball insert peg Regret@5 for OPE vs. ground truth Importance Sampling 0.54 ±0.11 0.54 ±0.23 0.83 ±0.05 0.22 ±0.03 0.54 Best DICE 0.24 ±0.07 0.55 ±0.06 0.44 ±0.07 0.75 ±0.04 0.44 0.73 ±0.01 0.30 FQE (L2) 0.04 ±0.02 0.00 ±0.02 0.37 ±0.07 0.74 ±0.01 0.30 Model based -FF 0.18 ±0.10 0.03 ±0.05 0.83 ±0.06 0.74 ±0.01 0.24 FQE (distributional) 0.03 ±0.03 0.01 ±0.02 0.50 ±0.30 0.73 ±0.01 0.11 Model based -AR 0.04 ±0.02 0.04 ±0.02 0.85 ±0.02 0.30 ±0.04 0.04 Halfcheetah Halfcheetah Halfcheetah Halfcheetah Door expert medium-expert medium-replay random cloned Rank Corr. Best DICE −0.44 ±0.30 −0.08 ±0.35 −0.15 ±0.41 −0.70 ±0.22 0.18 ±0.31 VPM 0.18 ±0.35 −0.47 ±0.29 −0.07 ±0.36 0.27 ±0.36 −0.29 ±0.36 FQE (L2) 0.78 ±0.15 0.62 ±0.27 0.26 ±0.37 −0.11 ±0.41 0.55 ±0.27 IS 0.01 ±0.35 −0.06 ±0.37 0.59 ±0.26 −0.24 ±0.36 0.66 ±0.22 Doubly Robust 0.77 ±0.17 0.62 ±0.27 0.32 ±0.37 −0.02 ±0.38 0.60 ±0.28 Door Hammer Hammer Maze2d Maze2d expert cloned expert large medium Rank Corr. Best DICE −0.06 ±0.32 0.35 ±0.38 −0.42 ±0.31 0.56 ±0.21 −0.64 ±0.23 VPM 0.65 ±0.23 −0.77 ±0.22 0.39 ±0.31 −0.26 ±0.33 −0.05 ±0.39 FQE (L2) 0.89 ±0.09 −0.15 ±0.33 0.29 ±0.34 0.30 ±0.36 0.16 ±0.38 IS 0.76 ±0.17 0.58 ±0.27 0.64 ±0.24 0.63 ±0.19 0.44 ±0.25 Doubly Robust 0.76 ±0.13 −0.70 ±0.20 0.49 ±0.31 0.31 ±0.36 0.41 ±0.35 Pen Relocate Ant Ant Ant expert expert expert medium medium-expert Rank Corr. Best DICE −0.53 ±0.30 −0.27 ±0.34 −0.13 ±0.37 −0.36 ±0.28 −0.33 ±0.40 VPM 0.08 ±0.33 0.39 ±0.31 −0.42 ±0.38 −0.20 ±0.31 −0.28 ±0.28 FQE (L2) −0.01 ±0.33 −0.57 ±0.28 −0.13 ±0.32 0.65 ±0.25 0.37 ±0.35 IS −0.45 ±0.31 0.52 ±0.23 0.14 ±0.41 −0.17 ±0.32 −0.21 ±0.35 Doubly Robust 0.52 ±0.28 −0.40 ±0.24 −0.28 ±0.32 0.66 ±0.26 0.35 ±0.35 Ant Ant Hopper Hopper Hopper medium-replay random expert medium random Rank Corr. Best DICE −0.24 ±0.39 −0.21 ±0.35 −0.08 ±0.32 0.19 ±0.33 −0.13 ±0.39 VPM −0.26 ±0.29 0.24 ±0.31 0.21 ±0.32 0.13 ±0.37 −0.46 ±0.20 FQE (L2) 0.57 ±0.28 0.04 ±0.33 −0.33 ±0.30 −0.29 ±0.33 −0.11 ±0.36 IS 0.07 ±0.39 0.26 ±0.34 0.37 ±0.27 −0.55 ±0.26 0.23 ±0.34 Doubly Robust 0.45 ±0.32 0.01 ±0.33 −0.41 ±0.27 −0.31 ±0.34 −0.19 ±0.36 Walker2d Walker2d Walker2d Walker2d Walker2d expert medium medium-expert medium-replay random Rank Corr. Best DICE −0.37 ±0.27 0.12 ±0.38 −0.34 ±0.34 0.55 ±0.23 −0.19 ±0.36 VPM 0.17 ±0.32 0.44 ±0.21 0.49 ±0.37 −0.52 ±0.25 −0.42 ±0.34 FQE (L2) 0.35 ±0.33 −0.09 ±0.36 0.25 ±0.32 −0.19 ±0.36 0.21 ±0.31 IS 0.22 ±0.37 −0.25 ±0.35 0.24 ±0.33 0.65 ±0.24 −0.05 ±0.38 Doubly Robust 0.26 ±0.34 0.02 ±0.37 0.19 ±0.33 −0.37 ±0.39 0.16 ±0.29 Median Rank Corr. Best DICE −0.19 VPM −0.05 FQE (L2) 0.21 IS 0.23 Doubly Robust 0.26 A.2.3 SCATTER PLOTS Code available at https://github.com/google-research/google-research/tree/ master/policy_eval. Code available at https://github.com/google-research/dice_rl. C Charu, Aggarwal, Recommender systems. Springer1Charu C Aggarwal et al. Recommender systems, volume 1. Springer, 2016. Distributional policy gradients. Gabriel Barth-Maron, Matthew W Hoffman, David Budden, Will Dabney, Dan Horgan, T B Dhruva, Alistair Muldal, Nicolas Heess, Timothy Lillicrap, International Conference on Learning Representations. Gabriel Barth-Maron, Matthew W. Hoffman, David Budden, Will Dabney, Dan Horgan, Dhruva TB, Alistair Muldal, Nicolas Heess, and Timothy Lillicrap. Distributional policy gradients. In International Conference on Learning Representations, 2018. The arcade learning environment: An evaluation platform for general agents. Yavar Marc G Bellemare, Joel Naddaf, Michael Veness, Bowling, Journal of Artificial Intelligence Research. 47Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environ- ment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47: 253-279, 2013.
[ "https://github.com/google-research/google-research/tree/", "https://github.com/google-research/dice_rl." ]
[ "Spin-polarized voltage probes for helical edge state: a model study", "Spin-polarized voltage probes for helical edge state: a model study", "Spin-polarized voltage probes for helical edge state: a model study", "Spin-polarized voltage probes for helical edge state: a model study", "Spin-polarized voltage probes for helical edge state: a model study", "Spin-polarized voltage probes for helical edge state: a model study" ]
[ "Vivekananda Adak \nDepartment of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia\n", "Krishanu Roychowdhury \nDepartment of Physics\nStockholm University\nSE-106 91StockholmSweden\n", "Sourin Das \nDepartment of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia\n", "Vivekananda Adak \nDepartment of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia\n", "Krishanu Roychowdhury \nDepartment of Physics\nStockholm University\nSE-106 91StockholmSweden\n", "Sourin Das \nDepartment of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia\n", "Vivekananda Adak \nDepartment of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia\n", "Krishanu Roychowdhury \nDepartment of Physics\nStockholm University\nSE-106 91StockholmSweden\n", "Sourin Das \nDepartment of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia\n" ]
[ "Department of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia", "Department of Physics\nStockholm University\nSE-106 91StockholmSweden", "Department of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia", "Department of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia", "Department of Physics\nStockholm University\nSE-106 91StockholmSweden", "Department of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia", "Department of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia", "Department of Physics\nStockholm University\nSE-106 91StockholmSweden", "Department of Physical Sciences\nIISER Kolkata\n741246MohanpurWest BengalIndia" ]
[]
Theoretical models of a spin-polarized voltage probe (SPVP) tunnel-coupled to the helical edge states (HES) of a quantum spin Hall system (QSHS) are studied. Our first model of the SPVP comprises NP spin-polarized modes (subprobes), each of which is locally tunnel-coupled to the HES, while the SPVP, as a whole, is subjected to a self-consistency condition ensuring zero average current on the probe. We carry out a numerical analysis which shows that the optimal situation for reading off spin-resolved voltage from the HES depends on the interplay of the probe-edge tunnel-coupling and the number of modes in the probe (NP ). We further investigate the stability of our findings by introducing Gaussian fluctuations in (i) the tunnel-coupling between the subprobes and the HES about a chosen average value and (ii) spin-polarization of the subprobes about a chosen direction of the net polarization of SPVP. We also perform a numerical analysis corresponding to the situation where four such SPVPs are implemented in a self-consistent fashion across a ferromagnetic barrier on the HES and demonstrate that this model facilitates the measurements of spin-resolved four-probe voltage drops across the ferromagnetic barrier. As a second model, we employ the edge state of a quantum anomalous Hall state (QAHS) as the SPVP which is tunnel-coupled over an extended region with the HES. A two-dimensional lattice simulation for the quantum transport of the proposed device setup comprising a junction of QSHS and QAHS is considered and a feasibility study of using the edge of the QAHS as an efficient spin-polarized voltage probe is carried out including disorder. arXiv:2104.00667v2 [cond-mat.mes-hall]
10.1016/j.physe.2021.115125
[ "https://export.arxiv.org/pdf/2104.00667v2.pdf" ]
232,478,449
2104.00667
8c823b5c1a293dac87581dfa1eef51f5d0b5d49e
Spin-polarized voltage probes for helical edge state: a model study Vivekananda Adak Department of Physical Sciences IISER Kolkata 741246MohanpurWest BengalIndia Krishanu Roychowdhury Department of Physics Stockholm University SE-106 91StockholmSweden Sourin Das Department of Physical Sciences IISER Kolkata 741246MohanpurWest BengalIndia Spin-polarized voltage probes for helical edge state: a model study Theoretical models of a spin-polarized voltage probe (SPVP) tunnel-coupled to the helical edge states (HES) of a quantum spin Hall system (QSHS) are studied. Our first model of the SPVP comprises NP spin-polarized modes (subprobes), each of which is locally tunnel-coupled to the HES, while the SPVP, as a whole, is subjected to a self-consistency condition ensuring zero average current on the probe. We carry out a numerical analysis which shows that the optimal situation for reading off spin-resolved voltage from the HES depends on the interplay of the probe-edge tunnel-coupling and the number of modes in the probe (NP ). We further investigate the stability of our findings by introducing Gaussian fluctuations in (i) the tunnel-coupling between the subprobes and the HES about a chosen average value and (ii) spin-polarization of the subprobes about a chosen direction of the net polarization of SPVP. We also perform a numerical analysis corresponding to the situation where four such SPVPs are implemented in a self-consistent fashion across a ferromagnetic barrier on the HES and demonstrate that this model facilitates the measurements of spin-resolved four-probe voltage drops across the ferromagnetic barrier. As a second model, we employ the edge state of a quantum anomalous Hall state (QAHS) as the SPVP which is tunnel-coupled over an extended region with the HES. A two-dimensional lattice simulation for the quantum transport of the proposed device setup comprising a junction of QSHS and QAHS is considered and a feasibility study of using the edge of the QAHS as an efficient spin-polarized voltage probe is carried out including disorder. arXiv:2104.00667v2 [cond-mat.mes-hall] I. INTRODUCTION Transport of electron spin with minimal loss of polarization and coherence over networks which are controllable by all electrical means is highly desirable for spintronics 1,2 and quantum information applications 3 . In particular, the surface states of two-dimensional and three-dimensional topological insulators, which possess spin-momentum locked spectra, can act as a resource for such applications [4][5][6] . Helical edge states (HES) 7-9 of quantum spin Hall state (QSHS) [10][11][12][13][14][15][16] , which are one-dimensional modes lying on the edges carrying electrons with their spin locked to their momenta (e.g., right movers being spin-up and left movers being spin-down along a chosen spin-quantization axis), is one such state that constitutes the main topic of discussion in this manuscript. If we intend to exploit these states for spintronics applications, it requires to devise a way for carrying our spin-resolved measurements on the edge. Experimental attempts of probing these states are already in place, for example, a less invasive detection of these edge states was carried out by using mesoscale SQUID loop in Ref. 17 whereas a more invasive one involving the injection of electrical current into the edge was carried out in Ref. 18. In this manuscript, we carry forward the latter idea and explore the possibility of using a spin-polarized voltage probe for reading off the local spin-resolved voltages on a helical edge. In particular, we formulate the problem to address a situation which is analogous to the six-probe Hall bar setup (two current probes and four voltage probes, as in Fig. 7) of Ref. 19, involving a quantum point contact (QPC) which is routinely used in measurements of Hall voltages. It should be noted that the idea of spin-polarized injection to HES using a ferromagnetic electrode has been theoretically explored in Ref. 20. In fact, an earlier experimental realization engaging a Hall bar type setup can be found in Ref. 21. We consider a situation where a single backscatterer is placed on the HES which is carrying a net current due to the FIG. 1. Schematic of the setup involving a SPVP, which is modelled as a collection of chiral, spin-polarized, one-dimensional modes, tunnel-coupled to the helical edge state subjected at a finite bias voltage (VR − VL). The red (blue) line represents the spin-up (down) edge. The shaded region in green represents the QSHS and the two gray patches on the two sides represents the leads across which the voltage bias is applied. electrical bias applied across the edge state. We then place four spin-polarized voltage probes, two on the right side, and two on the left side of the backscatterer. The polarization directions of the probes are such chosen that the pair of probes on each side of the backscatterer has one fully polarized along the z-axis (call it an "up-polarized") while the other fully polarized along the −z-axis (call it a "down-polarized"). We expect the up-polarized probe would couple primarily to the right movers (spin-up electrons on the edge) and the downpolarized probe will couple mostly to the left movers (spindown electrons on the edge) as we have assumed spin conserving electron tunneling between the probes and the helical edge. We will present a detailed analysis which explores the possibility of such spin-resolved coupling 22 of the z-polarized voltage probes. Note that a number of possible sources of backscattering on a HES have been studied, however, this work focuses primarily on the ballistic limit 9,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] . For a pictorial representation of the situation described above, see Fig. 7. It is straightforward to note the analogy between the situation described in Fig. 7, (a) and Fig. 7, (c) if one identifies the left (right) movers on the top and the bottom edge in the Hall setup with the left (right) movers of the HES. With such a setup, one can measure spin-resolved voltage drops (voltage drop between a pair of up or down probes) across the impurity which is nothing but the drop in voltages in the right and the left moving channels of the HES across the backscatterer. These voltages will be analogous to the longitudinal voltage drop (V lo ) in the Hall bar geometry measured across the QPC. We can also measure the voltage difference between the right and the left mover on the same side by using the up and the down-polarized probe on the same side of backscatterer and that will be analogous to the measurement of the Hall voltage (V H ) in the Hall bar geometry which should be a robust quantity, i.e., independent of the strength of the backscatterer provided the voltage probes are ideal [42][43][44][45][46] . In this manuscript, we will demonstrate via numerical calculations that our model for the SPVP, when optimized appropriately, can lead to measured values of V lo and V H which indeed correspond to the spin-resolved voltage drops across the backscatterer with the latter an analog of Hall voltage on the HES. This, in turn, implies that our theoretical model for the spin-polarized probe is capable of measuring the spinresolved voltages successfully and hence, could provide useful guidance to future experiments that are tuned to such objectives. The rest of the paper is organized as follows: In section II, we lay the concept of spin-resolved voltage measurements on a HES tunnel-coupled to a N P -subprobe SPVP and discuss the stability of the measurements in presence of Gaussian disorder in (i) the tunnel-coupling between the subprobes of the SPVP and the helical edge and (ii) the spin-polarization of the subprobes about a chosen direction. In section III, we demonstrate the six-probe setup discussed above and measure Hall response in presence of a backscatterer including Gaussian disorder in the individual SPVPs. Finally in section IV, we simulate a device setup using the KWANT package 47 which comprises a quantum anomalous Hall state (QAHS) acting as the SPVP for the HES formed at the edge of the QSHS and also present a feasibility study. We summarize the results and conclude in section V. II. SPIN-RESOLVED VOLTAGE MEASUREMENT ON HELICAL EDGE STATES As already discussed in the introduction, our setup to measure the spin-resolved voltage on the HES of a QSHS engages an extended SPVP tunnel-coupled to the HES. The SPVP consists of multiple modes each of which supports spin injection into the HES as shown in Fig. 1. We also assume that the QSHS is hosted on the x-y plane with the relevant HES of the QSHS laying along the x-axis and described by the Hamilto- nian H HES = −ı v F ∞ −∞ dx (ψ † R ∂ x ψ R − ψ † L ∂ x ψ L ), (1) where v F is the Fermi velocity, the operators ψ † R and ψ † L create electrons respectively in the right (R) and the left (L) propagating edge states with spinors |n R = [1 0] T and |n L = [0 1] T respectively implying the spin-polarization of the HES being along the z-axis, perpendicular to the plane hosting the QSHS as found in experiments 18 . We model the SPVP as a collection of one dimensional modes with linear spectrum each of which is, henceforth, referred to as a subprobe. Each of these subprobes is regarded as a right moving chiral mode (R ) with spin-polarization given by the spinor |n R ≡ [cos(θ/2 + π/4) sin(θ/2 + π/4)e iφ ] T (θ, φ are the polar and the azimuthal angle respectively of |n R on the Bloch sphere). As the tunneling between the subprobes and the HES is taken to be local, hence, the chirality of the subprobes (being right or left movers) is of no consequence as far as tunneling current is concerned. The linear spectrum of the subprobes ensures that the tunneling current, in the weak tunneling limit, does not develop an energy dependence due to the variation in the density of states of the probe at the Fermi level as desirable for an ideal probe. Also the modelling of the voltage probe as a collection of subprobes is motivated from the fact that this proves a minimal model for the probe accommodating a large number of electronic degrees of freedom. Hence, the Hamiltonian for a subprobe is given by H (j) subprobe = −ı v F ∞ −∞ dx ψ † R j ∂ x ψ R j ,(2)so that H SPVP = N P j=1 H (j) subprobe . Note that an offset of π/4 is introduced in the expression of |n R j merely to set the range of the polarization angle of the SPVP to be θ ∈ [−π/2, π/2], symmetric about θ = 0. The tunneling Hamiltonian for electrons between the SPVP with N P number of subprobes and the HES is taken to be H T = N P j=1 H (j) T such that, H (j) T = η t (j) ηR ψ † η (x j )ψ R j (0) + h.c.,(3) where the tunnel-coupling between the HES and the subprobe is taken to be such that for each subprobe, the tunneling is happening at x = 0 of the subprobe coordinates and at x = x j for the corresponding HES coordinates; t (j) ηR is the tunneling strength between the right or the left moving states [η ∈ (R, L)] of the HES and the chiral edge R representing the j-th subprobe, further expressed as t ηR = t (j) γ (j) ηR , t (j) being real. Note that the form of the tunneling Hamiltonian preserves the spin rotation symmetry of the electron. The quantity γ (j) ηR denotes the overlap between the spinors |n η and |n R j : γ (j) ηR = n η |n R j . For an extended SPVP with multiple subprobes, we consider the following cases -(i) All the subprobes have identical polarization set by an angle θ and also the tunneling strength t (j) are taken to be uniform across the junctions (with a magnitude t ). We refer to this as the uniform case. (ii) The tunneling strength t (j) is nonuniform across the junctions and is characterized by a Gaussian distribution with mean t and standard deviation σ t while the polarization angle θ of the subprobes fluctuates with a mean θ and standard deviation σ θ . We refer to this as the disordered case. Electron transport across a given tunneling point between a subprobe and the HES at x = x j (see Fig. 1) can be quantified in terms of a scattering matrix S j corresponding to the Hamiltonian H = H HES + H (j) subprobe + H (j) T . The wavefunctions associated with the incoming and the outgoing electrons at the tunneling point at x = x j (denoted Ψ in αj and Ψ out αj respectively) are related by the elements of S j : Ψ out αj = β s (j) αβ Ψ in βj ,(4) where α, β ∈ (R, L, R ) and the corresponding currents obey I out αj = β |s (j) αβ | 2 I in βj .(5) For the SPVP to act as an ideal voltage probe, the net current flowing through it must be zero i.e. I in SPVP − I out SPVP = 0 is the voltage probe condition, where I in SPVP = Np j=1 I in R j and I out SPVP = Np j=1 I out R j . For an SPVP consisting of one subprobe only (N P = 1, tunneling strength t , polarization θ), this condition yields a probe voltage given by V R = |s R R | 2 V R + |s R L | 2 V L |s R R | 2 + |s R L | 2 ,(6) when the HES is connected to a right and a left reservoir maintained at the voltages V R and V L respectively. In presence of a finite bias (V R − V L = 0), the HES develops a net magnetization along the z-axis, hence, in the weak tunneling limit (t v F ), the voltage measured by the SPVP is expected to be a sum of the average voltage V av = (V R + V L )/2 and an additional contribution which can be attributed to tunnel magnetoresistance 48 . Thus, we get V R | N P =1 = 1 2 (V R + V L ) − 1 2 (V R − V L ) sin θ. (7) Note that this expression is independent of the azimuthal angle φ as expected and the sin θ dependence of the magnetoresistance instead of the standard cos θ is due to our phase shifted definition of θ. This expression of V R is obtained by using the explicit forms of the scattering matrix elements (see Appendix A for details). In fact, the above form of V R is valid for all values of t owing to the fact that the ratios |s R R | 2 /(|s R R | 2 + |s R L | 2 ) = cos 2 (θ/2) and |s R L | 2 /(|s R R | 2 + |s R L | 2 ) = sin 2 (θ/2) are independent of FIG. 2. Zoomed-in pictorial representation of a subprobe tunnelcoupled to the HES where T represents the transfer matrix which connects the wavefunctions on the right side of the junction to that on its left side. t though t induces spin flip scattering in the HES via higher order processes when θ = ±π/2. This leads to the interesting fact that the form of the magnetoresistance contribution, which is expected to exist in the weak tunneling limit, actually remains intact even at intermediate and strong tunneling limits. This magnetoresistance contribution in V R is the key element for obtaining spin-resolved information of the HES by using the SPVP and it is the primary focus of this study. Evidently, when θ = −π/2, the spinor |n R aligns with |n R and the SPVP measures the voltage V R = V R , while for θ = π/2, |n R aligns with |n L yielding a voltage measurement V R = V L . However, these results holds only in the limit of few subprobes which we will discuss in detail later. For the case of multiple subprobes (N P > 1), the voltage probe condition implies N P j=1 (I out R j − e 2 V R /h) ≡ N P j=1 I (j) n = 0,(8) where I (j) n indicates the net current in the j-th subprobe. Also note that, as we have assumed all subprobes to be in equilibrium with a large reservoir, which is maintained at a voltage V R , hence, the incoming current in each of the subprobes is given by the Hall relation, I in R j = e 2 V R /h, ∀j and the information regarding the values of V L , V R enters Eq. 8 via I out R j due to Eq. 5. In general, to obtain an analytic expression for V R by solving Eq. 8 is fairly complicated for a given set of values for N P , V L , and V R . Hence, we solve Eq. 8 numerically and also set e 2 /h = 1 henceforth, unless stated otherwise. A. Transfer matrix approach for numerical analysis To obtain V R self-consistently for a given set of values for V R , V L , and t numerically, it is efficient to use a transfer matrix method which connects the left and right going currents on one side of the tunnel junction between the j-th subprobe and the HES at x = x j to that on the other side as shown in Fig. 2. To proceed further, we compartmentalize the HES into N P + 1 segments where the j-th segment is defined as the segment of the HES lying between the tunnel junction at x = x j−1 and x = x j (see Fig. 1). Note that the 1-st and the N P + 1-th segment are connected only to one tunnel junction being the first and the last one. Also the incoming current in each subprobe (in units of e 2 /h) is V R irrespective of the position of the subprobe. Now the relation between currents on different segments can be written as: I (j+1) R = T (j) RR I (j) R + T (j) RL I (j) L + T (j) RR V R I (j+1) L = T (j) LR I (j) R + T (j) LL I (j) L + T (j) LR V R I (j) R = T (j) R R I (j) R + T (j) R L I (j) L + T (j) R R V R ,(9) where T (j) ηη denotes the transfer matrix elements at the j-th junction for the currents while I (j) η (η ∈ R, L) corresponds to the right moving or the left moving current on the j-th segment. In this notation, the net current in the j-th subprobe is given by I (j) n = I (j) R − V R . The elements T (j) ηη can be expressed in terms of the elements of the scattering matrix, S (j) , and is given by T (j) RR = |s (j) RR | 2 − |s (j) RL | 2 |s (j) LR | 2 /|s (j) LL | 2 T (j) RL = |s (j) RL | 2 /|s (j) LL | 2 T (j) RR = |s (j) RR | 2 − |s (j) RL | 2 |s (j) LR | 2 /|s (j) LL | 2 T (j) LR = −|s (j) LR | 2 /|s (j) LL | 2 T (j) LL = 1/|s (j) LL | 2 T (j) LR = −|s (j) LR | 2 /|s (j) LL | 2 T (j) R R = |s (j) R R | 2 − |s (j) R L | 2 |s (j) LR | 2 /|s (j) LL | 2 T (j) R L = |s (j) R L | 2 /|s (j) LL | 2 T (j) R R = |s (j) R R | 2 − |s (j) R L | 2 |s (j) LR | 2 /|s (j) LL | 2 ,(10) and the expressions of the elements s (j) ηη in terms of t and θ are given in Appendix A. Also, we assume that there is no quantum coherence between the scattering of electrons at successive junctions between the HES and the subprobes. To obtain V R , the equations in Eq. 9 are solved recursively for j ∈ {1, . . . , N P } subject to the voltage probe condition in Eq. 8 and boundary conditions given by I N P +1 L = V L and I 1 R = V R . With this formulation discussed above, we can now study the cases mentioned previously, i.e., the uniform case and the disordered case. B. Uniform case The results for the uniform case with N P = 10 and N P = 30 subprobes are displayed in Fig. 3 (a) and (b) respectively for V R = −V L = 1. These values for V R , V L are taken for the numerical calculations as it corresponds to a zero V av hence leaving behind a neat magnetoresistance contribution in Eq. 7. From the plots in Fig. 3 we note that the value of V R decays monotonically towards the average voltage V av as we increase t . This can be understood as follows. In the weak tunneling limit (t v F ), spin-flip scattering induced by the subprobes in the HES is minimal and hence V R carries spin-resolved information, i.e., it follows Eq. 7. But as we increase the tunneling strength (t ), it leads to considerable spin-flip scattering in the HES at the junctions with the subprobes. The feedback between various subprobes amplifies the effect, hence, resulting in reduction of the magnetoresistance response appearing in Eq. 7 and eventually suppressing it completely in the large t limit. The rate at which V R drifts towards V av as a function of t depends on two factors: (i) the angle between the spin-polarization axis of the SPVP and the spin quantization axis of HES, i.e., θ, and (ii) the number of subprobes in the SPVP. Note that the case corresponding to θ = ±π/2 are pathological and in these two cases, V R does not decay towards V av as we increase t for the subprobes do not induce any spin-flip scattering in the HES and, hence, provides a perfect readout of the spin-resolved voltages. In Fig. 3 (c) we have plotted V R as a function of θ for different values of t and N P = 25. We note that the value of V R reduces to V av = 0 independent of t for θ = 0 owing to fact that the tunnel-coupling strengths of the SPVP with the spin-up (right mover) and the spin-down (left mover) channel are equal in this case (see the form of t (j) ηR below Eq. 3) hence, leading to a null magnetoresistance. We further note that larger values of t leads to stronger spin equilibration in the HES and hence, it forces V R to stay close to V av = 0 for most values of θ which are away from θ = ±π/2. But as we consider the value of θ close to −π/2 or π/2, the ratio between the tunneling strength for the spin-up and the spindown channel with the SPVP becomes very large or small respectively, hence, leading to a strong suppression of spin equilibration in the HES. Consequently, V R shows a large deviation from V av implying that it retains spin-resolved information. Before we move on to Fig. 3 (d), to elaborate further on the θ dependence of V R , we will discuss the plot of V R as a function of θ presented in Fig. 4 for different value of N P . We note that the plots for the case of N P = 2 for various values of t starting from a small value (t = 0.01) to a large one (t = 0.76) closely follows Eq. 3. This can be seen from the closeness of the plots for V R to that of sin θ. So it is clear that the case of two subprobes does not immediately changes the scenario as far as deviation from Eq. 3 is concerned. Only when we add multiple subprobes in the SPVP, it leads to a large deviation from Eq. 3 away from θ = ±π/2 and we observe this very clearly in the cases such as N P = 20 and N P = 30 in Fig. 4. In Fig. 3 (d), we further demonstrate that for θ = ±π/2, V R falls exponentially to V av with increasing number of subprobes and, in the large N P limit, is given by V R = V av + [ V R | N P =1 − V av ] e −(N P −1)/N0 .(11) This exponential decay is visible in the main panel of Fig. 3 (d) for various values of t . This indicates that the contribution to V R due to the magnetoresistance effect gets exponentially suppressed with N P ; N 0 multiplied by the average spacing between the subprobes defines a characteristic length scale for the decay. The decay length, obtained from N 0 , can be though of as a measure of the length scale over which the spin-up and the spin-down edge equilibrate leading to loss of spinpolarization of the HES induced by a voltage bias (V L -V R ). The plot in Fig. 3 (d) corresponds to a value of θ which is shifted from −π/2 by 0.2. It is obvious that the length over which the probe is coupled to the HES needs to be smaller than the equilibration scale dictated by N 0 so that the sensitivity of measured V R to the spin-polarization of the HES survives, which is equivalent to saying that this will ensure that V R contains a finite magnetoresistance contribution. Though it is clear that to measure spin-resolved voltages in a HES, we need a spin-polarized voltage probe, but to ensure that the probe indeed measures the spin-resolved voltages, we have to simultaneously optimize the tunneling strength and the physical extent of the coupling between the probe and the HES so that a complete equilibration of the spins on the HES induced by the probe does not obstruct the measurement of a spin-resolved V R . For the above study, we can conclude that the limit, in which an almost perfect readout of the spin-resolved voltage is possible, is when we are in the vicinity of θ = ±π/2 and the tunneling strength and the length of the entire junction (which is proportional to N P ) or alternatively, the physical extent of the coupling are chosen appropriately guided by Eq. 11 so that we are always away form the regime of complete spin equilibration. Also, the case of θ = ±π/2 with uniform t for the subprobes is too ideal to be realized in an experimental situation. This naturally compels us to consider the effects of fluctuations in θ and t across the subprobes in the next section where we analyze how such disorders would influence the readout of the SPVP. C. Disordered case In the disordered case, we first present a comparative study between different disorder averaging of V R sensed by a SPVP of N P = 10, performed over ensembles of various number of disorder configurations of t and θ. We denote the resultant voltage as V R and plot it against the average value of the tunneling strength t for different values of the average polarization of the SPVP, θ in Fig. 5 for ensembles of size 1, 5, 10, and 20. As evident from Fig. 5, we find that the averaging procedure results in a rapid convergence of the functional dependence of V R to that for the uniform case as predicted by Eq.11; in fact, an ensemble of size as few as 20 realizations is promising enough to obtain a close qualitative resemblance to the uniform case. Next we study the stability of the voltage measured by the SPVP as a function of fluctuations in the tunneling strength t and the polarization angle θ of the SPVP. In Fig. 6 (a), we plot the fluctuations in V R characterized by its standard deviation when the measurement is performed over many realizations of the subprobe disorders: σ V R = V 2 R − V R 2 , as a function of σ θ for various values of the tunneling strength t which is assumed to be uniform across all subprobes (the disorder averaging is performed over 10 3 realizations). Here σ θ represents the standard deviation in the values of θ for various subprobes about the chosen average value of θ. We note that the fluctuations in V R increase monotonically with the increase in σ θ as expected but the rate of increase is higher for larger values of the tunneling strength (t ), hence, indicating that small t limit is desirable for implementing such a voltage probe in presence of fluctuations in θ. Fig. 6 (b) shows the variation of σ V R as a function of σ t where σ t represents the standard deviation in the tunneling strength t across various subprobes about a chosen average value of t . If the value of θ corresponds to either the parallel or anti-parallel configuration (i.e., ±π/2), we expect σ V R to be strongly suppressed. We emphasize that in the plot corresponding to θ = −1.56 (a value close to θ = −π/2), increasing σ t has a little influence on σ V R and it stays close to zero. But as we move away from these polarizations, σ V R displays a behavior which is similar to that observed against σ θ discussed above. Also note that the rate of increase of σ V R as a function of σ t is the steepest at θ = 0 as expected owing to the fact that at θ = 0, the magnetoresistance response of each subprobe will be completely suppressed. Hence, for the above study, we conclude that a strong suppression in σ V R will take place if we take the average value of t to be small and the average value of θ to be close to θ = ±π/2. In what follows in the next section, we carry forward the discussion and explore the possibility of using a SPVP for reading off the local spin-resolved voltage drops on a HES using the model discussed above. In particular, we address a situation which is analogous to the six-probe Hall bar setup (two current probes and four voltage probes as in Fig.7 19 ) involving a quantum point contact (QPC) which represents an archetypal setup in the context of quantum Hall experiments. III. MEASURING HALL-TYPE RESPONSE IN A SIX-PROBE SETUP FOR HES Consider the FM barrier, localized at x = x 0 on the HES, is described by the Hamiltonian where |B| represents the coupling strength (proportional to the in-plane magnetic field produced by the FM barrier) and the Hamiltonian for the HES is already given in Eq. 1. In general, B can be a complex number, however, the transfer matrix that connects the currents across the FM barrier is independent of the phase of B, and thus, ±|B| is the only relevant input that enters the calculations for voltages measured by the probe. Hence, without loss of generality, we consider B to be real. The currents across the FM barrier are related as H FM = ∞ −∞ dx δ(x − x 0 ) Bψ † R ψ L + h.c. ,(12)I R (x + 0 ) = T (FM) RR I R (x − 0 ) + T (FM) RL I L (x − 0 ) I L (x + 0 ) = T (FM) LR I R (x − 0 ) + T (FM) LL I L (x − 0 ),(13) where the lower index of R, L represents the right and the left movers; x ± 0 = x 0 ± , being a vanishingly small positive number. The corresponding transfer matrix elements for the currents are LL | 2 ,(14) s (FM) ab being the elements of the scattering matrix for the FM barrier (full expressions given in Appendix B). Let us first consider the situation in which all the subprobes in each of the SPVPs, denoted by A, B, C, and D in Fig. 7 (a), are coupled to the HES with uniform tunneling strength (t ) with equal N P for each of them. The polarization angle of the SPVPs are further set according to θ A = θ C = −θ B = −θ D = −π/2 such that SPVP A and C measures the voltage for the ↑-channel only (note we have considered a shift in θ by π/2 at the starting of our formulation) and SPVP B and D do so for the ↓-channel only (see Eq. 7). In Fig. 7 (c) we have shown a Hall bar geometry where the Hall voltage can be defined as the voltage difference measured between the voltage probes 2, 6 or 3, 5. In the setup for the HES in Fig. 7 (a), the analog of Hall voltage V H is represented by the voltage difference between the ↑channel and the ↓-channel on the same side of the FM barrier, V H = V R − (rV R + tV L ) = t(V R − V L ) where the FM barrier plays the role of the QPC in the Hall bar geometry given in Fig. 7 (c) and further demonstrated in Fig. 7 (b). The net current in the HES is I = tV R + rV L − V L = t(V R − V L ) and hence, the Hall resistance can be defined as R H = V H /I = 1 (in the units of e 2 /h) which is quantized 42 . The longitudinal voltage drop V lo is the voltage difference along a given spinpolarized edge (↑ or ↓) across the barrier which is given by V lo = V R − (tV R + rV L ) = r(V R − V L ) and so, the longitudinal resistance is given by R lo = V lo /I = r/t which is nothing but the four-probe resistance 44 for the FM barrier. The sum of these two resistances satisfies R lo + R H = 1/t which is expected from the Landauer formula 49 . Now we will perform a self consistent numerical analysis to implement the setup described above in terms of our model for the SPVP and check if it reproduces the expected behaviors of R H and R lo . The numerical calculations are performed considering the coupled equations for the subprobes given in Eq. 9 for all the four SPVPs. These equations (each for SPVP A, B, C, and D) are further coupled to each other via transfer matrix for the current which, in the case between SPVP A and SPVP B, and between SPVP C and SPVP D, is an identity matrix while between SPVP B and SPVP C, is the transfer matrix for the FM barrier given in Eq. 14. Finally, the voltage probe condition is imposed simultaneously on SPVP A, B, C, and D and the full set of equations are solved numerically. As fluctuations in both t and θ for each of the SPVPs are expected to exist in a realistic situation, we perform our numerical analysis in their presence and check for the stability in the obtained value of V R for each probe. As discussed in the previous section, we model such fluctuations with a Gaussian distribution of θ among the subprobes in each of the SPVPs with the mean θ A = θ C = −θ B = −θ D = −π/2 and standard deviation σ θ ranging from 10 −4 to π/36 (∼ 5.56% of π/2). We further include such Gaussian disorder in the tunneling strength t as well with a mean t ∈ [0.01, 1] and standard deviation σ t = 0.01 in each of the SPVPs. For the coupling strength of the barrier, B = 0.3, which corresponds to a transmission probability of t = 91.39% (using Eq. B2 in the Appendix), we study the Hall conductance G H (this involves voltage difference between SPVP A and B) and the longitudinal conductance G lo (involving voltage difference between SPVP A and C) as a function of t and σ θ following a disorder averaging over 10 3 realizations. The results are shown in Fig. 8. For the FM barrier with transmission probability of 91.39%, the theoretical values for G H = 1 and G lo = t/r ≈ 10.6. We note from Fig. 8 that the our numerical analysis reproduces these predicted theoretical values very well in the limit of σ θ → 0 independent of the value of t where σ t is taken to be 0.01. The plot illuminates the robust nature of the disorder-averaged value of G H which shows significantly small variations in its value when plotted as a function of σ θ . An increase in t tends to stabilize the value of G H and forces it stay close to the value corresponding to σ θ = 0. But we must keep in mind that the standard deviation in G H should also increase as we increase t which is known from the results presented in Fig. 6, i.e., σ V R increases with increasing σ θ at a faster rate for larger values of t and hence, G H can develop large fluctuations due to its dependence on V R . Hence, though the average G H stays stable with t , the fluctuations in its values tend to increase in this limit implying that it will be safe to operate in the small t limit for reproducing the theoretically expected results. For the longitudinal conductance G lo , we observe approximately a constant value as we increase disorder in the polarization angle (σ θ ) at small values oft , however, the variation increases with increasingt as seen in Fig. 8 (b). This can be understood better by inspecting the output voltages in the individual SPVPs (measured with respect to their values for the uniform case, i.e., the disorder free case) as plotted in Fig. 9 at different values oft . At very small values oft [e.g.t = 0.1 as in Fig. 9 (a)], the profile of V m − V Fig. 8 (b) while it can have an adverse effect on G H though small [ Fig. 8 (a)]. But with increasingt [e.g.t = 0.5 or higher as in Fig. 9 (b)], the profile of V A − V To conclude, we note that the smallt limit is very important for implementing multi-terminal spin-resolved voltage measurements in presence of disorder (i.e., finite σ θ and σ t ) in the SPVPs with N P 1. Of course in the ideal limit of N P = 1, the value of t is of no consequence as long as θ A = θ C = −θ B = −θ D = −π/2 but that will correspond to an unrealistic situation as in any ohmic contact tunnel-coupled to HES, which could be used as a voltage probe, is expected to host multiple modes owing to its finite size. IV. SPIN POLARIZED CHIRAL EDGE STATE AS A SPVP In the study presented above, we have considered a theoretical model where the SPVP comprises a number of chiral one-dimensional spin-polarized modes which are mutually incoherent with each other and are tunnel-coupled to the HES via local tunneling. In this section, we consider the possibility of using a simpler model for the SPVP where a single FIG. 10. Model of a voltage probe consisting of a single chiral edge acting as a SPVP (constructed from a QAH system) tunnel-coupled to the HES of a QSHS as used in the simulation (L, WS, and WA specified in the main text). Lead-1 and Lead-2 represent the leads which act as the electron reservoirs used to apply a finite voltage bias across the HES while Lead-3 is used as a voltage probe with probe voltage given by V R . The FM is employed to open a finite gap in the spectrum of the electronic states lying on the lower HES. A square insulating region of dimension WI (specified in the main text) is used to facilitate a controlled tunneling between the QAH edge and the HES. The polarization of the HES is along Sz while that of the QAH edge is tilted from Sz at an angle θP which can be tuned continuously. spin-polarized chiral edge is employed for the purpose and the tunnel-coupling between the spin-polarized chiral edge and the HES is taken to be spread out over an extended region rather than being local which makes it complementary to our previous model. We simulate a lattice model involving a SPVP coupled to the HES of a QSHS using the software package KWANT 47 . The full lattice model consists of three parts (schematically shown in Fig. 10): (i) a quantum spin Hall (QSH) region described by the BHZ model 13 in its topological phase hosting HES at the edge, (ii) a quantum anomalous Hall (QAH) region described by the BHZ model with an applied exchange field which drives it into the QAH phase that supports a single spin-polarized chiral edge mode whose polarization can be controlled via rotating the direction of the applied field and, (iii) an insulating barrier between the QAH edge and the HES which is described by the BHZ model in its trivial phase. The dimensions of the QSH region are taken to be L × W S with L = 100a, W S = 147a, and a = 3 nm being the lattice constant. The dimensions of the QAH region is taken as L × W A with W A = 50a, and that of the insulating region is taken to be of dimensions 3a × 3a. The bulk of the QSH region is given by the BHZ Hamiltonian H BHZ = −Dk 2 + Ak x σ zσx − Ak yσy + (M − Bk 2 )σ z ,(15) where σ andσ denote the Pauli matrices to describe the spins (↑ or ↓ along S z ) and the orbitals (s or p type) respectively and, and k y = a −1 sin(k y a) (a being the lattice constant specified before), which reads 50,51 H tb = i (c † i H i,i+ax c i+ax +c † i H i,i+ay c i+ay +h.c.)+c † i H ii c i ,(16)where c † i ≡ (c † i,s,↑ , c † i,p,↑ , c † i,s,↓ , c † i,p,↓ ) denotes the set of creation operators for the electrons in s and p orbital with ↑ and ↓ spins at site i with coordinates i = (i x , i y ); a x = a(1, 0) and a y = a(0, 1) are the lattice vectors. Each of the terms, H ii and H i,i+ax(ay) , is a 4 × 4 block matrices defined by H ii = − 4D a 2 − 4B a 2σ z + Mσ z H i,i+ax = D + Bσ z a 2 + Aσ zσx 2ia H i,i+ay = D + Bσ z a 2 + iAσ y 2a .(17) The lattice constant is set to a = 3 nm to obtain a realistic band structure. As mentioned earlier, the spin-polarized chiral edge, that models the SPVP in this case, is obtained from the QSHS by inducing a topological phase transition via the application of an exchange field of strength g 0 in the BHZ Hamiltonian ( |g 0 | > |M | 52,53 ), where the direction of spinpolarization of the chiral edge can be tuned by tuning the direction of the applied exchange field and replacing the coefficient of A k x term with an appropriate term as done below. Hence the model Hamiltonian reads as H BHZ = −Dk 2 + Ak x (â.σ)σ x − Ak yσy + (M − Bk 2 )σ z + g 0 (â.σ)σ z ,(18) where we have replaced the term Ak x σ zσx in Eq. 15 with a new term Ak x (â.σ)σ x such that S a (the component of spin alongâ) is conserved and the unit vectorâ deciding the new direction of spin-polarization on the edge. The corresponding lattice model is given bỹ H ii = − 4D a 2 − 4B a 2σ z + Mσ z + g 0 (â.σ)σ z H i,i+ax = D + Bσ z a 2 + A(â.σ)σ x 2iã H i,i+ay = D + Bσ z a 2 + iAσ y 2a .(19) In our case, we consider g 0 = −15 meV in the QAH region which, for θ P = 0, yields a chiral ↑ edge in the QAH region and for θ P = π, a chiral ↓ edge. A lead (denoted as Lead-3 in Fig. 10) consisting of the same Hamiltonian as in Eq. 19 is attached to the bottom edge (the edge away from the QSH region) of the QAH region through which the spin-resolved probe voltage V R could be measured as a function of the polarization angle θ P ≡ cos −1 (â.σ) ∈ [0, π] as described below. The insulating region is a trivial insulator, characterized by the same Hamiltonian as in Eq. 19, however, g 0 = 0 and the sign of the mass term M flipped (taken M = 10.0 meV) in this region as required for driving a QSHS into a trivial insulating state. With this setup, we first study the current density which is proportional to the transmittance from the chiral (QAH) edge to the helical edge of the QSH region, denoted T ↑ and T ↓ for the ↑-channel and ↓-channel respectively (the reflectance along the chiral channel is R = 1 − T ↑ − T ↓ ). We plot the current density in Fig. 11 (a) for two values of θ P when electrons are incident at energy E = 10 −3 meV which clearly indicates the spin-polarization direction of QAH edge with respect to the two spin-momentum locked modes of the HES. As mentioned above, at a given value of θ P , the spin-polarization of the chiral edge has components along that of both the channels of the helical edge resulting in transmission into the ↑-channel with magnitude T ↑ and into the ↓ channel with magnitude T ↓ . An interesting feature to note in this plot is that when θ P is changed to π from 0, the positions of the two blocks characterized by σ z = ±1 (the eigenvalues of σ z being good quantum numbers as [σ z ,H BHZ ] = 0 at θ P = 0, π) in the modified BHZ HamiltonianH BHZ (Eq. 18) are switched, and as a result, when g 0 is applied with an appropriate magnitude to counter the mass term M , the spin on the (chiral) edge that survives gets flipped, however, the chirality is retained. This is evident from the direction of the current density (black arrows) in both the plots at θ P = 0 and θ P = π. The spin of the chiral (QAH) edge is flipped for θ P : 0 → π and consequently the transmittance changes from T ↑ (θ P = 0) = 0, T ↓ (θ P = 0) = 0 to T ↑ (θ P = π) = 0, T ↓ (θ P = π) = 0 as shown in Fig. 11 (b). The zeros in the transmission probabilities demonstrate a perfect suppression in electron tunneling between the spin orthogonal modes. The plot in Fig. 11 (b) shows the full behavior of T ↑ , T ↓ , and R as function of θ P . The inset displays the behaviors of T ↑ , T ↓ , and R for the model depicted in Fig. 2 for a single subprobe with a local tunnel-coupling between the HES and the subprobe obtained analytically from the scattering matrix at a tunneling strength of t = 0.5 (details given in Appendix A). Note that our twodimensional transport simulation shows a close qualitative resemblance to the ideal model for the subprobe discussed in the previous section. The discrepancy between the results obtained for lattice simulation and the one obtained from a simple minded analytical model given in Eq. 7 can be minimized by adjusting the size of the insulating region and the plots shown above in Fig. 11 are the closest we could obtain that exhibit a slight variation of R against θ P (albeit about a mean value that matches the analytics). Note that the simpleminded one-dimensional model for the subprobe assumes a tunnel-coupling which respects spin-rotation symmetry and this fact leads to the symmetric variation of T ↑ and T ↓ about θ P = π/2. But, on the other hand, the presence of finite spin-nonconserving tunneling events at the junction between the SPVP and the HES owing to the presence of an applied exchange field at the microscopic level in the lattice Hamiltonian constituting the junction always leads to the observed asymmetry in the main figure of Fig. 11 (b). Now let us consider a situation where V R > V L . Then the incoming current flowing into the probe is I ↑ = T ↑ (V R −V R ) while the outgoing one is I ↓ = T ↓ (V R − V L ). Assuming that the transmission probabilities T ↑ and T ↓ are independent of energy in the bias window V R − V L , the voltage probe condition, implying a zero net current in the probe, leads to V R = T ↑ V R + T ↓ V L T ↑ + T ↓ = V R + V L 2 + 1 2 T ↑ − T ↓ T ↑ + T ↓ (V R − V L ),(20) similar to Eq. 6, which we plot against θ P in Fig. 12 (a). Note that the expression for V R has two contributions: the first one is the average voltage of the left and the right moving edge which would have been the full contribution in absence of electron spin, while the second term, being proportional to (V R − V L ), is indeed the magnetoresistance contribution. In the plot presented in Fig. 12 (a), V R = −V L and hence, it features only the magnetoresistance response. Finally, it is important to identify an energy window in which the transmission probabilities have a very weak energy dependence for the SPVP to work and hence, we try to check for the presence of such an energy window in our lattice simulations. This is expected to work as we are working with edge states whose spectrum can be approximated to be linear to a large extent. We provide an estimate of an energy win- dow δE for our setup such that V R/L ∈ [−δE/2, δE/2] leads to treating T ↑ and T ↓ as constants under the variation of E. This window is shown in Fig. 12 (b) by the yellow shaded region in the plot of T ↑ against the incident energy E at various values of θ P . This plot clearly suggests that, within the band gap δE, T ↑ and T ↓ can indeed be approximated as constants but beyond this gap, the transmittances largely deviate from their constant values due to the contributions of the bulk bands gradually coming into play. To account for the effect due to disorder in the SPVP, a modified BHZ model is simulated which describes the QSH state in an inversion symmetry broken system such as InAs/GaSb quantum well. The Hamiltonian, in this case, accommodates spin-orbit interaction which breaks spin rotation symmetry about S z 54 H SO = c 0 σ yσy + k x c + 2 σ x + c − 2 σ xσz − c 3 2 σ y (1 +σ z ) − k y c − 2 σ y + c + 2 σ yσz − c 3 2 σ x (1 +σ z ) ,(21) where, c ± = c 1 ± c 2 , in addition to H BHZ in Eq. 15 but including the parameter g 0 and a chemical potential term (U ) as H BHZ = −Dk 2 + Ak x σ zσx − Ak yσy + (M − Bk 2 )σ z + g 0 σ zσz + U. (22) The chemical potential U takes random values on the lattice chosen from a uniform distribution with an energy cutoff: We note that to demonstrate prominent effects of disorder, we have considered the insulating region to be described by the HgTe system however its dimensions are 3a × 9a where a = 10 nm. The other relevant dimensions of the lattice as shown in Fig. 10 are L = 100a, W S = 141a, and W A = 50a. −E c ≤ U ≤ E c where E The results are shown in Fig. 13 which displays a plot of the transmission probability T ↑ (for the probe polarization θ P = 0) vs the incident energy E in presence of a disordered configuration of the chemical potential U , averaged over 50 such disorder configurations. In the absence of disorder, a plateau at T ↑ ≈ 0.2 is visible over an energy window −4 ≤ E ≤ 4 meV, similar to what is observed also for the HgTe system [see Fig. 12 (b)]. When disorder is taken into account, T ↑ shows a stable behavior, however, the plateauing transpires over a smaller energy window. This makes us conclude that the SPVP constructed from the chiral edge of the QAH system can yield robust measurements of spin-resolved voltages on a HES in presence of disorder as well. V. CONCLUSION Spin-momentum locked spectrum of surface states of two and three-dimensional topological insulators is a resource for a variety of spintronics applications. Though these states have no net polarization in equilibrium, once a finite bias is applied, they do develop finite polarization at the Fermi level and this fact can be exploited in a number of device applications. One way to measure such polarization is to employ spin-polarized voltage probes which could measure the spin-resolved voltages in these states subjected to a finite bias. In this work, we have explored two such possibilities for the surface states of quantum spin Hall system which is a two-dimensional topological insulator and also included disorder effects. Within our theoretical model, it is established that such measurements should be possible if the probes are designed appropriately 55 featuring robust signals even in the presence of disorder. VI. ACKNOWLEDGMENTS VA acknowledges support from IISER Kolkata in the form of a subsistence grant. VA also wants to thank Rafiqul Rahaman for helping him with job submission in the cluster computing facility. KR thanks the sponsorship, in part, by the Swedish Research Council. SD would like to acknowledge the MATRICS grant (MTR/ 2019/001 043) from the Science and Engineering Research Board (SERB) for funding. We acknowledge the central computing facility (DIRAC supercomputer) and the computational facility at the Department of Physics (KEPLER) at IISER Kolkata. ı ψ (j) η = [ψ (j) η , H (j) ],(A1) where H (j) = H HES + H (j) subprobe + H (j) T is specified in the main text (see Eq. 1-3). Integrating the e.o.m over a region from − and with the limit → 0 across x = x j , one obtains the required equations between the incoming and the outgoing amplitudes connected through the PC 56 . From now on, we drop the superscript j assuming all of the following relations pertain to the j-th PC. The scattering matrix elements s ηη , which appear in Eq. (4), are defined as   ψ R (0 + ) ψ L (0 − ) ψ R (0 + )   ψ out =   s RR s RL s RR s LR s LL s LR s R R s R L s R R   S−matrix   ψ R (0 − ) ψ L (0 + ) ψ R (0 − )   ψ in ,(A2) where s RR = ψ R (0 + )/ψ R (0 − ), s LR = ψ L (0 − )/ψ R (0 − ), s R R = ψ R (0 + )/ψ R (0 − ), and so on. The explicit expressions for these elements, after taking v F = 1, read s RR = [8 + ι(Γ RL Γ * RR Γ LR + Γ * RL Γ RR Γ * LR ) − 2{|Γ RL | 2 + |Γ RR | 2 − |Γ LR | 2 }]/D s RL = −8ιΓ RL − 4Γ RR Γ * LR /D s RR = [−8ιΓ RR − 4Γ RL Γ LR ] /D s LR = −8ιΓ * RL − 4Γ * RR Γ LR /D s LL = [8 + ι(Γ RL Γ * RR Γ LR + Γ * RL Γ RR Γ * LR ) − 2{|Γ RL | 2 − |Γ RR | 2 + |Γ LR | 2 }]/D s LR = [−8ιΓ LR − 4Γ * RL Γ RR ] /D s R R = −8ιΓ * RR − 4Γ * RL Γ * LR /D s R L = −8ιΓ * LR − 4Γ RL Γ * RR /D s R R = [8 + ι(Γ RL Γ * RR Γ LR + Γ * RL Γ RR Γ * LR ) − 2{−|Γ RL | 2 + |Γ RR | 2 + |Γ LR | 2 }]/D, (A3) where the common denominator, D is D = 8 − ι Γ RL Γ * RR Γ LR + Γ * RL Γ RR Γ * LR +2{|Γ RL | 2 + |Γ RR | 2 + |Γ LR | 2 }.(A4) The functional form of Γ ηη in terms of t and θ are noted in the main text from which all our results readily follow. the FM barrier The 2 × 2 scattering matrix defined at the FM barrier of the Fig. 7 can be easily obtained by following the procedure discussed in appendix A. as   ψ R (x + 0 ) ψ L (x − 0 )   ψ out =   s F M RR s F M RL s F M LR s F M LL   S−matrix   ψ R (x − 0 ) ψ L (x + 0 )   ψ in .(B1) The explicit expressions for the S-matrix elements are given by s F M RR = 8 − 2|Γ F M RL | 2 /D s F M RL = −8ιΓ F M RL /D s F M LR = −8ι(Γ F M RL ) * /D s F M LL = 8 − 2|Γ F M RL | 2 /D ,(B2) where the common denominator, D is D = 8 + 2|Γ F M RL | 2 .(B3) Similar functional forms of Γ F M ηη follow with t being replaced by B and the angular dependence having no relevance. FIG. 3 . 3Plot of V R against tunneling strength t for a uniform SPVP with NP = 10 in (a) and NP = 30 in (b), where the HES is maintained at a voltage bias given by VR = −VL = 1. Different curves correspond to different values of θ starting from θ = −π/2 (the top most plot) to θ = π/2 (the bottom most plot) in steps of π/40 (the offset along the t -axis of these two figures is t = 0.04). (c) Plot of V R as a function of θ at different values of t for NP = 25. (d) Plot of V R as a function of number of subprobes NP in the SPVP at different values of t for θ = −1.37. The inset shows the scale of this exponential decay, denoted N0, plotted as a function of t . The offset along the NP -axis for this plot is NP = 20. FIG. 4 . 4Variation of V R as a function of θ for NP = 2, 10, 20, and 30 in (a), (b), (c), and (d) respectively. FIG. 5 . 5Evolution of the disorder averaged variation of V R as a function of the tunneling strength (t ) for different values ofθ with NP = 10. The solid lines in each figure represent the uniform case: the variation of V R as a function of a uniform value of t for different values of θ. The scattered datapoints in (a) show the variation of V R for a single random disorder configuration of t and θ. Figure (b), (c), and (d) show the same variation after taking a disorder average over 5, 10, and 20 random configurations of t and θ about their respective average values. We have taken σt = σ θ = 0.2 and the offset along the x-axis of all the plots is taken to be 0.01. FIG. 6 . 6(a) Plot of σV R vs σ θ at different values of (uniform) t for a SPVP (NP = 10) with Gaussian disorder in θ across the subprobes with a mean value of θ to be −1.27. The fluctuations increase in the strong tunneling limit (averaging performed over 10 3 realizations).(b) Plot of σV R vs σ t at different values of (uniform) θ for the SPVP (NP = 10) with Gaussian disorder in t across the subprobes with a mean value of t to be 1.0. The fluctuations increase for the values of θ away from ±π/2, but the magnitude is much smaller compared to the disordered t case shown in (a). FIG. 7 . 7(a) Schematic of the six-probe (four voltage probes and two current probes) setup to measure the Hall conductance between the counterpropagating states of the QSHS and the longitudinal conductance across an FM barrier (shaded grey, not to confuse with the FM placed on the bottom edge). The Hall voltage and the longitudinal voltage are denoted as VH and V lo respectively. The polarization angle for the SPVP are set as θA = θC = −π/2 = −θB = −θD such that SPVP A and C sense the voltage on the ↑-channel (right movers) and SPVP B and D sense the voltage on the ↓-channel (left movers) only. A ferromagnet (FM) is employed to open a gap in the spectrum of the bottom edge hence, electrically disconnecting the left and the right leads via this edge. (b) The boxed figure on the right top shows the zoomed-in FM barrier with a transmission probability of t = 91.39% and the voltage drops across this barrier depending on t and the reflection probability r = 1 − t that characterize the scattering through the barrier. (c) The analogous schematic of a six-probe Hall bar setup involving a QPC in the middle as discussed in Ref. 19 where lead 1 and 2 represent the current probes and lead 3, 4, 5, and 6 represent the voltage probes. FIG. 8 . 8Plot of the Hall conductance GH in (a) and the longitudinal conductance G lo in (b) as a function oft and σ θ (defined in the main text). The variation of GH being small signifies its robust nature against disorder as illuminated in the main text. FIG. 9 . 9Plot of individual probe voltages with respect to their values for the uniform case, i.e., Vm − V (0) m (m = A, B, C, D), as function of σ θ for t = 0.1 in figure (a) and t = 0.5 in figure (b). A disorder averaging has been performed over 200 realizations in each of the plots. the value for the disorder free case) for m = A and m = C (and likewise, m = B and m = D) almost match resulting in small variation in G lo as observed in D ) tend to move away from each other resulting in G lo to fall off steeply from its uniform value with increasing σ θ . A, B, D, and M are material-dependent parameters. For HgTe/CdTe quantum wells, D = −512.0 nm 2 -meV, A = 364.5 nm-meV, B = −686.0 nm 2 -meV, and M = −10.0 meV, however, in our simulation, we set D = 0 to place the Dirac point at zero energy. A discrete form of the Hamiltonian in Eq. 15 can be identified on a square lattice with a basis of two sites (representing the two orbitals) using k 2 = 2a −2 [2 − cos(k x a) − cos(k y a)], k x = a −1 sin(k x a), FIG. 11 . 11(a) Plot of the current density on the entire lattice used in the simulation at two values of the probe polarization angle θP . The tick labels in the figure are given in nanometers. The bright regions indicate the presence of the (chiral) current with its flow guided by the black arrows. (b) Plots of T ↑ , T ↓ , and R (defined in the main text) as function of θP as obtained from the numerics for an incident energy of E = 10 −3 meV. The inset shows the ones obtained from an analytical calculation (details mentioned in the main text). FIG. 12 . 12(a) Plot of the probe voltage V R vs. θP as calculated from Eq. 20 showing its variation from V R = VR = 1 at θP = 0 to V R = VL = −1 at θP = π as expected. (b) A plot of T ↑ vs. the incident energy E at different values of θP reveals an energy window E ∈ δE = [−5, 5] meV within which the spin-resolved voltage measurement should be performed. FIG. 13 . 13Plot of T ↑ (empty circles) vs. the incident energy E in presence of disorder in the KWANT setup involving the InAs/GaSb system (details in the text). The disorder average is performed over 50 configurations and the uniform case (absence of disorder) is shown by the solid line. c can at most be equal to the bulk gap. A weak magnetic field B = (0, 0, B 0 ) is allowed to facilitate breaking of time reversal symmetry, however, the corresponding Zeeman energy is neglected. The magnetic field enters the Hamiltonian only via the Peierls phase Exp[i(e/ ) A · d ] of the hopping terms specified by the gauge potential A = (0, B 0 x, 0). For the InAs/GaSb system used in the simulation, the following parametric values are used: A = 37.0 meV-nm, D = 0.0 meV-nm 2 , B = −660.0 meV-nm 2 , M = −7.8 meV, g 0 = −12 meV, B 0 = 0.05 T, c 0 = 0.2 meV, c 1 = 0.066 meV-nm, c 2 = 0.06 meVnm, c 3 = −7.0 meV-nm, and E c = 7 meV. 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[ "Offline Time-Independent Multi-Agent Path Planning", "Offline Time-Independent Multi-Agent Path Planning", "Offline Time-Independent Multi-Agent Path Planning", "Offline Time-Independent Multi-Agent Path Planning" ]
[ "Keisuke Okmura [email protected] \nTokyo Institute of Technology\n\n", "François Bonnet [email protected] \nTokyo Institute of Technology\n\n", "Yasumasa Tamura [email protected] \nTokyo Institute of Technology\n\n", "Xavier Défago [email protected] \nTokyo Institute of Technology\n\n", "Keisuke Okmura [email protected] \nTokyo Institute of Technology\n\n", "François Bonnet [email protected] \nTokyo Institute of Technology\n\n", "Yasumasa Tamura [email protected] \nTokyo Institute of Technology\n\n", "Xavier Défago [email protected] \nTokyo Institute of Technology\n\n" ]
[ "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n", "Tokyo Institute of Technology\n" ]
[]
This paper studies a novel planning problem for multiple agents that cannot share holding resources, named OTIMAPP (Offline Time-Independent Multi-Agent Path Planning). Given a graph and a set of start-goal pairs, the problem consists in assigning a path to each agent such that every agent eventually reaches their goal without blocking each other, regardless of how the agents are being scheduled at runtime. The motivation stems from the nature of distributed environments that agents take actions fully asynchronous and have no knowledge about those exact timings of other actors. We present solution conditions, computational complexity, solvers, and robotic applications.
10.24963/ijcai.2022/645
[ "https://export.arxiv.org/pdf/2105.07132v3.pdf" ]
234,742,037
2105.07132
beb9f5135dfeb3f0b953fa5bf6304796ee22a8a2
Offline Time-Independent Multi-Agent Path Planning Keisuke Okmura [email protected] Tokyo Institute of Technology François Bonnet [email protected] Tokyo Institute of Technology Yasumasa Tamura [email protected] Tokyo Institute of Technology Xavier Défago [email protected] Tokyo Institute of Technology Offline Time-Independent Multi-Agent Path Planning This paper studies a novel planning problem for multiple agents that cannot share holding resources, named OTIMAPP (Offline Time-Independent Multi-Agent Path Planning). Given a graph and a set of start-goal pairs, the problem consists in assigning a path to each agent such that every agent eventually reaches their goal without blocking each other, regardless of how the agents are being scheduled at runtime. The motivation stems from the nature of distributed environments that agents take actions fully asynchronous and have no knowledge about those exact timings of other actors. We present solution conditions, computational complexity, solvers, and robotic applications. Introduction The eventual goal of collective path planning for multiple agents is to make each agent in a shared workspace be on their respective goal status. This problem becomes non-trivial when agents cannot pass through each other, i.e., each agent occupies some resources in the space while the others are blocked to access these resources at that time. We see such situations in fleet operations of warehouses [Wurman et al., 2008], intersection management for self-driving cars [Dresner and Stone, 2008], multi-robot 3D printing systems [Zhang et al., 2018], packet-switched networks with limited buffer spaces [Tel, 2000], and lock operations of transactions on distributed databases [Knapp, 1987], to name just a few. In such multi-agent systems, each agent inherently takes and finishes actions (or moves) at their own timings independently and unpredictably from other actors, regardless of centralized or decentralized controls. This is due to the nature of distributed environments such as message delay or clock shift/drift, as well as uncaptured individual differences between agents like frictions of physical robots. Nevertheless, the cutting-edge research on pathfinding for multiple agents, known as Multi-Agent Path Finding (MAPF) [Stern et al., 2019] that aims at finding a set of collision-free paths on graphs, heavily rely on timing assumptions. Typical MAPF * Contact Author i j i j Figure 1: Example of OTIMAPP. A graph is depicted with black lines. Two agents (i, j) and their paths are colored. left: Both agents stop progression permanently due to mutual exclusion (i.e., no collision) if i moved two steps before j moves. right: As long as each agent follows a respective path, both agents eventually reach their last vertex; these paths constitute an OTIMAPP solution. assumes that agents take actions just at the same time. Not to mention, such "timed" schedules contradict the nature of distributed environments. Even worse, on-time execution of offline planning is too optimistic with more agents. One counter approach to the timing uncertainties is runtime supports by online monitoring, re-planning, and intervention, e.g., [Van Den Berg et al., 2011;Ma et al., 2017;Atzmon et al., 2020b;Okumura et al., 2021]. This approach however requires runtime effort and additional infrastructures (e.g., steady network and monitoring systems) to manage agents' status in real-time. Moreover, how to realize such schemes in large systems is not trivial at all. Instead, this paper studies a novel planning problem in which agents spontaneously take actions without any timing assumptions. The problem requests a set of paths (i.e., solution) ensuring that all agents eventually reach their destinations without blocking each other permanently. To see this, consider the situation in Fig. 1(left). This plan runs a risk of execution failure; if the agent j gets delayed for any reason while the agent i moves two steps to the right, then each agent blocks each other and neither agent can progress on its respective path. In contrast, in Fig. 1(right), regardless of how the two agents are scheduled, both agents eventually reach their destinations unless they permanently stop the progression. We call the corresponding problem Offline Time-Independent Multi-Agent Path Planning (OTIMAPP). The contribution of this paper is to establish the foundation of OTIMAPP for both theory and practice. Specifically, the topics are categorized into two: We formalize and analyze OTIMAPP. Section 3 identifies a necessary and sufficient condition for a solution, i.e., a set of paths that makes all agents reach their goals without timing assumptions. This is based on characterization of deadlocks. Section 4 conducts a series of complexity analyses and reveals that (1) finding a solution is NP-hard on directed graphs, (2) finding a solution is NP-hard on undirected graphs when solutions are restricted to simple paths, and (3) verifying a solution is co-NP-complete. We present algorithms to solve OTIMAPP and demonstrate the utility of OTIMAPP via robotic applications. Section 5 presents two approaches to derive solutions: prioritized planning (PP) and deadlock-based search (DBS). Both algorithms are respectively derivative from basic MAPF algorithms [Erdmann and Lozano-Perez, 1987;Sharon et al., 2015] and rely on a newly developed procedure to detect deadlocks within a set of paths. Section 6 shows that either PP or DBS can compute large OTIMAPP instances to some extent. Furthermore, we show that solutions keep robots' moves efficient in an adverse environment for timing assumptions compared to existing approaches with runtime supports [Ma et al., 2017;Okumura et al., 2021]. Moreover, we demonstrate that solutions are executable with physical robots in both a centralized style and a decentralized style with only local interactions, without cumbersome procedures of online interventions. In the remainder, all omitted proofs including sketches are available in the appendix. The appendix, code, and movie are available on https://kei18.github.io/otimapp. Related work will be discussed at the end. Problem Definition An OTIMAPP instance is given by a graph G = (V, E), a set of agents A = {1, 2, . . . , N }, an injective initial state function s : A → V , and an injective goal state function g : A → V . An OTIMAPP instance on digraphs is similar to the undirected case. An execution schedule is an infinite sequence of agents. An OTIMAPP execution is defined by an OTIMAPP instance, an execution schedule E, and a set of paths {π 1 , . . . , π N } as follows. The agents are activated in turn according to E. Upon activation and until reaching the end of its path π i , an agent i takes a single step along π i if the vertex is vacant or stays at its current location otherwise. After reaching the end of the path, the agent only stays. E is called fair when every agent appears infinitely-many times in E. An OTIMAPP problem is to decide whether there is a set of paths {π 1 , . . . , π N } such that (1) each path for an agent i begins from s(i) and ends at g(i), (2) for any fair execution schedule, all agents reach the end of their paths (i.e., goals) in a finite number of activations. A solution is a set of paths satisfying these two. Other Notations Let s i and g i denote s(i) and g(i), respectively. A location for an agent i is associated with a progress index clock i ∈ {1, · · · , |π i |} and represented as π i [clock i ], where π i [j] is the j-th vertex in π i . Every progress index starts at one and is incremented each time the agent moves a step along its path. The progress index is non-decreasing and no longer increases after reaching the end of the path. We use S[−1] to denote the last element of the sequence S. ((i, j, k), (3, 1, 2)). right: terminal; (i, j, 2). Rationale and remarks Any solution must deal with all timing uncertainties because execution schedules are unknown when offline planning. We assume that agents are activated sequentially and that each activation is atomic. However, there is no loss of generality as long as an agent can atomically reserve its destination before each move. Indeed, several robots acted simultaneously in our demos. Throughout the paper, we assume that each path π i starts from s i and ends at g i to focus on analyses related to schedules. Solution Analysis Given a set of paths, our first question is to determine whether it is a solution. This section derives a necessary and sufficient condition for solutions. For this purpose, we introduce four types of deadlocks, categorized as; cyclic or terminal; potential or reachable. Informally, a cyclic deadlock is a situation where agent i wants to move to the current vertex of j, who wants to move to the current vertex of k, who wants to move to ... of i. A terminal deadlock is a situation where agent i reaches its destination and blocks the progress of another agent j. A potential deadlock is called reachable when there exists an execution schedule leading to the deadlock. Definition 3.1 (potential cyclic deadlock). Given an OTIMAPP instance and a set of paths {π 1 , . . . π N }, a potential cyclic deadlock is a pair of tuples ((i, j, k, . . . , l), (t i , t j , t k , . . . , t l )) such that π i [t i + 1] = π j [t j ] ∧ π j [t j + 1] = π k [t k ] ∧ . . . ∧ π l [t l + 1] = π i [t i ]. The elements of the first tuple are without duplicates. Definition 3.2 (potential terminal deadlock). Given an OTIMAPP instance and a set of paths {π 1 , . . . π N }, a potential terminal deadlock is a tuple (i, j, t j ) such that π i [−1] = π j [t j ] and i = j. Definition 3.3 (reachable cyclic deadlock). A potential cyclic deadlock ((i, j, . . . , l), (t i , t j , . . . , t l )) is reachable when there is an execution schedule leading to a situation where clock i = t i ∧ clock j = t j ∧ . . . ∧ clock l = t l . This deadlock is called a reachable cyclic deadlock. Definition 3.4 (reachable terminal deadlock). A potential terminal deadlock (i, j, t j ) is reachable when there is an execution schedule leading to a situation where clock i = |π i | ∧ clock j = t j − 1. This deadlock is called a reachable terminal deadlock. We refer to both reachable (or potential) cyclic/terminal deadlocks by reachable (resp. potential) deadlocks and simply use "deadlock" whenever the context is obvious. At least one execution schedule is required to verify whether a potential deadlock is reachable. For instance, in Fig. 1 (left), a schedule (i, i, . . .) is evidence. A potential deadlock is not always reachable as illustrated in Fig. 2. Theorem 3.5 (necessary and sufficient condition). Given an OTIMAPP instance, a set of path {π 1 , . . . , π N } is a solution if and only if there are (1) no reachable terminal deadlocks and (2) no reachable cyclic deadlocks. Proof sketch. Verifying that they are necessary is straightforward. To see that they are sufficient, consider a potential function φ := i∈A (|π i | − clock i ) defined over a configuration {clock 1 , . . . , clock N }. Observe that φ is non-increasing and φ = 0 means that all agents have reached their goals. Furthermore, when φ > 0, φ is guaranteed to decrease if each agent is activated at least once. Computational Complexity This section studies the complexity of OTIMAPP. In particular, we address two questions: the difficulty to find solutions (Sec. 4.1) and the difficulty to verify solutions (Sec. 4.2). Our main results are that both problems are computationally intractable; the former is NP-hard and the latter is co-NPcomplete. Both proofs are based on reductions from the 3-SAT problem, deciding satisfiability for a formula in conjunctive normal form with three literals in each clause. Finding Solutions We distinguish directed graphs and undirected graphs to analyze the complexity. The following proof is partially inspired by the NP-hardness of MAPF on digraphs [Nebel, 2020]. Theorem 4.1 (complexity on digraphs). OTIMAPP on directed graphs is NP-hard. Proof. The proof is a reduction from the 3-SAT problem. Figure 3 is an example of the reduction from a formula (x 1 ∨ x 2 ∨ ¬x 3 ) ∧ (¬x 1 ∨ x 2 ∨ x 3 ). A. Construction of an OTIMAPP instance. We introduce two gadgets, called variable decider and clause constrainer. The OTIMAPP instance contains one variable decider for each variable and one clause constrainer for each clause. The variable decider for a variable x i assigns x i to true or false. This gadget contains one agent χ i with two paths to reach its goal: left or right. Taking a left path corresponds to assigning x i to false, and vice versa. For the j-th clause C j in the formula, when its k-th literal is either x i or ¬x i , we further add one agent c j k to the gadget. Its start and goal are positioned on the right side from χ i when the literal is a negation; otherwise, on the left side. When several such agents are positioned on one side, let them connect (see the gadget for x 2 ). c j k has two alternate paths to reach its goal: a path within the variable decider or a path via a clause constrainer. The former is available only when χ i takes a path of the opposite direction to avoid a reachable cyclic deadlock. The clause constrainer for a clause C j connects the start and the goal of c j k . The gadget contains a triangle. Each literal c j k enters this triangle from a distinct vertex and exits from another vertex. As a result, this gadget prevents three literals in C j from being false simultaneously; if not so, three agents enter the gadget and there is a reachable cyclic deadlock. The number of agents, vertices, and edges are all polynomial with respect to the size of the formula. B. The formula is satisfiable if OTIMAPP has a solution: the use of one clause constrainer by three agents leads to a reachable cyclic deadlock. Thus, at least one literal for each clause becomes true in any OTIMAPP solution. C. OTIMAPP has a solution if the formula is satisfiable: If satisfiable, let χ i take a path that follows the assignment. Let c j k take a path within the variable decider when χ i takes the opposite direction; otherwise, use the clause constrainer. Since three agents never enter one clause constrainer due to satisfiability, those paths constitute a solution. For undirected graphs, we limit solutions to those containing only simple paths. 1 Theorem 4.2 (complexity on undirected graphs). For OTIMAPP on undirected graphs, it is NP-hard to find a solution with simple paths. Proof sketch. We add a new gadget called oneway constrainer, which transforms an undirected edge to a virtually directed one, to the proof of the NP-hardness on digraphs (Thm. 4.1). We derive the claim by replacing all directed edges, except for bidirectional edges, with this gadget. Figure 4 illustrates it, including two new agents: z 1 and z 2 . In this gadget, any agents outside of the gadget are allowed to move only in the direction from u to v. Verification The co-NP completeness of the verification relies on the following lemma, stating that finding cyclic deadlocks is computationally intractable. Its entire proof is delivered in the Appendix. Lemma 4.3 (complexity of detecting cyclic deadlocks). Determining whether a set of paths contains either reachable or potential cyclic deadlocks is NP-complete. We then derive the complexity result since a solution has no reachable deadlocks. Theorem 4.4 (complexity of verification). Verifying a solutions of OTIMAPP is co-NP-complete. Proof. Thm. 3.5 states that a solution has no reachable terminal/cyclic deadlocks. Verifying no terminal deadlocks is in co-NP; a terminal deadlock is verified in polynomial time with an execution schedule. Verifying no potential deadlocks is co-NP-complete according to Lemma 4.3. Solvers We now focus on how to solve OTIMAPP. In practice, it is difficult to use the necessary and sufficient condition (Thm. 3.5) because we have to find corresponding schedules. This motivates to build a relaxed sufficient condition. Theorem 5.1 (relaxed condition). Given an OTIMAPP instance, a set of path {π 1 , . . . , π N } is a solution when there are (1) no use of other goals, i.e., g j ∈ π i for all i = j except for s i = g j , and (2) no potential cyclic deadlocks. It is straightforward to see that the above conditions are respectively sufficient for the two conditions in Thm. 3.5. Given a set of paths, "no use of other goals" is easy to check while "no potential cyclic deadlocks" is intractable to compute (Lemma 4.3). Nevertheless, detecting potential cyclic deadlock is the heart of solving OTIMAPP. Thus, we first explain how to detect potential cyclic deadlocks. After that, two algorithms to solve OTIMAPP are presented. Detection of Potential Deadlocks Due to the space limit, we only describe the intuition behind the algorithm. The details are in the Appendix (Alg. 3). We first introduce a fragment, a candidate of potential cyclic deadlocks. Definition 5.2 (fragment). Given a set of paths {π 1 ,. . . ,π N }, a fragment is a tuple ((i, j, k, . . . , l), (t i , t j , t k , . . . , t l )) such that π i [t i + 1] = π j [t j ] ∧ π j [t j + 1] = π k [t k ] ∧ . . . = π l [t l ]. The elements of the first tuple are without duplicates. We say that a fragment starts from a vertex u when π i [t i ] = u and a fragment ends at a vertex v when π l [t l + 1] = v. A fragment that ends at its start (i.e., π l [t l + 1] = π i [t i ]) is a potential cyclic deadlock. Using fragments, we construct an algorithm to detect a potential cyclic deadlock in a set of paths if it exists. This is based on induction on π i . The induction hypothesis for i is that there are no potential cyclic deadlocks for induction key new fragments Table 1: Example of detecting potential cyclic deadlocks. We describe the update of Θs for π1 = (u, v, w), π2 = (v, x, y), π3 = (z, x, u). The table uses [(agents), (progress indexes), (path)] as a notation of fragment, where "path" is a corresponding sequence of vertices of the fragment. The algorithm halts with a blue-colored fragment, a detected potential cyclic deadlock. {π1} u [(1), (1), (u, v)] v [(1), (2), (v, w)] {π1, π2} u [(1, 2), (1, 1), (u, v, x)] v [(2), (1), (v, x)] x [(2), (2), (x, y)] {π1, π2, π3} u [(1, 2, 3), (1, 1, 2), (u, v, x, u)] v [(2, 3), (1, 2), (v, x, u)] x [(3), (2), (x, u)], [(3, 1), (2, 1), (x, u, v)] z [(3), (1), (z, x)], [(3, 2), (1, 2), (z, x, y)] {π 1 , . . . , π i−1 } and all fragments for them are identified. All new fragments about π i are categorized into three groups: (1) a fragment only with π i , (2) a fragment that extends existing fragments, and (3) a fragment that connects existing two fragments. In either case, if a newly created fragment ends at its start, this is a deadlock. The algorithm realizes this procedure by managing two tables that store fragments: Θ s and Θ t . Both tables take one vertex as a key. One entry in Θ s stores all fragments starting from the vertex. One entry in Θ t stores all fragments ending at the vertex. Table 1 presents an example to detect deadlocks. Prioritized Planning (PP) Prioritized planning [Erdmann and Lozano-Perez, 1987;Silver, 2005] is neither complete nor optimal, but it is computationally cheap hence a popular approach to MAPF. It plans paths sequentially while avoiding collisions with previously planned paths. Instead of inter-agent collisions, solvers for OTIMAPP have to care about potential cyclic deadlocks. Algorithm 1 is prioritized planning for OTIMAPP, named PP. When planning a single-agent path, PP avoids using (1) goals of other agents and (2) edges causing potential cyclic deadlocks [Line 3]. The latter is detected by storing all fragments created by previously computed paths. For this purpose, PP uses the adaptive version of Alg. 3 [Line 5] in the Appendix. A path satisfying the constraints can be found by ordinary pathfinding algorithms. If not, PP returns FAIL-URE. The correctness of PP is derived from Thm. 5.1. PP is simple but incomplete. In particular, the planning order of agents is crucial; an instance may be solved or may not be solved as illustrated in Fig. 5. One resolution is repeating PP with random priorities until the problem is solved; let call this PP + . However, finding good orders can be challenging because there are |A|! patterns. This motivates us to develop a search-based solver, described in the next. Deadlock-based Search (DBS) We present deadlock-based search (DBS) to solve OTIMAPP, based on a popular search-based MAPF solver called conflictbased search (CBS) [Sharon et al., 2015]. CBS uses a two- π i ← a path for agent i while avoiding the use of · g j , ∀j = i, except for s i · (u, v) ∈ E s.t. ∃θ ∈ Θ t [u] and θ starts from v avoiding cyclic deadlocks for π j , j < i 4: if π i is not found then return FAILURE 5: update Θ s and Θ t with π i using Algorithm 3 6: end for 7: return {π 1 , . . . , π N } i j Figure 5: Example that the planning order affects the solvability. When i plans prior to j, PP results in success with solid lines. PP fails if j plans first and takes the dotted line. level search. The high-level search manages collisions between agents. When a collision occurs between two agents at some time and location, two possible resolutions are depending on which agent gets to use the location at that time. Following this observation, CBS constructs a binary tree where each node includes constraints prohibiting to use space-time pairs for certain agents. In the low-level search, agents find a single path constrained by the corresponding high-level node. Instead of collisions, DBS considers potential cyclic deadlocks. When detecting a deadlock in a set of paths, a resolution is that one of the agents in the deadlock avoids using the edge. Thus, the constraints identify which agents prohibit using which edges in which orientation. Algorithm 2 describes the high-level search of DBS. Each node in the high-level search contains constraints, a list of tuples consisting of one agent and two vertices (representing "from vertex" and "to vertex"), and paths as a solution candidate. The root node does not have any constraints [Line 1]. Its paths are computed following "no use of other goals" of Thm. 5.1 [Line 2]. Then, the node is inserted into a priority queue OPEN Line 5: OPEN is a priority queue and needs the order of nodes. DBS works in any order but good orders reduce the search effort. As effective heuristics, we use the descending order of the number of deadlocks with two agents, which is if C = ∅ then return N.paths 8: for (i, u, v) ∈ C do 9: N ← {constraints : N.constraints + (i, u, v), paths : N.paths} 10: update π i in N .paths to follow N .constraints 11: if π i is found then insert N to OPEN 12: end for 13: end while 14: return FAILURE computed within a reasonable time. Line 6: Let ((i, j, k, . . . , l),(t i , t j , t k , . . . , t l )) be a returned deadlock by Alg. 3. Then, create constraints (i, π i [t i ], π i [t i + 1]), (j, π j [t j ], π j [t j + 1]), . . . , (l, π l [t l ], π l [t l + 1]). Line 10: forces one path π i in the node to follow the new constraints. This low-level search must follow "no use of other goals," furthermore, all edges in the constraints for i. If not found, DBS discards the corresponding successor. There is one potential cyclic deadlock in the paths then two constraints are created: either i or j avoids using the shared edge [Line 10]. Two child nodes are generated, however, the node that changes i's path is invalid because there is no such path without the use of the goal of j. Another one is valid; j takes the solid red line. Therefore, one node is added to OPEN from the root node. In the next iteration, this newly added node is expanded. There are no potential cyclic deadlocks in this node; DBS returns its paths as a solution. Optimization Although this paper focuses on a feasibility problem, DBS can adapt to optimization problems. As objective functions, total path length and maximum path length in a solution can be defined. Those optimization problems are solved optimally by DBS when it prioritizes high-level search nodes with smaller scores, as commonly done in CBS. Note that metrics that assess time aspects such as total traveling time used in MAPF studies are significantly affected by execution schedules; the adaptation is not trivial. Evaluation This section empirically demonstrates that OTIMAPP solutions are computable to some extent (Sec. 6.1) and they random-32-32-10 Figure 6: Stress test on 4-connected grids. The success rate is based on 25 identical instances. DBS * includes detected instances that are unsolvable for DBS before timeout, which is not possible for PP (+) . We also present accumulated runtime with a fixed number of agents over 100 instances, and runtime profiling (median) of each solver over success instances for both solvers. are useful in adverse environments about timings (Sec. 6.2) through the simulation experiments. We also present OTIMAPP execution with robots (Sec. 6.3). The simulator was coded in C++ and the experiments were run on a desktop PC with Intel Core i9 2.8 GHz CPU and 64 GB RAM. Stress Test Setup Each solver was tested with a timeout of 5 min on four-connected undirected grids picked up from [Stern et al., 2019], as a graph G. We also tested random graphs, shown in the Appendix. All instances were generated by setting a start s i and a goal g i randomly while ensuring that s i and g i have at least one path without the use of other goals; otherwise, it violates "no use of other goals" of Thm. 5.1. Note that unsolvable instances might still be included. Result Fig. 6 presents the results. The main findings are: (1) Both solvers can solve instances with tens of agents in various maps within a reasonable time. (2) PP often fails due to priority orders (e.g., Fig. 5) while PP + and DBS can overcome such limitations to some extent. (3) A bottleneck of each solver is the procedure of detecting potential cyclic deadlocks, an NP-hard problem (Lemma. 4.3). This also leads to similar success rates of PP + and DBS. Delay Tolerance We next show that OTIMAPP solutions (if found) are useful in a simulated environment with stochastic delays of agents' moves built on conventional MAPF, called MAPF-DP (with Delay Probabilities) [Ma et al., 2017]. Given a graph and start-goal pairs for each agent, the aim of MAPF is to move agents to their goals without collisions. Collisions occur when two agents occupy the same vertex or traverse the |A| = 35p = 0.2p = 0.5p = 0.8 MCPs+ECBS 1015 (1004,1026) 1422 (1404,1440) same edge simultaneously. Time is discrete. All agents synchronously take actions, i.e., either move to an adjacent vertex or stay at the current location. MAPF-DP emulates the imperfect execution of MAPF by introducing the possibility p i of unsuccessful moves for agent i (remaining there). Setup The delay probabilities p i were chosen uniformly at random from [0,p], wherep is the upper bound of p i . The higherp means that agents delay often, and vice versa. The metric is the total traveling time of agents; smaller values mean less wasting time at runtime. We tested the following two as baselines: (1) Result Table 2 shows that the execution of OTIMAPP solutions outperforms the alternatives. This is because: (1) Unlike MCPs, OTIMAPP solutions are free from temporal dependencies of offline plans that one agent delays are possibly fatal. (2) Unlike Causal-PIBT, agents follow long-term plans and avoid possible congested locations. Discussion Although finding OTIMAPP solutions is challenging, Table 2 motivates us to compute them. Meanwhile, the other approaches can solve larger instances with more agents (e.g., |A| = 200) and with much smaller planning time than solving OTIMAPP. Moreover, there are situations where OTIMAPP has no solutions while the others can find feasible plans because OTIMAPP assumes no intervention at runtime. One future direction pursues to fill these gaps. Robot Demonstrations We present two OTIMAPP execution styles: (1) a centralized control using the toio robots (https://toio.io) and (2) a decentralized one with only local interactions using a multi-robot platform [Kameyama et al., 2021]. A solution was obtained by DBS. Figure 7 is snapshots. A video is available online. In both cases, robots move without any synchronization procedures but are ensured to eventually reach their goals thanks to the nature of OTIMAPP. Moreover, for the latter, any actor has no methods to know the entire configuration, which cannot be addressed by conventional execution strategies. ]. Failing to represent the inherent uncertainty in the domain means the system behavior can be unpredictable. Alternative approaches are online intervention during execution, e.g., forcing agents to preserve temporal dependencies of offline planning via communication [Ma et al., 2017;Hönig et al., 2019;Atzmon et al., 2020b]. Another direction is online time-independent planning [Okumura et al., 2021] that incrementally moves agents based on current situations. OTIMAPP shares the concept of time independence but aims at offline planning without or less runtime effort. Related Work In graph theory, the (vertex) disjoint path problem and its variants [Robertson and Seymour, 1985] are partly related to ours in the sense that a set of disjoint paths clearly satisfies the solution condition of OTIMAPP, but the reverse does not. Conclusion This paper studied a novel path planning problem called OTIMAPP, motivated by the nature of distributed environments (i.e., timing uncertainties) that multi-agent systems must address. We focused on robotic applications in evaluation but believe that OTIMAPP can be leveraged to other resource allocation problems with mutual exclusion, e.g, distributed databases, which is our future direction. References Appendix We complement omitted proofs (Sec. A, B, and D), the detailed procedure of detecting potential cyclic deadlocks (Sec. C), additional results of stress test on random graphs (Sec. E), and the details of experimental setups (Sec. F). A Proof of Solution Analysis Theorem (3.5; necessary and sufficient condition). Given an OTIMAPP instance, a set of path {π 1 , . . . , π N } is a feasible solution if and only if there are: • No reachable terminal deadlocks. • No reachable cyclic deadlocks. Proof. Without "no reachable terminal deadlocks," there is an execution that one agent arrives at its goal and remains there; disturbing the progression of another agent. Without "no reachable cyclic deadlocks," a cyclic deadlock might occur and those agents stop the progression. Hence those two are necessary. We now prove that the two conditions are sufficient. Given a solution candidate {π 1 , . . . , π N } with no reachable deadlocks, consider the potential function φ := i∈A (|π i | − clock i ) defined over a configuration {clock 1 , . . . , clock N }. Observe that φ is non-increasing and φ = 0 means that all agents have reached their goals. Furthermore, when φ > 0, φ is guaranteed to decrease if each agent is activated at least once. We explain this as follows. Suppose contrary that φ( = 0) does not differ for the period. Since φ = 0, there are agents whose progress indexes are less than the maximum values. Let them B ⊆ A. For an agent i ∈ B, π i [clock i + 1] is occupied by another agent j ∈ B, according to "no reachable terminal deadlocks," otherwise, i moves there. This is the same for j, i.e., there is an agent k ∈ B such that π j [clock j + 1] = π k [clock k ]. By induction, this sequence of agents must form a cycle somewhere, i.e., occurring a cyclic deadlock; however, this contradicts "no reachable cyclic deadlocks." Each agent is activated at least once in a sufficiently long period due to the fair assumption, deriving the statement. B Proofs of Computational Complexity Theorem (4.2; complexity on undirected graphs). For OTIMAPP on undirected graphs, it is NP-hard to find a feasible solution with simple paths. Proof. We add a new gadget, which makes an undirected edge to a virtually directed one, to the proof of the NPhardness on digraphs (Thm. 4.1). We derive the claim by replacing all directed edges, except for bidirectional edges, with this gadget. Figure 3 (right) is it, including two new agents: z 1 and z 2 . In this gadget, any agents outside of the gadget are allowed to move only the direction from u to v. Assume contrary that, one agent ( = z 1 , z 2 ) takes a path through v to u within this gadget, and there is another path from v to u. To avoid cyclic deadlocks, z 1 and z 2 must move toward the left side, exit the gadget from u, use another path to enter v, and eventually reach their goals. In these paths, z 1 can arrive at its goal earlier than that of z 2 , contradicting "no reachable terminal deadlocks" in the necessary and sufficient condition (Thm. 3.5). Therefore, these paths are invalid. The size of the OTIMAPP instance is still polynomial on the 3-SAT formula. Thus, we conclude the statement. Note that, if we allow non-simple paths, it might be possible for other agents to move through the way from v to u. This is because, even though z 1 and z 2 temporarily leave from the gadget via u, we can construct paths that z 1 always arrive at its goal after z 2 's arrival, as illustrated in Fig. 8. Lemma (4.3; complexity of detecting cyclic deadlocks). Determining whether a set of paths contains either reachable or potential cyclic deadlocks is NP-complete. Proof. The proof is a reduction from the 3-SAT problem, i.e., constructing a combination of an OTIMAPP instance and a set of paths such that potential cyclic deadlocks exist if and only if the corresponding formula is satisfiable. We show the case of directed graphs. The proof procedure applies to the undirected case without modifications. In addition, all potential cyclic deadlocks are reachable in the translated problem. The reduction is done in polynomial time, deriving the NP-hardness of detecting both reachable and potential cyclic deadlocks. Since a potential cyclic deadlock can be verified in polynomial time, and since a reachable cyclic deadlock can be verified in polynomial time with an execution schedule, they are NP-complete. We now explain how to translate the 3-SAT formula to the OTIMAPP instance and the corresponding set of paths. Without loss of generality, we assume that all variables appear positively and negatively in the formula. Throughout the proof, we use the following example. (x 1 ∨ x 2 ∨ ¬x 3 ) ∧ (¬x 1 ∨ x 2 ∨ x 3 ) ∧ (x 1 ∨ ¬x 2 ∧ ¬x 3 ) Its outcome is partially depicted in Fig. 9. The complete version is presented in Fig. 10. A. Construction of an OTIMAPP instance and a set of paths For each literal in each clause, one literal agent is introduced. We denote by c j k a literal agent for the k-th literal in j-th clause C j in the formula. We also use one special agent z. Next, consider two gadgets: variable decider and clause constrainer. Note that they are different from those used in the proof of Thm. 4.1; however, their intuitions are similar and we use the same names. The variable decider determines whether a variable x i occurs positively or negatively. For each variable one gadget is introduced. All literal agents for x i (i.e., either x i or ¬x i ) start from vertices in this gadget. The gadget contains two paths: an upper path, corresponding to assign true to x i , and a lower path, corresponding to assign false to x i . Positive literals are connected to the upper path. Negative literals are connected to the lower path. For instance, x 2 has three literal agents: c 1 2 (x 2 ), c 2 2 (x 2 ), and c 3 2 (¬ x 2 ). In Fig. 9, we highlight the upper and the lower paths by bold lines. c 1 2 and c 2 2 are connected to the upper path while c 3 2 is connected to the lower path. Each literal agent uses one edge in the upper/lower path and moves to a clause constrainer via one vacation vertex. The clause constrainer contains all goals of the literal agents in the clause. Three edges are used to reach the goals. Each edge is for each literal agent. For instance, the clause constrainer of C 2 contains the goals of c 2 1 , c 2 2 , and c 2 3 . In Fig. 9, three edges are annotated with the agent's name. c 2 2 is supposed to use the colored middle one. Note that we use multiple edges for simplicity. It is not hard to convert the gadget to a simple graph version, as shown immediately later of this proof. As a result, all literal agents take six edges to reach their goals. This is visualized by colored edges in Fig. 9 and Fig. 10. The special agent z uses two edges to reach its goal, through ♣ marks in the figure. We finish the description of how to construct the OTIMAPP instance and the corresponding set of paths. The remaining part is to show these paths contain potential/reachable cyclic deadlocks if and only if the formula is satisfiable. This translation from the formula is clearly done in polynomial time. B. A potential cyclic deadlock exists if the formula is satisfiable. To see this, observe that if a potential cyclic deadlock exists, agents must try to use; (a) either an upper or a lower path for each variable decider, (b) one edge for each clause constrainer, and (c) the edge for z (♣). When the formula is satisfiable for one assignment, consider the following execution. 1. For each assigned value, move the corresponding clause agents to vacation vertices in each variable decider, i.e., one step before clause constrainers. Proof. The algorithm uses two tables that store fragments: Θ s and Θ t . Both tables take one vertex as a key. One entry in Θ s stores all fragments starting from the vertex. One entry in Θ t stores all fragments ending at the vertex. A fragment is registered in both tables. We now derive the statement by induction on π i . Base case: At the first iteration of the loop [Line 11-36], all fragments for {π 1 } are registered on Θ s and Θ t due to Line 14-15. There are no potential cyclic deadlocks for {π 1 }. Induction Hypothesis: Assume that there are no potential cyclic deadlocks for {π 1 , . . . , π i−1 } and all fragments for them are registered on Θ s and Θ t . Induction Step: We now show the property for i; otherwise, a potential cyclic deadlock exists for {π 1 , . . . , π i } and the algorithm returns it. All new fragments about π i are categorized into two: (1) a fragment only with π i or (2) a fragment that extends other fragments on Θ s and Θ t , using (u, v) ∈ π i . The former is preserved due to Line 14-15. The latter is further categorized into three cases: (a) a fragment ends at v, (b) a fragment starts from u, and (c) a fragment connecting two existing fragments that one ends at u and another starts from v. Each case corresponds to Line 16-21, Line 22-27, and Line 28-34, respectively. As a result, all fragments are to register on Θ s and Θ t ; otherwise, a potential cyclic deadlock exists and the algorithm returns it [Line 19, 25, and 32]. The time complexity does not contradict the NPcompleteness of detecting potential deadlocks (Lemma 4.3). Proposition C.2 (space and time complexity). Algorithm 3 requires Ω(2 |n| ) both for space and time complexity in the worst case. Proof. Consider an example in Fig. 13. In any solutions, the number of fragments starting from u becomes Ω 2 |n| ; this implies the statement. Although Alg. 3 does not run in polynomial time, it works sufficiently fast in a sparse environment such that not many paths use the same vertices. Proof. Assume that there is a solution π = {π 1 , . . . , π N } satisfying the relaxed sufficient condition (Thm. 5.1). At each cycle [Line 4-13], at least one node in OPEN is consistent with π, i.e., its constraints allow searching π. This is derived by induction: (1) the initial node R is consistent with π, Figure 14: Stress test of random graphs. The success rate is based on 25 identical instances. DBS * includes detected instances that are unsolvable for DBS before timeout, which is not possible for PP (+) . Results on random graphs G(n, p) are shown, where n is the number of vertices and every possible edge occurs independently with probability p. and (2) generated nodes from a consistent node with π must include at least one consistent node. The search space, i.e., which agents are prohibited using which edges in which directions, is finite. Therefore, DBS eventually returns π (or another solution); otherwise, such solutions do not exist. Figure 14 summarizes the results. The experimental setting is the same as Sec. 6.1. We can see that the difficulty of finding solutions is dominated by average degrees of graphs. E Stress Test on Random Graphs F Details of Experimental Setup F.1 Implementation of DBS An initial solution candidate is important for DBS. It is ideal to find solutions (i.e., a set of paths without deadlocks) from the beginning. Even if not, it is desired to obtain infeasible solutions with a small number of potential cyclic deadlocks, expected to expand a smaller number of nodes in the high-level search to reach feasible solutions. We thus made the low-level search for the initial solution take a path having fewer potential cyclic deadlocks with already planned paths, partially using Alg. 3. This is akin to tie-breaks in low-level search of CBS [Sharon et al., 2015]. F.2 Setup of MAPF-DP We carefully designed experiments to be fair as follows. Preliminaries MAPF-DP [Ma et al., 2017] emulates the imperfect execution of MAPF plans by introducing the possibility of unsuccessful moves, but still agents have to avoid collisions. Time is discrete. At each timestep, an agent i can either stay in place or move to an adjacent vertex with a probability p i of being unsuccessful. Solution quality is assessed by the total traveling time, where the time is the earliest time step that one agent reaches its goal and remains there. From OTIMAPP to MAPF-DP To adapt the execution of OTIMAPP to MAPF-DP, we introduce two changes for executions: (1) using modes to represent a state on edges, and, (2) an activation rule to represent the failure of movements. Mode: In reality, an agent i occupies two vertices simultaneously during a move from one vertex to another vertex. We introduce two modes in the execution of OTIMAPP to represent this state; • A mode contracted corresponds to when the agent i occupies one vertex. • A mode extended corresponds to when the agent i occupies two vertices. Agents move towards their goals by changing two modes alternately. Initially, they are in contracted . The names are from [Okumura et al., 2021]. Activation Rule: We repeated the following two phases: • Each agent i in extended is activated with probability 1 − p i . As a result, the agent i successfully moves to the adjacent vertex with probability 1 − p i and becomes contracted . • Choose one agent in contracted randomly then makes it activated. Repeat this until the configuration becomes stable, i.e, all agents in contracted do not change their states unless any agent in extended is activated. A pair of the two phases is regarded as one timestep. F.3 Setup of Robot Demonstrations Centralized Execution Platform We used the toio robots (https://toio.io/). The toio robots, connected to a computer via BLE (Bluetooth Low Energy), evolve on a specific playmat and are controllable by instructions of absolute coordinates. We informally confirmed that there is a non-negligible action delay between robots when sending instructions to several robots simultaneously (e.g., 10 robots, see the movie). Therefore, one-shot execution -robots move alone without communication after the receipt of plans -will result in collisions hence failure of the execution in a high possibility. The robots need some kinds of execution policies. Usage We created a virtual grid on the playmat and the robots followed the grid. A central server (a laptop) managed the locations of all robots and issued the instructions (i.e., where to go) to each robot step by step. The instructions were issued asynchrony between robots while avoiding collisions. The code was written in Node.js. Decentralized Execution Platform We used the AFADA platform [Kameyama et al., 2021]; an architecture that consists of mobile robots that evolve over an active environment made of flat cells each Figure 15: Components of AFADA. The system consists of two kinds of actors: cells and robots. A cell is covered by an acrylic roof patterned for line tracing (left cell), enabling robots to move following the grid structure. equipped with a computing unit (Fig. 15). Adjacent cells can communicate with each other via a serial interface. Cells form the environment in two ways: as a two-dimensional physical grid, and, as a communication network. In addition, a cell can communicate with robots on it via NFC (Near Field Communication). Usage Robots first receive the OTIMAPP solution from a laptop via Wi-Fi, then move following the plan. Cells achieve mutual exclusion of locations for robots, i.e., collision avoidance, using local communication as follows. Before moving to the next vertex (i.e., cell; denoted as v next ), a robot first asks the underlying cell v current the availability of v next . Then, v current asks v next its status. If v next is reserved by another robot, v current waits a while and asks the status of v next again; otherwise, v current makes v next reserved and notifies the robot to move to v next . When the robot reaches v next , then the robot releases v current via v next . Importantly, there is no central control at runtime. Any actor (robots, cells, and the laptop that sends the plan) has no methods to know the entire configuration. This also means that the system is fully asynchronous as for timing. Furthermore, there is no global communication; robots and cells decide their actions based on information from nearby actors. Additional References [Okumura et al., 2021] Keisuke Okumura, Yasumasa Tamura, and Xavier Défago. Iterative refinement for real-time multi-robot path planning. In Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2021. Figure 2 : 2Examples of unreachable potential deadlocks. left: cyclic; Figure 3 :Figure 4 : 34An OTIMAPP instance reduced from the 3-SAT formula (x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3)A oneway constrainer. This gadget transforms an undirected edge (u, v) to a directed one. Any agent is allowed to move only the way from u to v when limiting solutions to simple paths. [Line 3]. In the main loop [Line 4-13], DBS repeats; (1) Picking up one node [Line 5]. (2) Checking a deadlock and creating constraints [Line 6]. (3) Returning a solution if the paths contain no deadlocks [Line 7]. (4) If not, creating successors and inserting them to OPEN [Line 8-12]. DBS returns FAIL-URE when OPEN becomes empty [Line 14]. We complement several details below. Theorem 5.3 (DBS). DBS returns a solution when solutions satisfying Thm. 5.1 exist; otherwise returns FAILURE. Example We describe an example of DBS using Fig. 5. Assume that the initial path of i is the solid blue line and the path for j is the dashed red line [Line 2]. This node is inserted into OPEN [Line 3] and is expanded immediately [Line 5]. MCPs [Ma et al., 2017] force agents to preserve order relations of visiting one vertex in an offline MAPF plan at runtime. The plan was obtained by ECBS [Barer et al., 2014]. (2) Causal-PIBT [Okumura et al., 2021] is online time-independent planning, that is, each agent repeats one-step planning and action adaptively to surrounding current situations. The other details are in the Appendix. Figure 7 : 7An OTIMAPP execution with 10 robots in an 8 × 8 grid. Colored arrows represent an OTIMAPP solution. Figure 8 : 8Counterexample of oneway constrainer without assmptions of simple paths. z2 always arrives at its goal before the arrival of z1 if those two agents follow colored lines. DBS). DBS returns a solution when solutions satisfying Thm. 5.1 exist; otherwise returns FAILURE. Algorithm 1 PP: Prioritized Planning Input: an OTIMAPP instance Output: a solution {π 1 , . . . , π N } or FAILURE 1: Θ s , Θ t ← ∅ 2: for i = 1 . . . |A| do3: Algorithm 2 DBS: Deadlock-based Search Input: an OTIMAPP instance Output: a solution {π 1 , . . . , π N } or FAILURE 1: R.constraints ← ∅ 2: R.paths ← find paths with "no use of other goals" 3: insert R to OPEN OPEN : priority queue 4: while OPEN = ∅ do 5: N ← OPEN .pop() C ← get constraints of N using Algorithm 36: 7: Table 2 : 2Total traveling time on MAPF-DP.All settings used OTIMAPP is prevention. A non-deadlock state that is "inevitable" to reach deadlocks is called unsafe[Silberschatz et al., 2006]. Meanwhile, reachable deadlocks of OTIMAPP correspond to states that are "possible" to reach deadlocks. The notion of potential terminal deadlock is related to wellformed instances of MAPF[Čáp et al., 2015], that is, for each start-goal pair, a path exists that traverses no other starts and goals. The notion of reachable cyclic deadlock is mentioned as nonlive states/sets for deadlock management in automated manufacturing systems[Fanti and Zhou, 2004] or in a multirobot scheduling problem[Mannucci et al., 2021].The multi-agent pathfinding (MAPF) problem[Stern et al., 2019] aims at finding a set of collision-free paths on a graph. Many studies on MAPF consider timing uncertainties because they are inevitable in multi-agent scenarios. However, current methods largely rely on additional assumptions on the travel speed of agents or assume delays to follow some probability distributions[Wagner and Choset, 2017; Mansouri et al., 2019; Peltzer et al., 2020; Atzmon et al., 2020aA deadlock [Coffman et al., 1971] is a widely recognized phenomenon not limited to robotics; a system state that several components claim resources that others hold, then block each other permanently. Strategies to cope with dead- locks are categorized into prevention, detection/recovery, and avoidance [Silberschatz et al., 2006; Fanti and Zhou, 2004]; Atzmon et al., 2020b] Dor Atzmon, Roni Stern, Ariel Felner, Glenn Wagner, Roman Barták, and Neng-Fa Zhou. Robust multi-agent path finding and executing. Journal of Artificial Intelligence Research (JAIR), 2020. [Barer et al., 2014] Max Barer, Guni Sharon, Roni Stern, and Ariel Felner. Suboptimal variants of the conflict-based search algorithm for the multi-agent pathfinding problem. Coffman et al., 1971] Edward G Coffman, Melanie Elphick, and Arie Shoshani. System deadlocks. and Xavier Défago. Time-independent planning for multiple moving agents. In Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2021.[Atzmon et al., 2020a] Dor Atzmon, Roni Stern, Ariel Fel- ner, Nathan R Sturtevant, and Sven Koenig. Probabilistic robust multi-agent path finding. In Proceedings of Interna- tional Conference on Automated Planning and Scheduling (ICAPS), 2020. [In Proceedings of Annual Symposium on Combinatorial Search (SOCS), 2014. [Čáp et al., 2015] MichalČáp, Peter Novák, Alexander Kleiner, and Martin Seleckỳ. Prioritized planning al- gorithms for trajectory coordination of multiple mobile robots. IEEE Transactions on Automation Science and En- gineering (T-ASE), 2015. [ACM Com- puting Surveys (CSUR), 1971. [Dresner and Stone, 2008] Kurt Dresner and Peter Stone. A multiagent approach to autonomous intersection manage- ment. Journal of Artificial Intelligence Research (JAIR), 2008. [Erdmann and Lozano-Perez, 1987] Michael Erdmann and Tomas Lozano-Perez. On multiple moving objects. Al- gorithmica, 1987. [Fanti and Zhou, 2004] Maria Pia Fanti and MengChu Zhou. Deadlock control methods in automated manufacturing systems. IEEE Transactions on systems, man, and cybernetics-part A: systems and humans, 2004. [Hönig et al., 2019] Wolfgang Hönig, Scott Kiesel, Andrew Tinka, Joseph W Durham, and Nora Ayanian. Persistent and robust execution of mapf schedules in warehouses. IEEE Robotics and Automation Letters (RA-L), 2019. [Kameyama et al., 2021] Shota Kameyama, Keisuke Oku- mura, Yasumasa Tamura, and Xavier Défago. Active mod- ular environment for robot navigation. In Proceedings of IEEE International Conference on Robotics and Automa- tion (ICRA), 2021. [Knapp, 1987] Edgar Knapp. Deadlock detection in dis- tributed databases. ACM Computing Surveys (CSUR), 1987. [Ma et al., 2017] Hang Ma, TK Satish Kumar, and Sven Koenig. Multi-agent path finding with delay probabilities. In Proceedings of AAAI Conference on Artificial Intelli- gence (AAAI), 2017. [Mannucci et al., 2021] Anna Mannucci, Lucia Pallottino, and Federico Pecora. On provably safe and live multirobot coordination with online goal posting. IEEE Transactions on Robotics (T-RO), 2021. [Mansouri et al., 2019] Masoumeh Mansouri, Bruno Lac- erda, Nick Hawes, and Federico Pecora. Multi-robot plan- ning under uncertain travel times and safety constraints. In Proceedings of International Joint Conference on Arti- ficial Intelligence (IJCAI), 2019. [Nebel, 2020] Bernhard Nebel. On the computational com- plexity of multi-agent pathfinding on directed graphs. In Proceedings of International Conference on Automated Planning and Scheduling (ICAPS), 2020. [Okumura et al., 2021] Keisuke Okumura, Yasumasa Tamura, [Peltzer et al., 2020] Oriana Peltzer, Kyle Brown, Mac Schwager, Mykel J Kochenderfer, and Martin Sehr. Stt- cbs: A conflict-based search algorithm for multi-agent path finding with stochastic travel times. 2020. [Robertson and Seymour, 1985] Neil Robertson and Paul D Seymour. Disjoint paths-a survey. SIAM Journal on Al- gebraic Discrete Methods, 1985. [Sharon et al., 2015] Guni Sharon, Roni Stern, Ariel Felner, and Nathan R Sturtevant. Conflict-based search for opti- mal multi-agent pathfinding. Artificial Intelligence (AIJ), 2015. [Silberschatz et al., 2006] Abraham Silberschatz, Peter B Galvin, and Greg Gagne. Operating system concepts. John Wiley & Sons, 2006. [Silver, 2005] David Silver. Cooperative pathfinding. AIIDE, 2005. [Stern et al., 2019] Roni Stern, Nathan Sturtevant, Ariel Fel- ner, Sven Koenig, Hang Ma, Thayne Walker, Jiaoyang Li, Dor Atzmon, Liron Cohen, TK Kumar, et al. Multi- agent pathfinding: Definitions, variants, and benchmarks. In Proceedings of Annual Symposium on Combinatorial Search (SOCS), 2019. [Tel, 2000] Gerard Tel. Deadlock-free packet switching. In Introduction to distributed algorithms, chapter 5. 2000. [Van Den Berg et al., 2011] Jur Van Den Berg, Stephen J Guy, Ming Lin, and Dinesh Manocha. Reciprocal n-body collision avoidance. In Robotics Research. 2011. [Wagner and Choset, 2017] Glenn Wagner and Howie Choset. Path planning for multiple agents under uncer- tainty. In Proceedings of International Conference on Automated Planning and Scheduling (ICAPS), 2017. [Wurman et al., 2008] Peter R Wurman, Raffaello D'Andrea, and Mick Mountz. Coordinating hun- dreds of cooperative, autonomous vehicles in warehouses. AI magazine, 2008. [Zhang et al., 2018] Xu Zhang, Mingyang Li, Jian Hui Lim, Yiwei Weng, Yi Wei Daniel Tay, Hung Pham, and Quang- Cuong Pham. Large-scale 3d printing by a team of mobile robots. Automation in Construction, 2018. Other Experimental Setup The delay probabilities p i were chosen uniformly at random from [0,p], wherep is the upper bound of p i . The higherp means that agents delay often.p = 0 corresponds to perfect executions without delays. Implementations of the online time-independent planning, called Causal-PIBT, were obtained from the authors [Okumura et al., 2021]. The offline MAPF plans for MCPs [Ma et al., 2017] was obtained by ECBS [Barer et al., 2014], a bounded sub-optimal solver for MAPF. The sub-optimally was set to 1.1, which was adjusted to solve all instances in the experiment. The implementation of ECBS was obtained from [Okumura et al., 2021] (in the additional references). We recently proved that it is NP-hard for the general case of undirected graphs. The formal proof will appear soon. AcknowledgmentsWe thank the anonymous reviewers for their many insightful comments and suggestions. This work was partly supported by JSPS KAKENHI Grant Numbers 20J23011, 21K11748, and 21H03423. Keisuke Okumura thanks the support of the Yoshida Scholarship Foundation.: OTIMAPP instance and solution reduced from the 3-SAT instance (x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (x1 ∨ ¬x2 ∧ ¬x3). For visualization, we break a large circle; regard two ♣ marks as connected. Omitted vertices and edges are complemented inFig. 10.Each color corresponds to a path for each agent.2. Among the above agents, for each clause constrainer, there is at least one agent that can enter the clause constrainer due to satisfiability. Move them one step further. As a result, all clause constrainers have one agent at the first vertices. Vertices in upper/lower paths in the variable deciders must be vacant now. 3. Move all unassigned clause agents one step. As a result, all vertices in the unassigned paths are filled by the unassigned clause agents.We now have a cyclic deadlock, i.e., this deadlock is reachable thus potential.As an example, consider a satisfiable assignment x 1 = true, x 2 = true, x 3 = true. In the beginning, move assigned agents, c 1 1 , c 1 2 , c 2 2 , c 2 3 , and c 3 1 to vacation vertices in each variable decider (Fig. 11;Step 1). Next, move c 1 2 , c 2 2 , and c 3 1 to the first vertices of each clause constrainer of C 1 , C 2 , and C 3 , respectively (Fig. 11;Step 2). Then, move all unassigned agents, c 2 1 , c 3 2 , c 1 3 , and c 3 3 , one step (Fig. 11;Step 3). There is a cyclic deadlock with c 2 1 , c 3 2 , c 1 3 , c 3 3 , c 1 2 , c 2 2 , c 3 1 , and z, annotated with bold lines inFig. 11.C. The formula is satisfiable if a potential cyclic deadlock exists. To form a potential cyclic deadlock, for each variable decider, one or several agents try to move along either an upper or a lower path. Consider assigning an opposite value against the used path to the variable. For instance, if c 1 2 and c 2 2 involve in the deadlock at the variable decider (seeFig. 9), then assign false to x 2 . This assignment must satisfy the formula because at least one literal in each clause becomes true; otherwise, at least one clause constrainer exists such that the first vertex is empty, i.e., no deadlock.D. All potential cyclic deadlocks are reachable. So far, we established the claim that a potential cyclic deadlock exists if and only if the formula is satisfiable. Next, we claim that all potential cyclic deadlocks are reachable. According to the above discussion, given a potential cyclic deadlock, the corresponding satisfiable assignment exists. Consider the execution of Part B using this assignment, slightly changingStep 2. In this step, we can choose arbitrary agents for each clause constrainer. Therefore, it is possible to choose agents involved in the potential cyclic deadlock. As a result, this deadlock is reachable.In the proof of Lemma 4.3, we used multiple edges in a gadget clause constrainer for the reduction from 3-SAT. Since OTIMAPP assumes a simple graph (i.e., no multiple edges), we complement how to convert it to a correct OTIMAPP instance.Figure 12shows an example of the clause constrainer for C 2 . Recall that a clause constrainer contains all goals for the corresponding clause agents. In this new gadget, we add intermediate vertices for each edge that potentially leads to cyclic deadlocks. For each agent c j k , a new Step 1: Move assigned agents to vacation vertices Step 2: Fill clause constrainers Step 3: Move unassigned agents one stepFigure 11: Construction of a reachable deadlock. The formula isThe assignment is x1 = true, x2 = true, and x3 = true. Locations of all agents are colored. When an agent departs from its start, the corresponding vertex is filled by dark. Bold lines in Step 3 constitute a reachable deadlock. agentĉ j k is introduced. Its start is the intermediate vertex. Its goal is the original goal of c j k . We furthermore change a goal for c j k to starts ofĉ j k . Consider now replacing all old clause constrainers with this new gadgets. The translation is done in polynomial time. The rest of the proof is straightforward from Lemma 4.3.C Detecting Potential Cyclic DeadlocksUsing fragments, Alg. 3 detects a potential cyclic deadlock in a set of paths if exists. The intuition is the following: (1) the algorithm checks each path one by one, (2) it stores all fragments created so far, (3) for each edge in each path, it creates new fragments using existing fragments, and (4) if a fragment ends at its start, this is a potential cyclic deadlock. We describe the details in the proof of the completeness.Algorithm 3 Potential Cyclic Deadlock DetectionInput: a set of paths {π 1 , . . . , π n } Output: one potential cyclic deadlock or NONEFigure 13: Example that requires huge space and time to detect potential deadlocks. All starts are on the left. All goals are on the right. Two zones are connected by a sufficiently long path.
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[ "Panoptic-PolarNet: Proposal-free LiDAR Point Cloud Panoptic Segmentation", "Panoptic-PolarNet: Proposal-free LiDAR Point Cloud Panoptic Segmentation" ]
[ "Zixiang Zhou [email protected] \nDepartment of Computer Science\nUniversity of Central Florida\n\n", "Yang Zhang [email protected] \nDepartment of Computer Science\nUniversity of Central Florida\n\n", "Hassan Foroosh [email protected] \nDepartment of Computer Science\nUniversity of Central Florida\n\n" ]
[ "Department of Computer Science\nUniversity of Central Florida\n", "Department of Computer Science\nUniversity of Central Florida\n", "Department of Computer Science\nUniversity of Central Florida\n" ]
[]
Panoptic segmentation presents a new challenge in exploiting the merits of both detection and segmentation, with the aim of unifying instance segmentation and semantic segmentation in a single framework. However, an efficient solution for panoptic segmentation in the emerging domain of LiDAR point cloud is still an open research problem and is very much under-explored. In this paper, we present a fast and robust LiDAR point cloud panoptic segmentation framework, referred to as Panoptic-PolarNet. We learn both semantic segmentation and class-agnostic instance clustering in a single inference network using a polar Bird's Eye View (BEV) representation, enabling us to circumvent the issue of occlusion among instances in urban street scenes. To improve our network's learnability, we also propose an adapted instance augmentation technique and a novel adversarial point cloud pruning method. Our experiments show that Panoptic-PolarNet outperforms the baseline methods on SemanticKITTI and nuScenes datasets with an almost real-time inference speed. Panoptic-PolarNet achieved 54.1% PQ in the public SemanticKITTI panoptic segmentation leaderboard and leading performance for the validation set of nuScenes.
10.1109/cvpr46437.2021.01299
[ "https://arxiv.org/pdf/2103.14962v1.pdf" ]
232,404,586
2103.14962
9d96a5aa2631d22d6f1b48c5b1a5f8e6f5214969
Panoptic-PolarNet: Proposal-free LiDAR Point Cloud Panoptic Segmentation Zixiang Zhou [email protected] Department of Computer Science University of Central Florida Yang Zhang [email protected] Department of Computer Science University of Central Florida Hassan Foroosh [email protected] Department of Computer Science University of Central Florida Panoptic-PolarNet: Proposal-free LiDAR Point Cloud Panoptic Segmentation Panoptic segmentation presents a new challenge in exploiting the merits of both detection and segmentation, with the aim of unifying instance segmentation and semantic segmentation in a single framework. However, an efficient solution for panoptic segmentation in the emerging domain of LiDAR point cloud is still an open research problem and is very much under-explored. In this paper, we present a fast and robust LiDAR point cloud panoptic segmentation framework, referred to as Panoptic-PolarNet. We learn both semantic segmentation and class-agnostic instance clustering in a single inference network using a polar Bird's Eye View (BEV) representation, enabling us to circumvent the issue of occlusion among instances in urban street scenes. To improve our network's learnability, we also propose an adapted instance augmentation technique and a novel adversarial point cloud pruning method. Our experiments show that Panoptic-PolarNet outperforms the baseline methods on SemanticKITTI and nuScenes datasets with an almost real-time inference speed. Panoptic-PolarNet achieved 54.1% PQ in the public SemanticKITTI panoptic segmentation leaderboard and leading performance for the validation set of nuScenes. Introduction As a crucial step in applications such as autonomous driving and robotics, processing and analyzing 3D scanning data have received increasing attention in recent years in computer vision and deep learning. Panoptic segmentation is a recently introduced problem in the image domain [20] that presents a new challenge in unifying instance segmentation and semantic segmentation in a single training architecture. With the recent introduction of new LiDAR point cloud datasets [2,5,13] that include both pixel-wise semantic label annotation and object annotation, this problem can now be also explored for 3D scanning data as we propose * Contributed equally. † Now at Waymo LLC. Code at: https://github.com/edwardzhou130/Panoptic-PolarNet. in this paper. By definition, Panoptic segmentation requires that we identify both class labels and instance id's for points in the "thing" classes, and only the class labels for points in the "stuff" classes. To solve this problem, the first question to answer is: What information is needed in order to obtain a panoptic segmentation of data? It can be either the semantic label of all points and the instance clustering of the "thing" classes, or the instance segmentation of the "thing" classes and the class labels of remaining "stuff" classes. As a consequence, these two alternative designs would lead to two different categories of panoptic segmentation, known as proposal-free and proposal-based, the former being adapted from a semantic segmentation network [28] and the latter adapted from an object detection network [16]. 2D image panoptic segmentation faces two main problems. First, proposal-based ones segment instances independently within each individual object proposal. Such approaches require extra architectural modifications [26,51] to compensate for the impact of heavy object collision in the proposals. Second, semantic segmentation and instance segmentation are usually handled in two separate prediction heads in order to tailor the design of the dedicated network to each task. However, this may inevitably introduce either potential conflicts or redundant information since these two tasks clearly share common characteristics. For example, in the proposal-based methods, semantic and instance heads can yield different label predictions at the same pixel. And in the proposal-free methods, the features learned in the instance head have significant correlations with class labels. Both cases ultimately lead to inference inefficiency. 3D panoptic segmentation, on the other hand, is by and large at its infancy and still an open research problem. It is mainly motivated by LiDAR point cloud processing in applications such as self-driving cars, autonomous robot navigation, and environment mapping, all of which generally require real-time processing. On the other hand, compared to conventional 3D data in computer vision, LiDAR point clouds are irregularly sampled in the 3D space. These differences in terms of the nature of the 3D data, the need for real-time processing, and the level of accuracy needed for safety and security (e.g., in self-driving cars) are clearly creating new challenges, encouraging new innovative solutions. These challenges motivated us to find a more suitable architecture that takes into account the unique characteristics of LiDAR data, efficiently solves panoptic segmentation with minimum conflicts in predictions (instance versus class), and achieves real-time or near real-time speed without compromising accuracy. Given the speed limitation, proposal-free methods naturally seem to be a more favorable choice, since they are proven to perform better in computational time in the 2D case. Therefore, starting from a backbone semantic prediction network [56], our first goal is to integrate it with a network for class-agnostic instance clustering. We hypothesize that most "thing" class objects in the LiDAR point cloud are separable when projected onto the XY-plane. Instance separability implies that the discretized BEV representation [47] is highly suitable for LiDAR point cloud instance clustering. Therefore, we can use the same network of PolarNet also to generate discriminative features for separating instances in the BEV. Based on these observations and assumptions, we propose a panoptic segmentation framework that simultaneously learns semantic and instance features on the discretized BEV map. Therefore, we follow the backbone network design of PolarNet [56] to generate the 3D semantic prediction and use a lightweight 2D instance head inspired by Panoptic-DeepLab [7] on top of it. Predictions from semantic and instance heads are then fused through a majority voting to create the final panoptic segmentation. This results in a highly efficient proposalfree panoptic segmentation network design, which we refer to as Panoptic-PolarNet. We evaluated our approach on SemanticKITTI and nuScenes datasets. Panoptic-PolarNet achieves state-of-the-art performance. Compared to the PolarNet, our instance segmentation head only introduces 0.1M parameters and increases the inference time by only 0.027s. Our contributions are summarized as follows: • We propose a model taking into account the specific nature of LiDAR data and the applications in mind, to construct a proposal-free LiDAR panoptic segmentation network that can efficiently cluster instances on top of the semantic segmentation. • Unlike existing panoptic segmentation networks that generally use two entirely separate decoding modules for semantic and instance segmentation and rely on an attention module to connect the learned information, our networks share decoding layers among them, allowing for early fusion at feature extraction level. This early fusion strategy has two substantial impacts: (1) it reduces redundancy between the networks and therefore improves computational efficiency; (2) increases the PQ measure despite a smaller computation load. • Compared to existing proposal-based panoptic segmentation methods that suffer from class and instance prediction overlapping, we propose a proposal-free design and train the instance head without bounding box annotation, which allows us to avoid the conflict of class prediction. • We introduce two novel point cloud data augmentation methods that can apply to any other LiDAR segmentation networks. • Experiments show that our approach outperforms strong baselines on SemanticKITTI and nuScenes datasets with smaller and near-real-time latency, as shown in Figure 1. Related works Image based panoptic segmentation Current 2D panoptic segmentation methods normally divide panoptic segmentation into two subproblems: semantic segmentation and instance segmentation. They are trained in a single network with a shared feature encoding layer and separated heading layer. According to how they accurately separate different instances, panoptic segmentation methods can be categorized into top-down/ proposal-based and bottom-up/ proposal-free approaches. Top-down methods usually use Mask R-CNN [16] to get each object's instance mask first and then fill in the rest region with the semantic segmentation prediction. While this design gives a reliable instance segmentation result, it requires additional means to resolve the overlapping instances and the conflict between instance and semantic predictions. Liu et al. [26] propose a spatial ranking module to sort the overlapping masks. UPSnet [51] introduces a panoptic head to resolve the conflict between instance and semantic predictions by adding an unknown class label. EfficientPS [31] proposes a panoptic fusion module that dynamically adapts the fusion of instance and semantic heads according to their confidence. Recent research also focuses on designing either end-to-end training [26,51] or attention module bridging between semantic and instance learning [24,50,6]. On the contrary, bottom-up methods generally get semantic prediction and then fuse it with class-agnostic instance segmentation. The first bottom-up method, DeeperLab [54], proposes to separate the instance using bounding box corners and center. Panoptic-DeepLab [7] further simplifies this grouping method by predicting the instance center and offset. SSAP [10] uses cascaded graph partition to segment instance from a pixel-pair affinity pyramid. LiDAR point cloud object detection and semantic segmentation Compared to the conventional 3D point cloud data, Li-DAR point cloud is inherently 2.5D data since it is a perspective projection of the real world. This results in a sparse and imbalanced distribution of points among 3D geometrical space. Besides, most LiDAR point cloud tasks are targeted on the autonomous driving scenario, which creates even larger data size for the conventional point cloud method [37,38,35] to process. In addition to directly learning features on either the point level [41,45,18] or voxelized space [59,52], research on LiDAR point cloud also uses projected space like the bird's eye view [23,53,34] or spherical projection/range image [36,48,49,30,8], and sometimes the fusion of multiple aforementioned views [58,43]. Like its counterpart in 2D object detection, LiDAR point cloud object detection methods are also divided into proposal-based and proposal-free ones. Proposal-based methods [59,41,52] first generate region proposal from an encoded feature and use another head to select and refine object bounding box, while proposal-free methods directly predict object through vote clustering [35] or keypoint/center estimation [55]. For the segmentation problem, researchers pay more attention to efficiently extracting and recovering local and global context. KPConv [45] and RandLA [18] proposed to use kernel point convolution and local feature aggregation module to replace the conventional convolutional layer in an encoder-decoder structure to operate on the point cloud directly. However, it requires a time-consuming preprocessing to build the graph. Many other methods [48,8] choose to use 2D convolution to segment the point cloud on the 2D point projection. Rangenet++ [30] and KPRNet [21] introduce additional KNN and aligning processing to better recover label from a projected view to the original point cloud. Polar-Net [56] encodes point cloud into a polar BEV to compensate for the imbalanced distribution of points in the physical space. Point cloud panoptic segmentation As a rising task, LiDAR panoptic segmentation has not been well studied yet. However, many researchers have already explored indoor point cloud panoptic segmentation by combining instance segmentation and semantic segmentation methods. Most of them [46,22,33,27] use the discriminative loss [9] to learn a embedded feature space to cluster instances. Zhou et al. [57] extract the instance segmentation from the region proposals clustered from the semantic segmentation. SemanticKITTI [2] benchmarks the first panoptic segmentation LiDAR dataset by combining existing state-of-the-art object detection and semantic segmentation networks. MOPT [19] attaches a semantic head to Mask R-CNN to generate panoptic segmentation on the range image. Milioto et al. [29] proposes to solve LiDAR point cloud panoptic segmentation on the range image first then restore it to point cloud level through a tri-linear upsampling. Panoptic-PolarNet As shown in Figure 2, our Panoptic-PolarNet consists of the following four components: (1) a network that encodes the raw point cloud data to a fixed-size 2D polar BEV representation, (2) a shared encoder-decoder backbone network, (3) two independent heads for semantic and instance segmentation, (4) a fusion step that merges the aforementioned predictions into one final panoptic segmentation result. Preliminary Given a set of points P = {(x, y, z, r) n |n ∈ {1, . . . , N }}, where (x, y, z) are the 3D coordinates relative to the LiDAR scanner's reference frame and r is the intensity of reflection, and a set of ground truth class labels C GT = {l n |n ∈ {1, . . . , N }}, LiDAR point cloud semantic segmentation task aims to predict a unique set of class labels C p for the points P that minimizes the difference with C GT . Panoptic segmentation task extends this problem to requiring that points belonging to separate instances have different labels in some "thing" classes, e.g., car, bicycle, and human. The remaining classes are "stuff" classes, which do not require detailed separation and share the same label among all points. Polar BEV encoder To process a point cloud containing a random size of points, we need to create a fixed-size representation through projection and quantization. We use BEV for two main reasons. First, BEV provides a trade-off between computa- Figure 2: Our Panoptic-PolarNet framework. We first encode the raw point cloud data with K features into a fixed-size representation on the polar BEV map. Next, we use a single backbone encoder-decoder network [56] to generate semantic prediction, center heatmap and offset regression. Finally, we merge these outputs via a voting-based fusion to yield the panoptic segmentation result. tional cost and accuracy, enabling us to use the more efficient 2D convolutional networks to process the data. Second, since objects rarely overlap along the z-axis in the urban scene, BEV is empirically the best projection for object detection [47]. We also represent the points in the polar coordinates rather than conventional Cartesian coordinates to balance the distribution of points among different ranges [56]. The polar coordinate gives neural networks better potential to learn discriminative features at locations closer to the sensor and minimizes the information losses due to quantization. We adopt the original polar BEV encoder design from PolarNet [56]. More specifically, we first group a point cloud data P ∈ R N ×K to P ∈ R (H×W ×N )×K based on its position in the polar BEV map, where K is the input feature dimension, H and W are the grid size of the BEV map and N is the number of points in each BEV grid. Next, we encode this point cloud through a simplified PointNet [37], which only contains MLP. Then, a max-pooling layer is applied at each BEV grid to create a fixed-size representation M ∈ R H×W ×C , where C is the feature channel. We use C = 512 in our experiment. Semantic Segmentation After encoding LiDAR point cloud data into a feature matrix M , most 2D semantic segmentation backbone net-works are able to process it. We follow PolarNet to use Unet [39] with 4 encoding layers and 4 decoding layers as the backbone network. Unlike other panoptic segmentation networks that generally use two entirely separate decoding modules for semantic and instance segmentation, our network shares the first three decoding layers among them. Our semantic head generates multiple predictions at each pixel C p ∈ Z Z×H×W , which are later reshaped back to 3D voxels to separate labels at different heights along Z-axis. We calculate the loss at the voxel level during the training, where the groundtruth label for each voxel is decided by majority voting of points within the same voxel. Panoptic Segmentation According to [20], one big problem in 2D image panoptic segmentation is the difficulty to efficiently separate instances when the collision occurs, e.g., two people standing next to each other. We hypothesized that we could circumvent this challenge in LiDAR data based on two assumptions. First, objects rarely collide in 3D space even their masks overlap in 2D projection. Second, most "thing" class objects in the LiDAR point cloud of urban scenes are still separable when projected onto the XY-plane from 3D space. Such a claim is also supported by [47], who find the same object detection network has better performance in BEV in contrast to 2D projection. This suggests that the BEV rep-resentation has the potential to not only improve the performance but also reduce the problem of instance clustering in the LiDAR point cloud panoptic segmentation to a 2D problem. Therefore, we can use the same network of PolarNet to generate discriminative features for separating instances in the BEV. We follow the instance head design in Panoptic-DeepLab [7] to predict the center heatmap and the offset to the object center for each BEV pixel. Pixels that have the same nearest center are grouped together. Compared to other top-down methods with overlaps of class prediction between segmentation and instance branches, this bottomup design provides only class-agnostic instance grouping. This allows us to avoid the conflict of class prediction and train instance head without bounding box annotation. During the training phase, we encode the ground truth center map by a 2D Gaussian distribution around each instance's mass center. For each pixel p in the BEV map, the heatmap is H p = max i exp(− (p−ci) 2 2σ 2 ), where c i is the mass center of one instance in the polar BEV coordinates. To merge the 3D semantic segmentation and 2D instance grouping predictions, we propose a fusion step as shown in Figure 2. First, the top k centers are selected from the heatmap prediction after a non-maximum suppression. Next, we use the semantic segmentation prediction to create a foreground mask where at least one "thing" class is detected at one BEV pixel. Pixels in the foreground are then grouped together based on the minimal distance d(p, c i ) = p + offset(p) − c i 2 to one of the k centers. Lastly, "thing" class predictions in the semantic segmentation head are assigned a unique instance label L for each group G i in the BEV using a majority voting according to semantic segmentation probability P (v): L i = argmax v∈Gi P (v). All these operations are implemented in GPU, requiring little computational time. Augmenting Panoptic-PolarNet Instance augmentation: Training data augmentation on the instance level has proven to be an important technique for LiDAR object detection [23,15] without increasing inference computational cost. How the sensor samples points of an instance is determined by the sensor's angle interval, the relative pose, and distance of an instance to the sensor. Our instance augmentation aims to increase the variance of data without changing the projection properties of instance points. We summarise it as the following three steps: (1) Instance oversampling: We randomly choose 5 instances from the whole training set and paste them into the current training scan. The probability of each class being selected is in proportion to the reciprocal of its point distribution ratio. The imported points retain the same relative coordinates and reflection values as in their source. (2) Instance global augmentation: The goal here is to find a transformation to change an instance's location on XY-plane without altering its projection on the sensor. The need to preserve projection narrows the transformation to either rotation on the center or reflection on a certain view plane through the center. We apply those two transformations to each instance with a 20% probability for each transformation. (3) Instance local augmentation: We also apply small independent translation and rotation to each instance, which serves as measurement noise. We sample the translations [∆x, ∆y, ∆z] from a normal distribution N (0, 0.25), and the rotation angle ∆θ from a uniform distribution U (−20/π, +20/π). Point cloud self-adversarial pruning: Inspired by YOLO-v4 [4], we also use self-adversarial pruning on the point cloud after the training is almost converged. The idea of self-adversarial pruning is to find the most influential points through the network itself. Those points are likely to be either noise or key feature points. By omitting those points during training, we enforce the network to learn more general features from the overall point cloud instead of overfitting into some specific geometry patterns. More specifically, we use two forward-backward loops for each batch of input data. We use the gradient to select those highly influential points in the first forward-backward loop and feed the altered data to the second forward-backward loop to update the network weights after omitting those points. Similar to [32], we consider the gradient variance as the diagonal of the Fisher information matrix, which represents the importance of input to the panoptic loss. In the experiments, we deleted only the top 1% of the points according to validation results. Visibility feature: Visibility is a concept commonly used in the mapping problem to create an occupancy map of the environment through raycasting. Recently, Hu et al. [17] included the visibility in the detection problem to enrich the voxel representation of the point cloud. Given a point (x, y, z) in the LiDAR point cloud, the space along the same direction α(x, y, z) can be divided into visible if 0 < α < 1 and occluded if α > 1. However, computing the visibility for the whole 3D space requires traversing through all points, which is usually preprocessed offline before the training. Since the range at the z axis is much smaller than the other two axes in the LiDAR point cloud, we approximate this traversal as for each point at (d, θ, z), where d and θ are the distance and angle in the polar coordinates, the space of (αd, θ, z) is visible if 0 < α < 1 and occluded if α > 1. Hence we can compute the visibility for each voxel efficiently in the polar coordinate alongside the instance data augmentation during the training. We concatenate the visibility feature with the feature representation generated by the polar BEV encoder, then feed it into the backbone network in our implementation. Experiment In this section, we demonstrate our panoptic segmentation results on the SemanticKITTI [2] dataset and the nuScenes [5] dataset. Due to page limitations, please refer to our supplementary material for more details on the experiments, discussions, and qualitative examples. Datasets SemanticKITTI provides point-wise semantic and instance annotations for the well known KITTI [12] odometry dataset, which contains 10/1/11 training/validation/testing sequences, and a total of 43551 LiDAR scans of European city streets. Each SemanticKITTI scan has 104452 points on average and is annotated with 20 class labels, 8 of which are selected as "thing" classes. NuScenes is a large scale autonomous driving dataset created by Motional. It contains 1000 driving scenes, with 850 scenes for training and validation, and 150 scenes for testing. In each keyframe that is sampled every 0.5s, nuScenes provides bounding box annotations for 10 possible object classes and point-wise semantic labels for 16 classes. The first 10 are the same as object classes. Although nuScenes also provides image and radar data, we only used the LiDAR data in the keyframe during the training and validation. Unlike SemanticKITTI, nuScenes does not explicitly provide the instance label for each point. We manually created the panoptic instance annotation by assigning "thing" points to its closest detection bounding box. We remove outliers by omitting the "thing" points that are more than 5m apart from the nearest bounding box centroids. Since nuScenes does not provide panoptic segmentation metrics on the test set evaluation server, we only report panoptic segmentation results on the validation set. Baselines Our baselines include both dedicated panoptic LiDAR point cloud segmentation methods as well as combinations of state-of-the-art segmentation and detection pipelines. The method proposed by Milioto et al. [29], MOPT [19] and Panoster [11] are the only three methods specifically designed for LiDAR point cloud panoptic segmentation. The first two are trained on range images, while the third one is a variant of KPConv [45] at the point level. Due to the lack of public implementation, we use their reported results for comparison. For the combining methods, we pick the highest-ranking approaches with public implementation. We use PolarNet [56] and SalsaNext [8] to generate the semantic prediction. We use PV-RCNN [40] and PointR-CNN [41] to generate the object bounding box prediction for the SemanticKITTI dataset. In addition, we include two combining baselines (Rangenet++ [30]/KPConv [45] + PointPillar [23]) from [2] in our comparison. We generate all results of these baselines from their publicly available implementations and pretrained-networks. During the combination, we pick the points within and close to each bounding box prediction and assign a unique instance to all points that have the same class as the bounding box. We use the combined time of semantic segmentation and object detection as the total inference time for the combining method. On the nuScenes dataset, we use OpenPCDet's [44] pretrained nuScenes CBGS [60] model for the object detection. Since SalsaNext [8] has the best SemanticKITTI testing mIoU among all open-source LiDAR segmentation networks, we train the SalsaNext on the nuScenes dataset from scratch as there are no available pretrained nuScenes segmentation networks. Setup Metrics: We use mean intersection over union (mIoU) to evaluate the performance of semantic segmentation. For panoptic segmentation, we use the panoptic quality (PQ) metric [20], defined as P Q = TP IoU |TP| SQ |TP| |TP| + 1 2 |TN| + 1 2 |FP| RQ .(1) For an instance prediction to be considered as a TP, it needs at least 50% overlap with the groundtruth. Recognition quality (RQ) shows the accuracy of finding TP, while Semantic quality (SQ) shows the average IoU in all TPs. In addition, we report PQ † , which is proposed by Porzi et al. [34] to use only SQ as PQ in "stuff" classes. We also report the inference time to generate a single scan prediction. Implementation details: Following the same configuration of PolarNet [56], we discretize the 3D space within [distance : 3 ∼ 50m, z : −3 ∼ 1.5m] to [480, 360, 32] voxels in SemanticKITTI. We generate the groundtruth heatmap for the center prediction in a ±3 * 5 window around the mass center of points, and correspondingly use the NMS with kernel size σ = 5, threshold 0.1, and k = 100 during the panoptic fusion. Compared to SemanticKITTI, nuScenes uses the LiDAR sensor that contains 32 beams rather than 64 beams. Furthermore, each scan in nuScenes has 34720 points and 34 instances on average, whereas Se-manticKITTI has 104452 points and 5.3 instances. As a result, object points are more sparse in the nuScenes dataset. Hence, we consider an instance with a minimal of 20 points instead of 50 points as a valid instance during the panoptic segmentation evaluation. We use the same implementation setting as in SemanticKITTI to train Panoptic-PolarNet in nuScenes, except that we change the 3D space range to [distance : 0 ∼ 50m, z : −5 ∼ 3m]. We implemented Panoptic-PolarNet in Pytorch on a single NVIDIA TITAN Xp GPU. We use the Adam optimizer with the default configuration. We use the combination of [11] did not disclose their latency nor did they release their code. Since [11] is a variant of KPConv, its latency should be similar to KPConv which is stated to be 200ms [45]. cross-entropy loss (L ce ) and Lovasz softmax loss [3] (L ls ) to train our semantic segmentation head. For the instance head, we use the MSE loss (L hm ) for the heatmap regression and L1 loss (L os ) for the offset regression. The final loss is L = L ce + L ls + λ hm L hm + λ os L os ,(2) where we set λ hm = 100 and λ os = 10. In addition to instance augmentation, during the training, we also use data augmentations, which randomly reflects a point cloud along x, y and x + y axis and randomly rotates the point cloud around the Z axis. We apply dropblock [14] at the end of each up layer to further regularize the training of the proposed Panoptic-PolarNet. Unless specifically mentioned, all hyperparameters (percentage of points pruned per frame, etc.) are tuned on the validation dataset. Table 1 shows the comparison between Panoptic-PolarNet and the baselines on the test split of Se-manticKITTI. Our method outperforms the best baseline by 1.4% in PQ while having a near real-time inference speed, setting a new state-of-the-art performance for the LiDAR panoptic segmentation. It is noticeable that our method has a significant improvement for the "thing" classes compared with other dedicated state-of-the-art LiDAR object detector. We credit our superior instance prediction to the architecture design and augmentation methods. On the other hand, the results for "stuff" classes show a very close correlation to the semantic segmentation. Nevertheless, we still manage to achieve a better performance than the best combining baseline method because of a more well-balanced segmentation results among all classes. Our mini version of Panoptic-PolarNet with [320, 240, 32] grid size achieves a comparable result and only need 2/3 of inference time. Quantitative Results More detailed results with respect to each class will be presented in the supplemental materials. We present our SemanticKITTI validation results in Table 2. We additionally experimented with different settings of Panoptic-PolarNet with more variants of combining baselines. Similar to [56], we found that polar coordinate prevails Cartesian coordinate in terms of every metric while having a slower inference time. All three settings of Panoptic-PolarNet outperform the best baseline method by a large margin. We report the result on the validation set of nuScenes in Table 3. Our method outperforms the combining baseline method by 1.1% in PQ with only half of the time. However, due to the increase in the number of instances, the inference time in nuScenes is slightly higher than SemanticKITTI. Ablation Studies To further analyze the influence of each component, we conducted the ablation studies on the validation split of Se-manticKITTI, as shown in Table 4. We started by training Panoptic-PolarNet without any augmentation and used two independent decoding networks for semantic and instance heads. Rather than using an attention module to connect the learned information between semantic and instance heads, we found out that directly sharing the first three decoding layers can increase the PQ from 51.6% to 52.3% with an even smaller computation load. This indicates that the features learned by semantic and instance heads share plenty of similarities in our setting. Next, we tested the effect of different instance augmentation components on the segmentation results. Instance oversampling improves PQ by 2.8% and mIoU by 2.1%, which benefits the "thing" classes that seldom appear in a scan most. On the other hand, instance global augmentation and local augmentation both have improvements, and using all three instance augmenta- tion methods gives the best result in PQ. Self-adversarial pruning slightly improves the results in terms of PQ but helps to stabilize the semantic results, especially for "stuff" classes. Lastly, visibility feature improves the PQ by 1.6%. Those classes that are mostly surrounded by visible space, like bicyclist and motorcyclist, benefit most from the visibility feature. We also conducted an oracle test, as shown in Table 5 to investigate the room for improvement in Panoptic-PolarNet. We replaced some predictions in the semantic and instance heads to the ground truth for each experiment and generated the panoptic predictions using the same fusion step. It can be seen that our heatmap and offset prediction are both very close to the ground truth in our test setting and, when combined, have only 1.0% difference in PQ compared to the ground truth instance clustering. Conversely, ground truth semantic prediction greatly impacts the results and increases both the PQ and mIoU to above 90%. This matches the finding in [7] that the biggest bottleneck in proposal- free panoptic segmentation is the semantic segmentation. Lastly, Table 5 shows that PQ and mIoU are 96.8% and 96.4% when we use all three ground truth together. This shows that the discretization and projection errors are relatively small in our setting and also verifies our assumption that it is sufficient to separate instances directly on the BEV. Runtime We report the detailed runtime and model size of Panoptic-PolarNet with different settings in Table 6. Compared with PolarNet [56] that solves only for semantic segmentation, our method merely increases the parameter size by 0.1M, and prediction time by 0.02s. Such an insignificant increase reflects our method's high efficiency in generating instance prediction on top of a well-established semantic segmentation network. The inference time difference mostly comes from the fusion step, while it is worth noting that this part has a big room for improvement if better optimized. Both Panoptic-PolarNet and its mini version can process LiDAR data in real-time as a typical LiDAR sensor works at 10 FPS [12,42]. Conclusion In this paper, we present a real-time proposal-free Li-DAR point cloud panoptic segmentation framework named Panoptic-PolarNet. Our method builds upon the established semantic segmentation network and solves the instance segmentation by center regression on the polar BEV map. This design highly simplifies the complexity of the panoptic segmentation, requiring a negligible computation overhead on top of the semantic segmentation and achieving the state-of-the-art result on both SemanticKITTI and nuScenes datasets. We also propose several novel augmentation methods that can be generalized to any other LiDAR point cloud segmentation methods. We hope Panoptic-PolarNet can serve as a strong baseline for future research and a helpful framework for current semantic segmentation methods to migrate to the panoptic segmentation. Supplementary Material Discussion Choice of the loss: We adopted a combination of crossentropy loss and Lovasz softmax loss in the semantic head. Given the highly imbalanced class distribution in LiDAR point clouds, the cross-entropy loss will favor those classes that are the majority of points, like the road class and the building class. Conversely, Lovasz softmax loss optimizes directly on the mIoU Jaccard index, which treats all classes equally. Combining these two losses will force the network to optimize toward an overall accurate prediction while focusing more on hard classes. In the instance head, we chose the MSE loss instead of focal loss [25] for the heatmap regression. The reason is that we do not necessarily need a very accurate prediction of the center in the BEV due to the scarcity of instance overlaps. However, we need a monotonically decreasing heatmap from the center to the edge to have a proper keypoint selection in the NMS. Also, the experiments showed that focal loss decreases the PQ by 1.3%. Data augmentation: We apply instance oversampling to compensate for two imbalances in the LiDAR point cloud: (1) The imbalance between "thing" and "stuff". Points belong to "thing" classes usually consist of only a small portion of the point cloud. (2) The imbalance between different "thing" classes. For example, the most occurring class, car, has around 10 7 time more points than the least occurring class, motorcyclist in the SemanticKITTI dataset. Experiments show that even though this oversampling will decrease the segmentation accuracy in "stuff", the overwhelmingly increases in "thing" can still provide a huge improvement to the PQ and mIoU. Our experiments also show that either simply putting instance points at any place in a point cloud or transform it through its center will decrease the PQ. We conclude that such simple augmentation ignores projection properties, introduces inconsistency into the Li-DAR point cloud, and thus entangles BEV feature learning. Proposal-free vs. proposal-based: Even though proposal-based panoptic segmentation methods dominate in the 2D domain, there are only a few existing approaches for LiDAR point clouds. We think there are two reasons. First, proposal-based methods rely heavily on the annotation of bounding boxes, whereas point cloud datasets do not necessarily provide such annotations. Second, most current proposal-based object detection methods, like what we assume in our instance head, are not designed to represent the scene along the Z-axis. Lacking proper representation makes it more challenging to achieve a competitive result while maintaining speed when modified into a panoptic segmentation network. End-to-end training: We only train the network to get an intermediate result and use a majority voting fusion to generate the final panoptic segmentation. Making Figure 3: We highlight SAP-pruned points in color. Left: SAP prunes tangled vegetation and fence; Right: SAP prunes garden curbs that are annotated as the " fence " the proposal-free panoptic segmentation network end-toend trainable is still an open problem to explore in the future. Self-adversarial Pruning Visualization: SAP is designed to remove ambiguous, noisy or/ and informative points. Since SemanticKITTI is a well-annotated dataset, we visually find SAP tends to remove challenging or ambiguous cases. Some examples are shown in Fig. 3. Class-wise Results We show the class-wise results of Panoptic-PolarNet on SemanticKITTI and nuScenes in Table 7 and Table 8. Our method has a similar panoptic segmentation performance in the corresponding classes among these two datasets. The low performance comes from the class that either has a small physical shape (like bicycle) or has a small number of instances in the dataset (like truck and construction vehicle). Despite being a more challenging dataset due to its significantly higher number of instances, nuScenes has fewer classes than SemanticKITTI, which makes it more distinguishable and thus having higher PQ and mIoU. Qualitative Results We show the visualization examples of Panoptic-PolarNet on SemanticKITTI and nuScenes in Figure 4 and Figure 5 respectively. Our method can make accurate instance predictions regardless of the distance and point density variation. We can also visually verify that nuScenes has significantly more instances than SemanticKITTI. And most of those instances belong to some challenging classes that have a small number of points. There are also duplicated instance predictions within a short distance. This could be fixed by introducing class-wise prior knowledge in the grouping stage in the future. Figure 1 : 1SemanticKITTI[1] panoptic quality vs. single frame inference latency. The green line marks the sampling rate of the LiDAR scanner, which spins at 10 frames-persecond. Our proposed Panoptic-PolarNet outperforms other baselines in both speed and PQ. Figure 4 :Figure 5 : 45Visualization of Panoptic-PolarNet on the SemanticKITTI dataset. The red dots in the instance prediction represent the center for each instance.(a) Semantic Ground Truth (b) Semantic Prediction (c) Instance Ground Truth (d) Instance Prediction Visualization of Panoptic-PolarNet on the nuScenes dataset. The red dots in the instance prediction represent the center for each instance. Table 1 : 1Panoptic Segmentation results on the test split of SemanticKITTI.Method Latency PQ PQ † RQ SQ PQ T h RQ T h SQ T h PQ St RQ St SQ St mIoU Rangenet++ [30] + PointPillar [23] 0.409s 37.1% 45.9% 47.0% 75.9% 20.2% 25.2% 75.2% 49.3% 62.8% 76.5% 52.4% Milioto et al. [29] 0.085s 38.0% 47.0% 48.2% 76.5% 25.6% 31.8% 76.8% 47.1% 60.1% 76.2% 50.9% KPConv [45] + PointPillar [23] 0.514s 44.5% 52.5% 54.4% 80.0% 32.7% 38.7% 81.5% 53.1% 65.9% 79.0% 58.8% SalsaNext [8] + PV-RCNN [40] 0.255s 47.6% 55.3% 58.6% 79.5% 39.1% 45.9% 82.3% 53.7% 67.9% 77.5% 58.9% Panoster [11] -* 52.7% 59.9% 64.1% 80.7% 49.9% 58.8% 83.3% 55.1% 68.2% 78.8% 59.9% Panoptic-PolarNet-mini 0.057s 52.6% 59.4% 63.6% 80.9% 51.9% 59.5% 86.9% 53.1% 66.6% 76.5% 58.4% Panoptic-PolarNet 0.086s 54.1% 60.7% 65.0% 81.4% 53.3% 60.6% 87.2% 54.8% 68.1% 77.2% 59.5% * Table 2 : 2Panoptic Segmentation results on the validation split of SemanticKITTI.Method Latency PQ PQ † RQ SQ PQ T h RQ T h SQ T h PQ St RQ St SQ St mIoU MOPT [19] 0.146s 40.0% - 48.3% 73.0% 29.9% 33.6% 76.8% 47.4% 70.3% 59.1% 53.8% SalsaNext [8] + PointRCNN [41] 0.196s 47.5% 53.2% 58.2% 74.4% 42.5% 50.3% 73.4% 51.1% 64.0% 75.2% 59.0% PolarNet [56] + PointRCNN [41] 0.202s 47.9% 53.1% 58.7% 71.3% 39.5% 47.8% 71.0% 54.1% 66.7% 71.4% 58.2% PolarNet [56] + PV-RCNN [40] 0.261s 49.9% 55.0% 60.9% 71.4% 44.1% 52.9% 71.2% 54.1% 66.7% 71.4% 58.2% SalsaNext [8] + PV-RCNN [40] 0.255s 49.9% 55.6% 61.0% 74.4% 48.3% 56.7% 73.3% 51.1% 64.0% 75.2% 59.0% Panoptic-PolarNet in Cartesian coordinates 0.078s 54.3% 58.8% 65.5% 78.0% 58.4% 67.3% 85.3% 50.3% 64.2% 69.1% 58.6% Panoptic-PolarNet-mini 0.057s 57.1% 61.8% 68.1% 77.7% 63.8% 72.6% 87.4% 52.2% 64.9% 70.6% 61.9% Panoptic-PolarNet 0.086s 59.1% 64.1% 70.2% 78.3% 65.7% 74.7% 87.4% 54.3% 66.9% 71.6% 64.5% Table 3 : 3Panoptic segmentation results on the validation split of nuScenes.Method Latency PQ PQ † RQ SQ PQ T h RQ T h SQ T h PQ St RQ St SQ St mIoU PolarNet [56] + CBGS [60] 0.208s 66.6% 70.3% 78.0% 84.6% 63.8% 74.1% 85.1% 71.4% 84.5% 83.7% 71.8% SalsaNext [8] + CBGS [60] 0.207s 61.6% 66.3% 72.3% 84.5% 59.5% 68.1% 86.6% 65.1% 79.5% 81.0% 63.4% Panoptic-PolarNet 0.099s 67.7% 71.0% 78.1% 86.0% 65.2% 74.0% 87.2% 71.9% 84.9% 83.9% 69.3% Table 4 : 4Ablation study of Panoptic PolarNet on the valida- tion split of SemanticKITTI. 'SU','IO','IGA','ILA','SAP', 'Vis' stand for training with first three up layers shared among semantic and instance heads (SU), instance over- sampling (IO), instance global augmentation (IGA), in- stance local augmentation (ILA), self adversarial pruning (SAP), and visibility feature (Vis). SU IO IGA σ = 5 ILA SAP Vis PQ mIoU 51.6% 57.8% × 52.3% 58.1% × × 55.1% 60.2% × × × 57.0% 62.1% × × × × 57.3% 61.3% × × × × 57.3% 61.7% × × × × × 57.4% 61.6% × × × × × × 57.5% 62.1% × × × × × × 58.0% 62.6% × × × × × × × 59.1% 64.5% Table 5 : 5Oracle test of Panoptic-PolarNet on the validation split of SemanticKITTI.GT Heatmap GT Offset GT Semantic PQ mIoU 59.1% 64.5% × 59.5% 64.5% × 59.4% 64.5% × × 60.1% 64.5% × 91.9% 94.1% × × × 96.8% 96.4% Table 6 : 6Runtime and parameter size comparisons of Panoptic-PolarNet.Model Pred Fusion Params FPS PolarNet 0.059s - 13.6M 16.9 Panoptic-PolarNet 0.061s 0.025s 13.7M 11.6 Panoptic-PolarNet-mini 0.043s 0.014s 13.7M 17.5 Table 7 : 7Class-wise results on test split of SemanticKITTI. 8% 33.0% 51.8% 35.2% 37.6% 57.3% 69.9% 52.4% 88.6% 42.6% 61.2% 1.6% 85.7% 46.0% 75.7% 54.5% 41.5% 48.5% 56.4% 54.1% RQ 96.2% 46.7% 59.9% 38.7% 41.5% 65.7% 78.6% 57.7% 96.8% 56.2% 76.3% 2.8% 92.1% 62.3% 92.5% 73.6% 55.5% 66.0% 75.2% 65.0% SQ 92.3% 70.7% 86.4% 90.9% 90.5% 87.2% 88.9% 90.9% 91.5% 75.9% 80.3% 55.6% 93.0% 73.8% 81.9% 74.1% 74.9% 73.4% 75.0% 81.4% IoU 94.4% 38.7% 48.2% 46.2% 34.5% 51.1% 63.9% 24.9% 90.8% 61.3% 74.6% 16.5% 89.9% 61.1% 83.4% 66.7% 68.0% 56.8% 58.5% 59.5%metrics car bicycle motorcycle truck other-vehicle person bicyclist motorcyclist road parking sidewalk other-ground building fence vegetation trunk terrain pole traffic-sign mean PQ 88. Table 8 : 8Class-wise results on validation split of nuScenes. 5% 58.0% 70.4% 89.0% 36.4% 78.4% 85.1% 80.7% 49.7% 63.3% 95.7% 53.9% 67.7% 49.8% 84.3% 80.0% 67.7% RQ 54.4% 68.6% 76.1% 95.9% 44.9% 86.0% 95.2% 91.8% 58.2% 69.4% 100.0% 66.3% 85.3% 65.1% 98.2% 94.7% 78.1% SQ 76.3% 84.6% 92.5% 92.8% 81.2% 91.2% 89.4% 87.9% 85.3% 91.3% 95.7% 81.3% 79.4% 76.5% 86.0% 84.5% 86.0% IoU 52.3% 28.1% 88.0% 90.3% 32.4% 71.7% 72.2% 52.8% 58.3% 76.6% 95.9% 68.8% 74.3% 73.4% 87.3% 85.5% 69.3%metrics barrier bicycle bus car construction vehicle motorcycle pedestrian traffic cone trailer truck driveable surface other flat sidewalk terrain manmade vegetation mean PQ 41. Se-manticKITTI: A dataset for semantic scene understanding of lidar sequences. Jens Behley, Martin Garbade, Andres Milioto, Sven Behnke, Cyrill Stachniss, and Jurgen Gall. ICCVJens Behley, Martin Garbade, Andres Milioto, Jan Quen- zel, Sven Behnke, Cyrill Stachniss, and Jurgen Gall. 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Joint 3d instance segmentation and object detection for autonomous driving. In CVPR, 2020. 3 Jiquan Ngiam, and Vijay Vasudevan. End-to-end multi-view fusion for 3d object detection in lidar point clouds. Yin Zhou, Pei Sun, Yu Zhang, Dragomir Anguelov, Jiyang Gao, Tom Ouyang, James Guo, CoRLYin Zhou, Pei Sun, Yu Zhang, Dragomir Anguelov, Jiyang Gao, Tom Ouyang, James Guo, Jiquan Ngiam, and Vijay Va- sudevan. End-to-end multi-view fusion for 3d object detec- tion in lidar point clouds. In CoRL, 2020. 3 Voxelnet: End-to-end learning for point cloud based 3d object detection. Yin Zhou, Oncel Tuzel, CVPR. Yin Zhou and Oncel Tuzel. Voxelnet: End-to-end learning for point cloud based 3d object detection. In CVPR, 2018. 3 Class-balanced grouping and sampling for point cloud 3d object detection. Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, Gang Yu, arXiv6Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and Gang Yu. Class-balanced grouping and sampling for point cloud 3d object detection. In arXiv, 2019. 6, 8
[ "https://github.com/edwardzhou130/Panoptic-PolarNet.", "https://github.com/open-mmlab/OpenPCDet," ]
[ "The VGG Image Annotator (VIA)", "The VGG Image Annotator (VIA)" ]
[ "Abhishek Dutta [email protected] \nDepartment of Engineering Science\nVisual Geomtry Group (VGG)\nUniversity of Oxford\n\n", "Andrew Zisserman \nDepartment of Engineering Science\nVisual Geomtry Group (VGG)\nUniversity of Oxford\n\n" ]
[ "Department of Engineering Science\nVisual Geomtry Group (VGG)\nUniversity of Oxford\n", "Department of Engineering Science\nVisual Geomtry Group (VGG)\nUniversity of Oxford\n" ]
[]
Manual image annotation, such as defining and labelling regions of interest, is a fundamental processing stage of many research projects and industrial applications. In this paper, we introduce a simple and standalone manual image annotation tool: the VGG Image Annotator (VIA). This is a light weight, standalone and offline software package that does not require any installation or setup and runs solely in a web browser. Due to its lightness and flexibility, the VIA software has quickly become an essential and invaluable research support tool in many academic disciplines. Furthermore, it has also been immensely popular in several industrial sectors which have invested in adapting this open source software to their requirements. Since its public release in 2017, the VIA software has been used more than 500, 000 times and has nurtured a large and thriving open source community.
10.1145/3343031.3350535
[ "https://arxiv.org/pdf/1904.10699v1.pdf" ]
129,945,986
1904.10699
3f175fdd889dd2167c5aa36772bfd98caa96e5fc
The VGG Image Annotator (VIA) Abhishek Dutta [email protected] Department of Engineering Science Visual Geomtry Group (VGG) University of Oxford Andrew Zisserman Department of Engineering Science Visual Geomtry Group (VGG) University of Oxford The VGG Image Annotator (VIA) Manual image annotation, such as defining and labelling regions of interest, is a fundamental processing stage of many research projects and industrial applications. In this paper, we introduce a simple and standalone manual image annotation tool: the VGG Image Annotator (VIA). This is a light weight, standalone and offline software package that does not require any installation or setup and runs solely in a web browser. Due to its lightness and flexibility, the VIA software has quickly become an essential and invaluable research support tool in many academic disciplines. Furthermore, it has also been immensely popular in several industrial sectors which have invested in adapting this open source software to their requirements. Since its public release in 2017, the VIA software has been used more than 500, 000 times and has nurtured a large and thriving open source community. Introduction Manual image annotation is the process of defining and describing regions in an image. These regions can have any shape (e.g. rectangle, circle, ellipse, point, polygon, polyline, freehand drawn mask, etc.) and the description can contain any textual metadata. Manual image annotation software allows human annotators to define such regions in an image and describe those regions using textual metadata. In this paper, we present a simple and standalone manual image annotation tool, the VGG Image Annotator (VIA), that runs solely in a web browser and does not require any installation or setup. The complete VIA software is an offline and self-contained HTML page of size less than 400 kilobyte. VIA software is an open source project created solely using HTML, Javascript and CSS. This choice of platform has allowed us to build a flexible manual image annotation tool with the following capabilities: a) Up and running in a few seconds, b) No installation or setup required, c) Light weight, portable and offline, d) Simple and easy to use. Since VIA requires no installation or setup, non-technical users can begin annotating their images very quickly and consequently, we have seen widespread adoption of this software in a large number of academic disciplines and industrial sectors. The complete VIA software fits in a single self-contained HTML page of size less than 400 kilobyte that runs as an offline application in most modern web browsers. This light footprint of VIA software allows it to be easily shared (e.g. using email) and distributed amongst manual annotators. A minimalistic approach to user interface design and rigorous testing (both internally and by our vibrant open source community) has allowed the VIA software to become an easily configurable, simple and easy to use manual image annotation tool. Figure 1: Screenshot of the VIA software running as an offline and standalone application in a web browser. Human annotators can define image regions (e.g. polygon shaped region) and describe those regions using on-image annotation editor. Manually annotated images have a wide variety of applications in many academic disciplines and industrial projects. In Computer Vision, such images are essential for training and testing of many computer vision algorithms. Publicly available image datasets such as [1] and [2] contain such manual annotations and have played an important role in the progress of computer vision research. In the Humanities, manual annotation of early printed books and illustrations is a significant part of scholarly research. For example, [3] annotated hundreds of 15th-century printed illustrations and [4] annotated and classified 1.1 million 18th-century printed book ornaments using the VIA software. This paper is organised as follows. Section 2 and 3 describe the features of the VIA software. A brief overview of the software design and source code of VIA is included in Section 4, and an insight into the open source ecosystem thriving around VIA is included in Section 5. The impact of VIA on different academic disciplines and industrial sectors is described in Section 6. In Section 7, we review some existing manual image annotation tools. Finally, we describe our planned directions for extensions in Section 8. Single Image Annotation using VIA VIA is an open source software that allows human annotators to define and describe regions in an image. The manually defined regions can have one of the following six shapes: rectangle, circle, ellipse, polygon, point and polyline. Rectangular shaped regions are very common and are mostly used to define the bounding box of an object. Polygon shaped regions are used to capture the boundary of objects having a complex shape. The point shape is used to define feature points like facial landmarks, or keypoints in MRI images, location of particles in microscopy images, etc. The circle, ellipse and polyline shaped regions are less common but are essential for some projects. For example, the polyline shape has been used to annotate the railway tracks in satellite imagery and the circle shape is being used by [5] to annotate particles in cryo-electron microscopy images. In Figure 2, we show some sample annotations created using VIA in different academic disciplines. Textual description of each region is essential for many projects. Such textual descriptions often describe visual content of the region. VIA uses region attributes to define different properties (or attributes) of a region. For example, most common face annotation tasks requires human annotators to define rectangular regions delineating all the human faces present in an image, and to label the identity and pose of the face appearing in each region. Therefore, for the face annotation task, name Figure 2: The VIA software supports many types of region shapes which are useful for manual annotation in a wide range of projects. For example, (a) humanities researchers from [3] used rectangular shaped regions to annotate 15th-century printed illustrations; (b) the point shape is used by [6] to manually define location of particles in cryo-electron microscopy image; (c) boundary of arbitrarily shaped objects in scanning electron microscope image is defined by [5] using circle and polygon shapes; and (d) the polyline shaped regions can be used to define railyway tracks in satellite imagery. (a) (b) (c) (d) and pose are the required region attributes, and the anotators are required to set a value for these two region attributes. A plain text input element is sufficient to set a value for all types of region attributes. In addition to supporting text input, VIA supports the following additional input types to reduce the cognitive load of human annotators: checkbox, radio, image and dropdown. For example, in the face annotation task, the identity and pose of face contained in a region can be described using image and radio attribute types. Once the region attributes and their options are defined by the user, as shown in Figure 3 (left), human annotators can update the value of these attributes using on-screen annotation editor as shown in Figure 3 (right). The resulting on-screen annotation editor that allows human annotators to easily set a value for these region attributes. The VIA software can operate on images stored on a local computer and images hosted at remote servers as shown in Figure 4. Since the VIA software runs as an offline application, web browsers allow access to local images using the file:// URI scheme and remote images are accessed using the http:// URI scheme. The self contained package 1 demonstrates the capability of the VIA software to operate on remotely hosted images; it allows manual annotation of 9955 images remotely hosted by Wikimedia Commons. Image Group Annotation using VIA Annotation of large image datasets is rarely accompolished solely by human annotators. More usually, a two stage process is used to reduce the burden on human annotators: 1. Automatic Annotation: Computer vision algorithms are applied to the image dataset to perform a preliminary (but possibly imperfect) annotation of the images. 2. Manual Filtering, Selection and Update: Human annotators review the annotations produced by the Automatic Annotation process and perform manual filtering, selection and update to retain only the good quality annotations. This two stage process off-loads the burden of image annotation from human annotators and only requires them to perform filtering, selection and update of automatic annotations. The VIA software supports this two stage model of image annotation using its Image Grid View feature which is designed to help human annotators filter, select and update metadata associated with a group of images. The image groups are based on the metadata and regions defined by automatic computer vision algorithms. To illustrate the image grid view feature of VIA, let us consider the task of face track annotation which involves delineating and identifying face region of an individual in consecutive frames of a video -also called a face track. Such annotated dataset are often used to train face detection and recognition algorithms. For face track annotation, an automatic face detector (e.g. Faster R-CNN [7]) is used to detect face regions in consecutive frames of a video and a face track identification system (e.g. VGG Face Tracker [8]) identifies unique face tracks from the automatically detected face regions in consecutive frames. Automatically generated face track annotations are imported into the VIA software and human annotators -using the image grid view feature of VIA -review these automatically generated face tracks and select or filter the ones that are correct as shown in Figure 5. Furthermore, the VIA image grid view also allows bulk update of attributes associated with each group. This capability of VIA allows human annotators to quickly annotate large number of images that have been partially annotated by automatic computer vision algorithms. The grid view also enables annotators to easily remove erroneous images from a group, as shown in Figure 6. This functionality is very useful for re-training an existing image classifier by identifying images that have been incorrectly classified. (a) (b) (c) Figure 6: Removing incorrect group assignments using image grid view feature of VIA. (a) A group of 16 images were automatically assigned as containing a "cat" by an image classifier; (b) a human annotator removes two images that do not contain an image of a "cat" from this group; and (c) assigns these two images to the group named "other" to indicate that these two images do not contain a "cat". Software Design and Source Code The user interface of VIA is made using standard HTML components and therefore the VIA software looks familiar to most new users. These components are styled using CSS to achieve a greyscale colour scheme which helps avoid distractions and focus attention on the visual content that is being manually annotated using the VIA software. We follow the minimalist approach for the user interface, and strive for simplicity both in design and implementation. We resist adding new features or updating existing user interface components if we feel that such change leads to complexity in terms of usability and implementation. Most of our design decisions are influenced by feedback from the open source community thriving around the VIA software. The VIA application contains around 9000 lines of Javascript code that manages all aspects of user interactions, annotation import and export, file management, attribute and metadata editing. This codebase sprouted from an early prototype 2 of VIA which implemented a minimal -yet functional -image annotation tool using only 40 lines of HTML/CSS/Javascript code that runs as an offline application in most modern web browsers. This early prototype provides a springboard for understanding the current codebase of VIA which is just an extension of the early prototype. In this early prototype, an image is loaded and displayed by the <img> HTML element and user drawn regions are displayed using a HTML canvas element overlaid on the image -thereby creating an illusion that the regions are being drawn on the image. User interactions based on mouse events is captured by the mousedown and mouseup event handlers which uses the mouse click locations to define the top-left and bottom-right corners of user drawn rectangular regions. The full VIA software also uses similar strategy to record and display user drawn regions. Therefore, the 9000 lines of Javascript code in the full VIA application is simply an extension of this early prototype. For example, to support the resize and translation operations for user drawn regions, the full VIA application also handles the mousemove event in addition to the handling of mousedown and mouseup events. A more detailed source code documentation 3 is available for developers and contributors of the VIA open source project. Open Source Ecosystem We have nurtured a large and thriving open source community which not only provides feedback but also contributes code to add new features and improve existing features in the VIA software. The open source ecosystem of VIA thrives around its source code repository 4 hosted by the Gitlab platform. Most of our users report issues and request new features for future releases using the issue portal. Many of our users not only submit bug reports but also suggest a potential fix for these software issues. Some of our users also contribute code to add new features to the VIA software using the merge request portal. Thanks to the flexibility provided by our BSD open source software license, many representatives from commercial industry have contacted us through email to seek advice for their engineering team tasked with adapting the VIA software for internal or commercial use. Impact The VIA software has quickly become an essential and invaluable research support tool for many academic disciplines like Humanities, Anthropology, English Literature, History of Art, Geography and Environment, Geology, Zoology, Plant Science, Ecology. Using VIA, [3] annotated hundreds of 15th-century printed illustrations and [4] annotated and classified 1.1 million 18th-century printed book ornaments. In her DPhil thesis, [9] used VIA to manually annotate a multilayered 14thcentury cosmological diagram containing many different elements. For an ongoing project, the Oxford Anthropolgy department is using VIA to annotate chimpanzee faces in hundreds of hours of videos. Oxford's Geography and Environment department used VIA to annotate aerial imagery of buffalo in South Africa. A Norwegian project aiming to monitor plants and pollinators in the Arctic used the VIA software to annotate images gathered by monitoring cameras. A geological study in Russia annotated thin sections of rocks using VIA and a customised version of VIA is being used in Zoology to annotate fish in videos. VIA has also proved to be a useful support tool in Medicine. The organisers of EAD 2019 used VIA to prepare the annotations for the Endoscopy Artefact Detection (EAD) challenge dataset [10]. A customised version of VIA is being prepared to be used by clinicians to annotate intervertebral disc in MRI data. A training dataset of manually labelled 800 radiographs was created by [11] using VIA. This dataset was used to train a key points model. A non-profit platform created by [5] is using VIA to annotate arbitrarily shaped particles to create datasets for image-based particle analysis methods. The VIA software was adapted by [6] to prepare training data for particle detection in cryo-electron microscopy images using convolution neural networks. The development of VIA software began in August 2016 and the first public release was made in April 2017. As of March 2019, the VIA software has been used more than 500, 000 times (+120, 000 unique pageviews). Related Work VIA builds on a tradition of existing manual image annotation software systems. Many such systems already exist because each package is designed to address a set of specific use cases. The VIA software is a light-weight addition to this existing pool of image annotation tools. The manual image annotation tool developed by [12] is similar to the VIA software as both of them use the HTML, Javascript and CSS platform to create self-contained offline application that runs in any standard web browser. A limited number of image annotation tools support collaborative (public and private) annotation of image dataset. For example, [13] facilitates collaborative annotation of a large image dataset using client-server design where human annotators use a web browser based client user interface which is maintained and driven by a central server. Many image annotation tools can ease the burden of manual annotation by automatically performing image annotation based on cues from human annotators. For example, the FreeLabel [14] software aims to reduce cognitive load of manual annotators by automatically performing segmentation of an object based on seed freehand traces drawn on the object by a human annotator. There is a strong demand for offline image annotation tools that can be installed as standalone desktop applications across multiple platforms like Windows, Linux and MacOS. Such tools are often simple to setup and therefore are preferred by most non-technical users. For example, the manual image annotation tools developed by [15], [16], [17] and [18] are relatively simple to install on multiple platforms. Conclusions and Future Development In this paper, we described our manual image annotation tool called VGG Image Annotator (VIA). We continue to develop and maintain this software according to the principles of open source software development and maintenance. Due to its simplicity, flexibility and open source software code, the VIA software has quickly become an essential and invaluable software tool for many academic discplines and commercial industry. VIA is a continually evolving open source project which aims to be useful for image and video annotation tasks in many academic disciplines and industrial settings. This demands continuous improvement and introduction of advanced new features in VIA. Therefore, we are already working to add the following new features to the upcoming version 3 release of the VIA software: • Video Annotation: There are many tools for image annotation but only a limited number of tools for annotation of videos. The existing video annotation tools are either too complex or requires technical expertise to install, setup and use. We are adding support for video annotation capability in VIA to allow annotation of frame regions and temporal segments of a video. This feature can also be used for speaker diarisation in audio and video. • Collaborative Annotation: Annotating a large number of images (e.g. a million images) or videos (e.g. thousands of hours of videos) requires collaboration between a large number of human annotators. Therefore, we are upgrading VIA to add support for collaborative annotation which would allow multiple human annotators to incrementally and independently annotate a large collection of images and videos. • Plugins: The state-of-the-art computer vision models are becoming very accurate in common annotation tasks such as locating objects, detecting and recognising human faces, reading text, detecting keypoints on a human body and many other tasks commonly assigned to human annotators. We believe that these computer vision models -attached as plugin to VIA and running in background -can help speed up the manual annotation process by seeding an image with automatically annotated regions and then letting human annotators edit or update these detections to create the final annotation. Thanks to projects like TensorFlow.js [19], it is now possible to run many of these models in a web browser. We envisage such computer vision models attached as plugins to VIA and running in background to assist human annotators. Figure 3 : 3(left) Setting up the name and pose region attributes. (right) Figure 5 : 5A set of automatically detected face tracks in 165 consecutive video frames (left) from BBC Sherlock series is assigned metadata (is good track and name) quickly (right) by human annotators using the image grid view feature of VIA. A face track containing incorrect detections can also be easily filtered out by annotators by setting is good track to "No". VIA has also been immensely popular in several industrial sectors which have invested in adapting this open source software to their specific requirements. For example, Puget Systems (USA) and Larsen & Toubro Infotech Ltd. (India) have integrated the VIA software in their existing web application. Trimble Inc. (Colorado, USA) adapted VIA for large scale collaborative annotation by running VIA in the Amazon Mechanical Turk platform. Vidteq (Bangalore, India) have upgrade VIA to integrate it in their internal work flow. http://www.robots.ox.ac.uk/~vgg/software/via/via_wikimedia_demo.html http://www.robots.ox.ac.uk/~vgg/software/via/via-0.0.1.html (add ".txt" suffix to view source code) 3 https://gitlab.com/vgg/via/blob/master/CodeDoc.md 4 https://gitlab.com/vgg/via The pascal visual object classes challenge: A retrospective. 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The 15cbooktrade project. http://15cbooktrade.ox. ac.uk/. Accessed: Mar 2019. Fleuron: A database of eighteenth-century printers. Hazel Wilkinson, Hazel Wilkinson. Fleuron: A database of eighteenth-century printers' ornaments. https:// fleuron.lib.cam.ac.uk/, 2016. How-to: Generate primary object masks. Bigparticle, Cloud, BigParticle.Cloud. How-to: Generate primary object masks. https://www.bigparticle. cloud/index.php/how-to-generate-primary-object-masks/. Accessed: Mar 2019. Positive-unlabeled convolutional neural networks for particle picking in cryo-electron micrographs. Tristan Bepler, Andrew Morin, Julia Brasch, Lawrence Shapiro, Alex J Noble, Bonnie Berger, arXiv e-printsTristan Bepler, Andrew Morin, Julia Brasch, Lawrence Shapiro, Alex J. Noble, and Bonnie Berger. Positive-unlabeled convolutional neural networks for particle picking in cryo-electron micrographs. arXiv e-prints, Mar 2018. Faster R-CNN: Towards real-time object detection with region proposal networks. Kaiming Shaoqing Ren, Ross He, Jian Girshick, Sun, Advances in Neural Information Processing Systems (NIPS). Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster R-CNN: Towards real-time object detection with region proposal networks. In Advances in Neural Information Processing Systems (NIPS), 2015. Vgg face tracker. Qiong Cao, M Omkar, Mark Parkhi, Josef Everingham, Andrew Sivic, Zisserman, Qiong Cao, Omkar M. Parkhi, Mark Everingham, Josef Sivic, and Andrew Zisserman. Vgg face tracker. http://www.robots.ox.ac.uk/~vgg/software/face_tracker/. Accessed: Mar 2019. Diagram and Dimension: Visualising Time in the Drawings of Opicinus De Canistris (1296-c. 1352). Sarah Griffin, University of OxfordPhD thesisSarah Griffin. Diagram and Dimension: Visualising Time in the Drawings of Opicinus De Canistris (1296-c. 1352). PhD thesis, University of Oxford, 2018. Endoscopy artefact detection (EAD) dataset. Sharib, Sharib et al. Ali. 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C Bryan, Antonio Russell, Kevin P Torralba, William T Murphy, Freeman, International journal of computer vision. 771-3Bryan C Russell, Antonio Torralba, Kevin P Murphy, and William T Freeman. Labelme: a database and web-based tool for image annotation. International journal of computer vision, 77(1-3):157-173, 2008. Freelabel: A publicly available annotation tool based on freehand traces. Philipe Dias, Zhou Shen, Amy Tabb, Henry Medeiros, Winter Conference on Applications of Computer Vision (WACV). Philipe Dias, Zhou Shen, Amy Tabb, and Henry Medeiros. Freelabel: A publicly available annotation tool based on freehand traces. In Winter Conference on Applications of Computer Vision (WACV), January 2019. Image annotation tool with image masks. Alexander Kläser, Alexander Kläser. Image annotation tool with image masks. https://lear.inrialpes.fr/ people/klaeser/software_image_annotation, 2010. Mask editor: an image annotation tool for image segmentation tasks. 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[]
[ "DECOMPLETION OF CYCLOTOMIC PERFECTOID FIELDS IN POSITIVE CHARACTERISTIC", "DECOMPLETION OF CYCLOTOMIC PERFECTOID FIELDS IN POSITIVE CHARACTERISTIC" ]
[ "Laurent Berger ", "Sandra Rozensztajn " ]
[]
[]
Let E be a field of characteristic p. The group Z × p acts on E((X)) by a·f (X) = f ((1 + X) a − 1). This action extends to the X-adic completion E of ∪ n 0 E((X 1/p n )). We show how to recover E((X)) from the valued E-vector space E endowed with its action of Z × p . To do this, we introduce the notion of super-Hölder vector in certain E-linear representations of Z p . This is a characteristic p analogue of the notion of locally analytic vector in p-adic Banach representations of p-adic Lie groups.
10.5802/ahl.150
[ "https://export.arxiv.org/pdf/2201.09688v4.pdf" ]
246,240,139
2201.09688
e76b9f830e0968e9d8f7131d1e7a8fe3c3ee278f
DECOMPLETION OF CYCLOTOMIC PERFECTOID FIELDS IN POSITIVE CHARACTERISTIC 14 Nov 2022 Laurent Berger Sandra Rozensztajn DECOMPLETION OF CYCLOTOMIC PERFECTOID FIELDS IN POSITIVE CHARACTERISTIC 14 Nov 2022 Let E be a field of characteristic p. The group Z × p acts on E((X)) by a·f (X) = f ((1 + X) a − 1). This action extends to the X-adic completion E of ∪ n 0 E((X 1/p n )). We show how to recover E((X)) from the valued E-vector space E endowed with its action of Z × p . To do this, we introduce the notion of super-Hölder vector in certain E-linear representations of Z p . This is a characteristic p analogue of the notion of locally analytic vector in p-adic Banach representations of p-adic Lie groups. Introduction Let p be a prime number, and let E be a field of characteristic p. Let E = E((X)), and let E be the X-adic completion of ∪ n 0 E((X 1/p n )). Note that if E is perfect, the field E is perfectoid. The group Z × p acts on E by (a · f ) (X) = f ((1 + X) a − 1). This action extends to ∪ n 0 E((X 1/p n )) by (a · f ) (X 1/p n ) = f ((1 + X 1/p n ) a − 1), and by continuity to E. The question that motivated this paper is the following. Question. -Can we recover ∪ n 0 E((X 1/p n )) or even E((X)) from the data of the valued E-vector space E endowed with the action of Z × p ? In characteristic zero, it is possible to answer an analogous question by using Schneider and Teitelbaum's theory of locally analytic vectors in p-adic Banach representations of p-adic Lie groups. For characteristic p representations, there is no such theory. One of the main contributions of this article is to introduce a characteristic p analogue of locally analytic functions and vectors. Let M be an E-vector space, endowed with a valuation val M such that val M (xm) = val M (m) if x ∈ E × . We assume that M is separated and complete for the val M -adic topology. For example, we will consider M = E or E with the X-adic valuation. We say that a function f : Z p → M is super-Hölder if there exist constants λ, µ ∈ R such that val M (f (x) − f (y)) p λ · p i + µ whenever val p (x − y) i, for all x, y ∈ Z p and i 0. These super-Hölder functions are the characteristic p analogue of locally analytic functions Z p → Q p . We prove an analogue of Mahler's theorem for continuous functions f : Z p → M, and give a characterization of super-Hölder functions in terms of their Mahler expansions. This is a characteristic p analogue of a theorem of Amice. Assume now that Γ is a group that is isomorphic to Z p via a coordinate map c, and that M is endowed with an E-linear action of Γ by isometries. We say that m ∈ M is a super-Hölder vector if the orbit map z → c −1 (z) · m is a super-Hölder function Z p → M. This definition is a characteristic p analogue of the notion of locally analytic vector of a p-adic Banach representation of a p-adic Lie group. We let M Γ-sh,λ denote the space of super-Hölder vectors for a given constant λ as in the definition above. We also let M sh denote the set of super-Hölder vectors in M. Our main result is a complete answer to the question above. Consider M = E, endowed with the action of Γ = 1 + p k Z p for k 1 (or k 2 if p = 2). Theorem. -For all n 0, we have E (1+p k Zp)-sh,k−n = E((X 1/p n )). In particular, E sh = ∪ n 0 E((X 1/p n )). commutant of Aut(G m ), namely the set of u ∈ E val X >0 such that u • γ a = γ a • u for all a ∈ Z × p , where γ a (X) = (1 + X) a − 1. Using our main theorem, and a result of Lubin-Sarkis on the classical commutant of Aut(G m ), we prove that such a u is of the form γ b (X p n ) for some b ∈ Z × p and n ∈ Z. Next we study (ϕ, Γ)-modules over E. We prove that the action of Γ on a (ϕ, Γ)-module D is always super-Hölder, and deduce that ( E ⊗ E D) sh = (∪ n 0 E((X 1/p n ))) ⊗ E D. This allows us to extend our computation of super-Hölder vectors to the finite extensions of F p ((X)) provided by Fontaine and Wintenberger's theory of the field of norms. We finish this article with a computation that suggests that the theory of super-Hölder vectors could have some applications to the p-adic local Langlands correspondence. Super-Hölder functions and vectors In this section, we define super-Hölder functions Z p → M and super-Hölder vectors in M when M is a representation of a group isomorphic to Z p . We prove an analogue of Mahler's theorem for continuous functions Z p → M, and give a characterization of super-Hölder functions in terms of their Mahler expansions. x ∈ E × . We assume that M is separated and complete for the val M -adic topology. For example, we will consider M = E[[X]] with the X-adic valuation. Let C 0 (Z p , M) denote the space of continuous functions f : Z p → M. Definition 1.1. -We say that f : Z p → M is super-Hölder if there exist constants λ, µ ∈ R such that val M (f (x) − f (y)) p λ · p i + µ whenever val p (x − y) i, for all x, y ∈ Z p and i 0. We let H λ,µ (Z p , M) denote the space of functions such that val M (f (x)−f (y)) p λ ·p i +µ whenever val p (x − y) i, for all x, y ∈ Z p and i 0, and H λ ( Z p , M) = ∪ µ∈R H λ,µ (Z p , M) and H(Z p , M) = ∪ λ∈R H λ (Z p , M). For example, if M = E[[X]] with val M = val X , then [a → (1 + X) a ] ∈ H 0,0 (Z p , M). Indeed, (1 + X) a − (1 + X) a+p i b = (1 + X) a (1 − (1 + X p i ) b ) ∈ X p i E[[X]] if i 0. Remark 1.2. -The space H λ,µ (Z p , M) is closed in C 0 (Z p , M). Remark 1.3. -If α : Z p → Z p is an isometry, then f : Z p → M belongs to H λ,µ (Z p , M) if and only if f • α ∈ H λ,µ (Z p , M) Proposition 1.4. -Suppose that M is a ring, and that val M (mm ′ ) val M (m) + val M (m ′ ) for all m, m ′ ∈ M. If c ∈ R, let M c = M val M c . 1. If f ∈ H λ,µ (Z p , M c ) and g ∈ H λ,ν (Z p , M d ), and ξ = min(µ + d, ν + c), then f g ∈ H λ,ξ (Z p , M c+d ). 2. If λ, µ ∈ R, then H λ,µ (Z p , M 0 ) is a subring of C 0 (Z p , M). 3. If λ ∈ R, then H λ (Z p , M) is a subring of C 0 (Z p , M) 4. If d 1, we see GL d (M) as a subset of the valued E-vector space M d (M). If λ, ν ∈ R and Q ∈ H λ (Z p , GL d (M)) are such that val M (det Q(x)) ν for all x ∈ Z p , then Q −1 ∈ H λ (Z p , GL d (M)). Proof. -Items (2) and (3) follow from item (1), which we now prove. If x, y ∈ Z p , then (f g)(x) − (f g)(y) = (f (x) − f (y))g(x) + (g(x) − g(y))f (y), which implies the claim. We now prove (4). If d = 1, then Q −1 (y) − Q −1 (x) = Q(x) − Q(y) Q(x)Q(y) , which implies the claim. If d 1, we can write Q −1 = t co(Q) · det(Q) −1 , and the claim results from (3), and (4) applied to d = 1. Remark 1.5. -Take u ∈ X + X 2 E[[X]] , and let u •n be u composed with itself n times. Sen's theorem ([Sen69], theorem 1) implies that val X (u •p k (X)−X) p k if k 0, so that val X (u •x − u •y ) p i if val p (x − y) i. This implies that the map Z 0 → X + X 2 E[[X]], given by n → u •n , extends to a super-Hölder function on Z p . Super-Hölder vectors. - We now assume that M is endowed with an E-linear action by isometries of a group Γ, where Γ is isomorphic to Z p , via a coordinate map c. If m ∈ M, let orb m : Γ → M denote the function defined by orb m (a) = a · m, so that orb m •c −1 is a function Z p → M. This definition should be seen as a characteristic p analogue of the locally analytic vectors of a Banach representation of a p-adic Lie group, as defined in §7 of [ST03]. The requirement that Γ acts by isometries is the analogue of the condition that the norm be invariant. Definition 1.6. -Let M Γ-sh,λ,µ denote the set of m ∈ M such that orb m •c −1 ∈ H λ,µ (Z p , Remark 1.7. -We assume that Γ acts by isometries on M, but not that Γ acts con- tinuously on M, namely that Γ × M → M is continuous. However, let M cont denote the set of m ∈ M such that orb m •c −1 : Z p → M is continuous. It is easy to see that M cont is a closed sub-E-vector space of M, and that Γ × M cont → M cont is continuous (compare with §3 of [Eme17]). We then have M sh ⊂ M cont . Lemma 1.8. -We have m ∈ M Γ-sh,λ,µ if and only if val M (g · m − m) p λ · p i + µ for all g ∈ Γ such that c(g) ∈ p i Z p . Proof. -Since Γ acts by isometries, we have val M (hg · m − h · m) = val M (g · m − m) for all g, h ∈ Γ. Lemma 1.9. -The space M Γ-sh,λ,µ is a closed sub-E-vector space of M. Lemma 1.10. -If k 0 and Γ ′ = c −1 (p k Z p ), then g → c(g)/p k is a coordinate on Γ ′ , and M Γ-sh,λ = M Γ ′ -sh,λ+k . Proof. -It is clear that M Γ-sh,λ ⊂ M Γ ′ -sh,λ+k . Conversely, let C = {1, . . . , p k − 1}. If m ∈ M Γ ′ -sh,λ+k,µ , let ν = min c(h)∈C val M (h · m − m). If g ∈ Γ \ Γ ′ , we can write g = g k h with c(h) ∈ C and g k ∈ Γ ′ . We then have g · m − m = (g k · h · m − g k · m) + (g k · m − m) so that val M (g · m − m) min(µ, ν). This implies that m ∈ M Γ-sh,λ,µ ′ with µ ′ = min(µ, ν) − p k+λ . In particular, the space M Γ ′ -sh does not depend on the choice of open subgroup Γ ′ ⊂ Γ, and we denote it by M sh . Proposition 1.11. -Suppose that M is a ring, and that g( mm ′ ) = g(m)g(m ′ ) and val M (mm ′ ) val M (m) + val M (m ′ ) for all m, m ′ ∈ M and g ∈ Γ. 1. If v ∈ R and m, m ′ ∈ M Γ-sh,λ,µ ∩ M val M v , then m · m ′ ∈ M Γ-sh,λ,µ+v ; 2. If m ∈ M Γ-sh,λ,µ ∩ M × , then 1/m ∈ M Γ-sh,λ,µ−2 val M (m) . Proof. -Item (1) follows from prop 1.4 and lemma 1.8. Item (2) follows from We now show that m n (f ) → 0. If s 0, there exists t such that if val p (x − y) t then val M (f (x) − f (y)) s, as f is uniformly continuous. Take n p t and write n = qp t + r with 0 r < p t and q 1. Writing i = a + jp t , we get g 1 m − 1 m = m − g(m) g(m)m n (f ) = p t −1 a=0 q j=0 (−1) n+a+jp t n a + jp t f (a + jp t ). As we are in characteristic p, Lucas' theorem implies that n a+jp t = r a q j , so that: m n (f ) = p t −1 a=0 (−1) n+a r a   q j=0 (−1) j q j f (a + jp t )   . As q j=0 (−1) j q j · f (a) = 0, and val M (f (a + jp t ) − f (a)) s for all j, we get that val M (m n (f )) s if n p t . We now give a characterization of super-Hölder functions in terms of their Mahler expansions. Proposition 1.14. -If f ∈ C 0 (Z p , M), then f ∈ H λ,µ (Z p , M) if and only if for all i 0, we have val M (m n (f )) p λ · p i + µ whenever n p i . Proof. -Take f ∈ C 0 (Z p , M) such that val M (m n (f )) p λ · p i + µ whenever n p i . Recall that if a ∈ Z p and i 1, then for all j < p i we have a j = a+p i j in F p . If z ∈ Z p and i 1, then f (z + p i ) − f (z) = n 0 m n (f ) z + p i n − z n = n p i m n (f ) z + p i n − z n . Since val M (m n (f )) p λ · p i + µ whenever n p i , the formula above implies that val M (f (x + p i ) − f (x)) p λ · p i + µ. Iterating this, we get that val M (f (x + kp i ) − f (x)) p λ · p i + µ for all k ∈ Z 0 . By continuity, this implies that val M (f (y) − f (x)) p λ · p i + µ for all x, y ∈ Z p such that val p (y − x) i. Assume now that f ∈ H λ,µ (Z p , M). We prove that for all i 0 and n p i , we have val M (m n (f )) p λ · p i + µ. Fix i 0 and take a ∈ {0, . . . , p i − 1}. Define a function g on Z p by g(z) = f (a + p i z) − f (a) . By hypothesis, we have val M (g(z)) p λ · p i + µ for all z. This implies that val M (m n (g)) p λ · p i + µ for all n. We now compute m n (g). We have g(z) = Decompletion of cyclotomic perfectoid fields The action of Z × p . -The group Z × p acts continuously by isometries on each E + n by the formula a · X 1/p n = (1 + X 1/p n ) a − 1. This action is compatible when n varies, extends to the fields E n , and extends by continuity to E + and E. Remark 2.1. -If E = F p , then E is the tilt of Q p (µ p ∞ ) (see §3.3 for more details). The group Γ = Gal(Q p (µ p ∞ )/Q p ) is isomorphic to Z × p via the cyclotomic character χ cyc , and acts on E by g(f ) = χ cyc (g) · f . If k 1 (or k 2 if p = 2), let Γ k = 1 + p k Z p . The natural coordinate on Γ k is given by 1 + p k a → log p (1 + p k a)/p k . It differs from the coordinate 1 + p k a → a (which is not a group homomorphism) by an isometry. By remark 1.3, the definition of ( E + ) Γ k -sh,λ,µ does not depend on the choice of one of those coordinates, and we use 1 + p k a → a. Proposition 2.2. -We have E + n = (E + n ) Γ k -sh,k−n,0 . Proof. -We have (1 + X 1/p n ) 1+p k+i b = (1 + X 1/p n ) · (1 + X p k+i−n ) b , so that val X ((1 + X 1/p n ) 1+p k+i b − (1 + X 1/p n )) p k−n · p i . This implies that X 1/p n ∈ (E + n ) Γ k -sh,k−n,0 . The claim now follows from prop 1.11 and lemma 1.9. Proposition 2.4. -If ε > 0, then E[[X]] Γ k -sh,k+ε ⊂ E[[X p ]]. Proof. -Take f (X) ∈ E[[X]]. There is a power series h(Y, Z) ∈ E[[Y, Z]] such that f (Y + Z) = f (Y ) + Z · f ′ (Y ) + Z 2 · h(Y, Z). If m 0, this implies that f ((1 + X) 1+p m − 1) = f (X + X p m (1 + X)) = f (X) + X p m (1 + X) · f ′ (X) + O(X 2p m ). If f (X) / ∈ E[[X p ] ], then f ′ (X) = 0. Let µ = val X (f ′ (X)). The above computations imply that val X ((1 + p i+k ) · f (X) − f (X)) = p i+k + µ for i ≫ 0. This implies the claim. Corollary 2.5. -We have (E + ∞ ) Γ k -sh,k−n = E + n . Proof. -Take f (X 1/p m ) ∈ (E + ∞ ) Γ k -sh,k−n where f (X) ∈ E[[X]]. Since val X (h p ) = p · val X (h) for all h ∈ E + , we have f p m (X) ∈ (E + ∞ ) Γ k -sh,k+m−n , where f p m (X) ∈ E[[X]] is f p m (X) = f (X 1/p m ) p m . If m n + 1, then prop 2.4 implies that f p m (X) ∈ E[[X p ]] , so that f (X) = g(X p ), and f (X 1/p m ) = g(X 1/p m−1 ). This implies the claim. (1 + X) i a i (f ), with a i (f ) ∈ E + 0 , and a i (f ) → 0. Moreover, val X (f ) − 1 < inf i∈I val X (a i (f )) val X (f ). Proof. -See props 4.10 and 8.3 of [Col08]. In particular, for all i ∈ I, the map E + → E + 0 , given by f → a i (f ) is continuous. 1. The restriction of T n to E + n is the identity map. 2. We have T n (f ) → f as n → +∞. 3. We have val X (T n (f )) val X (f ) − 1 for all n. 4. Each T n is Z × p -equivariant. Proof. -If f = i∈I (1 + X) i a i (f ), let T n (f ) = i∈In (1 + X) i a i (f ). With this definition, the first property is immediate. The second and third one follow from prop 2.7. For the last one, observe that if i ∈ I and g ∈ Z × p , then g · (1 + X) i = (1 + X) ig so g ·(1+X) i can be written uniquely as (1+X) σg (i) u i,g (X) with σ g (i) ∈ I and u i,g (X) ∈ E + 0 . The map σ g induces a bijection from I m to itself for all m. Take f ∈ E + , and write f = i∈I (1 + X) i a i (f ). We have g · f = i∈I (1 + X) σg(i) u i,g (X)(g · a i (f )), so that T n (g · f ) = i∈In (1 + X) σg(i) u i,g (X)(g · a i (f )) = g · T n (f ). Decompletion of E. - We now prove that E sh = E ∞ . More precisely, we have the following result. a ∈ Z × p , theorem 6 of [LS07] implies that f ∈ Aut(G m ). Hence there exists b ∈ Z × p such that f (X) = γ b (X). This implies the theorem. 3.2. Decompletion of (ϕ, Γ)-modules. -Let Γ k = 1 + p k Z p with k 1, as in §2.1. Let M be a finite-dimensional E-vector space with a continuous semi-linear action of Γ k . E ⊗ E M) Γ k -sh,k−m = E m ⊗ E M. Proof. -By the same argument as in the proof of lemma 3.5, we see that for m m 0 , ( E ⊗ E M) Γ k -sh,k−m is a sub-E m -vector space of E ⊗ E M. The space ( E ⊗ E M) Γ k -sh,k−m contains M, and therefore also E m ⊗ E M. This proves one inclusion. We now prove that ( E ⊗ E M) Γ k -sh,k−m ⊂ E m ⊗ E M.x = n i=1 x i m i ∈ E ⊗ E M and write g(x) = n i=1 f i (g)m i . We have x ∈ ( E ⊗ E M) Γ k -sh,k−m if and only if f i ∈ H k−m (Γ k , E) for all i. In addition, g(x) = i,j g(x i ) Mat(g) j,i m j . Hence f j : g → n i=1 g(x i ) Mat(g) j,i belongs to H k−m (Γ k , E) for all j. We have g(x ℓ ) = n j=1 f j (g)(Mat(g) −1 ) ℓ,j . By props 3.6 and 1.4, [g → g(x ℓ )] ∈ H k−m (Γ k , E) and therefore x ℓ ∈ E Γ k -sh,k−m = E m for all ℓ. Corollary 3.8. -If M is super-Hölder, then ( E ⊗ E M) sh = E ∞ ⊗ E M. The field E = E((X)) is equipped with its action of Z × p and with the E-linear Frobenius map ϕ given by ϕ(f )(X) = f (X p ). Let Γ = Γ k with k 1. A (ϕ, Γ)-module D over E is a finite-dimensional E-vector space, endowed with commuting, semi-linear actions of ϕ and Γ, such that the action of Γ is continuous and such that Mat(ϕ) is invertible (in any basis of D). d 1 , . . . , X s d n ) for some s 0, and assume that P = Mat(ϕ) ∈ M n (E + ). Take r 1 such that X r P −1 ∈ X M n (E + ). Let G g be the matrix of g ∈ Γ. By continuity of the map Γ → GL n (E + ), g → G g , there exists ℓ k such that for all g ∈ Γ ℓ , we have val X (G g − Id) r. Write G g = Id +X r H g with H g ∈ M n (E + ). The p-adic local Langlands correspondence. - We now prove a result that suggests that the theory of super-Hölder vectors could have some applications to the p-adic local Langlands correspondence. In order to avoid too many technicalities, we consider only the simplest example. Recall that if f ∈ E + , there exist f 0 , . . . , f p−1 ∈ E + such that f = p−1 i=0 ϕ(f i )(1 + X) i . We define ψ(f ) = f 0 . The map ψ : E + → E + has the following properties: ψ(f ϕ(h)) = hψ(f ) if f, h ∈ E + and ψ • g = g • ψ if g ∈ Z × p . Let M = lim ← −ψ E + be the set of sequences m = (m 0 , m 1 , . . .) with m i ∈ E + and ψ(m i+1 ) = m i for all i 0. The space M is endowed with an action of Z × p given by (g · m) i = g · m i and the structure of an E + -module given by (f (X)m) i = ϕ i (f (X))m i . Following Colmez, we could extend these structures to an action of the Borel subgroup B 2 (Q p ) of GL 2 (Q p ) on M, and this idea is an important step in the construction of the p-adic local Langlands correspondence. The representation M is then the dual of most of the restriction to B 2 (Q p ) of a parabolic induction. However, we don't use this here. Let val X be the X-adic valuation on M: val X (m) is the max of the n 0 such that m ∈ X n M. The space M is separated and complete for the X-adic topology, although this is not the natural topology on M (the natural topology is induced by the product topology lim ← −ψ E + ⊂ E + . The action of Z × p on M is not continuous for the X-adic topology: M = M cont in the notation of remark 1.7). We Proof. -Recall that if m ∈ M and f (X) ∈ E, then (f (X)m) j = ϕ j (f (X))m j for all j 0. We have val X (ϕ j (f (X))) = p j val X (f (X)). In particular, if m ∈ M Γ k -sh,k , then m j ∈ (E + ) Γ k -sh,k+j . The results of §2.1 imply that m j ∈ ϕ j (E + ). If m j = ϕ j (f j ), the ψ-compatibility implies that f j = f 0 for all j 0. This implies the claim. A generalization of prop 3.17 to representations of B 2 (Q p ) obtained from (ϕ, Γ)-modules using Colmez' construction shows that using the theory of super-Hölder vectors, we can recover the (ϕ, Γ)-module giving rise to such a representation of B 2 (Q p ). One of the main results of [BV14] is that every infinite dimensional smooth irreducible E-linear representation of B 2 (Q p ) having a central character comes from a (ϕ, Γ)-module by Colmez' construction. Is it possible to reprove this result using super-Hölder vectors? 1. 1 . 1Super-Hölder functions. -We keep the notation of the introduction. Let M be an E-vector space, endowed with a valuation val M such that val M (xm) = val M (m) if M), and let M Γ-sh,λ and M Γ-sh be the corresponding sub-E-vector spaces of M. m . Remark 1.12. -One can extend the definition of super-Hölder vectors to the setting of a p-adic Lie group G acting by isometries on a valued E-vector space M as follows (the details are in our paper Super-Hölder vectors and the field of norms). Let P be a nice enough open pro-p subgroup of G. We say that m ∈ M is super-Hölder if and only if there exists λ, µ ∈ R and e > 0 such that val M (g · m − m) p λ+ei + µ whenever g ∈ P p i , for all i 0. Juan Esteban Rodríguez Camargo pointed out to us that there is a similar purely metric characterization of locally analytic vectors for a p-adic Lie group acting on a Banach space. 1.3. Mahler's theorem. -In this section, we prove a characteristic p analogue of Mahler's theorem for continuous functions Z p → Q p . We then give a characterization of super-Hölder functions in terms of their Mahler expansions. If z ∈ Z p and n 0, then z n ∈ Z p and we still denote by z n its image in F p . Theorem 1.13. -If {m n } n 0 is a sequence of M such that m n → 0, the function f : Z p → M given by f (z) = n 0 z n m n belongs to C 0 (Z p , M). We have m n = (−1) n n i=0 (−1) i n i f (i) and inf z∈Zp val M (f (z)) = inf n 0 val M (m n ). Conversely, if f ∈ C 0 (Z p , M), there exists a unique sequence {m n (f )} n 0 such that m n (f ) → 0 and such that f (z) = n 0 z n m n (f ). Proof. -Our proof follows Bojanic's proof (cf [Boj74]) of Mahler's theorem. The first part of the theorem is easy: f is continuous since it is a uniform limit of continuous functions, and if f (z) = n 0 z n m n , then val M (f (z)) inf n 0 val M (m n ). The fact that m n = (−1) n n i=0 (−1) i n i f (i) is a classical exercise, given that f (k) = k j=0 k j m j for all k 0, and it implies that val M (m n ) inf z∈Zp val M (f (z)) for all n. In order to show the converse, it is enough to show that if f is continuous and m n (f ) = (−1) n n i=0 (−1) i n i f (i), then m n (f ) → 0. Indeed, the functions f and z → n 0 z n m n (f ) are then two continuous functions on Z p with the same values on Z 0 , so that they are equal. a + p i z n − a n m n (f ) = n p i a + p i z n m n (f ), since a p i − 1. If we write n = t + p i ℓ, t+p i ℓ (f ), which gives m ℓ (g) = p i −1 t=0 a t m t+p i ℓ (f ) for all ℓ 1. λ · p i + µ for all ℓ 1 and a ∈ {0, . . . , p i − 1}. The matrix a t 0 a,t p i −1is unipotent with integral coefficients. Hence for a given ℓ 1, the above inequality implies that val M (m a+p i ℓ (f ))p λ · p i + µ for all a ∈ {0, . . . , p i − 1}. Writing n p i as n = a + p i ℓ, we get val M (m n (f )) p λ · p i + µ for all n p i . Remark 1.15. -Let W λ,µ (Z p , M) denote the set of f ∈ C 0 (Z p , M) such that val M (m n (f )) p λ n + µ for all n 0. Prop 1.14 implies that W λ,µ (Z p , M) ⊂ H λ,µ (Z p , M) ⊂ W λ−1,µ (Z p , M).Prop 1.14 and remark 1.15 strengthen the analogy between our definition of super-Hölder functions and the classical theory of locally analytic functions. Indeed, if f : Z p → Q p is a continuous function, and if f (z) = n 0 z n m n (f ) is its Mahler expansion, then by a result of Amice ([Ami64], see corollary I.4.8 of [Col10]), f is locally analytic if and only if there exists λ, µ ∈ R such that val p (m n (f )) p λ · n + µ for all n 0. Remark 1.16. -Daniel Gulotta pointed out to us that Gulotta (in §3 of [Gul19]), as well as Johansson and Newton (in §3.2 [JN19]), had defined a generalization of locally analytic functions, for functions valued in certain general Tate Z p -algebra. When p = 0 in the algebra, their definition is equivalent to our definition of super-Hölder functions. Taking n = 0 in prop 2.2, we find that E[[X]] = E[[X]] Γ k -sh,k . Let E = E + [1/X].Corollary 2.3. -We have E = E Γ k -sh,k . Proof. -This follows from prop 2.2 and prop 1.11. Proposition 2. 8 . 8-There exists a family {T n } n 0 of continuous maps T n : E + → E + n satisfying the following properties: Theorem 2.9. -We have E Γ k -sh,k−m = E m for all m 0, and E sh = E ∞ .Proposition 2.10. -If f ∈ ( E + ) Γ k -sh,λ,µ , then T n (f ) ∈ (E + n ) Γ k -sh,λ,µ−1 . Proposition 3. 4 . 4-There is an E + -lattice in M that is stable under Γ k . Proof. -Choose any lattice M + 0 of M. The map π : Γ k ×M → M is continuous, so there is an open subgroup H of Γ k and an n 0 such that π −1 (M + 0 ) contains H × X n M + 0 . In particular, h(m) ∈ X −n M + 0 for all h ∈ H and m ∈ M + 0 . Since H is open in the compact group Γ k , it is of finite index, and there exists d n such that g(m) ⊂ X −d M + 0 for all g ∈ Γ k and m ∈ M + 0 . The space M+ = g∈Γ k g(M + 0 ) is an E + -module such that M + 0 ⊂ M + ⊂ X −d M + 0 , so that M + is a lattice of M. It is clearly stable under Γ k .Choosing such an E + -lattice in M defines a valuation val M on M, such that Γ k acts on M by isometries. We make such a choice, and we can therefore define M sh and M Γ k -sh,λ as in definition 1.6. We say that the action of Γ k on M is super-Hölder if M = M sh . Lemma 3.5. -The space M Γ k -sh,λ does not depend on the choice of Γ k -stable lattice of M. If λ k then M Γ k -sh,λ is sub-E-vector space of M.Proof. -The first assertion results from the fact that if we choose two E + -lattices M + 1 and M + 2 in M, then there exists a constant C such that |val 1 − val 2 | C. Next, recall that by coro 2.3, E = E Γ k -sh,k . If m ∈ M sh,λ , f ∈ E, and g ∈ Γ k , theng(f m) − f m = g(f )(g(m) − m) + (g(f ) − f )m,so that f m ∈ M sh,λ by lemma 1.8. Lemma 3.5 implies that M sh is a sub-E-vector space of M. We say that a basis of M is good if it generates a lattice that is stable under Γ k . Proposition 3.6. -Take λ k and fix a good basis of M. We have M = M Γ k -sh,λ if and only if the map Γ k → M n (E + ), given by g → Mat(g), is in H λ (Γ k , M n (E + )). Proof. -We fix a good basis (m 1 , . . . , m n ) of M, and work with the corresponding valuation val M on M. By lemma 3.5, we have M = M Γ k -sh,λ if and only if m j ∈ M Γ k -sh,λ for all j. We have g · m j = n i=1 Mat(g) i,j m i by definition of Mat(g). Hence if g, h ∈ Γ k , then g ·m j −h·m j = n i=1 (Mat(g) i,j −Mat(h) i,j )m i . This implies that if ℓ 0 and µ ∈ R, then val M (g · m j − h · m j ) p λ+ℓ + µ if and only if val X (Mat(g) − Mat(h)) p λ+ℓ + µ. This implies the claim. If M is a finite-dimensional E-vector space with a semi-linear action of Γ k , then E⊗ E M is a finite-dimensional E-vector space with a semi-linear action of Γ k . If M is super-Hölder, there exists m 0 = m 0 (M) 0 such that M = M Γ k -sh,k−m 0 Proposition 3.7. -If M is super-Hölder and m m 0 (M), then ( Fix a good basis (m 1 , . . . , m n ) of M, the corresponding valuation val M on E ⊗ E M, and m m 0 . Take Proposition 3.9. -If D is a (ϕ, Γ)-module over E, then D = D Γ k -sh,k . Lemma 3.10. -If ℓ 1 and λ, µ ∈ R, then H λ,µ (Γ ℓ , M n (E + )) is a ring, that is stable under ϕ.Proof. -The first claim follows from prop 1.4. The second one follows from the fact that if M ∈ M n (E + ), then val X (ϕ(M)) val X (M).Proof of of proposition 3.9. -Choose a good basis (d 1 , . . . , d n ) of D. We can replace (d 1 , . . . , d n ) by (X s have an injection i : E + → M, given by i(f ) = (f, ϕ(f ), ϕ 2 (f ), . . .). Proposition 3.17. -We have M Γ k -sh,k = i(E + ). The main ingredients of the proof of this theorem are some simple computations inE[[X]], as well as Colmez' analogue of Tate traces for E.We give several applications of our main result. First, we compute the perfectoid Let E + = E[[X]]. For n 0, let E + n = E[[X 1/p n ]], so that E + = E + 0 . Let E + ∞ = ∪ n 0 E + and let E + be the X-adic completion of E + ∞ . We denote by E, E n , E ∞ , E the fields E + [1/X], E + n [1/X], E + ∞ [1/X], E + [1/X] respectively. The ring E + is the ring of integers of the field E = E + [1/X]. If E is perfect, then E is perfectoid.n Acknowledgements. We thank Juan Esteban Rodríguez Camargo for asking LB the question that motivated this paper, as well as Christophe Breuil, Daniel Gulotta, Gal Porat and the referee for their comments and questions.Proof. -If g ∈ Γ k , then g(T n (f )) − T n (f ) = T n (g(f ) − f ) so that val X (g(T n (f )) − T n (f )) = val X (T n (g(f ) − f )) val X (g(f ) − f ) − 1 by prop 2.8. This implies the claim.and this implies the theorem by prop 1.11.ApplicationsWe now give several applications of the fact that E sh = E ∞ .The perfectoid commutant ofRecall that a power serieswhich is equivalent to f being invertible for composition (denoted by •). We say that w(X) ∈ X · F p [[X]] is nontorsion if w •n (X) = X for all n 1. The following is a reformulation of lemma 6.2 of[Lub94].] be an invertible nontorsion series, and let f (X) ∈ X · F p [[X]] be a separable power series.Proof. -The group Z × p acts on E + by a·u = u•γ a , so we need to check that the function a → γ a • u is super-Hölder. This is clear since γ a (u) = n 1 a n u n and val X (u) > 0. Proof of theorem 3.1. -Take u ∈ E + such that val X (u) > 0 and u • γ a = γ a • u for all a ∈ Z × p . By lemma 3.3 and theorem 2.9, there exists m 0 such that u ∈ E + m . Hence there is an n ∈ Z such that f (X) = u(X 1/p n ) belongs to X · F p [[X]] and is separable.Take g ∈ 1 + pZ p such that g is nontorsion, and let w(X) = γ g (X) so that u • w = w • u.We also have f • w = w • f . By lemma 3.2, f is invertible. Since f • γ a = γ a • f for all Interpolation p-adique. Y Amice, Bull. Soc. Math. France. 92Y. Amice -"Interpolation p-adique", Bull. Soc. Math. France 92 (1964), p. 117-180. Théorie de Sen et vecteurs localement analytiques. L Berger, &amp; P Colmez, Ann. Sci. Éc. Norm. Supér. 4L. Berger & P. Colmez -"Théorie de Sen et vecteurs localement analytiques", Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 4, p. 947-970. A simple proof of Mahler's theorem on approximation of continuous functions of a p-adic variable by polynomials. R Bojanic, J. Number Theory. 6R. Bojanic -"A simple proof of Mahler's theorem on approximation of continuous functions of a p-adic variable by polynomials", J. Number Theory 6 (1974), p. 412-415. Irreducible modular representations of the Borel subgroup of GL 2 (Q p ). L Berger, &amp; M Vienney, Automorphic forms and Galois representations. Cambridge Univ. Press1L. Berger & M. Vienney -"Irreducible modular representations of the Borel sub- group of GL 2 (Q p )", in Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, 2014, p. 32-51. Espaces vectoriels de dimension finie et représentations de de Rham. P Colmez, Astérisque. 319P. Colmez -"Espaces vectoriels de dimension finie et représentations de de Rham", Astérisque (2008), no. 319, p. 117-186. Fonctions d'une variable p-adique. P Colmez, Astérisque. 330P. Colmez -"Fonctions d'une variable p-adique", Astérisque (2010), no. 330, p. 13-59. Locally analytic vectors in representations of locally p-adic analytic groups. M Emerton, Mem. Amer. Math. Soc. 2481175158M. Emerton -"Locally analytic vectors in representations of locally p-adic analytic groups", Mem. Amer. Math. Soc. 248 (2017), no. 1175, p. iv+158. Equidimensional adic eigenvarieties for groups with discrete series. D R Gulotta, Algebra Number Theory. 138D. R. Gulotta -"Equidimensional adic eigenvarieties for groups with discrete series", Algebra Number Theory 13 (2019), no. 8, p. 1907-1940. Extended eigenvarieties for overconvergent cohomology. C Johansson, &amp; J Newton, Algebra Number Theory. 131C. Johansson & J. Newton -"Extended eigenvarieties for overconvergent cohomol- ogy", Algebra Number Theory 13 (2019), no. 1, p. 93-158. Extrinsic properties of automorphism groups of formal groups. J Lubin, &amp; G Sarkis, J. Algebra. 3152J. Lubin & G. Sarkis -"Extrinsic properties of automorphism groups of formal groups", J. Algebra 315 (2007), no. 2, p. 874-884. Nonarchimedean dynamical systems. J Lubin, Compositio Math. 943J. Lubin -"Nonarchimedean dynamical systems", Compositio Math. 94 (1994), no. 3, p. 321-346. Perfectoid spaces. P Scholze, Publ. Math. Inst. Hautes Études Sci. 116P. Scholze -"Perfectoid spaces", Publ. Math. Inst. Hautes Études Sci. 116 (2012), p. 245-313. On automorphisms of local fields. S Sen, Ann. of Math. 2S. Sen -"On automorphisms of local fields", Ann. of Math. (2) 90 (1969), p. 33-46. Algebras of p-adic distributions and admissible representations. P Schneider, &amp; J Teitelbaum, Invent. Math. 1531P. Schneider & J. Teitelbaum -"Algebras of p-adic distributions and admissible representations", Invent. Math. 153 (2003), no. 1, p. 145-196. Le corps des normes de certaines extensions infinies de corps locaux; applications. J.-P Wintenberger, Ann. Sci. École Norm. Sup. 4J.-P. Wintenberger -"Le corps des normes de certaines extensions infinies de corps locaux; applications", Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 1, p. 59-89. UMR 5669 du CNRS E-mail : [email protected] • Url : perso.ens-lyon.fr/laurent.berger/ Sandra Rozensztajn, UMPA de l'ÉNS de Lyon, UMR 5669 du CNRS E-mail : [email protected] Url : perso. Laurent Berger, &apos; Éns De Lyon, Laurent Berger, UMPA de l'ÉNS de Lyon, UMR 5669 du CNRS E-mail : [email protected] • Url : perso.ens-lyon.fr/laurent.berger/ Sandra Rozensztajn, UMPA de l'ÉNS de Lyon, UMR 5669 du CNRS E-mail : [email protected] Url : perso.ens-lyon.fr/sandra.rozensztajn/
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[ "Time Associated Meta Learning for Clinical Prediction", "Time Associated Meta Learning for Clinical Prediction" ]
[ "Hao Liu [email protected] \nWashington University in St. Louis\n\n", "Muhan Zhang \nPeking University\n\n", "Zehao Dong [email protected] \nWashington University in St. Louis\n\n", "Lecheng Kong \nWashington University in St. Louis\n\n", "Yixin Chen ", "Bradley Fritz [email protected] \nWashington University in St. Louis\n\n", "Dacheng Tao [email protected] \nWashington University in St. Louis\n\n\nJD Explore Academy\n\n", "Christopher King [email protected] \nWashington University in St. Louis\n\n" ]
[ "Washington University in St. Louis\n", "Peking University\n", "Washington University in St. Louis\n", "Washington University in St. Louis\n", "Washington University in St. Louis\n", "Washington University in St. Louis\n", "JD Explore Academy\n", "Washington University in St. Louis\n" ]
[]
Rich Electronic Health Records (EHR), have created opportunities to improve clinical processes using machine learning methods. Prediction of the same patient events at different time horizons can have very different applications and interpretations; however, limited number of events in each potential time window hurts the effectiveness of conventional machine learning algorithms. We propose a novel time associated meta learning (TAML) method to make effective predictions at multiple future time points. We view time-associated disease prediction as classification tasks at multiple time points. Such closely-related classification tasks are an excellent candidate for model-based meta learning. To address the sparsity problem after task splitting, TAML employs a temporal information sharing strategy to augment the number of positive samples and include the prediction of related phenotypes or events in the meta-training phase. We demonstrate the effectiveness of TAML on multiple clinical datasets, where it consistently outperforms a range of strong baselines. We also develop a MetaEHR package for implementing both timeassociated and time-independent few-shot prediction on EHR data.
10.48550/arxiv.2303.02570
[ "https://export.arxiv.org/pdf/2303.02570v1.pdf" ]
257,365,113
2303.02570
b83fd54b6ddc578307aed524ff89bf3e20bcb628
Time Associated Meta Learning for Clinical Prediction Hao Liu [email protected] Washington University in St. Louis Muhan Zhang Peking University Zehao Dong [email protected] Washington University in St. Louis Lecheng Kong Washington University in St. Louis Yixin Chen Bradley Fritz [email protected] Washington University in St. Louis Dacheng Tao [email protected] Washington University in St. Louis JD Explore Academy Christopher King [email protected] Washington University in St. Louis Time Associated Meta Learning for Clinical Prediction Rich Electronic Health Records (EHR), have created opportunities to improve clinical processes using machine learning methods. Prediction of the same patient events at different time horizons can have very different applications and interpretations; however, limited number of events in each potential time window hurts the effectiveness of conventional machine learning algorithms. We propose a novel time associated meta learning (TAML) method to make effective predictions at multiple future time points. We view time-associated disease prediction as classification tasks at multiple time points. Such closely-related classification tasks are an excellent candidate for model-based meta learning. To address the sparsity problem after task splitting, TAML employs a temporal information sharing strategy to augment the number of positive samples and include the prediction of related phenotypes or events in the meta-training phase. We demonstrate the effectiveness of TAML on multiple clinical datasets, where it consistently outperforms a range of strong baselines. We also develop a MetaEHR package for implementing both timeassociated and time-independent few-shot prediction on EHR data. Introduction The use of rich Electronic Health Record (EHR) data has led to an explosion of clinical prediction for a wide variety of applications [Shickel et al., 2018]. EHR data contains extremely detailed information about patients, such as medical history, vital signs, surgical events, and demographic information [Birkhead et al., 2015]. Machine learning (ML) models have been proposed for many clinical outcomes, such as mortality [Lagu et al., 2016], chronic disease [Liu et al., 2018], heart failure [Panahiazar et al., 2015;Jin et al., 2018], and readmission [Rojas et al., 2018]. Because of certain characteristics of EHRs including heterogeneity and noise [Birkhead et al., 2015], the use of EHR data in healthcare predictive models can present some unique challenges. One challenging aspect of EHR data is the limited number of available annotated examples. Even though a huge number of records may be available, many important adverse events are uncommon, and as a result, training of ML models such as neural networks can be fragile and prone to overfitting. Label sparsity is aggravated by the many complex and high-dimensional features present in EHRs. In this paper, we address a very common few-shot problem in clinical prediction: prediction of future events in specific time-ranges (time-associated target prediction). Specifically, a clinical event (such as deterioration or improvement) can occur in different time periods, and the likely lead-time until these events greatly affects the interpretation and is vital for users. For example, predicted clinical deterioration could trigger immediate changes in therapy or consumption of scarce resources (such as ICU space) or watchful waiting and more tests depending on if the event is likely in the near or far term. In our example of postoperative survival time prediction, death soon after surgery has very different management implications from death in the subsequent weeks or months. Other events (such as discharge from hospital) are inevitable, and the appropriate planning and patient preparation steps depend on the likely lead-time. Time-associated target prediction can be approached using explicit time-to-event models [Wang et al., 2015]. However, directly learning to predict the time to an event is limited by (1) many patient records being censored by loss to follow up before the event, (2) a limited ability to tune the algorithm to focus on "more relevant" time windows or clinically equivalent time windows, (3) parametric assumptions on the distribution of event times built into loss functions, (4) difficulty interpreting the expected event time in the context of multimodal outcome distributions. Others have used multi-task learning to predict events at different times as joint tasks Suresh et al., 2018]. However, traditional multitask learning [Suresh et al., 2018] requires identical low-level representations of data across tasks, which may be too strong an assumption. To bridge the gap in learning ability between people and AI for small samples, a new paradigm was proposed named few-shot learning [Wang et al., 2020]. Meta-Learning has been shown to be an effective method to deal with few-shot problems, most of which rely on learning initialized parameters [Finn et al., 2017] and a metric space [Snell et al., 2017] across tasks. Meta learning assumes that an internal data representation is transferable between tasks [Suo et al., 2020], and as discussed in [Jose and Simeone, 2021], existing meta learning models tend to work better when meta-training tasks are similar to the target. However, in predicting the occurrence time of a clinical event under a regression framework, few tasks with similar distributions are usually available to directly apply meta learning. To address the above challenges, we propose Time Associated Meta Learning (TAML), a novel adaptation of modelagnostic meta-learning (MAML) [Finn et al., 2017] for temporal problems. We assume that the ultimate goal is to predict the occurrence of a target event in a specific time period. We split a time-predictive problem into multiple binary classification problems corresponding to events in different time periods, thus generating many similar tasks. To the best of our knowledge, this is the first study to transform the time-associated few-shot problem into a meta learning problem where we not only split the problem into multiple similar tasks, which are called time-associated tasks, but also include more time-independent tasks as meta learning tasks, which are called reference tasks. Considering the effect of task similarity on prediction performance, we distinguish these two kinds of tasks in the training process and apply different weighting strategies to emphasize the importance of time-associated tasks. This allows reference tasks to provide directions during training without overwhelming the time-associated tasks. Our approach keeps model weights for each task within a tunable distance of each other, encouraging joint learning but allowing differences, conceptually similar to fine-tuning each task during training, and weakening the necessary assumptions compared to multi-task learning. Such a setup results in fewer positive samples for each classification task. To overcome the drawback of sparsity and to incorporate inter-task timing information into this classification problem, we develop a novel task partitioning strategy named the Temporal Information Sharing Strategy (TISS) to augment the positive samples by exploiting the temporal persistence of the patient-state. We propose four situations for a patient-state in one time period, where we label the existence of patient-state in the current time period as a positive situation during training, thus augmenting positive labels. TISS is also refined to the different treatment of tasks belonging to different situations with the same label. Our main contributions are: 1) We are the first study to transform time-associated regression problems to a MAML setting with joint time-independent tasks. 2) We propose TISS to exploit temporal dependence of health states and augment positive labels in training. 3) We provide a general TAML framework with sufficient details to allow any timeassociated clinical prediction task to benefit from it. We show results on two public datasets and a real application on a local dataset. We use two clinical events as labels: mortality and intensive care unit discharge, where TAML outperforms stateof-art baselines. The TAML approach turns out to be remarkably insensitive to tuning parameters that are critical in other approaches; dividing time into very fine windows is harmless, and adding tasks of unclear relevance to the ensemble has very little risk. 4) We provide a package for few-shot clinical prediction named MetaEHR, which includes both the implementation of TAML for time-associated targets and other meta learning algorithms for time-independent targets. Related Works In contrast to traditional supervised machine learning algorithms, which learn a model for each label, meta-learning is a strategy of learning to learn. The goal is to train a model on a variety of tasks and use this experience to improve future learning performance. Because of the strong inductive bias created, a meta-trained algorithm can be applied to solve few-shot problems. Metric-based meta learning and optimization-based meta learning are the two main categories of meta learning. Metricbased meta learning aims to learn a metric or distance that compares training data with testing data. For example, [Koch et al., 2015] proposed a method to use a Siamese Neural Network for one-shot image classification. The Siamese Neural Network is composed of two twin networks, the outputs of which are combined with a function for learning the relationship between pairs of input data examples. Prototypical Networks [Snell et al., 2017] encode each input into a continuous latent space and carry out classification using the similarity of an example to the representation of latent classes. Optimization-based meta learning is essentially learning a good initialization of a neural network from which finetuning on a small number of additional training examples can be effective. MAML [Finn et al., 2017] is a widely used optimization-based meta-learning technique. Briefly, MAML trains in a nested loop. At each outer loop iteration, a set of tasks is sampled. In the inner loop, the current set of global parameters is updated with one or a few gradient steps independently for each task. The global parameters are then updated using a loss function which sums over all tasks including the adaptation computed in the inner loop. The global parameters therefore evolve to a point from which an acceptable model for any of the training tasks can be reached with a small step. This is made feasible by similar low-level representations and latent structures between tasks, which is also an assumption of the method. Since this method can be applied to diverse classes of models, it has been used in image classification [Raghu et al., 2019], reinforcement learning [Liu et al., 2019] and a variety of other domains. Multi-task learning addresses a similar need. By simultaneously training a base model and multiple "heads" corresponding to outcomes, common labels allow an algorithm to learn an effective data-representation and a relatively lowcomplexity head for rare outcomes. For example, [Liu et al., 2020] Time Associated Meta Learning (TAML) In this section, we describe the TAML framework. We introduce the meta-learning problem setup and the meta-train and meta-test phases, and explain the Temporal Information Sharing Strategy (TISS). Meta-Learning Problem Setup The purpose of TAML is to predict the occurrence of patient events during potentially multiple time periods of interest. We use survival time after surgery as an example. Since death shortly after most kinds of surgery is uncommon, it is hard to train an accurate prediction model, and our goal is to use adjacent time periods to improve model performance. However, viewing this few-shot problem from a regression perspective, i.e., calculating the expected event time, would leave the meta learning without enough relevant tasks. A limited number of patient events are time-associated and fully recorded, and MAML benefits from more closely related training tasks [Jose and Simeone, 2021]. We therefore generate new classification tasks by grouping patient events according to time of occurrence, where each task is a binary prediction of the occurrence of that event during a certain time period. This way, we effectively increase the number of available tasks. These time-associated tasks are highly related to each other, which is the setting in which meta learning has been successful. Additionally, since the problem has been switched into a binary classification form, other time-independent diseases or events related to the target with a larger number of positive labels can be added to the task space. We will refer to these additional tasks as reference tasks. The available tasks are depicted in Figure 1. In a fashion similar to MAML [Finn and Levine, 2017], in TAML we train a meta model which can quickly adapt to tasks that follow some distribution p(T ). The model is denoted by f with parameters θ, and the model f maps an input data x into a binary value y. There are two kinds of tasks involved in the TAML framework, time-associated tasks {T S } and reference tasks {T R }. Both groups of tasks will be used in our framework, but the time-associated tasks are what we really want to predict. The time-associated tasks, reference tasks, meta-train phase, and meta-test phase that will be used in our framework are defined below. Definition 1. T 0 is the true target patient event. We assume each example of T 0 is associated with a timestamp. We divide the given time span into J non-overlapping time periods. {T S j } (j = 1, 2, . . . , J) is the set of time-associated Sample data points D k and D k from T k 6: Compute Loss function with D k and θ 7: Use gradient descent to calculate θ k with α k 8: θ k = θ − α k θ L(D k , θ) 9: end for 10: Calculate weighted loss functions L k for each task {T k } using D k and θ k 11: L k (D k , θ k ) = w k L(D k , θ k ) 12: Update θ = θ − β θ k L k (D k , θ k ) 13: end while tasks, which includes the occurrence of T 0 in different time periods. For example, T S j describes whether T 0 occurs in the j th time period. {T R i } denotes the set of reference tasks, which includes other patient events potentially related to T 0 . We assume reference tasks do not contain timestamps. The framework consists of meta-train and meta-test phases. In the meta-train phase, parameters θ of a global-task model are trained with time-associated tasks and reference tasks. In the meta-test phase, we select one of the time-associated tasks as a target and fine-tune the meta-learner, evaluating the performance on a held-out test set. The Proposed Framework TAML training is very similar to MAML [Finn et al., 2017]. The parameters of the meta-model θ * are defined as θ * = argmin θ E T k ∼p(T ) L k (D k , θ k (θ)),(1) where tasks T k follow a distribution p(T ), and consist of both time-associated tasks T S j and reference tasks T R i . D k represents data points sampled from task T k . L k is a loss function for task T k over data points D k and parameters θ k , which may take various forms depending on the type of problem we focus on. θ k is the parameters adapted to task T k in the innerlevel loop using initialized parameters θ. The intuition is that we aim to find the meta parameters θ * that have a small loss on every task T k after adapting to that task. This requires θ * to absorb information from all the tasks and be adaptable to them. Algorithm 1 includes the architecture of the meta-train phase of TAML framework, which is composed of inner-level and outer-level loop updates. In each iteration of the outer-level loop, given a θ which is randomly initialized at the first time and later provided by the last iteration, we first uniformly sample a batch of tasks {T k } including both time-associated tasks {T S k } and reference tasks {T R k } from the task distribution p(T ). Then, in the inner-level loop, for each task, we sample two sets of data points, D k and D k , which form the support set and query set. D k is used in the inner-level loop update to obtain the adapted parameters θ k , where TISS labeling method is applied to D k . D k is used in the outer-level loop update to compute overall training objective for the update of global parameters θ. Formally, the inner-level loop update is given by, θ k = θ − α k ∇ θ L(D k , θ),(2) where α k is a task-specific update step size. The value of α k depends on the task category. After the inner loop, θ k is obtained for each task T k , which simulates a fine-tuning process to adapt the meta-learned θ to each specific task. Since the reference tasks and the time-associated tasks have different relationships to the true target, we give them different step sizes α k to reflect their relative importance. Considering the true target is time-associated, parameters of each time-associated task should provide more information than reference tasks, thus larger step size α k is assigned to time-associated tasks. The outer-level update optimize θ with the sum of weighted gradients on D k with respect to θ k . Similarly, we place a greater emphasis on the loss functions on k in T S than T R when updating θ. We use the following equation to calculate a loss function that is advantageous for updating parameters θ to a direction closer to that of time-associated tasks. L k (D k , θ k ) = w k L(D k , θ k ).(3) The selected weight w k is a hyper-parameter determined by both the task type and the time period. Specifically, w k is constant for all tasks in {T R }, but different (and larger) for tasks in {T S }, similar to when selecting the hyper-parameters α k . Then the sum of loss functions L k will be used to update the parameter θ by equation 4. θ = θ − β∇ θ k L k (D k , θ k ),(4) where β is the outer-level loop learning rate. Note that the derivative with respect to θ propagates through θ k , requiring an additional back-propagation through the computation graph for calculating θ k , which is supported by standard deep learning libraries. After obtaining a good initialization of parameters from the meta-train phase, the goal of the meta-test phase is to finetune a model that can predict the occurrence of the true target patient event in a specific time period. In the meta-test phase, the target event that occur in a certain time period is collected to form the test task. We adapt the initialized parameters learned from the meta-train phase to the test task. Temporal Information Sharing Strategy We previously described how to transform time-associated regression predictions into classification predictions suitable for MAML. In this new setting, a few time-associated tasks and reference tasks are available for training. However, these time-associated tasks essentially deal with a classification problem with extremely limited positive labels due to the time-based division strategy in Def 1. Hence, the problem inherently poses the challenge of label imbalance. To address the positive label sparsity issue, we propose a strategy to augment the number of positive labels, Temporal Information Sharing Strategy (TISS), which incorporates the temporal persistence of the patient event, thus taking the impact of other time periods on the target time period into consideration. Basically, TISS is a situation-based label setting strategy for time-associated tasks {T S j }(j=1,2,. . . , J). There are in total four situations discussed, and Figure 2 illustrates these situations with corresponding binary labels. Among the four situations, S 1 represents that the patient event precisely occurs during the target time period. S 2 indicates that the patient event occurred prior to the target time period and lasted at least until the target time period. S 3 means the patient event never occurs during the entire observation. S 4 which has two possible sub-situations, denotes the situation that the patient event begins after the target time period or ends before the target time period. Furthermore, we divide the four situations into two categories: the first category is referred to as the absolute real situations, including S 1 and S 3 ; the second category is referred to as the auxiliary situations, including S 2 and S 4 . In TISS, in addition to the label flipping of S 2 , we emphasize the importance of absolute real situations by assigning higher weights to the examples of absolute true situations. We refer to the weight ratio of the first category of situations over the second category as the augmentation ratio, where a larger augmentation ratio means the algorithm focuses more on the examples of absolute real situations, and vice versa. Some patient events (like ICU admission) have real associated durations. However, events like death never "resolve", and other patient states have unclear or unrecorded actual durations. We can adopt the same computational strategy with these kinds of outcomes by assigning them a pseudoduration; effectively, smoothing events from one time period into adjacent ones. For S 2 , while giving it a positive label, we limit its ability to influence the model. For S 4 , although we retain its negative label, the fact that the patient event has occurred in other time periods also reflects the patient's health status, so the two negative label situations, S 3 and S 4 , are distinguished. With durations set to a large number, the method reduces to estimating the cumulative event rate at each time. The TISS strategy will be applied to the training process in both the meta-train (inner-loop) and meta-test (fine-tuning) phases, and the original occurrence labeling method will be used in the outer-level loop of the meta-train phase and evaluation of the meta-test phase. Experimental Results We first introduce the Python package for few-shot clinical prediction named MetaEHR. Then, we conduct experiments to predict two clinical events: time to death (survival time) and the time to discharge of intensive care unit (ICU) patients (length of stay = LOS). We compare the performance to state-of-the-art baselines, conduct ablation studies, and explore sensitivity to tuning parameters. MetaEHR: a Python Package for Few-shot Clinical Prediction We develop an easy-to-use Python package: MetaEHR, for few-shot clinical classification prediction. MetaEHR provides both TAML implementation on time-associated targets and several meta learning algorithm implementations on time-independent targets. The implementation of metalearning on EHR data differs from other domains in that each task comes from a separate outcome where all the outcomes share common patients. Please find it in https://github.com/ Haoliu-cola/Meta-EHR. MIMIC III is a publicly accessible single-center critical care ICU database from the Beth Israel Deaconess Medical Center (BIDMC) [Johnson et al., 2016]. It includes information on 46,521 patients admitted to ICU from 2001 to 2012. We extract both time-dependent statistical features and discrete features. MIMIC IV [Johnson et al., 2022] is a partially overlapping dataset with MIMIC III; it excludes early years of MIMIC III (before a transition in the EHR) and includes several years after MIMIC III closed. The eICU Collaborative dataset [Pollard et al., 2019; contains EHR abstracts for more than 200,000 patients from 2014 and 2015. Labels and Dataset Description In ICU LOS experiment, a set of overlapping features are selected between MIMIC IV, eICU, and local hospital datasets, including demographic and statistical features of vital signs. The available number of EHRs is 34,925, 73,908, and 25,591 respectively. Data Processing and Implementation To evaluate our model on multiple time horizons, we selected different time lengths on local hospital and MIMIC III datasets as survival time prediction targets. In the dataset from local hospital, mortality in the first 90 days is most clearly related to immediate postoperative care. Mortality is divided into four groups: 0-6 days, 7-18 days, 19-30 days, and 31-90 days. For ICU datasets, death in the following day highlights potential immediately needs or patient deterioration. We therefore divide the MIMIC III mortality events into 0-6 hours, 6-12 hours and 12-24 hours after ICU admission as time-associated tasks. For ICU LOS, our applied question revolves around resource utilization and potentially avoidable ICU admissions, so we divide discharge times into 0-1 days, 1-2 days, and 2-4 days with 0-1 days as the main target. We include patients who die before discharge in the "long stay" category. For the purpose of selecting reference tasks, we use correlation analysis to identify several outcomes associated with the target. For each above datasets, models are trained using 70% vs. 30% train-test split. Networks used in our evaluation are four-layer fully-connected neural networks. The experiments are repeated five times with this division ratio, and the average performance is reported. 5-fold cross validation is used to fine-tune the network configuration and hyperparameters including augmentation ratio, weight ratio (w k of time-associated tasks over w k of reference tasks) and α k . Adam [Kingma and Ba, 2014] is the optimizer for inner and outer level updates. For three sets of experiments: survival time on local hospital, survival time on MIMIC III, and ICU LOS, we choose 1e-3, 3e-4 and 3e-4 as the corresponding step size; 1.5, 1.2, 1.4 as the augmentation ratio; 1.3, 1.4, 1.4 as the weight ratio. Performance Evaluation We present AUROC and average recall as evaluation metrics. Baselines include a Deep Neural network (DNN), DNNbased semi-parametric survival analysis model (Survival), Multi-task Learning network (MTL), Pre-trained Model (Pretrain), Model agnostic meta learning (MAML), and a prototypical network (ProtoNet). The description of baselines is provided in the appendix. The performance comparison is shown in Table 1, 2, and 3, where TAML consistently outperforms all the other models. The deep neural network and survival analysis, which are trained separately for each time slot, has no advantages compared to other baselines. We observe that meta learning models (especially ProtoNet) and multi-task learning models always outperform other baselines. The reason why multi-task learning and prototypical networks can achieve the advantage is thanks to the division of tasks and the shared information between tasks. Compared to other baselines that make separate predictions for each time period, these two algorithms both jointly learn. The improvement of TAML in the second and third experiments is generally larger than that in the first experiment. As the time slot division in these two experiments contributes to much closer relationships between time periods, tasks in other time periods can provide more information under the TAML framework. We also observe that, compared to DNN, some predictions targeting the middle time period can be improved more than the two ends of the time period, for instance, the case of 19-30 days in the first experiment. Ablation Study In TAML, we propose several strategies to improve the MAML model for predicting time-associated tasks. The following ablation study is provided to evaluate the contribution of different components. The strategies include a weighted gradient update to emphasize the difference between timeassociated tasks and reference tasks in the meta-train phase and the TISS label setting applied in the training process. We aim to test whether both strategies are beneficial by removing each and redoing the experiments. We also explore the sensitivity of the model to task selection. Results are shown in Table 4, 5 and 6. Since the results are consistent across the three experiments, the ablation study of the second and third experiments is in the appendix. In TAML w/o TISS scenario, we omit TISS in both metatrain and meta-test phases, and only keep the weight assignment to time-associated tasks and reference tasks. In TAML w/o weight scenario, weights assigned to different tasks are the same. In TAML w/o TISS train scenario, TISS is omitted only in the meta-train phase. Besides, to assess the sensitivity of the model to task selection, the mutual information between tasks and the true target is calculated to filter out a few tasks that are least relevant to the current target. In TAML w/o unrelated tasks scenario, some tasks with low relevance are removed from training (See appendix for low relevance task list). We can observe that each part of our framework has an enhancing effect on the results, but the enhancement of just assigning weights to different tasks is limited. The most powerful part of the TAML framework is the combination of temporal information sharing strategy in the meta-test phase and weight assignment among tasks. It gives an insight into the TAML framework: for targets of different time periods, the model can also have good performance if implementing the meta-test based on the same meta-train results. When some of the tasks with low relevance are removed, the change in the model performance is relatively small in all four time periods. This shows that although using some low- relevance tasks may induce noise in the model, it also ensures the model's generalization, so that model performance may not be hurt even if some low-relevance tasks are included in the selection of tasks. Sensitivity Analysis To evaluate the performance under different model settings and hyperparameters, we study the impact of the size of support set, augmentation ratio, train-test split ratio, and time window width and provide some rules for parameter selection. We show the results of the analysis on the first two parameters in Figure 3 and the rest of analysis in appendix. Size of Support Set: The first column of Figure 3 shows that as the size of the support set increases from 5 to 20, the performance improves; however the smaller support set still has acceptable performance. The additional improvement from support tasks does not appear to saturate. We choose 15 as the size of support set in all experiments. Augmentation Ratio: Observed from the second column, on the local hospital dataset, the curve of augmentation ratio is nearly flat around 1.5, which indicates that paying more attention to "real" label tasks improves performance. However, on the MIMIC III dataset, the AUC drops when the value of the augmentation ratio is larger than 1.2, suggesting that the auxiliary label samples are helping. The reason might be that in the MIMIC III data set the time periods are contained in a fairly narrow window, thus more smoothing is helpful compared to the local hospital dataset. Thus, a larger augmentation ratio can be chosen for the setup with weak time period correlation. Additionally, we add a line of the best criterion in the form of a dashed line, which represents the best baseline result of the same color target. Although there are small fluctuations in AUC as the augmentation ratio changes, the advantage of TAML over the best baseline is still significant. Target time window width: We vary the width of the target window in ICU LOS from 0.2 to 1 day, showing the results in Figure 4. With small target windows, TAML outperforms other baselines, although there is a small decrease in performance. Compared to MAML, TAML has more improvement in terms of learning ability for the middle time widths, which benefit from the TISS. Conclusions In this paper, we focus on a specific few-shot problem in clinical predication: predicting future events in specific timeranges (time-associated target prediction). We propose a time-associated meta learning (TAML) framework. We increase the choice of available tasks by transforming the regression problem into multiple binary classification problems, adding not only many closely related time-associated tasks, but also some time-independent reference tasks. To deal with the sparsity caused by task splitting, the temporal information shared strategy (TISS) is designed to augment positive labels and smooth the relationship between adjacent event categories during training. We validate our model on public datasets and local hospital datasets to predict two clinical events: survival time and ICU length of stay (LOS), on which our model shows strong performance over baselines. Additionally, TAML turns out to be insensitive to tuning parameters and unrelated tasks and can achieve excellent performance regardless of the length of the time window. Thus, the TAML framework can be applied to clinical time-associated target prediction, thereby providing a reference for decision-making. Besides, we develop an Python package MetaEHR for few-shot clinical prediction, which includes the implementation of TAML on timeassociated prediction as well as other meta learning algorithms for time-independent prediction. A Appendix A.1 Baseline Description Simple DNN Traditional deep neural network is implemented as a baseline, where the patient EHR data is used as input, mortality or ICU LOS in a selected time period is considered as target. We train a separate DNN for each target and report the results in turn. MAML The MAML framework is used as a baseline, where we treat time-associated tasks and reference tasks as indistinguishable. In the training process, different situations are given the same weight and TISS is also not used. Multi-task Learning A shared-layer multi-task learning network is considered as a baseline, where the prediction targets are mortality in different time periods. Survival Analysis DNN-based semiparametric survival analysis which predict the event time is taken to evaluate. For a comparable evaluation, we divide the predicted time into the same groups in TAML and evaluate the performance in each group. Prototypical Neural Network Prototypical neural network is used as a baseline representing metric-based meta learning. The number of prototypes equals to the number of time periods in each dataset. Pretrained Model We also pre-train a model to compare the performance, where all the tasks involved in TAML are used in pre-training process. Then we fine-tune the model on the time-associated tasks. A.2 Ablation study on the second and third experiment The ablation study on the second and third experiments is provided in Table 5 and 6. A.3 Parameter Sensitivity: Train and test split The parameter sensitivity of the parameter: train and test split ratio is provided in Figure 5. A.4 Reference Tasks In survival time experiment on local hospital, we use 13 reference tasks including: ARF, Heart Attack, Cardiac Arrest, CHF, Stroke, LegBloodClot, LungBloodClot, Respiratory Arrest, Pneumonia, GIBleed, Abnormal Heart Rythmn, VTE, disposition, where we select four low-relevance tasks by calculating mutual information. These four tasks are Leg-BloodClot, LungBloodClot, Respiratory Arrest, Stroke. In survival time experiment on MIMIC III dataset, we use 12 reference tasks including: Speticemia, Cardiac dysrhythmias, AKI, heart failure, peripheral vascular, hypertension, diabetes, liver disease, myocardial infarction (MI), coronary artery disease (CAD), cirrhosis, and jaundice, where we select four low-relevance tasks. These four tasks are peripheral vascular, hypertension, diabetes, and jaundice. In ICU LOS experiment, we use 13 reference tasks including: Arrhythmia, Stroke, Hyperglycemia, HepaticDisease, AnemiaBleed, Coagulopathy, Thrombocytopenia, Pneumonia, UTI, BloodInfection, Sepsis, AlteredMentalStatus, Resp-Failure, where we select four low-relevance tasks. These four low relevance tasks are BloodInfection, Thrombocytopenia, Coagulopathy, and HepaticDisease. Figure 1 : 1Task Decomposition. Reference tasks are on the top. They are time-independent and related to the true target. Time-associated tasks are at the bottom. Each target is represented by a block over the timeline. Figure 2 : 2TISS Strategy. In the top three rows, the event occurrence is in the second time period and lasts until the fourth time period. The dark blue block represents the target time period. Bottom row: no events occur. Figure 3 : 3Parameter Sensitivity Study. Top: Survival time on local hospital; Mid: Survival time on MIMIC III; Bottom: ICU LOS. First column: Size of Support Set; Second column: Situation Ratio. The dotted line is the best baseline of the same color target. Figure 4 : 4Short Time Slot experiments on ICU LOS Figure 5 : 5Parameter Sensitivity Study. Left: Survival time on local hospital, Right: Survival time on MIMIC III uses MTL to predict mortality of rare diseases. Other meta-learning approaches include MetaPred [Zhang et al., 2019] which uses a simulated target during the meta-train process, and MetaCare++ [Tan et al., 2022] which proposes a specialized clinical meta-learner with a hierarchical subtyping strategy to capture temporal relations. Algorithm 1 Time-associated meta-train Input: Time-associated tasks {T S }, Reference tasks {T R } Parameter: Inner step sizes α k , outer step size β, outer task weights w k1: Randomly initialize θ 2: while Outer Loop not done do 3: Sample batch tasks {T k } including {T S k } and {T R k } 4: for all {T k } do 5: Table 2 : 2Performance of Survival Time on MIMIC III Table 3 : 3Performance Evaluation of ICU LOS generalization performance by meta-testing on local hospital dataset.The dataset from local hospital includes patient demographics, height and weight, comorbidities, preoperative vital signs, laboratory results, and medications for all adults undergoing inpatient surgery from September 2012 to June 2018. 42,853 patients with preoperative features are included in the experiment. It has been described previously in anonymized references. Table 4 : 4Ablation Study on local hospital dataset Table 5 : 5Ablation Study of Survival Time on MIMIC III dataset /o unrelated 0.7875 0.4495 0.7993 0.4662 0.7723 0.4422 TAML(Ours) 0.7892 0.4418 0.7946 0.4617 0.7715 0.4436Model 0-1 Days 1-2 Days 2-4 Days AUROC Recall AUROC Recall AUROC Recall MAML 0.7458 0.4209 0.7642 0.4278 0.7486 0.4236 w/o TISS 0.7625 0.4353 0.7781 0.4489 0.7628 0.4337 w/o weight 0.7639 0.4317 0.7725 0.4458 0.7623 0.4346 w/o train 0.7801 0.4412 0.7876 0.4548 0.7684 0.4392 w Table 6 : 6Ablation Study of ICU LOS Uses of electronic health records for public health surveillance to advance public health. Birkhead, 25581157Annual Review of Public Health. 361References [Birkhead et al., 2015] Guthrie S. Birkhead, Michael Klom- pas, and Nirav R. Shah. Uses of electronic health records for public health surveillance to advance public health. Annual Review of Public Health, 36(1):345-359, 2015. PMID: 25581157. 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[ "BOWERS-STEPHENSON'S CONJECTURE ON THE CONVERGENCE OF INVERSIVE DISTANCE CIRCLE PACKINGS TO THE RIEMANN MAPPING", "BOWERS-STEPHENSON'S CONJECTURE ON THE CONVERGENCE OF INVERSIVE DISTANCE CIRCLE PACKINGS TO THE RIEMANN MAPPING" ]
[ "Yuxiang Chen ", "Yanwen Luo ", "X U Xu ", "Siqi Zhang " ]
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Bowers and Stephenson [3]introduced the notion of inversive distance circle packings as a natural generalization of Thurston's circle packings[28]. They conjectured that discrete conformal maps induced by inversive distance circle packings converge to the Riemann mapping. Motivated by the recent work of Luo-Sun-Wu [21], we prove Bowers-Stephenson's conjecture for Jordan domains by establishing a maximal principle, an infinite rigidity theorem and a solvability theorem of certain prescribing combinatorial curvature problems for inversive distance circle packings.CONTENTS
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[ "https://export.arxiv.org/pdf/2211.07464v1.pdf" ]
253,510,777
2211.07464
119b023351bd47088645de4ed3384499927b2c05
BOWERS-STEPHENSON'S CONJECTURE ON THE CONVERGENCE OF INVERSIVE DISTANCE CIRCLE PACKINGS TO THE RIEMANN MAPPING 14 Nov 2022 Yuxiang Chen Yanwen Luo X U Xu Siqi Zhang BOWERS-STEPHENSON'S CONJECTURE ON THE CONVERGENCE OF INVERSIVE DISTANCE CIRCLE PACKINGS TO THE RIEMANN MAPPING 14 Nov 2022 Bowers and Stephenson [3]introduced the notion of inversive distance circle packings as a natural generalization of Thurston's circle packings[28]. They conjectured that discrete conformal maps induced by inversive distance circle packings converge to the Riemann mapping. Motivated by the recent work of Luo-Sun-Wu [21], we prove Bowers-Stephenson's conjecture for Jordan domains by establishing a maximal principle, an infinite rigidity theorem and a solvability theorem of certain prescribing combinatorial curvature problems for inversive distance circle packings.CONTENTS In [29], Thurston proposed a constructive approach to the Riemann mapping theorem by approximating conformal mappings in simply connected domains using circle packings. Thurston conjectured that the discrete conformal maps induced by circle packings converge to the Riemann mapping. Thurston's conjecture has been proved elegantly by Rodin-Sullivan [25]. Since then, there have been lots of important works on the convergence of discrete conformal maps to the Riemann mapping. See [4,12,15,16,21,30] and others. Motivated by Thurston's circle packings [28], Bowers-Stephenson [3] introduced the notion of inversive distance circle packings and conjectured that the Riemann mapping could be approximated by inversive distance circle packings. In this paper, we prove Bowers-Stephenson's conjecture for Jordan domains as a counterpart of Thurston's conjecture [29] in the setting of circle packings. The main idea comes from the recent work of Luo-Sun-Wu [21]. Suppose S is a topological surface possibly with boundary and T is a triangulation of S. We use V = V (T ), E = E(T ) and F = F (T ) to denote the set of vertices, edges, and faces of T respectively. A piecewise linear metric d (PL metric for simplicity) on (S, T ) is a flat cone metric on S such that each face in F in the metric d is isometric to a non-degenerate Euclidean triangle. In this case, one can represent the PL metric on (S, T ) as a length function l : E → R >0 , which satisfies the strict triangle inequality for any face in F . Conversely, given a function l : E → R >0 satisfying the strict triangle inequality, one can construct a PL metric on (S, T ) by isometrically gluing Euclidean triangles along edges in pairs. Hence, we also refer to a PL metric on (S, T ) as a function l : E → R >0 satisfying the strict triangle inequality for any face in F . For a PL metric l : E → R >0 on (S, T ), the combinatorial curvature is a map K : V → (−∞, 2π) sending an interior vertex v ∈ V to 2π minus the sum of angles at v and a boundary vertex v ∈ V to π minus the sum of angles at v. The combinatorial curvature K for a PL metric on (S, T ) satisfies the discrete Gauss-Bonnet formula A PL metric l : E → R >0 on (S, T ) is an inversive distance circle packing metric on the weighted triangulated surface (S, T , I) if there exists a function u : V → R such that for any edge e ∈ E with vertices v and v ′ , the length l(e) is given by (2) l(e) = e 2u(v) + e 2u(v ′ ) + 2I(e)e u(v)+u(v ′ ) . The function u : V → R is called a label on (S, T , I). Two inversive distance circle packing metrics (S, T , I, l) and (S, T ,Ĩ,l) are conformally equivalent if I =Ĩ. In this case, we set w =ũ − u and denote this relation as l * =l = w * l. The function w is called a discrete conformal factor on (S, T , I, l). If we set r(v) = e u(v) for v ∈ V , then the weight I(e) in (2) is the inversive distance of the two circles centered at v and v ′ with radii r(v) and r(v ′ ) respectively. The map r : V → (0, +∞) is referred as an inversive distance circle packing on the weighted triangulated surface (S, T , I). Thurston's circle packing [28] is a special type of inversive distance circle packing with I ∈ [0, 1] in (2). An excellent source for the comprehensive theory of circle packings is [27]. The main focus of this paper is to provide an affirmative answer to Bowers-Stephenson's conjecture on the convergence of discrete conformal maps induced by inversive distance circle packings to the Riemann mapping for Jordan domains. Specifically, let Ω be a Jordan domain in the plane with three distinct boundary points p, q, r specified. By the Riemann mapping theorem, there exists a conformal map from Ω to the interior of an equilateral Euclidean triangle 2 △ABC with unit edge length, which could be uniquely extended to be a homeomorphism g from Ω to △ABC with p, q, r sent to A, B, C respectively by Caratheodory's extension theorem [24]. The map g and g −1 are referred as the Riemann mapping for (Ω, (p, q, r)). Let (D, T , I) be an oriented weighted polygonal disk in the plane with three distinct boundary vertices p, q, r and l be a flat inversive distance circle packing metric on (D, T , I). Suppose that there exists a function w : V → R such that l * = w * l is an inversive distance circle packing metric on (D, T , I) with total area √ 3 4 , combinatorial curvature 2π 3 at p, q, r, and flat at other vertices. Then (D, T , l * ) is isometric to a triangulated unit equilateral triangle (△ABC, T ′ ) with some triangulation T ′ and the standard flat metric. Let f be the orientation-preserving piecewise linear map induced by the map sending the vertices of T to the corresponding vertices of T ′ such that f (A) = p, f (B) = q and f (C) = r. The map f is called the discrete conformal map associated to (D, T , I, l, {p, q, r}). We prove the following theorem on the convergence of discrete conformal maps induced by a specific sequence of inversive distance circle packings on Ω. Theorem 1.2. Let Ω be a Jordan domain in the complex plane with three distinct boundary points p, q, r specified. Let f be the Riemann mapping from the equilateral triangle △ABC to Ω such that f (A) = p, f (B) = q, f (C) = r. Then there exists a sequence of weighted triangulated polygonal disks (Ω n , T n , I n , (p n , q n , r n )) with inversive distance circle packing metrics l n , where T n is a triangulation of Ω n , I n : E n → (1, +∞) is a weight defined on E n = E(T n ) and p n , q n , r n are three distinct boundary vertices of T n , such that (a) Ω = ∪ ∞ n=1 Ω n with Ω n ⊂ Ω n+1 , and lim n p n = p, lim n q n = q, lim n r n = r. (b) discrete conformal maps f n from △ABC to (Ω n , T n , I n , l n ) with f n (A) = p n , f n (B) = q n , f n (C) = r n exist. (c) discrete conformal maps f n converge uniformly to the Riemann mapping f . In comparison with Rodin-Sullivan's convergence theorem for circle packings in [25], which allows the approximating triangulated polygonal disks to be arbitrarily selected, Theorem 1.2 requires that the approximating weighted triangulated polygonal disks should be carefully selected. The key difference is that the discrete conformal map does not exist for general inversive distance circle packings on weighted triangulated polygonal disks with inversive distance I : E → (1, +∞), while Koebe-Andreev-Thurston theorem ensures the existence of discrete conformal maps for any circle packings on triangulated polygonal disks. In the rest of this paper, we assume that I : E → (1, +∞) unless otherwise stated. This condition corresponds to the "S-packings" introduced by Bowers-Stephenson [3]. The paper is organized as follows. In Section 2, we give some preliminaries on inversive distance circle packings and weighted Delaunay triangulations. In Section 3, we derive a maximal principle and a ring lemma for inversive distance circle packings. We also study the properties of inversive distance circle packings on spiral hexagonal triangulations in this section. In Section 4, we prove the rigidity of infinite inversive distance circle packings on the hexagonal triangulated plane. In Section 5, we solve some prescribing combinatorial curvature problem for inversive distance circle packings and prove Theorem 1.2. Acknowledgement. In this section, we collect some basic properties of inversive distance circle packings and weighted Delaunay triangulations. We first describe the admissible space of inversive distance circle packings on a triangle and the variation of inner angles in this space. Then we discuss a notion of generalized weighted Delaunay triangulations and their relationships with inversive distance circle packings. 2.1. Basic properties of inversive distance circle packings. Let (S, T , I) be a weighted triangulated surface. We use v i to denote a vertex in V , e ij = v i v j to denote an edge in E and △v i v j v k to denote a face in F . We will denote f i = f (v i ) if f is a function defined on V , f ij = f (v i v j ) = f (e ij ) if f is a function defined on E, and f ijk = f (△v i v j v k ) if f is a function defined on F . For any function u : V → R, the formula (2) produces a positive function l on E. However, for a face △v i v j v k in (S, T , I), the positive numbers l ij , l ik , l jk may not satisfy the strict triangle inequality (3) l rs < l rt + l st , {r, s, t} = {i, j, k}. The label u : V → R is said to be admissible if the function l : E → (0, +∞) determined by u : V → R via the formula (2) satisfies the strict triangle inequality (3) for every face in (S, T , I). We also say that the corresponding inversive distance circle packing r : V → R >0 on (S, T , I) with r i = e u i is admissible, if it causes no confusion in the context. The admissible space of inversive distance circle packings on (S, T , I) consists of all the admissible inversive distance circle packings on (S, T , I). For an admissible inversive distance circle packing r on (S, T , I), every face in (S, T , I) is isometric to a non-degenerate Euclidean triangle with edge lengths given by (2). We also say that r : V → R >0 generates a PL metric on (S, T , I) for simplicity in this case. If three positive numbers l ij , l ik , l jk satisfy the triangle inequality (4) l rs ≤ l rt + l st , {r, s, t} = {i, j, k}, then l ij , l ik , l jk generate a generalized Euclidean triangle △v i v j v k . If l ij = l ik + l jk , the generalized triangle △v i v j v k is flat at v k , and the inner angle at v k is defined to be π. In this case, the generalized triangle is referred as a degenerate triangle. A function l : E → R >0 is called a generalized PL metric on (S, T ) if the triangle inequality (4) is satisfied for every face in (S, T ). A PL metric is a special type of generalized PL metric with the strict triangle inequality (3) for every face in (S, T ). The combinatorial curvature of generalized PL metrics is defined the same as that of PL metrics and still satisfies the discrete Gauss-Bonnet formula (1). A generalized PL metric l : E → R >0 is called a generalized inversive distance circle packing metric on a weighted triangulated surface (S, T , I) if there exists a map u : V → R such that l is determined by u via the formula (2). In this case, the map r : V → R >0 with r i = e u i is said to be a generalized inversive distance circle packing on (S, T , I). We will denote it as (S, T , I, l), (S, T , I, u), or (S, T , I, r) interchangeably. We have a characterization of the admissible space of inversive distance circle packings on a weighted triangle and an extension of inner angles for generalized triangles generated by generalized inversive distance circle packings. 4 Lemma 2.1 ([13,31,32]). Let △v 1 v 2 v 3 be a face in (S, T ) with three weights I 1 , I 2 , I 3 ∈ (1, +∞) defined on edges opposite to the vertices v 1 , v 2 , v 3 respectively. Let u : {v 1 , v 2 , v 3 } → R be a function defined on the vertices, inducing edge lengths by (5) l ij = e 2u i + e 2u j + 2e u i +u j I k = r 2 i + r 2 j + 2r i r j I k , where r i = e u i , {i, j, k} = {1, 2, 3}. (a) l 12 , l 13 , l 23 generate a non-degenerate Euclidean triangle if and only if (6) Q := κ 2 1 (1 − I 2 1 ) + κ 2 2 (1 − I 2 2 ) + κ 2 3 (1 − I 2 3 ) + 2κ 1 κ 2 γ 3 + 2κ 1 κ 3 γ 2 + 2κ 2 κ 3 γ 1 > 0, where γ i := I i + I j I k and κ i := r −1 i . They generate a degenerate Euclidean triangle if and only if Q = 0. (b) The admissible space Ω 123 of inversive distance circle packings (r 1 , r 2 , r 3 ) ∈ R 3 >0 on △v 1 v 2 v 3 is Ω 123 = R 3 >0 \ ⊔ 3 i=1 V i , where ⊔ 3 i=1 V i is a disjoint union of V i = (r 1 , r 2 , r 3 ) ∈ R 3 >0 |κ i ≥ −B i + √ ∆ i 2A i with (7) A i =I 2 i − 1, B i = − 2(κ j γ k + κ k γ j ), ∆ i =4(I 2 1 + I 2 2 + I 2 3 + 2I 1 I 2 I 3 − 1)(κ 2 j + κ 2 k + 2κ j κ k I i ) . Let θ i be the inner angle of △v 1 v 2 v 3 at v i , then the inner angles of △v 1 v 2 v 3 could be uniquely continuously extended by constants as follows θ i (r 1 , r 2 , r 3 ) =    θ i , if (r 1 , r 2 , r 3 ) ∈ Ω 123 ; π, if (r 1 , r 2 , r 3 ) ∈ V i ; 0, otherwise. Corollary 2.2 ( [13,31,32]). If v i is the flat vertex of the degenerate triangle △v 1 v 2 v 3 generated by (r 1 , r 2 , r 3 ) ∈ R 3 >0 , then (r 1 , r 2 , r 3 ) ∈ ∂V i , i.e. κ i = −B i + √ ∆ i 2A i . The following lemma describes the change of inner angles along PL metrics generated by smooth families of labels on (S, T , I). (a) Suppose that the label u ∈ R 3 induces a non-degenerate Euclidean triangle △v 1 v 2 v 3 , then (8) ∂θ i ∂u j = ∂θ j ∂u i = h ij,k l ij , ∂θ i ∂u i = − ∂θ i ∂u j − ∂θ i ∂u k < 0, where (9) h ij,k = r 2 1 r 2 2 r 2 3 A 123 l ij [κ 2 k (1 − I 2 k ) + κ j κ k γ i + κ i κ k γ j ] = r 2 1 r 2 2 r 2 3 A 123 l ij κ k h k with A 123 = l 12 l 13 sin θ 1 and (10) h k = κ k (1 − I 2 k ) + κ i γ j + κ j γ i . 5 (b) If u = (u 1 , u 2 , u 3 ) ∈ R 3 is not admissible, then one of h 1 , h 2 , h 3 is negative and the other two are positive. Specially, if u ∈ R 3 generates a degenerate triangle △v 1 v 2 v 3 having v 3 as the flat vertex, then h 1 > 0, h 2 > 0, h 3 < 0 at u. Moreover, in this case, h 12,3 → −∞, h 13,2 → +∞, h 23,1 → +∞ as (r 1 ,r 2 ,r 3 ) ∈ Ω 123 tends to (r 1 , r 2 , r 3 ) = (e u 1 , e u 2 , e u 3 ) ∈ ∂Ω 123 . Note that h ij,k is only defined for non-degenerate inversive distance circle packings (r 1 , r 2 , r 3 ) ∈ Ω 123 ⊆ R 3 >0 , while h i is defined for any (r 1 , r 2 , r 3 ) ∈ R 3 >0 . For a non-degenerate inversive distance circle packing metric l on (S, T , I), set η k ij = h ij,k /l ij and define the conductance η : E → R for (S, T , I, l) by (11) η ij = η k ij + η m ij , v i v j is an interior edge contained in △v i v j v k and △v i v j v m ; η k ij , v i v j is a boundary edge contained in △v i v j v k . As a direct corollary of formula (8), we have the following variation of combinatorial curvatures. Corollary 2.4 ( [13,31,32]). Suppose w(t) * l is a family of inversive distance circle packing metrics on (S, T , I) induced by a smooth family of discrete conformal factor w(t) ∈ R V . Let K(t) and η(t) be the combinatorial curvature and the conductance of (S, T , I, w(t) * l). Then (12) dK i (t) dt = j∼i η ij (t)( dw i dt − dw j dt ). We prove the following results on inversive distance circle packings. (a) For any fixed r i , r j ∈ (0, +∞), the set of r k ∈ (0, +∞) such that (r i , r j , r k ) is an admissible inversive distance circle packing on △v 1 v 2 v 3 is an open interval. As a result, if (r i , r j ,r k ) and (r i , r j ,r k ) are two generalized inversive distance circle packings on △v 1 v 2 v 3 withr k <r k , then for any r k ∈ (r k ,r k ), (r i , r j , r k ) generates a non- degenerate triangle △v 1 v 2 v 3 . (b) If △v 1 v 2 v 3 generated by (r 1 , r 2 , r 3 ) ∈ R 3 >0 is a degenerate triangle having v 3 as the flat vertex, then there exists ǫ > 0 such that (r 1 , r 2 , r 3 + t) ∈ Ω 123 and ∂h 12,3 ∂r 3 (r 1 , r 2 , r 3 + t) > 0 for t ∈ (0, ǫ). Proof. To prove part (a), without loss of generality, set {i, j} = {2, 3}, k = 1 and f (κ 1 ) = (1 − I 2 1 )κ 2 1 + 2κ 1 (κ 2 γ 3 + κ 3 γ 2 ) + κ 2 2 (1 − I 2 2 ) + κ 2 3 (1 − I 2 3 ) + 2κ 2 κ 3 γ 1 . By Lemma 2.1 (a), we need to show that the solution of f (κ 1 ) > 0 with κ 1 ∈ (0, +∞) is an open interval. The inequality f (κ 1 ) > 0 is equivalent to the following quadratic inequality (I 2 1 − 1)κ 2 1 − 2κ 1 (κ 2 γ 3 + κ 3 γ 2 ) − κ 2 2 (1 − I 2 2 ) − κ 2 3 (1 − I 2 3 ) − 2κ 2 κ 3 γ 1 < 0. By I > 1, we have − b 2a = κ 2 γ 3 + κ 3 γ 2 I 2 1 − 1 and the discriminant of the quadratic polynomial defined in (7) satisfies ∆ = 4(I 2 1 + I 2 2 + I 2 3 + 2I 1 I 2 I 3 − 1)(κ 2 2 + κ 2 3 + 2κ 2 κ 3 I 1 ) > 0. This implies that the solution of f (κ 1 ) > 0 with κ 1 > 0 is an open interval in (0, +∞). To prove part (b), recall that the triangle △v 1 v 2 v 3 is degenerate if and only if Q = 0 by Lemma 2.1 (a), where Q is defined by (6). By direct calculations, we have ∂Q ∂κ 3 = 2h 3 < 0 at (r 1 , r 2 , r 3 ) by Lemma 2.3 (b), which implies that ∂Q ∂r 3 = ∂Q ∂κ 3 ∂κ 3 ∂r 3 = − 1 r 2 3 ∂Q ∂κ 3 > 0 around (r 1 , r 2 , r 3 ). Therefore, for small t > 0, Q(r 1 , r 2 , r 3 + t) > 0 and (r 1 , r 2 , r 3 + t) generates a non-degenerate triangle. Using the identities Q = κ 1 h 1 + κ 2 h 2 + κ 3 h 3 and A 2 123 = r 2 1 r 2 2 r 2 3 Q, we can deduce from the definition (9) of h 12,3 that (13) ∂h 12,3 ∂κ 3 = r 2 1 r 2 2 r 2 3 A 3 123 l 12 [r 2 1 r 2 2 r 2 3 (κ 1 h 1 + κ 2 h 2 )h 3 − A 2 123 (κ 1 γ 2 + κ 2 γ 1 )]. Note that v 3 is the flat vertex of the degenerate triangle △v 1 v 2 v 3 generated by (r 1 , r 2 , r 3 ), then A 123 = 0 and h 1 > 0, h 2 > 0, h 3 < 0 at (r 1 , r 2 , r 3 ) by Lemma 2.3 (b), which implies that ∂h 12,3 ∂κ 3 < 0 around (r 1 , r 2 , r 3 ) in the admissible space Ω 123 by (13). Note that It has wide applications in computational geometry. See [2,5] and others. In this subsection, we propose an alternative characterization of weighted Delaunay triangulations for inversive distance circle packing metrics and generalize weighted Delaunay triangulations for non-degenerate inversive distance circle packing metrics to generalized inversive distance circle packing metrics. Assume r : V → (0, +∞) is a non-degenerate inversive distance circle packing on a weighted triangulated surface (S, T , I). Let △v 1 v 2 v 3 be a Euclidean triangle in the plane isometric to a face in (S, T , I, r). Then there exists a unique geometric center C 123 such that its power distances to v i , defined by |C 123 − v i | 2 − r 2 i , are equal for i = 1, 2, 3. Projections of the geometric center C 123 to the lines v 1 v 2 , v 1 v 3 , v 2 v 3 give rise to the geometric centers of these edges, which are denoted by C 12 , C 13 , C 23 respectively. Please refer to Figure 1. One can refer to [6,7,8,9] for more information on the geometric center generated by discrete conformal structures on manifolds. Denote d ij as the signed distance of C ij to the vertex v i and h ij,k as the signed distance of C 123 to the edge v i v j . Glickenstein [7] obtained the following identities (14) d ij = r 2 i + r i r j I ij l ij , h ij,k = d ik − d ij cos θ i sin θ i . Note that d ij ∈ R >0 could be defined by (14) independent of the existence of the geometric center C ijk , and h ij,k is symmetric in the indices i and j, while d ij is not. For a weighted triangulated surface with a non-degenerate inversive distance circle packing (S, T , I, r), an interior edge v i v j is weighted Delaunay if h ij,k + h ij,l ≥ 0, where △v i v j v k and △v i v j v l are two triangles in F sharing the common edge v i v j . And (S, T , I, r) is weighted 7 v 1 v 2 C 12 d 12 d 21 d 13 C 123 d 31 h 13,2 h 12,3 h 23,1 C 23 v 3 d 32 d 23 C 13 FIGURE 1. Sign distances of the geometric center. Delaunay if all the interior edges are weighted Delaunay. Note that weighted Delaunay triangulations are only defined for non-degenerate inversive distance circle packings. We need to introduce the definition of weighted Delaunay triangulations for generalized inversive distance circle packing metrics. To this end, we introduce the following notion. Definition 2.6. Let r ∈ R V >0 be a generalized inversive distance circle packing on a weighted triangulated surface (S, T , I). Suppose that △v 1 v 2 v 3 is a generalized triangle in (S, T , I, r). If △v 1 v 2 v 3 is non-degenerate, define θ ij,k = arctan h ij,k d ij . If △v 1 v 2 v 3 is degenerate, define θ ij,k as θ ij,k = π 2 , if v i or v j is the flat vertex, − π 2 , if v k is the flat vertex. Note that for a non-degenerate triangle △v 1 v 2 v 3 in (S, T , I, r), θ ij,k is in fact the signed angle ∠v j v i C ijk , which is negative if h ij,k < 0 and non-negative otherwise. Please refer to Figure 2 for this. v i v j d ij v k θ ij,k h ij,k C ijk h ij,k d ij θ ij,k C ijk v k v i v j FIGURE 2. The angle θ ij,k when h ij,k < 0 (left) and h ij,k > 0 (right). For non-degenerate inversive distance circle packings on a weighted triangle △v 1 v 2 v 3 , θ ij,k is a continuous function of (r 1 , r 2 , r 3 ) ∈ Ω 123 and satisfies θ ij,k + θ ik,j = θ i . We further have 8 the following property on θ ij,k for generalized inversive distance circle packings on a weighted triangle. Lemma 2.7. Suppose △v 1 v 2 v 3 is a face in a weighted triangulated surface (S, T , I). Then θ ij,k (r 1 , r 2 , r 3 ) is a continuous function defined on Ω 123 and satisfies (15) θ ij,k + θ ik,j = θ i . Proof. We just need to prove that θ ij,k (r 1 , r 2 , r 3 ) → θ ij,k (r 1 ,r 2 ,r 3 ) as (r 1 , r 2 , r 3 ) ∈ Ω 123 tends to a point (r 1 ,r 2 ,r 3 ) ∈ ∂Ω 123 . If v k is the flat vertex of the degenerate triangle △v 1 v 2 v 3 generated by (r 1 ,r 2 ,r 3 ), then h ij,k (r 1 , r 2 , r 3 ) → −∞ as (r 1 , r 2 , r 3 ) → (r 1 ,r 2 ,r 3 ) by Lemma 2.3. As a result, we have θ ij,k (r 1 , r 2 , r 3 ) = arctan h ij,k d ij → − π 2 = θ ij,k (r 1 ,r 2 ,r 3 ) by Definition 2.6. If v i is the flat vertex of the degenerate triangle △v 1 v 2 v 3 generated by (r 1 ,r 2 ,r 3 ), then h ij,k (r 1 , r 2 , r 3 ) → +∞ as (r 1 , r 2 , r 3 ) → (r 1 ,r 2 ,r 3 ). As a result, we have θ ij,k (r 1 , r 2 , r 3 ) → π 2 = θ ij,k (r 1 ,r 2 ,r 3 ) as (r 1 , r 2 , r 3 ) → (r 1 ,r 2 ,r 3 ) by Definition 2.6. The same argument applies to the case that v j is the flat vertex. Q.E.D. Weighted Delaunay triangulations for non-degenerate inversive distance circle packings have a simple characterization using θ ij,k . Corollary 2.8. Suppose r ∈ R V >0 is a non-degenerate inversive distance circle packing on a weighted triangulated surface (S, T , I). An edge v i v j ∈ E is shared by two adjacent non- degenerate triangles △v i v j v k and △v i v j v l in (S, T , I, r). Then the edge v i v j is weighted Delaunay in the inversive distance circle packing r if and only if θ ij,k + θ ij,l ≥ 0. Proof. Since θ ij,k = arctan h ij,k d ij ∈ (− π 2 , π 2 ) and θ ij,l = arctan h ij,l d ij ∈ (− π 2 , π 2 ) by Definition 2.6, we have h ij,k + h ij,l d ij = tan θ ij,k + tan θ ij,l = sin(θ ij,k + θ ij,l ) cos θ ij,k cos θ ij,l , which implies h ij,k + h ij,l ≥ 0 is equivalent to θ ij,k + θ ij,l ≥ 0 by d ij > 0. Q.E.D. Remark 2.9. Under the conditions in Corollary 2.8, we further have that h ij,k + h ij,l > 0 is equivalent to θ ij,k + θ ij,l > 0 for non-degenerate inversive distance circle packings. Note that h ij,k is only defined for non-degenerate inversive distance circle packings, while θ ij,k could be defined for generalized inversive distance circle packings. We introduce the following definition of weighed Delaunay triangulations for generalized inversive distance circle packings, which generalizes the classical definition of weighed Delaunay triangulations for non-degenerate inversive distance circle packings. Let v i v j ∈ E be an edge shared by two adjacent triangles △v i v j v k and △v i v j v l in T . An interior edge v i v j ∈ E is weighted Delaunay in the generalized inversive distance circle packing r if θ ij,k + θ ij,l ≥ 0. The triangulation T is weighted Delaunay in the generalized inversive distance circle packing r if every interior edge is weighed Delaunay. 9 For simplicity, we also say that r is a generalized weighted Delaunay inversive distance circle packing on (S, T , I) if T is weighted Delaunay in r. We further have the following monotonicity for the angle θ ij,k in Definition 2.6. Lemma 2.11. Suppose △v 1 v 2 v 3 is a face in a weighted triangulated surface (S, T , I). Let (r 1 , r 2 ,r 3 ) and (r 1 , r 2 ,r 3 ) be two generalized inversive distance circle packings on △v 1 v 2 v 3 withr 3 <r 3 . If r 1 and r 2 are fixed, then θ 12,3 is strictly increasing in r 3 ∈ [r 3 ,r 3 ]. Proof. By Proposition 2.5 (a), (r 1 , r 2 , r 3 ) generates a non-degenerate triangle △v 1 v 2 v 3 for r 3 ∈ (r 3 ,r 3 ). For r 3 ∈ (r 3 ,r 3 ), h 12,3 and θ 12,3 are smooth functions of r 3 . By the definition of h i and γ i , we can deduce that ∂h 12,3 ∂κ 3 = r 2 1 r 2 2 r 2 3 A 3 123 l 12 [r 2 1 r 2 2 r 2 3 (κ 1 h 1 + κ 2 h 2 )h 3 − A 2 123 (κ 2 γ 1 + κ 1 γ 2 )] = r 4 1 r 4 2 r 3 3 A 3 123 l 12 (1 − I 2 12 − I 2 13 − I 2 23 − 2I 12 I 13 I 23 )(κ 2 1 + κ 2 2 + 2κ 1 κ 2 I 12 ) < 0. This implies ∂θ 12,3 ∂r 3 = − d 12 κ 2 3 d 2 12 + (h 12,3 ) 2 · ∂h 12,3 ∂κ 3 > 0, ∀r 3 ∈ (r 3 ,r 3 ) by the definition of θ 12,3 . Note that θ 12,3 is a continuous function of r 3 ∈ [r 3 ,r 3 ] by Lemma 2.7, then θ 12,3 is strictly increasing in r 3 ∈ [r 3 ,r 3 ]. Q.E.D. A MAXIMAL PRINCIPLE, A RING LEMMA AND SPIRAL HEXAGONAL TRIANGULATIONS 3.1. A maximal principle. Let P n be a star-shaped n-sided polygon in the plane with boundary vertices v 1 , · · · , v n ordered cyclically (v n+i = v i ). Assume v 0 is an interior point of P n and it induces a triangulation T of P n with triangles △v 0 v i v i+1 . Then an assignment of radii r : V (T ) → R >0 is a vector in R n+1 . For any two vectors x = (x 0 , . . . , x n ) and y = (y 0 , . . . , y n ) in R n+1 , we use x ≥ y to denote x i ≥ y i for all i ∈ {0, . . . , n}. v j+1 v j v j−1 v 0 FIGURE 3. A star triangulation of a polygon. We have the following maximal principle for inversive distance circle packings. Theorem 3.1. Let T be a star triangulation of P n with boundary vertices v 1 , . . . , v n and a unique interior vertex v 0 . Let I : E → (1, +∞) be a weight. Suppose r and r are two generalized inversive distance circle packings on (P n , T , I) such that 10 (a) r and r are generalized weighted Delaunay inversive distance circle packings, (b) the combinatorial curvatures K 0 (r) and K 0 (r) at the vertex v 0 satisfy K 0 (r) ≤ K 0 (r), (c) max{ r i r i |i = 1, 2, · · · , n} ≤ r 0 r 0 . Then there exists a constant c > 0 such that r = cr. We use the following notations to prove Theorem 3.1. For i ∈ {1, · · · , n}, we denote I 0i as I i for simplicity. For two adjacent triangles △v 0 v j v j±1 in T , set θ 0 j,j±1 to be the inner angle at v 0 in the triangle △v 0 v j v j±1 . Moreover, set h − j = h 0j,j−1 , h + j = h 0j,j+1 , θ − j = θ 0j,j−1 , θ + j = θ 0j,j+1 . The proof of the maximal principle is based on the following key lemma. Lemma 3.2. If r, r : {v 0 , v 1 , . . . , v n } → R >0 satisfy (a), (b), (c) in Theorem 3.1 and there exists j ∈ {1, 2, . . . , n} such that r j r j < r 0 r 0 , then there existsr ∈ R n+1 >0 such that (a)r i ≥ r i for i ∈ {1, · · · , n}, (b)r i r i ≤r 0 r 0 = r 0 r 0 for all i = 1, 2, . . . , n, (c)r is a generalized weighted Delaunay inversive distance circle packing on (P n , T , I), (d) if α(r) is the cone angle of the inversive distance circle packing r at v 0 , then (16) α(r) > α(r). Proof. Up to a scaling, we may assume that r 0 =r 0 . Then the condition (c) in Theorem 3.1 is equivalent to r i ≤r i for all i ∈ {1, 2, . . . , n}. Set J ={j ∈ {1, 2, . . . , n}|r j <r j }, K ={k ∈ {1, 2, . . . , n}|r k =r k }, γ(r) = j∈J (θ 0j,j+1 + θ 0j,j−1 ) = j∈J (θ + j + θ − j ), β(r) = k∈K (θ 0k,k+1 + θ 0k,k−1 ) = k∈K (θ + k + θ − k ). Then J = ∅ by assumption. By (15), we have α(r) = β(r) + γ(r), α(r) = β(r) + γ(r), which further implies (17) β(r) + γ(r) ≤ β(r) + γ(r) by the condition K 0 (r) ≤ K 0 (r). Claim 1: For any j ∈ J, θ 0 j−1,j (r) < π and θ 0 j,j+1 (r) < π. We will prove that for any j ∈ J, v 0 is not the flat vertex if the triangle △v 0 v j v j−1 is degenerate. Otherwise, suppose that for some j ∈ J, v 0 is the flat vertex of the degenerate triangle △v 0 v j v j−1 generated by r. By Corollary 2.2, r satisfies κ 0 = f (κ j−1 , κ j ), where f (κ j−1 , κ j ) = 1 I 2 j,j−1 − 1 [(κ j γ j−1 + κ j−1 γ j ) + (I 2 j + I 2 j−1 + I 2 j,j−1 + 2I j I j−1 I j,j−1 − 1) 1/2 (κ 2 j + κ 2 j−1 + 2I j,j−1 κ j κ j−1 ) 1/2 ]. Note that κ j > κ j and κ j−1 ≥ κ j−1 . Then we have This implies that (r 0 , r j , r j−1 ) is in the complement of the space of generalized inversive distance circle packings on △v 0 v j v j−1 in R 3 >0 by Lemma 2.1 (b), which contradicts the assumption that r is a generalized inversive distance circle packing on (P n , T , I). Claim 2: There exists j ∈ J such that θ + j (r) + θ − j (r) > 0. We will consider the two cases K = ∅ and K = ∅. Case 1: K = ∅. By Lemma 2.11, for any i ∈ K, θ − i and θ + i are strictly increasing in r i−1 and r i+1 respectively, which implies that β(r) ≤ β(r). As J = ∅, there exists i ∈ K such that i − 1 or i + 1 is in J. Say i − 1 ∈ J, then r i−1 <r i−1 and then θ − i (r) < θ − i (r) by Lemma 2.11. Thus, β(r) < β(r), which implies 0 ≤ γ(r) < γ(r) by (17). Therefore, there exists j ∈ J such that θ + j (r) + θ − j (r) > 0 by the definition of γ(r). Case 2: K = ∅. If K = ∅, we have J = {1, . . . , n}, γ(r) = j∈J (θ + j (r) + θ − j (r)) = α(r) ≥ 0. If α(r) > 0, there exists j ∈ J such that θ + j (r) + θ − j (r) > 0. If α(r) = 0, for any tri- angle △v 0 v j v j−1 , j = 1, . . . , n, the inner angle at v 0 is equal to zero. Thus all triangles are degenerate. For any triangle △v 0 v j v j−1 , the flat vertex is v j or v j−1 by Claim 1. Then {θ − j (r), θ + j−1 (r)} = { π 2 , − π 2 }, ∀j ∈ {1, · · · , n}. Without loss of generality, we may assume v 1 is the flat vertex of △v 0 v 1 v 2 . Then θ + 1 (r) = π 2 , θ − 2 (r) = − π 2 by Definition 2.6 and l 02 (r) = l 01 (r) + l 12 (r) > l 01 (r). By the weighted Delaunay condition (a) in Theorem 3.1, θ + 2 (r) = π 2 , which implies θ − 3 (r) = π 2 and l 03 (r) = l 02 (r) + l 23 (r) > l 02 (r). By induction, we have a contradiction l 01 (r) < l 02 (r) < · · · < l 0n (r) < l 01 (r). This completes the proof of Claim 2. Now we fix j ∈ J in Claim 2. Then we have (18) θ + j (r) + θ − j (r) > 0. In the following, we show that there exists ǫ > 0 such thatr = (r 0 , . . . , r j + t, . . . , r n ) satisfies Lemma 3.2 for t ∈ (0, ǫ). It is easy to check that for t ∈ (0, r j − r j ),r satisfies Lemma 3.2 (a) and (b). To see part (c) of Lemma 3.2, we first show that there exists ǫ > 0 such thatr is a generalized inversive distance circle packing on (P n , T , I) for t ∈ (0, ǫ). Furthermore, we will show that the triangles △v 0 v j v j±1 generated byr are non-degenerate. The triangle △v 0 v j v j−1 generated by r is non-degenerate or degenerate with v j or v j−1 as the flat vertex by Claim 1. By Proposition 2.5, we just need to prove that v j−1 is not the flat vertex of the triangle △v 0 v j v j−1 generated by r if it is degenerate. Otherwise, we have θ − j (r) = − π 2 by Definition 2.6, which implies θ + j (r) > π 2 by (18). However, this is impossible since θ + j (r) ∈ [− π 2 , π 2 ] by Definition 2.6. Therefore, v j−1 can never be the flat vertex of the triangle △v 0 v j v j−1 if it is degenerate. Similar arguments applying to the triangle △v 0 v j v j+1 show that v j+1 can never be the flat vertex of the triangle △v 0 v j v j+1 if it is degenerate. Therefore, by Proposition 2.5 (b), there exists ǫ > 0 such that for t ∈ (0, ǫ),r is a generalized inversive distance circle packing on (P n , T , I) and the triangles △v 0 v j v j±1 generated byr are non-degenerate. Next, we show thatr satisfies the weighted Delaunay condition. Asr differs from r only at the j-th position, we just need to consider the edges v 0 v j and v 0 v j±1 . For the edge v 0 v j , since 12 θ + j (r) + θ − j (r) > 0, we have θ + j (r) + θ − j (r) > 0 for small t > 0 by the continuity of θ ± j in Lemma 2.7. For the edge v 0 v j−1 , θ − j−1 (r) = θ − j−1 (r). We further have θ + j−1 (r) < θ + j−1 (r) for t ∈ (0, r j − r j ) by Lemma 2.11, which implies θ + j−1 (r) + θ − j−1 (r) > θ + j−1 (r) + θ − j−1 (r) ≥ 0. This implies that the edge v 0 v j−1 satisfies the weighted Delaunay condition forr. The same arguments apply to the edge v 0 v j+1 . To see part (d) of Lemma 3.2, by the arguments for part (c), there exists ǫ > 0 such that the triangles △v 0 v j v j±1 are non-degenerate inr and θ + j (r) + θ − j (r) > 0 for t ∈ (0, ǫ), which implies h + j (r) + h − j (r) > 0 for t ∈ (0, ǫ) by Remark 2.9. Note that α(r) is continuous for t ∈ [0, ǫ], smooth for t ∈ (0, ǫ), and ∂α ∂t (r) = h + j (r) + h − j (r) l 0j > 0, t ∈ (0, ǫ) by Lemma 2.3 (a), we have α(r) > α(r) for t ∈ (0, ǫ). Q.E.D. Now we can prove Theorem 3.1. Proof for Theorem 3.1: Without loss of generality, we assume r 0 =r 0 and r i ≤r i for all i = 1, 2, . . . , n. We prove the theorem by contradiction. Suppose that there exists a weighted Delaunay inversive distance circle packing r on (P n , T , I) such that r 0 =r 0 , r i ≤r i for all i = 1, 2, . . . , n with one r i 0 <r i 0 and α(r) ≤ α(r). By Lemma 3.2, we may assume that (19) α(r) < α(r). On the other hand, consider the set X := {x ∈ R n+1 |r ≤ x ≤r, and x is a generalized weighted Delaunay inversive distance circle packing on (P n , T , I)}. Obviously, r ∈ X and X is bounded. By Lemma 2.7, X is a closed subset of R n+1 . Therefore, X is a nonempty compact set and α(x) has a maximum point on X. Let t ∈ X be a maximum point of the continuous function f (x) = α(x) on X. If t =r, then by Lemma 3.2, we can find a weighted Delaunay inversive distance circle packingt on (P n , T , I) such thatt ≥ t,t 0 =r 0 , t ≤r and α(t) > α(t), which implies that t is not a maximum point of f (x) = α(x) on X. So, t =r and then α(r) = α(t) ≥ α(r) > α(r), where the last inequality comes from (19). This is a contradiction. Q.E.D. Remark 3.3. For simplicity, we only present the maximal principle for the inversive distance I : E → (1, +∞), which is enough for the application in the convergence of inversive distance circle packings in this paper. For a much more general version of the maximal principle for inversive distance circle packings with I : E → (−1, +∞) and generic discrete conformal structures on surfaces [9,33], please refer to [22]. 3.2. A ring lemma. Proof. Without loss of generality, we assume r 0 = 1, otherwise applying a scaling to the weighed triangulated polygon (P n , T , I, r). Then we just need to prove Cr i ≥ 1 for some C ∈ R >0 , which is equivalent to κ i ≤ C, for all i ∈ {1, 2, · · · , n}. We prove Lemma 3.4 by contradiction. If the result in Lemma 3.4 is not true, then there exists a sequence of generalized inversive distance circle packings {r (m) } ∞ m=1 with r (m) 0 = 1 on (P n , T , I) such that lim m→∞ κ (m) i = +∞ for some i ∈ {1, 2, · · · , n}. Without loss of generality, we can assume i = 1. For the triangle △v 0 v 1 v 2 , we set I 0 = I 12 , I 1 = I 02 and I 2 = I 01 for simplicity. As r (m) is a generalized inversive distance circle packing on (P n , T , I), we have (I 2 2 − 1)(κ (m) 2 ) 2 + (I 2 1 − 1)(κ (m) 1 ) 2 + (I 2 0 − 1) − 2κ (m) 1 γ 2 − 2κ (m) 2 γ 1 − 2κ (m) 1 κ (m) 2 γ 0 ≤ 0 by Lemma 2.1 (a), which implies (20) κ (m) 2 ≥ 1 I 2 2 − 1 [κ (m) 1 γ 0 + γ 1 − (I 2 0 + I 2 1 + I 2 2 + 2I 0 I 1 I 2 − 1)((κ (m) 1 ) 2 + 2I 2 κ (m) 1 + 1)] = (I 2 1 − 1)(κ (m) 1 ) 2 − 2I 2 κ (m) 1 + I 2 0 − 1 κ (m) 1 γ 0 + γ 1 + (I 2 0 + I 2 1 + I 2 2 + 2I 0 I 1 I 2 − 1)((κ (m) 1 ) 2 + 2I 2 κ (m) 1 + 1) . Note that lim m→∞ κ v i v i+1 , i = 2, 3, · · · , n subsequently give lim m→∞ (θ 0 i,i+1 ) (m) → 0 for all i = 1, 2, · · · , n, which implies lim m→∞ K (m) 0 = 2π. This contradicts the assumption that v 0 is a flat interior vertex of (P n , T , I, r (m) ). Q.E.D. 3.3. Spiral hexagonal triangulations and linear discrete conformal factors. We first recall the definition of developing maps in [21]. Let l be a flat polyhedral metric on a simply connected triangulated surface (S, T ). Then S is homeomorphic to its universal covering, so the developing map φ : (S, T , l) → C for this polyhedral metric can be constructed by starting with any isometrically embedding in C of a Euclidean triangle t ∈ F . This defines an initial map φ|t : (t, l) → C, which can be extended to any adjacent triangle s ∈ F with e = t ∩ s ∈ E by isometrically embedding s in C such that φ(e) = φ(s) ∩ φ(t). Since S is simply connected, we can repeat this extension for all triangles, which induces a well-defined developing map up to isometries of (S, T , l). (b) Let φ be the developing map for (C, T st , I, w * l). If there exists a non-degenerate triangle in (C, T st , I, w * l), then there are two different non-degenerate triangles t 1 and t 2 in (C, T st , I, w * l) such that φ(int(t 1 )) ∩ φ(int(t 2 )) = ∅. In other words, φ does not produce an embedding of (C, T st , I, w * l) in the plane. (c) If all the triangles in (C, T st , I, w * l) are degenerate, then there exists an automorphism ψ of the triangulation T st and two positive constantsλ =λ(I, u) andμ =μ(I, u) such that w(ψ(m v 1 + n v 2 )) = m lnλ + n lnμ. Proof. The proof is a modification of the proof for Proposition 3.4 in [21]. For completeness, we include the proof here. To prove part (a), consider two translations τ 1 and τ 2 of the triangulation T st defined by v 2 − v 1 − v 2 v 1 − v 2 v 2 v 1 α 1 α 2 γ 1 β 2 β 2 t 1 t 2 t 1 t 2 t 1 t 1 − v 1 β 1 γ 2 β 1 γ 1 γ 2 α 1 γ 2τ 1 (v) = v + v 1 , τ 2 (v) = v + v 2 , v ∈ V. The lattice L is isometric to the abelian subgroup of the automorphism group of T st generated by τ 1 and τ 2 . Up to the action to this subgroup, all the triangles in (C, T st , I, w * l) belong to two equivalent classes, one of which is equivalent to t 1 with vertices 0, v 1 , v 2 and the other of which is equivalent to t 2 with vertices 0, v 2 − v 1 , v 2 . Please refer to Figure 4. Since I is invariant under the translations generated by { v 1 , v 2 }, w * l(τ 1 (e)) = λw * l(e) and w * l(τ 2 (e)) = µw * l(e) for any edge e ∈ E by (21) and (2). It is straightforward to check that the generalized triangle τ 1 (t) (τ 2 (t) respectively) is a scaling of the generalized triangle t by a factor λ (µ respectively). As a result, the triangles in the same equivalence class are similar to each other. Assume that the inner angles in t i are α i , β i and γ i for i = 1, 2. Then it is clear from Figure 4 that the curvature K(v) = 0 for all v ∈ V by α i + β i + γ i = π. Therefore, (C, T st , I, w * l) is flat. To prove part (b), note that (λ, µ) = (1, 1) by the assumption that w is not a constant. Without loss of generality, assume λ = 1. By the similarity of triangles in (C, T st , I, w * l) under the action of τ 1 and τ 2 , the two translations τ 1 and τ 2 induce two affine transformations η and ζ of the plane when composed with the developing map φ. Taking τ 1 for example, by the similarities of triangles in (C, T st , I, w * l) under the action of τ 1 and τ 2 , there exists θ ∈ [0, 2π) such that φ(τ 1 (v)) − φ(v) = λe iθ [φ(v) − φ(τ −1 1 (v))] for any v ∈ V , which implies φ(τ 1 (v)) − λe iθ φ(v) = φ(v) − λe iθ φ(τ −1 1 (v)). Therefore, φ(τ 1 (v)) − λe iθ φ(v) is a constant for any v ∈ V , denoted by c ∈ C, which implies φ(τ 1 (v)) = λe iθ φ(v) + c. Similar arguments 15 apply to τ 2 . Therefore, there exists an affine transformation η(z) = λ * z + c with |λ * | = λ = 1 such that φ • τ 1 = η • φ. Since λ = 1, η has a unique fixed point p ∈ C. By τ 1 τ 2 = τ 2 τ 1 , we have ηζ = ζη, which implies ζ(p) = p by the uniqueness of the fixed point of η. Set φ(v) = φ(v) − p, thenφ is still a developing map. For simplicity, we still denoteφ by φ. Then φ(τ 1 (v)) = ηφ(v) = λ * φ(v) and φ(τ 2 (v)) = ζφ(v) = µ * φ(v) for some linear transformations η(z) = λ * z and ζ(z) = µ * z. Notice that the group G =< η, ζ > generated by η and ζ is either a non-trivial cyclic group or an abelian group isomorphic to Z 2 . By the assumption of part (b), U = φ(C) has non-empty interior containing the interior of a triangle t 1 . If G =< η, ζ > is a cyclic group, then there exists (k, j) = (0, 0) such that η k ζ j is the identity map. Set t 2 = τ k 1 τ j 2 (t 1 ). Then φ(t 2 ) = φ(τ k 1 τ j 2 (t 1 )) = η k ζ j φ(t 1 ) = φ(t 1 ), which implies φ(t 1 ) ∩ φ(t 2 ) = ∅. If G =< η, ζ > is isometric to Z 2 , then η k ζ j is never identity and their action in the plane is not discontinuous. Specifically, for W := φ(int(t 1 )), there exists (k, j) = (0, 0) so that η k ζ j (W ) ∩ W = ∅. Set t 2 = τ k 1 τ j 2 (t 1 ), then φ(t 1 ) ∩ φ(t 2 ) = η k ζ j (W ) ∩ W = ∅. To see part (c), since all the triangles in (C, T st , I, w * l) are degenerate, the inner angles of the triangles t 1 and t 2 are 0 or π. Composing with an automorphism of the triangulation T st , we may assume α 1 = γ 2 = π, where the angles are marked in Figure 4. For the degenerate triangle with vertices 0, − v 1 and − v 2 , it is flat at − v 2 by assumption. By Corollary 2.2, if we use κ * to denote the reciprocal of radii in the metric w * l, then (22) κ * (− v 2 ) = 1 I 2 0,− v 1 − 1 {γ − v 1 ,− v 2 ,0 κ * (0) + γ 0,− v 1 ,− v 2 κ * (− v 1 ) + ∆ 0,− v 1 ,− v 2 [(κ * (0)) 2 + (κ * (− v 1 )) 2 + 2I 0,− v 1 κ * (0)κ * (− v 1 )]}, where γ v i ,v j ,v k = I v j v k + I v i v k I v i v j and ∆ 0,− v 1 ,− v 2 = I 2 0,− v 1 + I 2 0,− v 2 + I 2 − v 1 ,− v 2 + 2I 0,− v 1 I 0,− v 2 I − v 1 ,− v 2 − 1. Note that κ * (0) = κ(0), κ * (− v 1 ) = κ(− v 1 )λ and κ * (− v 2 ) = κ(− v 2 )µ, we have (23) κ(− v 2 )µ = 1 I 2 0,− v 1 − 1 {γ − v 1 ,− v 2 ,0 κ(0) + γ 0,− v 1 ,− v 2 κ(− v 1 )λ + ∆ 0,− v 1 ,− v 2 [(κ(0)) 2 + κ 2 (− v 1 )λ 2 + 2I 0,− v 1 κ(0)κ(− v 1 )λ]} by (22). Denote the right hand side of the equation (23) as f 1 (λ). Then f 1 (λ) is a strictly increasing function of λ. Furthermore, we have lim λ→0+ f 1 (λ) = C 1 > 0 and lim λ→+∞ f 1 (λ) = +∞. Dividing both sides of (23) by λ gives (24) κ(− v 2 ) µ λ = 1 I 2 0,− v 1 − 1 {γ − v 1 ,− v 2 ,0 κ(0)λ −1 + γ 0,− v 1 ,− v 2 κ(− v 1 ) + ∆ 0,− v 1 ,− v 2 [(κ(0)) 2 λ −2 + κ 2 (− v 1 ) + 2I 0,− v 1 κ(0)κ(− v 1 )λ −1 ]}, which implies that µ λ is a strictly decreasing function of λ with lim λ→0+ µ λ = +∞ and lim λ→+∞ µ λ = C 2 > 0. 16 On the other hand, for the triangle with vertices 0, − v 2 and v 1 − v 2 , it is flat at − v 2 by assumption. Applying Corollary 2.2 to this triangle gives (25) κ * (− v 2 ) = 1 I 2 0, v 1 − v 2 − 1 {γ 0,− v 2 , v 1 −v 2 κ * ( v 1 − v 2 ) + γ v 1 − v 2 ,0,− v 2 κ * (0) + ∆ 0,− v 2 , v 1 − v 2 [(κ * (0)) 2 + (κ * ( v 1 − v 2 )) 2 + 2I 0,− v 1 κ * (0)κ * ( v 1 − v 2 )]}. Note that κ * (0) = κ(0), κ * (− v 2 ) = κ(− v 2 )µ and κ * ( v 1 − v 2 ) = κ( v 1 − v 2 ) µ λ , we have (26) κ(− v 2 )µ = 1 I 2 0, v 1 − v 2 − 1 {γ 0,− v 2 , v 1 − v 2 κ( v 1 − v 2 ) µ λ + γ v 1 − v 2 ,0,− v 2 κ(0) + ∆ 0,− v 2 , v 1 − v 2 [(κ(0)) 2 + κ 2 ( v 1 − v 2 ) µ 2 λ 2 + 2I 0,− v 1 κ * (0)κ( v 1 − v 2 ) µ λ ]} by (25). Denote the right hand side of the equation (26) as f 2 (λ). Then f 2 (λ) is a strictly decreasing function of λ by the fact that µ λ is a strictly decreasing function of λ. Furthermore, lim λ→0+ f 2 (λ) = +∞ and lim λ→+∞ f 2 (λ) = C 3 > 0. Set f (λ) = f 1 (λ) − f 2 (λ), then f (λ) is a strictly increasing continuous function of λ ∈ R >0 with lim λ→0+ f (λ) = −∞ and lim λ→+∞ f 2 (λ) = +∞, which implies that there exists a unique numberλ =λ(I, u) ∈ R >0 such that f 1 (λ) = f 2 (λ). As a result, the system (23) and (26) has a unique solutionλ =λ(I, u) andμ =μ(I, u) in R >0 . This completes the proof for part (c). Q.E.D. We call the weight I in Proposition 3.5 translation invariant on T st since I(e) = I(e + δ) for any δ ∈ V = L = {m v 1 + n v 2 } and e ∈ E. It is in fact determined by the three weights on the edges of any triangle in T st . RIGIDITY OF INFINITE INVERSIVE DISTANCE CIRCLE PACKINGS The conformality of the limit of discrete conformal maps f n in Theorem 1.2 is a consequence of the rigidity of infinite inversive distance circle packings in the plane, which is also conjectured by Bowers-Stephenson [3]. The main result of this section confirms this conjecture. The rigidity of inversive distance circle packings with prescribed combinatorial curvatures on weighted triangulated compact surfaces has been proved in [13,20,31,32] based on variational principles. Theorem 4.1 provides a result on the rigidity of infinite inversive distance circle packings in the non-compact plane. In the case of Thurston's circle packings, the rigidity of infinite circle packings in the plane has been explored in [14,25,26]. To prove Theorem 4.1, recall the following definition and properties of embeddable flat polyhedral surfaces in [21]. is said to be embeddable into C if for every simply connected finite subcomplex P of T , there exist a sequence of flat PL metrics on P whose developing maps φ n : P → C are topological embeddings and converge uniformly to φ| P . (1) Suppose φ is embeddable. If two simplices s 1 , s 2 represent two distinct non-degenerate triangles or two distinct edges in T , then φ(int(s 1 )) ∩ φ(int(s 2 )) = ∅. (2) If φ is the pointwise convergent limit lim n→∞ ψ n of the developing maps ψ n of embeddable flat polyhedral metrics (X, T , l n ), then (X, T , l) is embeddable. The standard hexagonal geodesic triangulations of open sets in C are embeddable. On the other hand, the generic Doyle spirals produce circle packings with overlapping disks, so the corresponding polyhedral metrics are not embeddable. Proof. Suppose e ∈ E is an edge with vertices v and v ′ . By Definition 1.1, we have u * ((w − w 0 ) * l 0 )(e) =[e 2w(v)+2u(v) + e 2w(v ′ )+2u(v ′ ) + 2I(e)e w(v)+u(v)+w(v ′ )+u(v ′ ) ] 1/2 =[e 2w(w n (v) := w(v + v n ) − w(v n ) satisfies (a) for all v ∈ V , the limit w ∞ (v) = lim n→∞ w n (v) exists. (b) w n * l 0 and w ∞ * l 0 are flat generalized weighted Delaunay inversive distance circle packing metrics on (S, T st , I). (c) w ∞ (v + δ) − w ∞ (v) = a := sup{w(v + δ) − w(v)|v ∈ V } for all v ∈ V . (d) the normalized developing maps φ wn * l 0 of w n * l 0 converges uniformly on compact subcomplex of (S, T st ) to the normalized developing maps φ w∞ * l 0 of w ∞ * l 0 . As a result, if (S, T st , I, w * l 0 ) is embeddable, then (S, T st , I, w ∞ * l 0 ) is embeddable. Proof. Since w 0 is a constant function, without loss of generality, we can assume that w 0 = 0, otherwise applying a scaling to l 0 . To see part (a), notice that Lemma 3.4 implies that there exists a positive constant M such that (27) M = M(V, I) = sup{|w(v + δ) − w(v)||v ∈ V, δ ∈ {±u 1 , ±u 2 , ±(u 1 − u 2 )}. Then for fixed δ ∈ {±u 1 , ±u 2 , ±(u 1 − u 2 ), we have a := sup{w(v + δ) − w(v)|v ∈ V } ≤ M. Therefore, there exist a sequence {v n } in V such that (28) a − 1 n ≤ w n (δ) = w(v n + δ) − w(v n ) ≤ a. Furthermore, we have w n (0) = 0 and (29) w n (v + δ) − w n (v) = w(v + δ + v n ) − w(v + v n ) ≤ a, ∀v ∈ V by the definition of w n and a. By Lemma 3.4, if v ∈ V is of combinatorial distance m to 0, then |w n (v)| =|w n (v) − w n (0)| ≤ m i=1 |w n (v i ) − w n (v i−1 )| = m i=1 |w(v i + v n ) − w(v i−1 + v n )| = m i=1 |w(v i + v n ) − w(v i−1 + v n )| ≤ mM, where v m = v, v 0 = 0 and v 0 ∼ v 1 ∼ · · · ∼ v m is a path of combinatorial distance m between 0 and v. By the diagonal argument, there exists a subsequence of {v n }, still denoted by {v n } for simplicity, such that w ∞ (v) := lim n→∞ w n (v) exists for all v ∈ V . To see part (b), for any fixed n ∈ N and any edge e ∈ E, we have (30) w n * l 0 (e) = e −w(vn) w * l 0 (e + v n ) by the translation invariance I(e) = I(e + δ) for the weight I. This implies that w n * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on (S, T st , I) by the assumption that w * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on (S, T st , I). By w ∞ (v) = lim n→∞ w n (v) and continuity, we have w ∞ * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on (S, T st , I). (29), which implies Moreover, we have w ∞ (v + δ) − w ∞ (v) ≤ a for any v ∈ V by(31) sup{w ∞ (v + δ) − w ∞ (v)|v ∈ V } ≤ a. To see part (c), by w n (0) = 0, (28) and (31), we have w ∞ (0) = 0 and If (S, T st , I, w * l 0 ) is embeddable, then (S, T st , I, w n * l 0 ) is embeddable by (30). The rest of the proof is an application of Lemma 4.3. w ∞ (δ) − w ∞ (0) = w ∞ (δ) = a ≥ sup{w ∞ (v + δ) − w ∞ (v)|v ∈ V }, which implies that w ∞ (v + δ) − w ∞ (v) attains the maximal value sup{w ∞ (v + δ) − w ∞ (v)|v ∈ V } at v = 0. Note Q.E.D. Theorem 4.1 follows from the following more general result. Theorem 4.6. Let (S, T st , I) be a weighted hexagonal triangulated plane with the weight I translation invariant, and l 0 be a weighted Delaunay inversive distance circle packing metric on (S, T st , I) generated by a constant label w 0 : V → R such that the vertex set is a lattice L = V = {mu 1 + nu 2 |m, n ∈ Z}. Suppose w * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on the plane (S, T st , I) and (S, T st , I, w * l 0 ) is embeddable into C. Then w is a constant function. Proof. The idea of proof can be summarized as follows. Assume w is not a constant, we will construct a sequence of discrete conformal factor w n by extracting "directional derivatives" of w at different base points. This construction relies heavily on the symmetric structure of the lattice V (T st ) = L generated by I and w 0 , which implies that the limit of this sequence produce a linear discrete conformal factor w ∞ . By Lemma 4.5, (S, T st , I, w ∞ * l 0 ) is embeddable. However, by Proposition 3.5, if w ∞ is not a constant, (S, T st , I, w ∞ * l 0 ) contains overlapping triangles under the developing maps. This leads to a contradiction. Step 1: construct a linear discrete conformal factorŵ. Since w is assumed to be different from a constant function, then there exists a δ 1 ∈ {±u 1 , ±u 2 , ±(u 1 − u 2 )} such that a 1 = sup{w(v + δ 1 ) − w(v)|v ∈ V } > 0. By Lemma 3.4, a 1 ∈ (0, ∞). Applying Lemma 4.5 to w * l 0 in the direction δ 1 , we can find a function w ∞ : V → R such that (S, T st , I, w ∞ * l 0 ) is embeddable and w ∞ (v + δ 1 ) − w ∞ (v) = a 1 , ∀v ∈ V. Further applying Lemma 4.5 to w ∞ * l 0 in the direction δ 2 ∈ {±u 1 , ±u 2 , ±(u 1 − u 2 )} − {±δ 1 } gives rise to a functionŵ = (w ∞ ) ∞ : V → R such that (S, T st , I,ŵ * l 0 ) is embeddable. Moreover,ŵ (v + δ 1 ) −ŵ(v) = a 1 ,ŵ(v + δ 2 ) −ŵ(v) = a 2 , ∀v ∈ V, which shows thatŵ(v) is an affine function of the formŵ(nδ 1 + mδ 2 ) = na 1 + ma 2 + a 3 with a 1 ∈ (0, +∞), a 2 , a 3 ∈ R. Without loss of generality, we can assumeŵ(nδ 1 + mδ 2 ) = na 1 + ma 2 , because the weighted Delaunay property of generalized inversive distance circle packing metrics is invariant under scaling. Furthermore, (S, T st , I,ŵ * l 0 ) is embeddable. Step 2: Overlapping of (S, T st , I,ŵ * l 0 ). By Step 1, there are two positive numbersλ ∈ (1, +∞) andμ ∈ (0, +∞) so that w(mδ 1 + nδ 2 ) = m logλ + n logμ and (S, T st , I,ŵ * l 0 ) is embeddable. Then there is no non-degenerate triangle in the image of the developing mapφ for (S, T st , I,ŵ * l 0 ). Otherwise by Proposition 3.5, there are two triangles with overlapping interiors. Therefore, all the triangles in the image of (S, T st , I,ŵ) underφ are degenerate. All the inner angles are either 0 and π. Up to obvious automorphisms of T st , there are three cases in Figure 5 showing triangles in the star of the origin. Case 1 and Case 2 are differed by a rotation γ, and Case 1 and Case 3 are differed by an orientationreversing automorphism ρ of T st such that ρ(0) = 0, ρ( v 1 ) = v 2 , and ρ( v 2 ) = v 2 − v 1 . Therefore, we only need to consider Case 1. By Proposition 3.5 (c), the constantsλ andμ depend only on I and w 0 . 0 0 0 π 0 0 0 π 0 π 0 0 π π π 0 0 0v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 l 1 l 2 l 3 FIGURE 6. Intersecting edges in the developing maps. Consider the lengths of edges e 1 = v 0 v 3 , e 2 = v 0 v 6 , e 3 = v 6 v 7 and their respective lengths l 1 , l 2 , l 3 inŵ * l 0 in Figure 6. Notice that l 3 = (λ/μ)l 2 and l 1 = (μ/λ)l 2 , then l 1 + l 3 ≥ 2l 2 > l 2 . Since (S, T st , I,ŵ * l 0 ) with a developing mapφ is embeddable, there exist a sequence of flat polyhedral metrics with developing maps φ n which are embeddings such that φ n converge toφ uniformly on compact sets. Then for n large enough, the images of e 1 and e 3 under φ n intersect by the inequality above. This is because the angle condition at v 6 forces e 3 to rotate clockwise and the angle condition at v 0 forces e 1 to rotate counterclockwise. Then the intersection contradicts the fact that (S, T st , I,ŵ * l 0 ) is embeddable. Q.E.D. THE CONVERGENCE OF INVERSIVE DISTANCE CIRCLE PACKINGS 5.1. Proof of the main theorem. Recall the main theorem of this paper. Theorem 5.1. Let Ω be a Jordan domain in the complex plane bounded by a Jordan curve ∂Ω with three distinct points p, q, r ∈ ∂Ω. Let f be the Riemann mapping from the unit equilateral triangle △ABC to Ω such that f (A) = p, f (B) = q, f (C) = r. There exists a sequence of weighted triangulated polygonal disks (Ω n , T n , I n , (p n , q n , r n )) with inversive distance circle 21 packing metrics l n , where T n is an equilateral triangulation of Ω n , I n : E n → (1, +∞) is a weight defined on E n = E(T n ) and p n , q n , r n are three boundary vertices of T n , such that (1) Ω = ∪ ∞ n=1 Ω n with Ω n ⊂ Ω n+1 , and lim n p n = p, lim n q n = q and lim n r n = r, (2) the discrete conformal maps f n from △ABC to (Ω n , T n , I n , l n ) with f n (A) = p n , f n (B) = q n , f n (C) = r n exist, (3) the discrete conformal maps f n converge uniformly to the Riemann mapping f . To prove Theorem 5.1, we need to establish the existence of discrete conformal maps induced by inversive distance circle packings from a flat polyhedral disk to an equilateral triangle. In the case of Thurston's circle packings, the Koebe-Andreev-Thurston theorem guarantees the existence of a circle packing of the unit disk with any given triangulation of a disk as the nerve of the packing. In the case of vertex scaling [19], the discrete uniformization theorem [10,11] gives the existence of a discrete conformal factor with prescribed combinatorial curvature on marked closed surfaces. Unfortunately, there is no known existence theorem for inversive distance circle packings on arbitrary triangulations. Theorem 5.2 below establishes such an existence theorem for inversive distance circle packings when the triangulations of flat polyhedral disks are subdivided in a scheme as follows. Let (P, T , l) be a flat polyhedral disk with an equilateral triangulation, in which all triangles are equilateral. Then the length function l is a constant function on E. Given an equilateral Euclidean triangle △ in the plane, the n-th standard subdivision of △ is the equilateral triangulation of △ by n 2 equilateral triangles. Applying this subdivision to each triangle in an equilateral triangulation of a flat polyhedral disk (P, T , l), we obtain its n-th standard subdivision (P, T (n) , l (n) ). Furthermore, if l is an inversive distance circle packing metric induced by a constant label u and a constant weight I : E → (1, +∞), we require that l (n) is also an inversive distance circle packing metric induced by a constant label u (n) and a constant weight I n : E (n) → (1, +∞) taking the same value as I : E → (1, +∞). Theorem 5.2. Suppose (P, T , l) is a flat polyhedral disk with an equilateral triangulation T such that exactly three boundary vertices p, q, r have curvature 2π 3 , and the metric l is an inversive distance circle packing metric induced by a constant label u and a constant weight I : E → (1, +∞). Then for sufficiently large n, there is a discrete conformal factor w : V (n) → R for the n-th standard subdivision (P, T (n) , I (n) , l (n) ) such that (1) K i (w * l (n) ) = 0 for all v i ∈ V (n) − {p, q, r}, (2) K i (w * l (n) ) = 2π 3 for all v i ∈ {p, q, r},(3) there is a constant θ 0 = θ 0 (I) > 0 independent of n such that all inner angles of triangles in (T (n) , w * l (n) ) are in the interval [θ 0 , π/2 + θ 0 ]. Note that the underlying metric space of (P, T (n) , I (n) , w * l (n) ) is an equilateral triangle, and (P, T (n) , I (n) , l (n) ) is weighted Delaunay for each n. Assuming Theorem 5.2, the proof of Theorem 5.1 is a standard argument using the properties of quasiconformal maps in the plane. To achieve this aim, we first recall the following three theorems on the extension and convergence of quasiconformal maps. The following theorem is a simple consequence of Lemma 2.1 and Theorem 2.2 in [17]. 22 Theorem 5.4. If f n : D → Ω n is a sequence of K-quasiconformal maps such that Ω n is uniformly bounded, then every subsequence of f n contains a subsequence that converge locally uniformly. Moreover, the limit of this subsequence is a K-quasiconformal map or a constant map. A sequence of Jordan curves J n in C converge uniformly to a Jordan curve J in C if there exist homeomorphisms φ n : S 1 → J n and φ : S 1 → J such that φ n converge uniformly to φ. Theorem 5.5 ([23], Corollary 1). Assume that Ω n is a sequence of Jordan domains such that ∂Ω n converge uniformly to ∂Ω. If f n : D → Ω n is a K-quasiconformal map for each n, and the sequence {f n } converge to a K-quasiconformal map f : D → Ω uniformly on compact sets of D, then f n converge to f uniformly on D. Proof of Theorem 5.1. By taking the intersection of scalings of the standard hexagonal triangulation in the plane with Ω, we can construct a sequence of nested polygonal disks Ω n such that ∂Ω n converge uniformly to ∂Ω and there are three boundary vertices p n , q n , r n ⊂ ∂Ω n such that lim n p n = p, lim n q n = q and lim n r n = r. By adding or subtracting boundary vertices if necessary, we can assume that the curvatures at p n , q n , r n ∈ ∂Ω n are 2π 3 and the curvatures at all other boundary vertices of Ω n are not 2π 3 . By Theorem 5.2, we produce some standard subdivision T n of Ω n and some discrete conformal factors w n such that (Ω n , T n , w n * l st ) is isometric to the unit equilateral triangle (△ABC, T n ), where A, B, C correspond to p n , q n , r n respectively. Let f n : (△ABC, T n , (A, B, C)) → (Ω n , T n , (p n , q n , r n )) be the discrete conformal map induced by the correspondence of triangulations. Letf be the Riemann mapping from △ABC to Ω sending A, B, C to p, q, r respectively. We claim that f n converges uniformly tof on △ABC. By Theorem 5.2, all angles of triangles in (△ABC, T n , w * l (n) ) are at least ǫ 0 > 0. Then the discrete conformal maps f n are K-quasiconformal from int(△ABC) to int(Ω n ) for some constant K independent of n and continuous from △ABC to Ω n . Letf n be the restriction of f n in int(△ABC). Theorem 5.4 implies that every convergent subsequence of {f n } converge to a K-quasiconformal mapg from int(△ABC) to int(Ω). Since Ω = ∪ n Ω n ,g is onto int(Ω). Theorem 5.3 implies thatg extends to a homeomorphism g : △ABC → Ω. Theorem 5.5 implies that f n converge uniformly to g on △ABC. It is straightforward to check that g(A) = p, g(B) = q, and g(C) = r. Notice that the Riemann mappingf is the only continuous extension of a conformal map from int(△ABC) to Ω withf (A) = p,f (B) = q, andf (C) = r. This means that if we can show g is conformal, then g =f and all limits of convergent subsequences of {f n } aref . This will complete the proof of f n →f uniformly on △ABC. The conformality of g follows from Theorem 4.6 by the same argument as the Hexagonal Packing Lemma in [25]. We briefly repeat the arguments here for completion. For a vertex v 0 ∈ T st , let B n be the n-ring neighborhood of v 0 in T st . Then B n is a finite simplicial complex whose underlying space is a topological disk D. Assume that l n is a flat inversive distance circle packing on B n with the constant weight I. Let s n be the maximal ratio of radii of two adjacent circles in l n of B n . Lemma 3.4 implies that s n is uniformly bounded by some constant C(I). As n → ∞, we can pick a convergent subsequence of (D, B n , I, l n ), still indexed with n, such that all circles converge geometrically. We claim that lim n s n = 0. Otherwise, as n → ∞, the limit produces an inversive distance circle packing on T st such that circles have different sizes. This contradicts the fact that w is a constant in Theorem 4.6. 23 As n → ∞, the arguments above show that s n of T n goes to zero. Equilateral triangles in T n contained in a compact subset of Ω are mapped by f −1 n to triangles in (△ABC, T n ) which are close to be equilateral. Then f n restricted in each triangle converge to a similarity map. The dilatations K n of f n converge to 1. Therefore, g is 1-conformal, which is equivalent to be conformal. Q.E.D. The rest of this paper is devoted to prove Theorem 5.2. To find the discrete conformal factors in Theorem 5.2, we will construct a system of ordinary differential equations proposed in [12] to deform the discrete conformal factors via discrete curvatures at vertices. We first consider such a flow on a standard subdivision of an equilateral triangle in Theorem 5.10, then use the flow to construct the discrete conformal factor required in Theorem 5.2. In the rest of this section, we assume that any initial polyhedral metric l on (P, T ) is an inversive distance circle packing metric induced by a constant weight I > 1 and a constant label, which is weighted Delaunay. For any discrete conformal factor w on V , denote the angle at v k of a triangle △v i v j v k in the metric w * l as θ k ij (w). Similarly, the conductance of w * l defined by the formula (11) is denoted as η(w), and the curvature of w * l is denoted as K(w). The notation a = O(b) refers to the fact that |a| ≤ C|b| for some constant C = C(I) > 0. 5.2. Inversive distance circle packings along flows. In this subsection, we will solve the following prescribing curvature problem: assume V 0 ⊂ V and the initial curvature of (P, T , l) is K 0 . Given a prescribed curvature K * on V − V 0 , find a discrete conformal factor w such that w| V 0 = 0 and K(w) = K * . Consider a smooth family of discrete conformal factors w(t) satisfying (32) K i (w(t)) = (1 − t)K 0 i + tK * i , v i ∈ V − V 0 , w i (t) = 0, v i ∈ V 0 , and w(0) = 0. This family of w(t), if it exists in the interval [0, 1], provides a linear interpolation between the initial curvature K 0 and the prescribed curvature K * . Therefore, w(1) * l has curvature K * . Hence, we need to show that this flow exists on [0, 1] for some standard subdivisions of T on V − V 0 . To this end, we recall some basic notions of analysis on graphs. Given a graph (V, E), the set of all oriented edges in (V, E) is denoted byĒ. If v i v j is an edge in E, we denote it as i ∼ j. A conductance on G is a function η :Ē → R ≥0 so that η ij = η ji . The following definitions and results are well-known. See [18] for details. Definition 5.6. Given a finite graph (V, E) with a conductance η, the gradient ▽ : R V → RĒ is defined by (∇f ) ij = η ij (f i − f j ), the Laplace operator associated to η is the linear map ∆ : R V → R V defined by (∆f ) i = j∼i η ij (f i − f j ). Given a set V 0 ⊂ V and a function g : V 0 → R, the solution to the Dirichlet problem is a function f : V → R satisfying (∆f ) i = 0, ∀v i ∈ V − V 0 , and f | V 0 = g. Proposition 5.7. Suppose (V, E) is a finite connected graph with a conductance η(e) > 0 for any edge e ∈ E. Given a nonempty V 0 ⊂ V and g : V 0 → R, the solution f to the Dirichlet problem exists. Moreover, (a) (Maximum principle) max v i ∈V f i = max v i ∈V 0 f i . (b) (Strong maximum principle) If V − V 0 is connected and max v i ∈V −V 0 f i = max v i ∈V 0 f i , then f | V −V 0 is a constant function. Recall that the formula (11) defines a conductance η for any inversive distance circle packing on (S, T , I). If it is weighted Delaunay, then η ij ≥ 0. In the rest of this paper, we assume that the Laplace operator ∆ is induced from this conductance η for an inversive distance circle packing. By the variation formula of curvatures in (12), we have the following system of ODEs by taking derivative of equation (32) with respect to t (33) (∆w ′ ) i = j∼i η ij (w ′ i − w ′ j ) = K * i − K 0 i , v i ∈ V − V 0 , w ′ i (t) = 0, v i ∈ V 0 , with the initial value w(0) = 0 and w ′ i = dw i dt . We will show that the solution to the system (33) exists for all t ∈ [0, 1] if (P, T , l) is chosen carefully. Prior to the existence, we first characterize the maximal interval for the existence of the solution to (33). Given a weighted triangulated surface (S, T , I) with an inversive distance circle packing metric l, consider the set of discrete conformal factors W ⊂ R V defined by (34) W = {w ∈ R V |w * l is an inversive distance circle packing metric on (S, T , I) such that η ij > 0 for all edges}. Lemma 5.8. Let (P, T , I) be a weighted triangulated surface with an inversive distance circle packing metric l generated by a label u. The initial valued problem (33) defined on W has a unique solution in a maximum interval [0, t 0 ) with t 0 > 0 if V 0 = ∅ and 0 ∈ W . Moreover, if t 0 < ∞, then either lim inf t→t − 0 θ i jk (w(t)) = 0 for some angle θ i jk or lim inf t→t − 0 η ij (w(t)) = 0 for some edge v i v j . Proof. The ODE system (33) can be written as A(w) · w ′ (t) = b, w(0) = 0, where A(w) is a square matrix valued smooth function of w, b is a column vector determined by curvature, and w ′ (t) is a column vector. Then A(w) is an invertible matrix for a fixed w ∈ W . Indeed, consider the following system of linear equations for a fixed w (35) A(w) · f = 0. From (33) we know that equation (35) is equivalent to (∆f ) i = 0, v i ∈ V − V 0 , f i = 0, v i ∈ V 0 , where η ij > 0 for all edges since w ∈ W . The maximal principle in Proposition 5.7 implies that f = 0. Therefore, A(w) is invertible. As a result, (33) can be written as w ′ (t) = A(w) −1 b. Picard's existence theorem for solutions to the ODE systems implies that there exists an interval [0, t 0 ) on which (33) has a solution. If t 0 < ∞ and t ր t 0 , then w(t) leaves every compact set in W . Consider subsets W δ = {w ∈ W |θ i jk ≥ δ, |w i | ≤ 1 δ , η ij ≥ δ}. It is straightforward to check that W δ is compact. Since w(t) leaves every W δ for each δ > 0, one of the following three cases occurs: 25 (1) lim inf t→t − 0 θ i jk (w(t)) = 0 for some θ i jk , or (2) lim inf t→t − 0 η ij (w(t)) = 0 for some edge , or (3) lim sup t→t − 0 |w i (t)| = +∞ for some v i ∈ V . We claim that the case (3) implies the case (1). Otherwise, there exists δ > 0 such that lim inf t→t − 0 θ i jk (w(t)) > δ for all θ i jk . Since w ′ i (t) = 0 for v i ∈ V 0 along the flow (33), the radius r i = e w i +u i does not change along the flow. Then for any triangle △v i v j v k with v i as a vertex, the sine law implies that l 2 ij l 2 ik ≤ 1 sin 2 δ , l 2 ik l 2 ij ≤ 1 sin 2 δ , which further implies that r j ≤ √ I sin δ (r i +r k ) and r k ≤ √ I sin δ (r i +r j ) by I > 1 and (2). Therefore, r j and r k are of the same order. Specially, r j → +∞ if and only if r k → +∞. If w k and w j go to infinity, then cos θ i jk = l 2 ij + l 2 ik − l 2 jk 2l ij l ik = r 2 i + r i r j I + r i r k I − r j r k I l ij l ik → −I < −1, which is impossible. Since the 1-skeleton of T is a finite connected graph, we can show inductively that w i is bounded for all v i ∈ V , which contradicts the assumption in case (3). This completes the proof for the claim. Q.E.D. 5.3. Standard subdivisions of an equilateral triangle. In this subsection, we consider the ODE system (33) when the polyhedral surface is an equilateral triangulation of an equilateral triangle. We will prescribe special curvatures at the boundary vertices such that the discrete conformal maps approximate the power functions in complex analysis. To apply the estimates in network theory, we need to bound the conductance of a weighted Delaunay triangulation as follows. Lemma 5.9. Let △v 1 v 2 v 3 be a weighted triangle generated by an inversive distance circle packing (r 1 , r 2 , r 3 ) and the weight I > 1 is a constant. There exists a constant θ 0 = θ 0 (I) ∈ (0, π 6 ) such that if the three inner angles of the triangle are bounded in [π/6 − θ 0 , π/2 + θ 0 ], then (a) r j /r i ≤ 20 for any two radii r i and r j , (b) C ≤ η k ij ≤ M for some constants C = C(I) > 0 and M = M(I) > 0. Proof. Set θ 0 = min{ π 1000 , arcsin 1 10(20 + I) }. To prove part (a), by the angle bound and the sine law, (36) l ij /l ik ≤ 1/ sin(π/6 − π/1000) < √ 5 for any two edges in the triangle. Without loss of generality, assume that r i = 1. We will prove that r j ≤ 20 by contradiction in the following two cases. If r j > 20 and r k /r j ≤ 1/5, then l 2 ij l 2 ik ≥ r 2 j /5 + 2Ir j + 1 + 4r 2 j /5 (r j /5) 2 + 2Ir j /5 + 1 ≥ 5, which contradicts (36). If r j > 20 and r k /r j > 1/5, then r k > 4 and the inner angle θ i jk at v i is the largest inner angle in △v i v j v k . Note that in this case, we have I(r k + r j − r k r j ) + 1 < 0, l ij < √ I(r j + 1) and l ik < √ I(r k + 1). As a result, by the cosine law, we have cos θ i jk = l 2 ij + l 2 ik − l 2 jk 2l ij l ik = I(r k + r j − r k r j ) + 1 l ij l ik < −I(r k + r j + r k r j + 1) + 2I(r k + r j ) + I + 1 I(r k + r j + r k r j + 1) ≤ −1 + 2I(r k + r j ) + 2I I(r k + r j + r k r j + 1) ≤ −11 21 . t ∈ [0, s 0 ). Therefore, Lemma 5.8 implies that 1 ≤ s 0 < t 0 . Then w(1) ∈ W . By continuity, the metric w(1) * l is a non-degenerated inversive distance circle packing metric satisfies (1)-(4) in Theorem 5.10. Finally, we use Lemma 5.12 to prove the last statement in Theorem 5.10. Notice that by Lemma 5.9, 0 < C ≤ η ij ≤ M, where C = C(I), M = M(I). Applying Lemma 5.12 to the function f = dw(t) dt /|α − π/3|, we obtain dK i (t) dt = |(∆w ′ ) i | = |α − π/3| · |(∆f ) i | = O( 1 ln(n) ), v i ∈ V 0 . Then for v i = A, we have for t ∈ [0, 1], |K i (t) − K i (0)| ≤ t 0 dK i (t) dt dt = O( 1 ln(n) ). Moreover, if α = π/3, then (38) is automatically true since the flow would be a constant flow by Proposition 5.7. Hence we assume α = π/3. We claim that w ′ A (t) = 0 for t ∈ [0, t 0 ). Otherwise, w ′ A (s) = 0 for some s ∈ [0, t 0 ). Applying the maximum principle, i.e. Proposition 5.7, to the following Dirichlet problem (∆w ′ (s)) i = 0, v i ∈ V − {A} ∪ V 0 , w ′ i (s) = 0, v i ∈ {A} ∪ V 0 , we obtain w ′ i (s) = 0 for all v i ∈ V , which implies (∆w ′ ) A (s) = 0. This contradicts (∆w ′ ) A (s) = α − π 3 = 0. This completes the proof of the claim. Furthermore, applying the maximal principle, i.e. Proposition 5.7, to (40) again shows that w ′ A (t) and w ′ i (t) have the same sign. Note that for v i ∈ V 0 , w ′ i (t) = 0. We have w ′ A (t)K ′ i (t) = w ′ A (t) i∼j η ij (w ′ i − w ′ j ) = − i∼j η ij w ′ j (t)w ′ A (t) ≤ 0, which implies (K i (t) − K i (0))w ′ A (t) ≤ 0. By the discrete Gauss-Bonnet formula (1), we have K A (t) + v i ∈V 0 K i (t) = K A (0) + v i ∈V K i (0) = 2π. Since K i (t) − K i (0) have the same sign for all v i ∈ V 0 , we conclude that for t ∈ [0, 1], v i ∈V 0 |K i (t) − K i (0)| = | v i ∈V 0 (K i (t) − K i (0))| = |K A (t) − K A (0)| = |t(α − π 3 )| ≤ π 6 . Q.E.D. 5.4. Proof of Theorem 5.2. There are two steps to find the discrete conformal factor required in Theorem 5.2. In the first step, we construct a discrete conformal factor by Theorem 5.10 where the triangles contain boundary vertices of nonzero curvatures. This step will diffuse the curvature of boundary vertices of polyhedral disk P to interior vertices such that curvatures are small if the subdivision is sufficiently dense. In the second step, we eliminate these small curvatures using a flow similar to (33). We need the following lemma in the second step. χ(S) is the Euler characteristic of the surface. A vertex v is flat in a PL metric if K(v) = 0. A PL metric is flat if all interior vertices are flat. Definition 1.1 ([3]). Suppose (S, T ) is a triangulated surface with a weight I : E → (−1, +∞). Lemma 2.3 ([13,31,32]). Let △v 1 v 2 v 3 be a face in (S, T , I) given by Lemma 2.1. Proposition 2. 5 . 5Let △v 1 v 2 v 3 be a face in (S, T , I) given by Lemma 2.1. . Weighted Delaunay triangulations. Weighted Delaunay triangulations are natural generalizations of the classical Delaunay triangulations, where the sites generating the corresponding Voronoi decomposition are disks instead of points. Definition 2 . 10 . 210Suppose r : V → (0, +∞) is a generalized inversive distance circle packing on a weighted triangulated surface (S, T , I). Lemma 3. 4 . 4Let T be a star triangulation of an n-sided polygon P n with boundary vertices v 1 , . . . , v n and a unique interior vertex v 0 . Let I : E → (1, +∞) be a weight and r be a flat generalized inversive distance circle packing on (P n , T , I). Then there exists a constant C = C(I, n) > 0 such that r 0 ≤ Cr i for all i ∈ {1, 2, · · · , n}. 13 (m) 1 =(m) 2 = 12+∞. We have lim m→∞ κ +∞ by(20), which is equivalent to lim m→∞ r have lim m→∞ (θ 0 12 ) (m) → 0, where θ 0 12 is the inner angle of the triangle △v 0 v 1 v 2 at v 0 . The same arguments applying to the triangles △v 0 Proposition 3. 5 . 5Let T st be the standard hexagonal triangulation of the plane. Let l be a weighted Delaunay inversive distance circle packing metric determined by a label u :V → R on (C, T st , I) such that the vertex set is a lattice V = L = {m v 1 + n v 2 }, where { v 1 , v 2 } isa geometric basis of the lattice L, and I is invariant under the translations generated by { v 1 , v 2 }. Suppose w : V → R is a non-constant linear function defined by two positive numbers λ and µ via 1 + n v 2 ) = m log λ + n log µ and w * l is a generalized weighted Delaunay inversive distance circle packing metric on (C, T st , I). Then the following statements hold.(a) (C, T st , I, w * l) is flat. FIGURE 4 . 4Angles of spiral triangulations. Theorem 4 . 1 . 41Let (C, T st , I) be a weighted hexagonal triangulated plane such that the weight I : E → (1, +∞) is translation invariant. Assume l is a weighted Delaunay inversive distance circle packing metric on (C, T st , I) induced by a constant label. If (C, T st , I, w * l) is a weighted Delaunay triangulated surface isometric to an open set in the plane, then w is a constant function. Definition 4.2 ([21], Definition 4.1). Suppose (S, T ) is a simply connected triangulated surface with a generalized PL metric l and φ is a developing map for (S, T , l). Then (S, T , l, φ) Let (S, T , l) be a flat polyhedral metric on a simply connected surface with a developing map φ. Lemma 4 . 4 . 44Let (S, T st , I) be a weighted hexagonal triangulated plane with the weight I translation invariant, and l 0 be a weighted Delaunay inversive distance circle packing metric on (S, T st , I) generated by a label w 0 : V → R such that the vertex set is a lattice L = V . Suppose (w − w 0 ) * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on the plane (S, T st , I). For any δ ∈ V , set u(v) = w(v + δ) − w(v). Then u * ((w −w 0 ) * l 0 ) = (u+w −w 0 ) * l 0 is a flat generalized weighed Delaunay inversive distance circle packing metric on (S, T st ). Furthermore, if u(v 0 ) = max v∈V u(v), then u is a constant. v+δ) + e 2w(v ′ +δ) + 2I(e + δ)e w(v+δ)+w(v ′ +δ) ] 1/2=(w − w 0 ) * l 0 (e + δ),where I(e) = I(e + δ) is used in the third line. By the condition that (w − w 0 ) * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on (S, T st , I), we have u * ((w − w 0 ) * l 0 ) is a flat generalized weighted Delaunay inversive distance circle packing metric on (S, T st , I). The rest of the proof is an application of the discrete maximal principle, i.e. Theorem 3.1.Q.E.D. The function u in Lemma 4.4 could be taken as a discrete version of the directional derivative of w. Lemma 4. 5 . 5Let (S, T st , I) be a weighted hexagonal triangulated plane with the weight I translation invariant, and l 0 be a weighted Delaunay inversive distance circle packing metric on (S, T st , I) generated by a constant label w 0 : V → R such that the vertex set is a lattice L = V = {mu 1 + nu 2 |m, n ∈ Z}. Suppose w * l 0 is a flat generalized weighted Delaunay inversive distance circle packing metric on the plane (S, T st , I). Then for any δ ∈ {±u 1 , ±u 2 , ±(u 1 − u 2 )}, there exists a sequence {v n } ⊂ V such that that for a fixed δ and u(v) := w ∞ (v + δ) − w ∞ (v), u * (w ∞ * l 0 ) is a flat generalized weighted Delaunay inversive distance circle packing metric on (S, T st , I) by Lemma 4.4. By Theorem 3.1, we have w ∞ (v + δ) − w ∞ (v) = a for any v ∈ V . 19 FIGURE 5 . 5Three cases of degenerate triangulations. Theorem 5.3 ([1], Corollary in Page 30). If f : D → Ω is a K-quasiconformal map from the open unit disk D onto a Jordan domain Ω, then f extends continuously to a homeomorphism f : D → Ω. Lemma 5 . 513 ([21], Proposition 5.10). Suppose (P, T ′ , l) is polygonal disk with an equilateral triangulation and T is the n-th standard subdivision of the triangulation T ′ with n ≥ e 10 6 . Let η : E = E(T ) → [ 1 M , M] be a conductance function with M > 0 and ∆ : R V → R V be the associated Laplace operator. Let V 0 ⊂ V (T ) be a thin subset such that for all v ∈ V 30 The research of Xu Xu is supported by the Fundamental Research Funds for the Central Universities under Grant No. 2042020kf0199. 3 2. INVERSIVE DISTANCE CIRCLE PACKINGS AND WEIGHTED DELAUNAYTRIANGULATIONS ∂h 12,3 ∂κ 3 . Therefore, there exists ǫ > 0 such that ∂h 12,3 ∂r 3 (r 1 , r 2 , r 3 + t) > 0 for t ∈ (0, ǫ).Q.E.D. κ 0 = κ 0 = f (κ j−1 , κ j ) < f (κ j−1 , κ j ). This contradicts that the angle bound is [π/6 − θ 0 , π/2 + θ 0 ].To prove part (b), the definition of η k ij and the formula (9) implies η k ij = h ij,kwhere A = l ij l ik sin θ i jk . The sign of η k ij is determined by h k . We will show (37)We just need to check the case that r k /r i ≤ 1 and r k /r j ≤ 1. If both r k /r i ≥ 1/2 and r k /r j ≥ 1/2, then r k h k ≥ 1+I. Hence, we only need to consider the situation that r k /r i < 1/2 or r k /r j < 1/2. By the angle bound and cosine law, we have − 1 5(20 + I) l 2 ij ≤ l 2 jk + l 2 ik − l 2 ij .This is equivalent to I(r i r k + r j r k − r i r j ) ≥ −r 2 k − 1 10(20 + I) (r 2 i + r 2 j + 2Ir i r j ), which implieswhere the results in part (a) of Lemma 5.9 is used in the last inequality. Then by the formula (10) of h k , we haveTherefore, under the assumption that r k /r i ≤ 1 and r k /r j ≤ 1, we have r k h k ≥ 3(1 + I)/10 > (1 + I)/4 when r k /r i < 1/2 or r k /r j < 1/2. The sine law implies that l 2 ij /50 ≤ A ≤ 5l 2 ij . Combining with part (a) of Lemma 5.9, we can find two constants M = M(I) and C = C(I) such thatTheorem 5.10. Let P = △ABC be an equilateral triangle, T (n) be the n-th standard subdivision of P, l be an inversive distance circle packing metric on (P, T (n) ) induced by a constant weight I and a constant label. SetGiven any α ∈ [ π 6 , π 2 ], there exists a smooth family of discrete conformal factors w(t) ∈ R V for t ∈ [0, 1] such that w(0) = 0 and w(t) * l is an inversive distance circle packing metric on T (n) with curvature K(t) = K(w(t) * l) satisfyingall inner angles θ i jk (t) in the metric w(t) * l are in the intervalMoreover,Notice that the angle at the vertex A is tα + (1 −t)π/3 along w(t), and curvatures of vertices stay zero except vertices in BC and the vertex A. The piecewise linear map from (P, T (n) , l) to (P, T (n) , w * l) determined by Theorem 5.10 is a discrete analogue of the analytic function f (z) = z 3α/π . This construction works for any n-th subdivision of equilateral triangulations.To prove Theorem 5.10, we need the following two estimates for solutions to the Dirichlet problem when the graph is an equilateral triangulation of a polygonal disk.Lemma 5.11 ([21], Lemma 5.8). Assume ∆ABC, n, T , V 0 are as given in Theorem 5.10. Let τ : T → T be the involution induced by the reflection of ∆ABC about the angle bisector of ∠BAC and η : E → R ≥0 be a conductance so that ητ = η and η ij = η ji . Let ∆ :Proof of Theorem 5.10. We will prove Theorem 5.10 by considering the ODE system (33) for (△ABC, T (n) , l) when n is sufficiently large. The prescribed curvature K * isBowers-Stephenson's conjecture on convergence of inversive distance circle packingsThe initial curvature K 0 isC}. Then the ODE system (33) could be written aswith the initial value w i (0) = 0. It is straightforward to check that w(0) ∈ W , where W is defined by (34). Then there exists a maximum t 0 > 0 such that a solution w(t) to (40) exists. Moreover, Lemma 5.8 implies that there exists a maximal time s 0 such that w(t) ∈ W and the statement (4) holds for t ∈ [0, s 0 ). We will prove s 0 ≥ 1. Moreover, w(1) exists and w(1) * l is an inversive distance circle packing metric satisfying (1)-(4) in Theorem 5.10. Without loss of generality, we assume that s 0 < ∞.Claim: For any inner angle θ i jk and t ∈ [0, s 0 ), we haveTo prove this claim, notice that α ∈ [π/6, π/2] implies θ i jk (t) ∈ [π/6, π/2] for t ∈ [0, s 0 ) by the statement (4). By Lemma 5.9, η k ij (t) > C(I) > 0 for anyBy the formula (8), we haveThen for all t ∈ [0, s 0 ),Now it is not hard to show that s 0 ≥ 1 from the claim. Notice that lim inf t→s − 0 θ i jk (w(t)) ≥ π/6 and lim inf t→s − 0 η ij (w(t)) > 0 for all edges by Lemma 5.9. Therefore, as t → s − 0 , for some θ i jk ,by the definition of s 0 and Lemma 5.8. If s 0 < 1, then for all the inner angles, we have lim supby (41), which contradicts (42). Therefore, s 0 ≥ 1.Notice that θ i jk (t) ∈ [π/6, π/2] for all inner angles and η ij (t) ≥ C(I) > 0 by Lemma 5.9 when t ∈ [0, s 0 ). This means that w(t) * l is non-degenerate and strictly weighted Delaunay for 29ln(ln(n)) .Proof of Theorem 5.2. We call each boundary vertex of P other than p, q, r corner if it has nonzero curvature. Denote the set of corners as V c . Notice that by the assumption on P, each vertex in V c has degree m = 3, 5 or 6. Moreover, the standard subdivision of each triangle of P does not introduce new corners. Thus, the cardinality |V c | of V c is independent of the subdivision Step 1: For every v ∈ V c , we will deform its curvature to zero. In particular, we apply Theorem 5.10 to △ v 1 , . . . , △ v m−1 with α = π/(m−1) ∈ [ π 6 , π 2 ]. It produces a discrete conformal factor w i on △ v i for each i = 1, . . . , m − 1. Notice that the discrete conformal factor on T (n) in Theorem 5.10 depends only on α. Then discrete conformal factor w i are identical on each △ i . By symmetry, we can glue them together to form a discrete conformal factorw on T (n) . Specifically, the value ofw on B [n/3] (v) for v ∈ V c is determined by Theorem 5.10, and the values ofw on other vertices are zero.LetK be the curvature of inversive distance circle packing metricl =w * l. Let K be the curvature of the target equilateral triangle with K i = 0 for all v i ∈ V (n) − {p, q, r} and K i = 2π 3 for all v i ∈ {p, q, r}. Then Theorem 5.10 implies thatwhere N denote the number of corners.Notice that the set V k is the union of the sets V 0 given by Theorem 5.10 for each v ∈ V c . Statement(1)and(2)are immediate by the construction. Statement (3), (4), and (5) are immediate by Theorem 5.10.Step 2: we construct a flow to deform the curvatures of vertices in V k to be zero when the subdivision is sufficiently dense. Specifically, consider the following ODE system on T (n)with initial value w(0) = 0. The idea is the same as that of(33). Namely, we linearly interpolate the initial curvatureK and the target curvature K. By Lemma 5.8, there exists a maximal s 0 > 0 such that the solution w(s) to (43) exists and w(s) * l satisfies that on [0, s 0 ), all inner angles at v ∈ V , θ v ij ∈ [ π 6 − θ 0 , π 2 + θ 0 ], where θ 0 is the parameter given by Lemma 5.9. Now we apply Lemma 5.13 to estimate the angle deformation along the flow (43). Set V B = V (n) \V k . First notice that V B is a thin set in V (n) of T (n) . In particular, |B r (v i )∩V B | ≤ 10r 31Yuxiang Chen, Yanwen Luo, Xu Xu, Siqi Zhang for any v i ∈ V (n) and any r ≤ n/3. Moreover, by(4)and(5) inStep 1, we obtainandLemma 5.9 implies that f = w ′ along the flow (43) satisfies the conditions in Lemma 5.13. Therefore, we obtain that if i ∼ j, then1 ln(ln(n)) ).As a result, We claim that s 0 > 1. 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Introduction to circle packing. The theory of discrete analytic functions. K Stephenson, Cambridge University PressCambridgeK. Stephenson, Introduction to circle packing. The theory of discrete analytic functions. Cambridge Univer- sity Press, Cambridge, 2005. W Thurston, Geometry and topology of 3-manifolds. Princeton lecture notes 1976W. Thurston, Geometry and topology of 3-manifolds, Princeton lecture notes 1976, http://www.msri.org/publications/books/gt3m. The finite Riemann mapping theorem. W Thurston, An International Symposium at Purdue University on the Occasion of the Proof of the Bieberbach Conjecture. W. Thurston, The finite Riemann mapping theorem. An International Symposium at Purdue University on the Occasion of the Proof of the Bieberbach Conjecture, 1985. The convergence of discrete uniformizations for closed surfaces. T Wu, X Zhu, arXiv:2008.06744v2to appear in J. Differential Geom. math.GTT. Wu, X. Zhu, The convergence of discrete uniformizations for closed surfaces, to appear in J. Differential Geom. arXiv:2008.06744v2 [math.GT]. Rigidity of inversive distance circle packings revisited. X Xu, Adv. Math. 332X. Xu, Rigidity of inversive distance circle packings revisited, Adv. Math. 332 (2018), 476-509. A new proof of Bowers-Stephenson conjecture. X Xu, Math. Res. Lett. 284X. Xu, A new proof of Bowers-Stephenson conjecture, Math. Res. Lett. 28 (2021), no. 4, 1283-1306. X Xu, arXiv:2103.05272v2Rigidity and deformation of discrete conformal structures on polyhedral surfaces. math.DGX. Xu, Rigidity and deformation of discrete conformal structures on polyhedral surfaces. arXiv:2103.05272v2 [math.DG].
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[ "Alternative Technique to Asymmetry Analysis-Based Overlapping for Foot Ulcer Examination: Scalable Scanning", "Alternative Technique to Asymmetry Analysis-Based Overlapping for Foot Ulcer Examination: Scalable Scanning" ]
[ "Naima Kaabouch \nElectrical Engineering Department\nUniversity of North Dakota\n58202NDUSA\n", "Wen-Chen Hu \nComputer Science Department\nUniversity of North Dakota\n58202NDUSA\n", "Yi Chen \nElectrical Engineering Department\nUniversity of North Dakota\n58202NDUSA\n" ]
[ "Electrical Engineering Department\nUniversity of North Dakota\n58202NDUSA", "Computer Science Department\nUniversity of North Dakota\n58202NDUSA", "Electrical Engineering Department\nUniversity of North Dakota\n58202NDUSA" ]
[ "J Diabetes Metab" ]
Asymmetry analysis based on the overlapping of thermal images proved able to detect inflammation and, predict foot ulceration. This technique involves three main steps: segmentation, geometric transformation, and overlapping. However, the overlapping technique, which consists of subtracting the intensity levels of the right foot from those of the left foot, can also detect false abnormal areas if the projections of the left and right feet are not the same. In this paper, we present an alternative technique to asymmetry analysis-based overlapping. The proposed technique, scalable scanning, allows for an effective comparison even if the shapes and sizes of the feet projections appear differently in the image. The tested results show that asymmetry analysis-based scalable scanning provides fewer false abnormal areas than does asymmetry analysis -based overlapping.
10.4172/2155-6156.s5-003
[ "https://arxiv.org/pdf/1606.03578v1.pdf" ]
16,771,877
1606.03578
2edd272854e6456076d31a25d9366d505b9af507
Alternative Technique to Asymmetry Analysis-Based Overlapping for Foot Ulcer Examination: Scalable Scanning 2011. 2011. 2011. 2011 Naima Kaabouch Electrical Engineering Department University of North Dakota 58202NDUSA Wen-Chen Hu Computer Science Department University of North Dakota 58202NDUSA Yi Chen Electrical Engineering Department University of North Dakota 58202NDUSA Alternative Technique to Asymmetry Analysis-Based Overlapping for Foot Ulcer Examination: Scalable Scanning J Diabetes Metab 532011. 2011. 2011. 201110.4172/2155-6156.S5-003Received October 31, 2011; Accepted December 17, 2011; Published December 21, 2011Research Article Open Access J Diabetes Metab Diabetic Neuropathy Page 6 of 6 J Diabetes Metab Diabetic NeuropathyDiabetesFoot UlcersInfrared Imaging Asymmetry analysis based on the overlapping of thermal images proved able to detect inflammation and, predict foot ulceration. This technique involves three main steps: segmentation, geometric transformation, and overlapping. However, the overlapping technique, which consists of subtracting the intensity levels of the right foot from those of the left foot, can also detect false abnormal areas if the projections of the left and right feet are not the same. In this paper, we present an alternative technique to asymmetry analysis-based overlapping. The proposed technique, scalable scanning, allows for an effective comparison even if the shapes and sizes of the feet projections appear differently in the image. The tested results show that asymmetry analysis-based scalable scanning provides fewer false abnormal areas than does asymmetry analysis -based overlapping. Introduction Foot ulceration is an important precursor to amputation, having been identified as a component in 84% of lower extremity amputations [1,2]. Obesity and Chronic conditions associated with an aging U.S. population -such as diabetes, neuropathy, circulatory insufficiency, or a combination of these pathologies -mirror the current increased incidence of foot ulcers. The heel is the second most common site of pressure ulcers, producing 28% of all reported pressure ulcers [3]. These ulcers are among the most difficult to heal. In 1995 alone, lowerextremity ulcers cost Medicare $1.5 billion [4]. Patients with foot ulcers also can suffer from secondary conditions, including severe pain, immobility, increased infection risks, embarrassment and worry, and a dramatic impact on daily quality of life. One of the most common mechanisms that produce foot ulceration involves a cumulative effect of unrecognized repetitive trauma at pressure points on the sole of the foot over the course of several days [5][6][7]. Areas that are likely to ulcerate have also been associated with increased local skin temperatures due to inflammation and enzymatic autolysis of tissue [8][9][10]. Identifying precise areas of injury by the presence of inflammation can allow patients or healthcare providers to take early action to decrease the inflammation before a wound or ulcer actually develops. Such inflammation is mainly characterized by five signs: Redness, pain, swelling, loss of function, and heated tissue. Some of these signs are difficult to assess objectively. In a neuropathic extremity, pain and disturbance of function may be absent because of neuropathy, and thus, these signs are poor indicators of oncoming inflammation [11]. In addition, swelling and redness are difficult to grade precisely even for experienced clinicians. Conventional noninvasive methods to assess skin, including visual inspection and palpation, can be valuable diagnostic methods, but usually they do not detect complete enough changes in skin integrity until skin breakdown has actually occurred. However, temperature measurements can provide quantitative information that can be more precisely predictive of impending ulceration [8,[12][13][14]. In this research work, thermal imaging is used to monitor the temperature distribution of foot skin. However, there is no standard distribution for the skin-surface temperature of a healthy foot because that temperature can be affected by many factors, such as ambient and internal thermal conditions, age, sex, weight, etc. One way to eliminate this variability is to compare the thermal skin distributions of both feet on the same subject [10,11]. This comparison, called asymmetry/ symmetry analysis, has been widely used by researchers and clinicians to identify pathological conditions in the brain, breast, and other body parts that present similar symmetric characteristics. In a previous research [15][16][17][18], asymmetry analysis was combined with a genetic algorithm to detect inflammation and, hence, predict ulcers before they can develop. The methodology involves three steps: • Segmentation-isolate the feet and remove as much noise as possible, • Geometric transformation-adjust the left and the right foot so the two are in the same position in the image, • Asymmetry analysis-subtract the intensity level of each pixel in the left foot from the intensity level for the symmetric pixel of the right foot to detect abnormal areas. In each foot, an abnormal area is detected if the intensity level (temperature) of that area is higher than a specific designated threshold. Although this technique shows high efficiency in identifying inflammation when the feet present as the same size and shape in an image, it tends to detect false abnormal areas when the feet are not the same size or shape in the image. In this paper, an alternative technique to asymmetry analysis-based overlapping is offered. The proposed technique, asymmetry analysis-based scalable scanning, allows for a more reliable comparison even if the shapes and sizes of the two foot projections are different. Methodology A high resolution thermal camera, FLIR A320 (FLIR Systems Inc., Boston, USA) with a thermal sensitivity less than 0.05 o C, was used to record the temperature of foot skin distributions 5 minutes after the patients have removed their socks and shoes. Successive images of the patient's feet are taken during 15 minutes. However, in this paper only the first image of the patient's feet is analyzed. The resulting thermal image is then analyzed using an approach, implemented using MATLAB as a platform involving the following steps: In the first step, a genetic algorithm was used to extract the feet from the image background and remove as much noise as possible. Genetic algorithms are optimization algorithms, based on biological mechanics of natural selection, such as chromosome, population size, cross rate, mutation rate, and maximum generation [16,[19][20]. The main steps of the genetic algorithm are summarized below: 1. Assign the length of chromosome, population size, cross rate, mutation rate, and maximum generation. 2. Initialize population of the first generation, with each individual being a random eight bits binary string, representing a specific intensity level. 3. Evaluate the fitness of the whole population. The fitness is evaluated by ( ) ( ) 2 b f b f M M Num Num x Fitness − ⋅ ⋅ = (1) Where Num f and Num b are the numbers of foreground and background pixels, respectively. M f , the mean intensity of foreground pixels, is given by Where I f is the sum of intensities of foreground pixels. M b is the mean intensity of background pixels given by b b b Num I M =(3) Where I b is the sum of intensities of background pixels. 4. Generate the next population by performing selection, crossover and mutation operations. 5. Go to step 3 if the desired number of generations is not reached. 6. Reduce the cross rate and mutation rate after half of the desired generation number is reached. 7. Segment the image using the optimal threshold level when the desired number of generation is reached. The best threshold is determined by the following equation: ( ) ( ) { } x Fitness x Fitness max = * (4) Where x represents a population The output image of the genetic algorithm is then divided in two images, containing the left and the right foot, respectively. After this process, the centroid and the farthest point from the centroid to the heel edge are identified in each image and then used as feature points for a clear reference line. In the fifth step, the resolution of the scanning is defined. This resolution depends on two parameters: 1) the number of radial lines and 2) the distance between two comparison points. The first parameter, the number of radial lines, represents how many radial lines are compared in a foot with the same corresponding radial lines in the other foot. For each foot, the location of each line is determined by the angle step given as: anglestep=2*pi/lines(5) The number of lines is set in such a way that the anglestep will not exceed five degrees to scan the smallest area possible. The second parameter, the distance between two comparison points, serves as the scale adopted for our comparison. It represents the number of pixels between the two comparison points in the same radial line. A standard distance is used on the left foot, and the distance is scaled line by line on the right foot to guarantee the same number of comparison points. The edge points of each foot are sorted counter-clockwise with the heel point as the initial point. The radial lines are constructed counterclockwise with each edge point representing the limit of each line. As shown in Figure 1, a reference line is set up between the two feature points, the centroid and the heel point. The second line is located at one angle step from the reference line, the third line is located at two angle steps from the reference line, and so on. To limit the scanning within the feet, the edge points of the feet are used as the limits of the radial lines. Figure 1 shows examples of radial line point limits for a specific angle step. While it is easy to find the coordinates of points within a continuous line, it is difficult to determine discrete points (pixels) along a line in computer graphics. In our approach, a well-known algorithm called the Bresenham line drawing was used. This algorithm computes the discrete best-fit line from (0, 0) to (X, Y), where the point (X, Y) lies in the NE half-quadrant, i.e., 0 < Y =<X. The best-fit line is one that does not deviate more than half a pixel away from the real line. For efficiency, the algorithm computes the pixel values (x, y) of this best-fit line using only linear operations. By comparing the centroid and the foot edges point with the matrix of the Bresenham points, the pixels belonging to each radial line can be determined. As shown in Figure 1, the Bresenham points for each radial line are defined between the centroid, and the intersection edge point is clipped for a specific angle step. In the initialization section, the left foot always uses a standard distance which is assigned manually. The distance of the right foot varies line by line as follows: stepR(i)=stepL* Ratio(i)(6) Here, Ratio(i) is given by Ratio(i) = len_Right(i)/len_Left(i)(7) The len_Right(i) and the len_Left(i) represent the lengths of the corresponding segments of the left foot and right foot in the image, respectively. After the grid matrices of the left and the right feet are generated in a corresponding fashion, any abnormalities are identified by comparing each point in the radial line to its corresponding point in the line of the other foot. This comparison is made from the initial line to the assigned number of lines and from the centroid to the edge point along the line. A threshold is assigned manually to determine which of the two compared points is abnormal. If the difference in the intensities of the left foot and the right foot exceed this threshold, the point with the higher intensity is considered as an abnormal point and then marked out on the corresponding foot. Results and Discussion To test the proposed algorithm, 140 thermal images were used. These images were divided into two sets. The first set included 80 images that corresponded to healthy feet with 40 images that represented feet with visibly different size projections and 40 images representing feet with the same size projections. The second set included 60 images corresponding to feet with visible thermal abnormalities, with 30 images having visibly different sizes of feet projections; the other 30 images had the same size of feet projections. Table 1 summarizes the images used for each set and category. Examples of images from these sets are shown in Figures 2a, 2b. Figure 2a shows a thermal image of healthy feet, but with different feet projections in the image; Figure 2b shows an example of a thermal image in which the right foot contains a high temperature area, indicating the presence of inflammation. As one can see, these images have strong non-uniform backgrounds, mainly due to heat transfer from the legs and the feet. Results after segmenting the images with the genetic algorithm, using Step 1 of the proposed algorithm, are shown in Figures 3a,3b, which correspond to the input images shown in Figures 2a,2b. As observed from the two images of Figures 3a,3b, the genetic algorithm efficiently segmented the feet from the background and removed much of the noise. The efficiency of this genetic algorithm was extensively studied in previous works [15,16]. Figure 4 shows the results after applying Steps 2 to 8 on the image of Figure 2b. The red spots in the images correspond to the grid matrices that correspond to the left and right foot, respectively. The spots give the locations of the points used for a comparison of the right and left feet for a specific angle step. Figure 5 offers the results after applying the proposed technique to the thermal image shown in Figure 2b. The red points in each foot represent those points that have higher intensity levels than those for the corresponding points in the other foot. As expected, an abnormal area was detected on the right foot. In addition, other small abnormal temperature areas, especially those of the foot edges, were identified. These small areas, however, do not correspond to inflammations. Such false detection occurs when the comparison threshold is set to an intensity equal to 5 and corresponding to less than 1 o C, a difference not significant enough to distinguish between a normal and an abnormal area according to several studies, including those of Armstrong et al. [21] and Lavery et al. [22], a difference of 2.2 o is more clinically indicative of impending ulceration, . Therefore any increase of the threshold should eliminate such false small areas without affecting the precision of the proposed technique. Figures 6 and 7 show the output images resulting from an increase in the threshold of the intensity values. As one can see, there is a decrease of false abnormal points on the foot edges. These false abnormal points will easily decrease when not involved with foot edge points in the comparison process. To assess the performance of the proposed algorithm, we used the 140 images described in this section and compared the results of the proposed technique to those of asymmetry analysis-based overlapping. Figure 6 shows these results after applying both techniques on the image in Figure 2b. As one can see, both the overlapping and scalable scanning methods can identify abnormalities. However, the first technique detects false abnormal points on foot edges because the feet sizes are not perfectly equal, while the second technique detects fewer false abnormal points that correspond to the points located at the foot edges. This false detection can be minimized by eliminating any point comparison with the points located on the edges of the foot. Figure 7 shows the results after applying the overlapping technique and the proposed technique, respectively, on Figure 2a, which contains healthy feet with different projections. One can see that the asymmetrybased overlapping technique indicates many false abnormal areas located on the edges, while the scanning technique offers a limited number of false abnormal points, located mainly on the foot edges. As mentioned earlier, such false detection is easily decreased by eliminating any comparison that involves foot edge points. An assessment of the proposed technique utilizes a comparison using three criteria: the ability to detecting abnormal areas, the ability to analyze images with different feet projections without false detection, and the number of total false abnormal points. This latter number, however, depends on the image itself, i.e. the movement of the feet before recording the image. Table 2 provides a summary of this evaluation. One can see that while the overlapping technique works well only when the feet projections sizes and shapes are the same, the scalable scanning technique works well for all types of feet projections. Conclusion In this research, we developed an alternative technique to using asymmetry analysis-based overlapping to investigate the thermal distributions of feet to detect inflammation and predict foot ulcers. While, the overlapping technique works well when feet projection sizes are the same, it fails when feet projections are different. However, the scalable scanning technique works for all types of feet projections. The experimental results show that the proposed technique gives fewer false abnormal areas and hence is more reliable to use to identify inflammation. In the future, our research objectives will include 1) studying the decay rates of temperature distributions over time for successive images that correspond to the same feet, and 2) combining temperature and pressure distributions to predict foot ulceration. Divide the image into separate images, each containing the left and the right foot 3. Find the centroid of each foot and the furthest point from the centroid to the heel edge in each image 4. Initialize • Put edge points into a queue and index each point • Determine the angle step and the number of lines 5. Calculate the angle of i th line 6. Find the intersection point of the i th line with the foot edge 7. Find the straight line running from the centroid to the intersection point, using the Bresenham algorithm 8. Determine the comparison points along the i th line 9. Scan the whole foot, and compare the points on every line in the left foot to the points of the corresponding line in the right foot 10. Identify abnormalities, if any, in each foot. Figure 1 : 1Illustration Figure 2 :Figure 3 : 23Original foot images for a) healthy feet with different projections and b) feet with different projections and with an abnormal area evident in the right foot. Output images after applying the genetic algorithm on: a) the image illustrated in Figure 2a. b) the image illustrated in Figure 2b. J Diabetes Metab Diabetic Neuropathy ISSN:2155-6156 JDM, an open access journal Figure 4 : 4Location of the comparison points after applying steps 1 to 8 on the left and right feet images corresponding to the image of Figure 2b. Figure 5 : 5Red spots correspond to the identified abnormal areas in each foot after applying the scalable scanning to the original image illustrated in Figure 2b. J Diabetes Metab Diabetic Neuropathy ISSN:2155-6156 JDM, an open access journal Figure 6 : 6Comparison results after applying a) the asymmetry-based overlapping and b) the scalable scanning on the original image illustrated inFigure 2b. Figure 7 : 7Comparison results after applying a) the asymmetry-based overlapping and b) the scalable scanning on the original image illustrated inFigure 2a. *Corresponding author: Naima Kaabouch, Electrical Engineering Department, University of North Dakota, USA, E-mail: [email protected] Diabetes Metab Diabetic Neuropathy ISSN:2155-6156 JDM, an open access journal Table 1 : 1Number of images in each set used for the assessment. Table 2 : 2Results of the assessment.Total number of images without abnormal areas-- Healthy feet same sizes and shapes 40 Total number of images without abnormal areas-- Healthy feet different sizes and shapes 40 Total number of images with abnormal areas--Feet of same sizes and shapes: 60 Total number of images with abnormal areas--Feet of different sizes and shapes 30 Technique- Number of images with false abnormal areas detected Number of images with false abnormal areas Number of images with abnormal areas detected Number of false abnormal points Asymmetry Based Overlapping 0 40 60 High Proposed Technique - Scalable Scanning 0 0 60 Small Pathways to diabetic limb amputation. Basis for prevention. R E Pecoraro, G E Reiber, E M Burgess, Diabetes Care. 13Pecoraro RE, Reiber GE, Burgess EM (1990) Pathways to diabetic limb amputation. Basis for prevention. Diabetes Care 13: 513-521. Chronology and determinants of tissue repair in diabetic lower extremity ulcers. R E Pecoraro, J H Ahroni, E J Boyko, V L Stensel, Diabetes. 40Pecoraro RE, Ahroni JH, Boyko EJ, Stensel VL (1991) Chronology and determinants of tissue repair in diabetic lower extremity ulcers. Diabetes 40: 1305-1313. Fourth national pressure ulcer prevalence survey. C A Barczak, R I Barnett, E J Childs, L M Bosley, Advan Wound Care. 10Barczak CA, Barnett RI, Childs EJ, Bosley LM (1997) Fourth national pressure ulcer prevalence survey. Advan Wound Care 10: 18-26. A cost analysis of diabetic lower-extremity ulcers. C Harrington, M J Zagari, J Corea, J Klitenic, Diabetes Care. 23Harrington C, Zagari MJ, Corea J, Klitenic J (2000) A cost analysis of diabetic lower-extremity ulcers. Diabetes Care 23: 1333-1338. Practical criteria to screen patients at risk for diabetic foot ulceration. L A Lavery, D G Armstrong, S Vela, J G Fleishli, Arch Intern Med. 158Lavery LA, Armstrong DG, Vela S, Fleishli JG (1998) Practical criteria to screen patients at risk for diabetic foot ulceration. 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The insensitive foot (including leprosy. P W Brand, Disorders of the Foot and Ankle, Jahss M. Philadelphia, SaundersBrand PW (1991) The insensitive foot (including leprosy) In Disorders of the Foot and Ankle, Jahss M, Ed. Philadelphia, Saunders, pp. 2173-2175. Thermography and pedobarography in the assessment of tissue damage in neuropathic and atherosclerotic feet. R P Clark, M R Goff, J Hughes, L Klenerman, Thermology. 3Clark RP, Goff MR, Hughes J, Klenerman L (1988) Thermography and pedobarography in the assessment of tissue damage in neuropathic and atherosclerotic feet. Thermology 3: 15-20. Home monitoring of foot skin temperatures to prevent ulceration. L A Lavery, K R Higgins, D R Lanctot, G P Constantinides, R G Zamorano, Diabetes Care. 27Lavery LA, Higgins KR, Lanctot DR, Constantinides GP, Zamorano RG, et al. (2004) Home monitoring of foot skin temperatures to prevent ulceration. Diabetes Care 27: 2642-2647. 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Early detection of foot ulcers through asymmetry analysis. N Kaabouch, Y Chen, W C Hu, J Anderson, F Ames, Proc SPIE. 724372621Kaabouch N, Chen Y, Hu WC, Anderson J, Ames F, et al. (2009) Early detection of foot ulcers through asymmetry analysis. Proc SPIE 7243: 72621L. Asymmetry analysis based on Genetic Algorithms for the Prediction of Foot Ulcers. N Kaabouch, Y Chen, J Anderson, F Ames, R Paulson, Proc SPIE. 7243724304Kaabouch N, Chen Y, Anderson J, Ames F, Paulson R (2009) Asymmetry analysis based on Genetic Algorithms for the Prediction of Foot Ulcers. Proc SPIE 7243: 724304. Predicting Neuropathic Ulceration: An Analysis of Static Temperature Distributions in Thermal Images. N Kaabouch, W C Hu, Y Chen, W C Hu, J Anderson, J Biomed Opt. 1561715Kaabouch N, Hu WC, Chen Y, Hu WC, Anderson J, et al. (2010) Predicting Neuropathic Ulceration: An Analysis of Static Temperature Distributions in Thermal Images. J Biomed Opt 15: 061715. Enhancement of the Asymmetry-Based Overlapping Analysis through Features Extraction. N Kaabouch, W C Hu, Y Chen, W C Hu, J Anderson, J Electron Imaging. 2013012Kaabouch N, Hu WC, Chen Y, Hu WC, Anderson J, et al. (2011) Enhancement of the Asymmetry-Based Overlapping Analysis through Features Extraction. J Electron Imaging 20: 013012. Genetic Algorithms in Search, Optimization and Machine Learning. D E Goldber, Addison-WesleyGoldber DE (1989) Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley. An Introduction to Genetic Algorithms. M Melanie, MIT-PressMelanie M (1999) An Introduction to Genetic Algorithms. MIT-Press, 1999. Skin temperature monitoring reduces the risk for diabetic foot ulceration in high-risk patients. D G Armstrong, K Holtz-Neiderer, Wendel C Mohler, M J Kimbriel, H R , Am J Med. 120Armstrong DG, Holtz-Neiderer K, Wendel C, Mohler MJ, Kimbriel HR, et al. (2007) Skin temperature monitoring reduces the risk for diabetic foot ulceration in high-risk patients. Am J Med 120: 1042-1046. Preventing Diabetic Foot Ulcer Recurrence in High-Risk Patients Use of temperature monitoring as a self-assessment tool. L A Lavery, K R Higgins, D R Lanctot, G P Constantinides, R G Zamorano, Diabetes Care. 30Lavery LA, Higgins KR, Lanctot DR, Constantinides GP, Zamorano RG, et al. (2007) Preventing Diabetic Foot Ulcer Recurrence in High-Risk Patients Use of temperature monitoring as a self-assessment tool. Diabetes Care 30: 14-20. UK Submit your next manuscript and get advantages of OMICS Group submissions Unique features: • User friendly/feasible website-translation of your paper to 50 world's leading languages • Audio Version of published paper • Digital articles to share and explore Special features: • 200 Open Access Journals • 15,000 editorial team • 21 days rapid review process • Quality and quick editorial, review and publication processing • Indexing at PubMed (partial). Diabetic Neuropathy handled by Editor(s). Dr. Yuriy K. Bashmakov, Cambridge Theranostics Ltd.Scopus, DOAJ, EBSCOIndex Copernicus and Google Scholar etc • Sharing Option: Social Networking Enabled • Authors, Reviewers and Editors rewarded with online Scientific Credits • Better discount for your subsequent articles. Submit your manuscript at: www.editorialmanager.com/acrgroupThis article was originally published in a special issue, Diabetic Neuropathy handled by Editor(s). Dr. Yuriy K. Bashmakov, Cambridge Theranostics Ltd., UK Submit your next manuscript and get advantages of OMICS Group submissions Unique features: • User friendly/feasible website-translation of your paper to 50 world's leading languages • Audio Version of published paper • Digital articles to share and explore Special features: • 200 Open Access Journals • 15,000 editorial team • 21 days rapid review process • Quality and quick editorial, review and publication processing • Indexing at PubMed (partial), Scopus, DOAJ, EBSCO, Index Copernicus and Google Scholar etc • Sharing Option: Social Networking Enabled • Authors, Reviewers and Editors rewarded with online Scientific Credits • Better discount for your subsequent articles Submit your manuscript at: www.editorialmanager.com/acrgroup
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[ "Multistage Entanglement Swapping", "Multistage Entanglement Swapping" ]
[ "Alexander M Goebel \nPhysikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany\n", "Claudia Wagenknecht \nPhysikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany\n", "Qiang Zhang \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Yu-Ao Chen \nPhysikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany\n\nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Kai Chen \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Jörg Schmiedmayer \nAtominstitut derösterreichischen Universitäten\nTU-Wien\nA-1020ViennaAustria\n", "Jian-Wei Pan \nPhysikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany\n\nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n" ]
[ "Physikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany", "Physikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Physikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Atominstitut derösterreichischen Universitäten\nTU-Wien\nA-1020ViennaAustria", "Physikalisches Institut\nRuprecht-Karls-Universität Heidelberg\nPhilosophenweg 1269120HeidelbergGermany", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina" ]
[]
We report an experimental demonstration of entanglement swapping over two quantum stages. By successful realizations of two cascaded photonic entanglement swapping processes, entanglement is generated and distributed between two photons, that originate from independent sources and do not share any common past. In the experiment we use three pairs of polarization entangled photons and conduct two Bell-state measurements (BSMs) one between the first and second pair, and one between the second and third pair. This results in projecting the remaining two outgoing photons from pair 1 and 3 into an entangled state, as characterized by an entanglement witness. The experiment represents an important step towards a full quantum repeater where multiple entanglement swapping is a key ingredient.Entanglement swapping is arguably one of the most important ingredients for quantum repeaters and quantum relays, which lays at the heart of quantum communication[1,2,3,4]. For photonic quantum communication, the distance is largely limited due to decoherence from coupling to the environment and an increasing loss of photons in a quantum channel. This leads to an exponential decay in the fidelity of quantum information. This drawback can eventually be overcome by subdividing larger distances into smaller sections over which entanglement or quantum states can be distributed. The sections are then bridged by entanglement swapping processes [2, 3]. The swapping procedure therefore constitutes one of the key elements for a quantum relay [3], and a full quantum repeater [2] if combined with quantum purification [5, 6] and quantum memory[7]. As a result, quantum communication becomes feasible despite of realistic noise and imperfections. At the same time, the overhead for the used resources and communication time only increase polynomially with the distance [2, 3, 4].Experimentally, photonic entanglement swapping has so far been successfully achieved for the case of discrete variables[8,9], and for continuous variable [10], both via a single stage process. However, only after successful multiple swapping, will we be able to have a fully functional quantum repeater. There are additional advantages utilizing a multiple swapping process. For a quantum relay with many segments, it is equivalent to significantly lower the dark-count rate, which is a substantial factor limiting the transmission distance of successful quantum communication[3]. For quantum information carriers possessing mass, multiple swapping processes can speed up the distribution of entanglement by a factor that is proportional to the number of segments used[11]. Moreover, multistage entanglement swapping can improve the protection of quantum states against noise from amplitude errors[11].We report in this letter an experimental demonstration of a multiple entanglement swapping over two stages. This is achieved by utilizing three synchronous spatially independent pairs of polarization entangled photons, and performing BSMs among the three segments between the two communication parties. Two successful BSMs yield a final maximally entanglement pair distributed between the two parties. To quantitatively evaluate the performance, we have observed the quality of the output state by the characterization of an entanglement witness, which confirms genuine entanglement generation. Our experiment implements an entanglement distribution over two distant stations which are initially independent of each other and have never physically interacted in the past. This proof-of-principle demonstration constitutes an important step towards robust long-distance quantum relays, quantum repeaters and related quantum protocols based on multiple entanglement swapping.The principle for multistage entanglement swapping is sketched inFig. 1. Consider three independent stations, each simultaneously emitting a pair of Einstein-Podolsky-Rosen (EPR) maximally entangled photons. In our experiments, we generate these states through the process of spontaneous parametric down-conversion[12]. By post-selecting events with only one photon in each output arm, we obtain polarization entangled photons in the statewhere |Ψ − ij is one of the four maximally entangled Bell states, which form a complete orthonormal basis for the joint state of two entangled photons
10.1103/physrevlett.101.080403
[ "https://arxiv.org/pdf/0808.2972v1.pdf" ]
2,308,038
0808.2972
9e3c3a4e7f73520d4ffd7f670a8b39bb3beebe4d
Multistage Entanglement Swapping 21 Aug 2008 (Dated: May 30, 2008) Alexander M Goebel Physikalisches Institut Ruprecht-Karls-Universität Heidelberg Philosophenweg 1269120HeidelbergGermany Claudia Wagenknecht Physikalisches Institut Ruprecht-Karls-Universität Heidelberg Philosophenweg 1269120HeidelbergGermany Qiang Zhang Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Yu-Ao Chen Physikalisches Institut Ruprecht-Karls-Universität Heidelberg Philosophenweg 1269120HeidelbergGermany Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Kai Chen Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Jörg Schmiedmayer Atominstitut derösterreichischen Universitäten TU-Wien A-1020ViennaAustria Jian-Wei Pan Physikalisches Institut Ruprecht-Karls-Universität Heidelberg Philosophenweg 1269120HeidelbergGermany Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Multistage Entanglement Swapping 21 Aug 2008 (Dated: May 30, 2008)numbers: 0367Bg0367Mn4250Dv4250Xa We report an experimental demonstration of entanglement swapping over two quantum stages. By successful realizations of two cascaded photonic entanglement swapping processes, entanglement is generated and distributed between two photons, that originate from independent sources and do not share any common past. In the experiment we use three pairs of polarization entangled photons and conduct two Bell-state measurements (BSMs) one between the first and second pair, and one between the second and third pair. This results in projecting the remaining two outgoing photons from pair 1 and 3 into an entangled state, as characterized by an entanglement witness. The experiment represents an important step towards a full quantum repeater where multiple entanglement swapping is a key ingredient.Entanglement swapping is arguably one of the most important ingredients for quantum repeaters and quantum relays, which lays at the heart of quantum communication[1,2,3,4]. For photonic quantum communication, the distance is largely limited due to decoherence from coupling to the environment and an increasing loss of photons in a quantum channel. This leads to an exponential decay in the fidelity of quantum information. This drawback can eventually be overcome by subdividing larger distances into smaller sections over which entanglement or quantum states can be distributed. The sections are then bridged by entanglement swapping processes [2, 3]. The swapping procedure therefore constitutes one of the key elements for a quantum relay [3], and a full quantum repeater [2] if combined with quantum purification [5, 6] and quantum memory[7]. As a result, quantum communication becomes feasible despite of realistic noise and imperfections. At the same time, the overhead for the used resources and communication time only increase polynomially with the distance [2, 3, 4].Experimentally, photonic entanglement swapping has so far been successfully achieved for the case of discrete variables[8,9], and for continuous variable [10], both via a single stage process. However, only after successful multiple swapping, will we be able to have a fully functional quantum repeater. There are additional advantages utilizing a multiple swapping process. For a quantum relay with many segments, it is equivalent to significantly lower the dark-count rate, which is a substantial factor limiting the transmission distance of successful quantum communication[3]. For quantum information carriers possessing mass, multiple swapping processes can speed up the distribution of entanglement by a factor that is proportional to the number of segments used[11]. Moreover, multistage entanglement swapping can improve the protection of quantum states against noise from amplitude errors[11].We report in this letter an experimental demonstration of a multiple entanglement swapping over two stages. This is achieved by utilizing three synchronous spatially independent pairs of polarization entangled photons, and performing BSMs among the three segments between the two communication parties. Two successful BSMs yield a final maximally entanglement pair distributed between the two parties. To quantitatively evaluate the performance, we have observed the quality of the output state by the characterization of an entanglement witness, which confirms genuine entanglement generation. Our experiment implements an entanglement distribution over two distant stations which are initially independent of each other and have never physically interacted in the past. This proof-of-principle demonstration constitutes an important step towards robust long-distance quantum relays, quantum repeaters and related quantum protocols based on multiple entanglement swapping.The principle for multistage entanglement swapping is sketched inFig. 1. Consider three independent stations, each simultaneously emitting a pair of Einstein-Podolsky-Rosen (EPR) maximally entangled photons. In our experiments, we generate these states through the process of spontaneous parametric down-conversion[12]. By post-selecting events with only one photon in each output arm, we obtain polarization entangled photons in the statewhere |Ψ − ij is one of the four maximally entangled Bell states, which form a complete orthonormal basis for the joint state of two entangled photons We report an experimental demonstration of entanglement swapping over two quantum stages. By successful realizations of two cascaded photonic entanglement swapping processes, entanglement is generated and distributed between two photons, that originate from independent sources and do not share any common past. In the experiment we use three pairs of polarization entangled photons and conduct two Bell-state measurements (BSMs) one between the first and second pair, and one between the second and third pair. This results in projecting the remaining two outgoing photons from pair 1 and 3 into an entangled state, as characterized by an entanglement witness. The experiment represents an important step towards a full quantum repeater where multiple entanglement swapping is a key ingredient. Entanglement swapping is arguably one of the most important ingredients for quantum repeaters and quantum relays, which lays at the heart of quantum communication [1,2,3,4]. For photonic quantum communication, the distance is largely limited due to decoherence from coupling to the environment and an increasing loss of photons in a quantum channel. This leads to an exponential decay in the fidelity of quantum information. This drawback can eventually be overcome by subdividing larger distances into smaller sections over which entanglement or quantum states can be distributed. The sections are then bridged by entanglement swapping processes [2,3]. The swapping procedure therefore constitutes one of the key elements for a quantum relay [3], and a full quantum repeater [2] if combined with quantum purification [5,6] and quantum memory [7]. As a result, quantum communication becomes feasible despite of realistic noise and imperfections. At the same time, the overhead for the used resources and communication time only increase polynomially with the distance [2,3,4]. Experimentally, photonic entanglement swapping has so far been successfully achieved for the case of discrete variables [8,9], and for continuous variable [10], both via a single stage process. However, only after successful multiple swapping, will we be able to have a fully functional quantum repeater. There are additional advantages utilizing a multiple swapping process. For a quantum relay with many segments, it is equivalent to significantly lower the dark-count rate, which is a substantial factor limiting the transmission distance of successful quantum communication [3]. For quantum information carriers possessing mass, multiple swapping processes can speed up the distribution of entanglement by a factor that is proportional to the number of segments used [11]. Moreover, multistage entanglement swapping can improve the protection of quantum states against noise from amplitude errors [11]. We report in this letter an experimental demonstration of a multiple entanglement swapping over two stages. This is achieved by utilizing three synchronous spatially independent pairs of polarization entangled photons, and performing BSMs among the three segments between the two communication parties. Two successful BSMs yield a final maximally entanglement pair distributed between the two parties. To quantitatively evaluate the performance, we have observed the quality of the output state by the characterization of an entanglement witness, which confirms genuine entanglement generation. Our experiment implements an entanglement distribution over two distant stations which are initially independent of each other and have never physically interacted in the past. This proof-of-principle demonstration constitutes an important step towards robust long-distance quantum relays, quantum repeaters and related quantum protocols based on multiple entanglement swapping. The principle for multistage entanglement swapping is sketched in Fig. 1. Consider three independent stations, each simultaneously emitting a pair of Einstein-Podolsky-Rosen (EPR) maximally entangled photons. In our experiments, we generate these states through the process of spontaneous parametric down-conversion [12]. By post-selecting events with only one photon in each output arm, we obtain polarization entangled photons in the state |Ψ 123456 = |Ψ − 12 × |Ψ − 34 × |Ψ − 56 ,(1) where |Ψ − ij is one of the four maximally entangled Bell states, which form a complete orthonormal basis for the joint state of two entangled photons Here |H (|V ) denotes the state of a horizontally (vertically) polarized photon. Note that photon pairs 1-2, 3-4 and 5-6 are entangled in an antisymmetric polarization state. The states of the three pairs are factorizable from each other, namely, there is no entanglement among photons from different pairs. As a first step we perform a joint BSM on photons 2 and 3, that is, photons 2 and 3 are projected onto one of the four Bell states. This measurement also projects photons 1 and 4 onto a Bell state, in a form depending on the result of the BSM of photons 2 and 3. Close inspection shows that for the initial state given in Eq. (1), the emerging state of photons 1 and 4 is identical to the one that photons 2 and 3 collapse into. This is a consequence of the fact that the state of Eq. (1) can be rewritten as |Ψ ± ij = 1 √ 2 (|H i |V j ± |V i |H j ) |Φ ± ij = 1 √ 2 (|H i |H j ± |V i |V j ).|Ψ 123456 = 1 2 [|Ψ + 14 |Ψ + 23 − |Ψ − 14 |Ψ − 23 −|Φ + 14 |Φ + 23 + |Φ − 14 |Φ − 23 ] ×|Ψ − 56(2) In all cases photons 1 and 4 emerge entangled despite the fact that they never interacted with one another in the past. The joint measurement of photons 2 and 3 tells about the type of entanglement between photons 1 and 4. Without loss of generality, we assume in the first step that photons 2 and 3 have collapsed into the state |Φ + 23 as a result of the first BSM. The remaining four-photon state is then of the form |Ψ 1456 = 1 2 [|Ψ + 16 |Φ − 45 + |Ψ − 16 |Φ + 45 −|Φ + 16 |Ψ − 45 − |Φ − 16 |Ψ + 45 ] (3) In a similar manner we perform a second BSM on photons 4 and 5. Again a detection of the state |Φ + 45 results in projecting the remaining photons 1 and 6 onto the Bell UV PBS23 Pol. Pol. D5 D4 Pol. BBO Delay F F F D6 l/4 PBS45 Pol. Pol. D1 D2 Pol. In order to achieve indistinguishability at the interference PBS23 and PBS45 the spatial and temporal overlap are maximized by adjusting the delays and observing 'Shih-Alley-Hong-Ou-Mandel'-type interference fringes [19] behind the PBS23 (PBS45) in the ± basis [20]. With the help of polarizers and half/quarter wave plates, we are able to analyze the polarization of photons in arms 1 and 6. All photons are spectrally filtered by narrow band filters with ∆λFWHM ≈ 2.8nm and are monitored by silicon avalanche single-photon detectors [21]. Coincidences are counted by a laser clocked fieldprogrammable gate array based coincidence unit. state |Ψ − 16 = 1 √ 2 (|H 1 |V 6 − |V 1 |H 6 )(4) A schematic diagram of our setup for multistage entanglement swapping is illustrated in Fig. 2. We use a pulsed high-intensity ultraviolet (UV) laser with a central wavelength of 390nm, a pulse duration of around 180 fs and a repetition rate of 76 MHz. The beam successively passes through two β-Barium-Borate (BBO) crystals, and is reflected to pass again through the second BBO to generate three polarization entangled photon pairs via type-II parametric down conversion [12]. Due to the high average power of 1W UV-light and improvements in collection efficiency and stability of the photon sources [13], we are able to observe up to 10 5 photon pairs per second from each source. With this bright-ness of the entangled photon sources we could obtain around 4.5 six-photon events per minute in our setup. For the joint BSM of photons 2 and 3 (photons 4 and 5), we choose to analyze the case of detecting the projection onto a |Φ + state. Using a polarizing beam splitter (PBS) allows the projection of photons 2 and 3 (4 and 5) onto the state |Φ + upon detecting a |+ |+ or |− |− coincidence at detectors D2 and D3 (D4 and D5) (with |± = (|H ± |V )/ √ 2). In our experiment only the |+ |+ coincidences were registered, which reduces the overall success probability by a factor of 1/64. This could be improved by installing a half wave plate (HWP) at 22.5 • , which corresponds to a polarization rotation of 45 • , and a PBS after each output arm of PBS23 (PBS45). This configuration would also allow to detect the state |Φ − , which results in a |+ |− or |− |+ coincidence [14]. Thus, a factor of 1/4 for the overall success probability could be achieved in an ideal case. As shown in equations Eq. (2,3,4) the projection measurements onto |Φ + 23 and |Φ + 45 leave photons 1 and 6 in the maximally entangled state |Ψ − 16 . In contrast to quantum state tomography, the measurement of witness operators does not provide a complete reconstruction of the original quantum state, it however allows to check with a minimal number of local measurements for a entanglement character of a quantum state. To verify that the two photons are really in an entangled state, and thus the swapping operation is successful, the expectation value of the corresponding witness operator [15,16] is expected to take a negative value. In our case, the applied witness operator W is the most efficient one since it involves only the minimal number of local measurements [15]. It can be measured locally by choosing correlated measurement settings, that involve only the simultaneous detection of linear, diagonal and circular polarizations for both photons. We have performed local measurements on the outgoing state of photons 1 and 6 in the three complementary bases; linear (H/V), diagonal (+/-) and circular (R/L) (with |L = (|H + i|V )/ √ 2 and |R = (|H − i|V )/ √ 2). The entanglement witness is given by W = 1 2 (|HH HH| + |V V V V | + | + + + + | +| − − − − | − |RL RL| − |LR LR|). (5) In the experiment, we perform measurements for each correlation function of the witness. The expectation values are shown in Fig. 3. Experimental integration time for each local measurement took about 60 hours and we recorded about 180 events of desired two-qubit coincidences. Every expectation value for a correlation function is obtained by making a von Neumann measurement along a specific basis and computing the probability over all the possible events. For example, for a HH correlation Tr(ρ|HH HH|), we perform measurements along the H/V basis. Then its value is given by the number of coincidence counts of HH over the sum of all coincidence counts of HH, HV, VH and VV. We proceed likewise for the other correlation settings. The witness can then directly be evaluated to Tr(ρW ) = −0.16 ± 0.03. The negativity of the measured witness implies clearly that entanglement has indeed been swapped. The imperfection of our data is due to the non-ideal quality of entangled states generated from the high power UV beam, as well as the partial distinguishability of independent photons at PBS23 and PBS45, which leads to non-perfect interferences and a degrading of entanglement output quality [17]. Moreover, double pair emission by a single source causes noise of an order of 10 spurious six-fold coincidences in 60 hours and was not subtracted in calculating the expectation value of the witness operator. To ensure that there is no entanglement between photons 1 and 6 before either of the entanglement swapping process, we have performed a complete quantum state tomography. The experimental expectation values for various bases are illustrated in Fig. 4. Concurrence [18] is a monotonic function of entanglement, ranging from 0 for a separable state to 1 for a maximally entangled state. In terms of concurrence, we can thus quantify the degree of entanglement through a reconstructed density matrix ρ init for the initial combined state from the data shown in Fig. 4. The concurrence C init derived from ρ init is C init = max(0, −0.39 ± 0.01) = 0. As expected the concurrence is indeed 0, therefore photons 1 and 6 did not reveal any entanglement whatsoever before the swapping. Ideally, for a completely mixed state the expectation values for all local measurements should be 0, FIG. 4: Complete quantum state tomography on photon 1 and 6 before entanglement swapping. Label X corresponds to measurement setting σx, while Y and Z are for σy and σz, respectively. The result shows that the photons didn't reveal any entanglement whatsoever before the swapping operation. except for the unity operator, which should be 1. The contributions of the measurement settings other than the unity operator are mainly due to noise caused by scattered light of the UV beam at the BBO crystal. For convenience of comparison, we also performed the same witness measurement of Eq. (5) to give W = 0.28±0.01, which is safely above the bound W < 0 needed to reveal entanglement. However, after the two-stage entanglement swapping, entanglement arises as unambiguously confirmed by negativity of expectation value for the witness W = −0.16 ± 0.03 as discussed above. In conclusion, we have for the first time provided a proof-of-principle demonstration of a two-stage entanglement swapping using photonic qubits. The feasibility and effectiveness of this process has been verified by a successful distribution of genuine entanglement after two simultaneously independent swapping process. This result yields the possibility of immediate near-future applications of various practical quantum information processing tasks. If combined with narrow-band entanglement sources, the implementation of quantum relays (without quantum memory) and quantum repeaters (with quantum memory) would become within current reach [2,7,9], as well as quantum state transfer and quantum cryptography networks in a more efficient way and over much larger distances of around hundreds of kilometers [3]. Our demonstration also allows for the possibility of utilizing multi-party, multiple stages entanglement swapping to achieve global quantum communication networks though with significant challenges ahead [11]. This work was supported by the Marie Curie Excel- PACS numbers: 03.67.Bg, 03.67.Mn, 42.50.Dv, 42.50.Xa FIG. 1 : 1Principle of multistage entanglement swapping: three EPR sources produce pairs of entangled photons 1-2, 3-4 and 5-6. Photon 2 from the inial state and photon 3 from the first ancillary pair are subjected to a joint BSM, and so are photon 4 from the first ancillary and photon 5 from the second acillary pair. The two BSMs project outgoing photons 1 and 6 onto an entangled state. Thus the entanglement of the initial pair is swapped to an entanglement between photons 1 and 6. FIG. 2 : 2The focused ultraviolet laser beam passes the first BBO generating photon pair 1-2. Refocussed, it passes the second BBO generating the ancillary pair 5-6 and again retroreflected through the second BBO generating pair 3-4. FIG. 3 : 3Experimental expectation values for every correlation function of the entanglement witness for the swapped state. 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[ "EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces", "EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces" ]
[ "Vernon J Lawhern \nHuman Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD\n\nDepartment of Computer Science\nUniversity of Texas at San Antonio\nSan AntonioTX\n", "Amelia J Solon \nHuman Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD\n\nDCS Corporation\nAlexandriaVA\n", "Nicholas R Waytowich \nHuman Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD\n\nDepartment of Biomedical Engineering\nColumbia University\nNew YorkNY\n", "Stephen M Gordon \nHuman Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD\n\nDCS Corporation\nAlexandriaVA\n", "Chou P Hung \nHuman Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD\n\nDepartment of Neuroscience\nGeorgetown University\nWashingtonDC\n", "Brent J Lance \nHuman Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD\n" ]
[ "Human Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD", "Department of Computer Science\nUniversity of Texas at San Antonio\nSan AntonioTX", "Human Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD", "DCS Corporation\nAlexandriaVA", "Human Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD", "Department of Biomedical Engineering\nColumbia University\nNew YorkNY", "Human Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD", "DCS Corporation\nAlexandriaVA", "Human Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD", "Department of Neuroscience\nGeorgetown University\nWashingtonDC", "Human Research and Engineering Directorate\nArmy Research Laboratory\nU.S\nAberdeen Proving Ground\nMD" ]
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Objective: Brain-Computer Interface technologies (BCI) enable the direct communication between humans and computers by analyzing brain measurements, such as electroencephalography (EEG). These technologies have been applied to a variety of domains, including neuroprosthetic control and the monitoring of epileptic seizures. Existing BCI systems primarily use a priori knowledge of EEG features of interest to build machine learning models. Recently, convolutional networks have been used for automatic feature extraction of large image databases, where they have obtained state-of-the-art results. In this work we introduce EEGNet, a compact fully convolutional network for EEG-based BCIs developed using Deep Learning approaches. Methods: EEGNet is a 4-layer convolutional network that uses filter factorization for learning a compact representation of EEG time series. EEGNet is one of the smallest convolutional networks to date, having less than 2200 parameters for a binary classification. Results: We show state-of-the-art classification performance across four different BCI paradigms: P300 event-related potential, error-related negativity, movement-related cortical potential, and sensory motor rhythm, with as few as 500 EEG trials. We also show that adding more trials reduces the error variance of prediction rather than improving classification performance. Conclusion: We provide preliminary evidence suggesting that our model can be used with small EEG databases while improving upon the state-of-the-art performance across several tasks and across subjects. Significance: The EEGNet neural network architecture provides state-of-the-art performance across several tasks and across subjects, challenging the notion that large datasets are required to obtain optimal performance.
10.1088/1741-2552/aace8c
[ "https://arxiv.org/pdf/1611.08024v1.pdf" ]
3,849,381
1611.08024
a7ab6fe31ee11a6b59e4d0c15de9f81661ef0d58
EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces November 28, 2016 Vernon J Lawhern Human Research and Engineering Directorate Army Research Laboratory U.S Aberdeen Proving Ground MD Department of Computer Science University of Texas at San Antonio San AntonioTX Amelia J Solon Human Research and Engineering Directorate Army Research Laboratory U.S Aberdeen Proving Ground MD DCS Corporation AlexandriaVA Nicholas R Waytowich Human Research and Engineering Directorate Army Research Laboratory U.S Aberdeen Proving Ground MD Department of Biomedical Engineering Columbia University New YorkNY Stephen M Gordon Human Research and Engineering Directorate Army Research Laboratory U.S Aberdeen Proving Ground MD DCS Corporation AlexandriaVA Chou P Hung Human Research and Engineering Directorate Army Research Laboratory U.S Aberdeen Proving Ground MD Department of Neuroscience Georgetown University WashingtonDC Brent J Lance Human Research and Engineering Directorate Army Research Laboratory U.S Aberdeen Proving Ground MD EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces November 28, 2016 Objective: Brain-Computer Interface technologies (BCI) enable the direct communication between humans and computers by analyzing brain measurements, such as electroencephalography (EEG). These technologies have been applied to a variety of domains, including neuroprosthetic control and the monitoring of epileptic seizures. Existing BCI systems primarily use a priori knowledge of EEG features of interest to build machine learning models. Recently, convolutional networks have been used for automatic feature extraction of large image databases, where they have obtained state-of-the-art results. In this work we introduce EEGNet, a compact fully convolutional network for EEG-based BCIs developed using Deep Learning approaches. Methods: EEGNet is a 4-layer convolutional network that uses filter factorization for learning a compact representation of EEG time series. EEGNet is one of the smallest convolutional networks to date, having less than 2200 parameters for a binary classification. Results: We show state-of-the-art classification performance across four different BCI paradigms: P300 event-related potential, error-related negativity, movement-related cortical potential, and sensory motor rhythm, with as few as 500 EEG trials. We also show that adding more trials reduces the error variance of prediction rather than improving classification performance. Conclusion: We provide preliminary evidence suggesting that our model can be used with small EEG databases while improving upon the state-of-the-art performance across several tasks and across subjects. Significance: The EEGNet neural network architecture provides state-of-the-art performance across several tasks and across subjects, challenging the notion that large datasets are required to obtain optimal performance. Introduction A Brain-Computer Interface (BCI) is a mechanism for communicating with a machine via brain signals, bypassing normal neuromuscular outputs by using neural activity [1]. Traditionally, BCIs leverage implanted electrodes for medical applications, such as neural control of prosthetic artificial limbs [2]. However, recent research has opened up the possibility for novel BCIs focused on enhancing performance of healthy users, often focused on noninvasive approaches based on electroencephalography (EEG) [3][4][5][6]. In both the medical and non-medical domains, there has been a proliferation of BCI paradigms. These paradigms range from directly controlling devices such as mouse cursors [7,8], to passive brain monitoring for estimating states such as alertness or fatigue [9][10][11] or the occurrence of epileptic seizures [12][13][14], to paradigms for spelling [15] or image analysis [16]. Despite this wide range of paradigms, the general approach to desigining BCI technologies remain the same. A BCI consists of five main processing stages [17]: a data collection stage, where neural data is recorded; a signal processing stage, where the recorded data is preprocessed and cleaned; a feature extraction stage, where meaningful information is extracted from the neural data; a classification stage, where a decision is interpreted from the data; and a feedback stage where the result of that decision is provided to the user. Each distinct BCI paradigm relies on different aspects of the neural signal obtained from the user, meaning that each paradigm uses distinct and specialized methods for signal processing [18], feature extraction [19] and classification [20]. Many feature extraction and classification methods have been used for EEG-based BCI, including spatially filtering the EEG data (e.g. Common Spatial Patterns, or CSP), spectrally filtering the data (e.g. Fourier or autoregressive methods), dimension reduction methods such as Principal Component Analysis (PCA), blind source separation methods such as Independent Components Analysis (ICA), and source localization methods such as LORETA [21][22][23][24][25]. Unfortunately, the optimal combination of feature extraction and classification approach has to be defined manually for each BCI application, a process which often requires significant subject-matter expertise and a priori knowledge about the expected EEG signal [20]. There is also no guarantee that the same combination of feature extraction and classifier will provide acceptable performance across BCI applications, limiting the overall utility of these systems. Deep Learning has largely alleviated the need for manual feature extraction, achieving stateof-the-art performance in fields such as computer vision and speech recognition [26,27]. The term Deep Learning refers to a class of multi-layered artificial neural network models aimed at learning abstract feature representations and classifiers from raw data, with the derived feature representation depending on the architecture used. The use of convolutional neural networks (CNNs) in particular has increased significantly, due in part to the success of the first large-scale CNN for image classification that won the ImageNet 2012 competition [28]. This model obtained gains of over 10% in top-5 accuracy, where gains in previous years managed only a few percentage points. Subsequent models published by various research groups have improved upon this approach, with current classification performance rivaling that of human labelers [27,29,30]. The main advantage of this approach lies in the ability of the network to automatically extract relevant features for the problem at hand, rather than relying on manual feature extraction approaches. Recently, Deep Learning approaches have been used to model neurophysiological data, in functional magnetic resonance imaging (fMRI) [31] as well as EEG [32][33][34][35]. For example, Längkvist et. al. used deep belief networks (DBNs) and restricted boltzmann machines (RBMs), together with hidden Markov models, to temporally model different stages of sleep using EEG [36]. Mirowski et al. used a temporal convolutional network on EEG signals for predicting epileptic events [37]. Bashivan et. al. used convolutional recurrent networks for classification of mental workload by temporally stacking EEG "images" using frequency transforms to form a "movie", where they then used recurrent neural network approaches, originally developed for video classification, to predict mental workload [32]. Stober et. al. investigated using convolutional networks on frequency-transformed EEG signals for classification among different classes of listened music [34,38]. Temporal convolutional networks have also been used for the detection of visual-evoked potentials in [39] and in [33]. However, a comprehensive analysis of the performance of a single model architecture across multiple EEG modalities has not been previously explored. In this work we introduce EEGNet, a compact fully convolutional network for EEG-based BCIs developed using Deep Learning methodologies. EEGNet differs from previous convolutional models for EEG in several ways. First, our model removes the dependence on the channel layout through the use of spatial filtering in the first layer [21]. This operation serves to both improve the signal-to-noise (SNR) ratio and reduce the dimensionality of the EEG signal of interest. Second, we focus on spatiotemporal convolutions in the spatial filter space to capture both spatial and temporal relationships in the EEG. Finally, our model omits fully-connected layers in an effort to reduce the total number of parameters, a strategy inspired by the work of [40]. We evaluate our model against the current state-of-the-art approaches for four data collections representing four different BCI paradigms: P300 visual-evoked potential (P300), error-related negativity (ERN), movement-related cortical potential (MRCP) and the sensory motor rhythm (SMR). For binary classification problems, EEGNet has approximately 2200 parameters, which is half the total number of parameters compared to the model proposed by [41] and has much fewer parameters than the ∼ 10K parameters of the smallest model proposed by [32]. We fit several models, all with the same number of free parameters, to statistically control for the effect of model capacity versus performance. Our models are also trained through a cross-subject cross-validation procedure, implying that they are user-independent, without requiring any test subject information [42,43], whereas previous approaches have mainly trained within subject [33,39,41]. We provide preliminary evidence suggesting that our model can be used with small EEG databases while still obtaining, and in some cases improving upon, the state-of-the-art performance across several tasks and across subjects, challenging the notion that large datasets are required to obtain optimal performance. Background BCIs are generally categorized into two types, depending on the EEG feature of interest [42]: event-related and oscillatory. Event-Related Potential (ERP) BCIs are designed to detect an EEG response to a known, time-locked external stimulus. They are generally robust across subjects and contain well-stereotyped waveforms, enabling the exact time course of the ERP to be modeled through machine learning efficiently [44]. BCI systems can also leverage desynchronization/synchronization of EEG oscillations, such as those which might occur during a self-paced mental task or a change in user mental state. In contrast to ERP-based BCIs, which rely mainly on the detection of the ERP waveform from some external event or stimulus, Oscillatory BCIs use the signal power of specific EEG frequency bands for external control and are generally not time-locked to an external stimulus [45]. When oscillatory signals are time-locked to an external stimulus, they can be represented through event-related spectral perturbation (ERSP) analyses [46]. Oscillatory BCIs are more difficult to train, generally due to the lower SNR as well as greater variation across subjects [45]. Oscillatory BCIs are also more susceptable to external noise sources than ERP BCIs, and thus require more data and/or more advanced signal processing approaches to design effective systems [42]. An overall summary of the datasets used to evaluate EEGNet can be found in Table Paradigm Feature The P300 event-related potential is a stereotyped neural response to novel visual stimuli [47]. It is most commonly elicited with the visual oddball paradigm, where participants are shown repetitive "nontarget" visual stimuli that are interspersed with infrequent "target" stimuli at a fixed presentation rate (for example, 1 Hz). Observed over the parietal cortex, the P300 waveform is a large positive deflection of electrical activity observed approximately 300 ms post stimulus onset, the strength of the observed deflection being inversely proportional to the frequency of the target stimuli. The P300 ERP is one of the strongest neural signatures observable by EEG, especially when the target presentation rate is infrequent [47]. When the image presentation rate increases to 2 Hz or more, it is commonly referred to as rapid serial visual presentation (RSVP), which has been used in several BCIs for large image database triage [43,48,49]. The EEG data used here has been previously described in [49]; a brief description is given below. 18 participants volunteered for an RSVP BCI study. Participants were shown images of natural scenery at 2 Hz rate, with images either containing a vehicle or person (target), or with no vehicle or person present (nontarget). The target/nontarget ratio was 20%/80%. Data from 15 participants (9 male and 14 right-handed) who ranged in age from 18 to 57 years (mean age 39.5 years) were further analyzed. EEG recordings were digitally sampled at 512 Hz from 64 scalp electrodes arranged in a 10-10 montage using a BioSemi Active Two system (Amsterdam, The Netherlands). Continuous EEG data were referenced offline to the average of the left and right earlobes, digitally bandpass filtered to 1-40 Hz and downsampled to 128 Hz. EEG trials of target and nontarget conditions were extracted at [0, 1]s post stimulus onset, and used for a two-class classification. Paradigm 2: Feedback Error-Related Negativity (ERN) Error potentials are perturbations of the EEG following an erroneous or unusual event in the subject's environment or task. They can be observed in a variety of tasks, including time interval production paradigms [50] and in forced-choice paradigms [51,52]. Here we focus on the feedback error-related negativity (ERN), which is an amplitude perturbation of the EEG following the perception of an erroneous feedback produced by a BCI. The feedback ERN is characterized as a large negative deflection approximately 300ms after feedback, followed by a positive component 500ms to 1s after feedback (see Figure 7 of [53] for an illustration). The detection of the feedback ERN provides a mechanism to infer, and to possibly correct in real-time, the incorrect output of a BCI. This two-stage system has been proposed as a hybrid BCI in [54,55] and has been shown to improve the performance of a P300 speller in online applications [56]. Layer Input (C x T ) Operation Output Number of Parameters 1 C x T 16 x Conv1D (Cx1) 16 x 1 x T 16C + 16 16 x 1 x T BatchNorm 16 x 1 x T 32 16 x 1 x T Reshape 1 x 16 x T 1 x 16 x T Dropout (.25) 1 x 16 x T 2 1 x 16 x T 4 x Conv2D (2x32) 4 x 16 x T 4 × 2 × 32 + 4 = 260 4 x 16 x T BatchNorm 4 x 16 x T 8 4 x 16 x T Maxpool2D (2,4) 4 x 8 x T /4 4 x 8 x T /4 Dropout (.25) 4 x 8 x T /4 3 4 x 8 x T /4 4 x Conv2D (8x4) 4 x 8 x T /4 4 × 4 × 8 × 4 + 4 = 516 4 x 8 x T /4 BatchNorm Table 2: Convolutional Network Architecture, where C = number of channels, T = number of time points and N = number of classes, respectively. For Layers 1-3, the Exponential Linear Unit (ELU) activation function is used. 4 x 8 x T /4 8 4 x 8 x T /4 Maxpool2D (2,4) 4 x 4 x T /16 4 x 4 x T /16 Dropout (.25) 4 x 4 x T /16 4 4 x 4 x T /16 Softmax Regression N T N + N Total 16C + N (T + 1) + 836 The EEG data used here comes from [53] and was used in the "BCI Challenge" hosted by Kaggle (https://www.kaggle.com/c/inria-bci-challenge); a brief description is given below. 16 healthy participants participated in a P300 speller task, a system which uses a random sequence of flashing letters, arranged in a 6 × 6 grid, to elicit the P300 response [15]. EEG data was recorded at 600Hz using 56 passive Ag/AgCl EEG sensors (VSM-CTF compatible system) following the extended 10-20 system for electrode placement. The EEG data was referenced offline to an electrode placed on the nose, band-pass filtered to 1-40 Hz and down-sampled to 128Hz. EEG trials of correct and incorrect feedback were extracted at [0, 1]s post feedback presentation and used as features for a two-class classification. Datasets: Oscillatory Paradigm 3: Movement-Related Cortical Potential (MRCP) Some neural activities contain both an ERP component as well as an oscillatory component. One particular example of this is the movement-related cortical potential (MRCP), which can be elicited by voluntary movements of the hands and feet and is observable through EEG along the central and midline electrodes, contralateral to the hand or foot movement [57]. The oscillatory component of the MRCP can be seen both before movement onset (an early desynchronization in the 10-12Hz frequency band) as well as after movement onset (a late synchronization of 20-30Hz activity approximately 1s after movement execution). The ERP component of the MRCP occurs at the start of the movement, with a duration of approximately 800ms. The MRCP has been used previously to develop motor control BCIs for both healthy and physically disabled patients [58]. The EEG data used here has been previously described in [59]; a brief description is given below. In this study, 13 subjects performed self-paced finger movements using the left index, left middle, right index, or right middle fingers. This produced the well-known alpha and beta synchronizations (i.e. increases in power) and desynchronizations (i.e. decreases in power), most clearly observed over the contralateral motor cortex [60][61][62]. The data was originally recorded using a 256 channel BioSemi Active II at 1024 Hz. Due to extensive signal noise present in the data, the EEG data were first processed with the PREP pipeline [63]. The data was referenced to linked mastoids, bandpass filtered between 0.3 Hz and 50 Hz, and then downsampled to 128 Hz. We further downsampled the channel space to the standard 64 channel BioSemi montage. The index and middle finger blocks for each hand were combined for binary classification of movements originating from the left or right hand. The classes are approximately balanced, with each subject having about 500 trials per class. EEG trials of left and right hand finger movements were extracted at [−.5, 1]s around finger movement onset and used for a two-class classification. Paradigm 4: Sensory Motor Rhythm (SMR) A common control signal for oscillatory-based BCI is the sensorimotor rhythm (SMR), wherein mu (8-12Hz) and beta (18-26Hz) bands desynchronize over the sensorimotor cortex contralateral to an actual or imagined movement. The SMR is very similar to that of the oscillatory component of the MRCP. While SMR-based BCIs can facilitate nuanced, endogenous BCI control (enabling high dimensional control of cursors or even prosthetic limbs) they are not without their practical challenges. As signals, SMRs tend to be weak and highly variable across and within subjects, conventionally demanding user-training (neurofeedback) and long calibration times (20 minutes) in order to achieve reasonable performance [42]. The EEG data used here is from the PhysioNet EEG motor movement/imagery dataset (N=109) [64,65]. The EEG data was recorded using 64 channels following the aforementioned 10-10 system for electrode placement, at a sampling rate of 160 Hz. The EEG data were band-pass filtered at 0.1-40Hz. A visual cue appeared to the subjects for 4.1s indicating which task to perform. Periods of activity were separated by 4.1s of rest. To avoid confounding neural activity due to the onset of the visual cue, we omitted from analysis the data 1s after the onset of the cue box. For the remaining 3.1s of data, we extracted two trials of 2 seconds length, with a 0.9 second overlap. The goal of the analysis is to predict the imagined movements of the right or left hand. Methods Convolutional Network Architecture Our EEGNet model is shown in Table 2, for EEG trials having C channels and T time samples. All layers used the Exponential Linear Unit (ELU) non-linear activation function [66]. The model was estimated using Adam, a stochastic optimization algorithm using adaptive moment estimation [67], optimizing the binary cross-entropy criterion. All models were trained on an NVIDIA Quadro K6000 GPU, with CUDA 7.5 and cuDNN v5, in Theano [68], using the Keras API [69]. • In Layer 1, we learn 16 convolutional kernels of size (C, 1). This operation estimates a set of spatial filters over the entire period of the trial and is similar to that of previous approaches such as Common Spatial Patterns (CSP) [21], xDAWN spatial filtering [70] and independent component analysis (ICA) [24]. Note that these approaches are specifically designed to produce spatial filters that either maximize the variance difference among two classes (CSP), enhance the signal to signal-plus-noise ratio of the EEG signal of interest (xDAWN) or produce spatial filters that are as mutually independent as possible (ICA). In contrast, the spatial filters used here are optimized to minimize the categorical cross-entropy of the predicted outputs and are not required to achieve either optimal variance separation or mutual independence. The spatial filters are regularized with an elastic-net (L 1 + L 2 ) constraint, with L 1 = 0.01 and L 2 = 0.01. We apply Batch Normalization [71] together with Dropout [72] as this improved model robustness. • In Layer 2, we learn 4 2-dimensional convolutional kernels of size (2 × 32). Here, we use zeropadding to preserve the original dimension of the data after the convolution. 2D max-pooling is applied to the data; while this operation is traditionally done to induce invariance to image transformation in computer vision [73], we find that for processing these EEG tasks this operation is beneficial primarily for dimension reduction. The total reduction in parameter size due to max-pooling is by a factor of 8 (2 x 4). As in Layer 1, Batch Normalization and Dropout are used. • In Layer 3, we learn 4 2-dimensional convolutional kernels of size (8 × 4). All the operations in Layer 2 are also applied here. • In Layer 4, the features are passed to a softmax classification with N units, N being the number of classes in the data. We omit the use of a fully connected layer prior to the softmax classification layer to reduce the number of free parameters in the model, as was done in [40]. Convolutional Network Model Comparison In order to statistically separate the effects of the number of parameters of the model versus classification performance, we generated 12 different models, all having the same number of parameters, using the model framework described in Table 2. We generated the 12 models by setting the convolutional kernel sizes in Layer 2 to be one of 4 different values: (16,4), (8,8), (4,16) and (2,32) and the kernel sizes in Layer 3 to be one of 3 different values: (8,4), (4,8) and (2,16), respectively. Each configuration emphasizes different spatial/temporal aspects of the EEG signal. We also compare EEGNet to a recent convolutional network model for EEG presented by Manor and Geva in [33] with some minor modifications. First, since the model in [33] was designed using EEG data sampled at 64Hz, we appropriately scaled the length of the temporal kernels to match the rate of our data (128Hz for the P300, Feedback ERN and MRCP, 160Hz for SMR) to model the same amount of time temporally. Second, [33] uses Dropout after fully-connected layers only, however the dropout fraction was not specified. Therefore, we set it to 0.5 for our comparison. Also note that the rectified linear unit (ReLU) activation function was used in [33], in contrast to the ELU that is used in EEGNet. Note that this model has approximately 11.8M parameters, which represents more than a 5000-fold increase in parameter size when compared to EEGNet. This model will be referred to as the "MG" model for the remainder of the manuscript. Comparison with Traditional Approaches We compare the performance of our CNN model to that of the best performing reference algorithm for each individual paradigm. For all event-related data analyses (P300 VEP, ERN, MRCP) the reference algorithm is xDAWN Spatial Filtering + Bayesian LDA [70]. For oscillatory-based classification of SMR, the reference algorithm is a CSP with covariance matrix regularization [74] with a Regularized LDA as used in [42]. ERP: xDAWN Spatial Filtering + Bayesian LDA Here we provide a summary of the xDAWN algorithm; more details can be found in [70,75]. xDAWN is a spatial filtering algorithm designed to enhance the ERP through maximizing the signal to signal plus noise ratio (SSNR). The result of this algorithm are the spatial filters U , which are ranked in terms of their SSNR. The EEG signal X is decomposed into three parts as: XU = (D 1 A 1 + D 2 A 2 + N ) U(1) where X ∈ R T ×C , where T and C denote the number of time samples and channels, respectively. D 1 A 1 denotes the ERP neural response, D 2 A 2 denotes the neural response that is confounded with the ERP neural response whenever any stimulus is presented, and N is a residual noise term. xDAWN optimizes the SSNR, given as: SSN R(U ) = argmax U Tr U T 1 T D T 1 D 1Â1 U Tr (U T X T XU )(2) where 1 is the solution to the least squares problem A =  1 A 2 = D 1 D 2 T D 1 D 2 −1 D 1 D 2 T X(3) where [D 1 ; D 2 ] ∈ R T ×F , F denotes the number of filters, and Tr(·) denotes the trace operator. The spatially filtered EEG signals are then classified using Bayesian Linear Discriminant Analysis (BLDA) (see Appendix B of [76] for more details). Oscillatory (SMR): Covariance Shrinkage Regularized CSP The best performing user-independent model for SMR-based BCI, which was evaluated on the BCI Competition IV competition dataset and compared against 11 other approaches, is the approach outlined in [77], wherein all available subjects are pooled to train CSP filters and a linear discriminant analysis (LDA) classifier. In addition to the pooled design, the covariance matrices in both the CSP and LDA algorithms are regularized by diagonal loading, which shrinks the covariance matrix towards the identity matrix [77]. A single parameter, λ, must be identified, for which there is an automatic method outlined in [74]. EEG Data Processing We split each data collection into a training, testing, and validation set. For Paradigms 1, 2 and 3, we sample, without replacement, N − 2 subjects for the train set and 1 for the test and validation sets, respectively, where N is the total number of subjects the dataset (see Table 1). Due to the relatively few trials per subject in Paradigm 4, we sample, without replacement, 64 subjects for the train set, 15 for the test and 30 for the validation set. In all cases, the data were balanced such that the number of trials in each class is equal; this was done by randomly downsampling the size of the larger class to match the size of the smallest class. For each paradigm we generated 30 unique datasets of training/testing/validation with this procedure. The train, test and validation sets were z-score normalized at each (channel x time) pair relative to the training set; if the EEG data is arranged in a 3-dimensional matrix as (channels, time, trials), then the z-score is calculated along the third dimension. Statistical Analysis We calculate ∆AU C, the difference between the AU C performance between EEGNet and the reference algorithm for each paradigm for each of 30 folds, where positive values of ∆ denote that EEGNet outperformed the reference algorithm. For each EEGNet model configuration a one-way t-test was performed, testing if the model significantly outperformed the reference algorithm. We use the False Discovery Rate (FDR) [78] procedure to correct for multiple comparisons. We also conduct two additional analyses where model performance is compared relative to (1) the number of subjects in the training set and (2) the number of EEG trials in the training set. For the first analysis, we sample, without replacement, N subjects to be in the training set and increased N up to the total number of training subjects available (see Table 1). For the P300, ERN and MRCP analyses we started with N = 1 subject and increased by 1 subject up to the total number of training subjects available, while for SMR we started with N = 8 subjects and increased by 8 subjects up to the total number of training subjects available. This procedure was repeated 10 times for each value of N . For the second analysis, from the full training set we randomly sample, without replacement, K trials and set this random sample to be the training set. For the P300 analysis we start with K = 500 trials, then increase by 2000 trials up to the total number of available trials; the ERN analysis, we start with K = 500 trials, then increase by 500 ; the MRCP analysis we start with K = 500 trials, then increase by 2000; finally for the SMR analysis we start with approximately K = 800 trials, then increase by approximately 700 trials. This procedure was repeated 10 times for each value of K. Figure 1 shows the AUC difference in mean classification performance between the EEGNet model and the reference algorithm for each paradigm. For classification of both the P300 and SMR, we see that all models, regardless of kernel configuration, produced statistically significant improvements over that of the reference algorithm. For classification of the ERN, we see that only a few of the models were significantly better, with the (2, 32) x (8, 4) configuration having the highest performance improvement. While gains were observed for classification of the MRCP, these gains were not statistically significant (p > .05). Results Of the 12 models tested, either the (4, 16) × (8, 4) or the (2, 32) × (8, 4) model configurations were among the top four models across all data paradigms (see Figure 1). This is true for the ERN paradigm, whose best performing model is the (2,32) x (8,4) configuration. While the best model for P300 was the (8,8) x (4,8) configuration, the next best configuration is the (4,16) x (8,4), which is the best configuration observed in MRCP. For SMR, the fourth-best model is the (2,32) x (8,4) configuration. This trend suggests that a general model configuration can be used when lack of apriori knowledge of model architecture exists for a particular analysis, although specific configurations are better for each individual paradigm. Figure 2 illustrates the classification performance relative to the number of trials in the training set when using the (2, 32) × (8, 4) model configuration, with different colors denoting the number of subjects available in the training set. As expected, when the size of the training set increases, the classification performance also increases while also having a smaller error variance in the prediction. The classification performance with less than 20% of the training set is often times very competitive when compared to having the full training set, suggesting that our compact model can accurately capture the dynamics of EEG across subjects with very little data. For example, for P300 classification, having 2000 training trials (slightly more than 20% of all available trials) produces a classification performance equivalent to having all trials available, albeit with a slightly larger variance in the prediction. For SMR, the classification performance of the model trained with approximately 2200 trials, which is about 33% of the training set, produces a classification performance within 5% of the model using all available training data. This trend is also observed when sampling a subset of data from all available subjects (orange open circles) and using this subset as the training set. For the Feedback ERN classification, having only 500 trials sampled across 14 subjects (which constitutes less than 20% of all available trials) produces a classification performance of approximately 0.78 AUC, which is within 5% of the classification performance from having all trials available (0.8 AUC, with ∼ 2700 trials). For the MRCP, however, we found that we needed significantly more trials (at least 2500) to obtain reasonable classification performance. We believe this is due primarily to the MRCP having lower SNR than that of the P300 and the Feedback ERN. Across all paradigms we found that having at least 1 EEG trial per parameter produced consistently good performance, and that one could have fewer trials depending on the strength of the expected neural response. Beyond this threshold, increases in the training data size primarily has the effect of reducing the error variance as opposed to improving classification performance. Figure 3 shows the difference in AUC performance of EEGNet when compared to both the MG model [33] and the reference algorithm (xDAWN+BLDA for ERP and Regularized CSP for SMR) for each of 10 folds across several different training set sizes. EEGNet outperforms both the MG model and the reference algorithm in nearly all cases across all training set sizes. For the P300 paradigm, xDAWN has both larger errors as well as larger error variances at the smaller training set sizes, both of which decrease as the training set size increases, converging to an average of approximately 0.05 AUC difference. Generally speaking, the MG model outperforms xDAWN at all training set sizes while underperforming EEGNet. This is interesting to note, as MG contains more than 5000 times the parameters than EEGNet (∼11.8M for MG vs. ∼2200 for EEGNet). In contrast, for classification of the SMR, the MG model performance does not exceed chance level (0.5 AUC) at any training set size, whereas EEGNet was able to classify better than chance with only 800 trials (see Figure 2). We suspect this is due to having insufficient data given the larger model size of the MG model. Figure 3: Difference in AUC performance of EEGNet when compared to both the MG model and the reference algorithm for the P300 and SMR paradigms for the data shown in Figure 2. Solid lines denote the mean performance at each training set size. The reference algorithm is xDAWN + BLDA for the P300 and a Regularized CSP for SMR. Discussion In this work we proposed EEGNet, a compact fully-convolutional network for EEG-based BCIs. We showed that our model can outperform the state-of-the-art approach over many BCI paradigms with as few as 500 training samples, and that adding more training samples primarily reduces the prediction variance as opposed to obtaining improved performance. We note that the number of parameters in our model is directly tied to the sampling rate of the EEG (see Table 2). The EEG data used in this manuscript have a sampling rate of at least 128 Hz. This gives us a sufficient sampling rate to measure frequency content of up to 64 Hz by the Nyquist theorem, encapsulating the EEG frequency bands of interest for most BCI applications. A higher sampling rate would incur more parameters in the model, which could be offset by increasing the dimension of the maxpool used in Layers 2 and 3. In contrast, there is generally less influence on the number of channels of the EEG due to the spatial filter that is used in Layer 1; the spatial filter operation removes any channel dependence after the first layer, while the effect of the sampling rate propagates throughout all layers. Our model employs multiple forms of regularization at each layer, which is an approach that is generally uncommon in other fields such as computer vision and natural language processing. For example, the computer vision models proposed by [29] use Dropout only after dense layers, while [71] suggests that using Batch Normalization may remove the need for Dropout entirely. In contrast, we found that Batch Normalization with Dropout improved model performance over each approach individually, and that combining these regularizers together with elastic net regularization further improved robustness. We believe that, given the low SNR of EEG together with traditionally small datasets, that this is necessary to learn the true underlying neural signal, as opposed to learning on signal noise. Our model comparison analysis showed that a general model configuration (shown in Table 2) can be used whenever knowledge of a particular EEG feature of interest is not known a priori. Because all the models have exactly the same number of parameters, statistically speaking the performance of the model can be primarily attributed to the kernel configuration itself. The model alternates between aggregating information along spatial (Layers 1 and 3) and temporal (Layer 2) dimensions. This makes intuitive sense in terms of efficient model parametrization using filter factorization. For example, to represent an EEG segment of size [C, T ], for C channels and T time points, one could model the kernel size as [C, T ] directly, incurring C × T total parameters, or one could model the EEG segment with two layers, with the first layer having a kernel size of [C, 1] and the second layer having a kernel size of [1, T ], incurring a total of C + T parameters. It is easy to see that both approaches model the same segment of data; however, the second approach represents a significant reduction in the total number of parameters. This parametrization trick was used by [29] as a way to efficiently build deep convolutional networks for image analysis; by replacing a 7 × 7 kernel size with 3 layers of 3 × 3 kernels, one can represent the same amount of data with about a 44% reduction in parameters. We essentially use the same strategy here to represent EEG time series in a compact manner by aggregating information over the spatial and temporal dimensions iteratively. Because our model was validated using a cross-subject cross-validation procedure, our model is essentially user-independent; the model requires no observations from the test subject to achieve reasonable classification performance. As a result, this model has immediate implications for practical BCI systems, as the development of reliable user-independent methods is still an active area of research interest [42,43]. One question that arises in this situation is how to effectively update a model using trials obtained from the test subject, especially when the training dataset used to train the model is large. We will explore how to update our model as EEG trials are recorded in real time in future research. The first three layers of EEGNet can potentially be used as a preprocessing step prior to using recurrent neural networks for modeling long temporal sequences in the EEG. Recently, a recurrent convolutional network was proposed by [32] to predict mental workload. Their approach first used fourier transforms to convert the EEG time series into a time-frequency representation. This representation was then used in a recurrent neural network to model temporal dependencies in the time-frequency space. In our future work we will explore a similar approach, where we will replace the manually derived representation (time-frequency space) with the automatically-derived representation from the EEGNet model as the initial feature extraction, with the goal of analyzing non-time-locked EEG events temporally. We will also test the generalizability of this approach for describing other kinds of EEG phenomena, for example performance or alertness monitoring, in future research. Conclusion In this manuscript we present EEGNet, a compact convolutional network for EEG-based BCIs. The model is computationally efficient, and for some paradigms, can be estimated with as few as 500 EEG trials. We showed that the model can outperform the current state-of-the-art across several BCI paradigms. Because our model is validated using a cross-subject cross-validation procedure, our model is user-independent, which has immediate applications for practical BCI technologies. We show significant gains in classification accuracy of EEGNet across multiple paradigms when compared to a recent CNN model proposed by [33]. Finally, this model can potentially serve as the initial feature extraction to more advanced neural network architectures such as recurrent neural networks to efficiently model long EEG sequences. Figure 1 : 1Classification performance for all 12 EEGNet models across all paradigms. 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[ "https://github.com/fchollet/keras," ]
[ "Optimal Multi-robot Formations for Relative Pose Estimation Using Range Measurements", "Optimal Multi-robot Formations for Relative Pose Estimation Using Range Measurements" ]
[ "Charles Champagne Cossette ", "Mohammed Ayman Shalaby ", "David Saussié ", "Jérôme Le Ny ", "James Richard Forbes " ]
[]
[]
In multi-robot missions, relative position and attitude information between agents is valuable for a variety of tasks such as mapping, planning, and formation control. In this paper, the problem of estimating relative poses from a set of inter-agent range measurements is investigated. Specifically, it is shown that the estimation accuracy is highly dependent on the true relative poses themselves, which prompts the desire to find multi-agent formations that provide the best estimation performance. By direct maximization of Fischer information, it is shown in simulation and experiment that large improvements in estimation accuracy can be obtained by optimizing the formation geometry of a team of robots.
10.1109/iros47612.2022.9981301
[ "https://arxiv.org/pdf/2205.14263v1.pdf" ]
249,192,259
2205.14263
666969b826d8f685c9d8b353b71c6ece3080ae27
Optimal Multi-robot Formations for Relative Pose Estimation Using Range Measurements Charles Champagne Cossette Mohammed Ayman Shalaby David Saussié Jérôme Le Ny James Richard Forbes Optimal Multi-robot Formations for Relative Pose Estimation Using Range Measurements In multi-robot missions, relative position and attitude information between agents is valuable for a variety of tasks such as mapping, planning, and formation control. In this paper, the problem of estimating relative poses from a set of inter-agent range measurements is investigated. Specifically, it is shown that the estimation accuracy is highly dependent on the true relative poses themselves, which prompts the desire to find multi-agent formations that provide the best estimation performance. By direct maximization of Fischer information, it is shown in simulation and experiment that large improvements in estimation accuracy can be obtained by optimizing the formation geometry of a team of robots. I. INTRODUCTION The ability for a robot, or agent, to determine the relative position and attitude, collectively called pose, of another robot is an important prerequisite in multi-robot team applications. Tasks such as collaborative mapping and planning, as well as formation control, usually require relative position or pose information between the robots. This functionality has been achieved using various sensors, such as cameras with object detection [1], or with infrared emitters/receivers [2]. Ultra-wideband (UWB) is a type of radio signal that can be timestamped with sub-nanosecond-level accuracy at both transmission and reception [3]. As such, UWB is commonly used to obtain about 10-cm-accurate range (distance) measurements between a pair of UWB transceivers called tags. The transceivers' small size, weight, and cost make them an attractive sensor for many robotics applications, including relative position estimation in multi-robot scenarios. By placing one or more tags on each robotic agent, a completely self-contained relative positioning solution is possible [4,5], which does not depend on any external infrastructure such as static UWB tags, called anchors, or a motion capture system. In this theme of infrastructure-free relative position estimation, a wide variety of approaches exist in the literature. For example, visual odometry or optical flow have been used along with a single UWB tag on each agent [6][7][8][9][10]. However, these single-tag-per-agent approaches typically have a *This work was supported by the FRQNT under grant 2018-PR-253646, with funding also acknowledged from CFI JELF, the William Dawson Scholar Program, and the NSERC Discovery Grant Program. C. C. Cossette persistency of excitation (POE) requirement. That is, agents must be under persistent relative motion for relative states to be observable [11,12]. This can be energy intensive and impractical, as a static or slowly-moving team of agents will have drifting position estimates. One way to eliminate the POE requirement is to use visual detection of other agents, as in [13], which also uses visual odometry and UWB ranging. Although their solution is accurate, deep-learningbased object detection can be computationally expensive, and the agents must periodically enter each other's camera fieldof-view. Another class of approaches that do not require computer vision or POE is to have multiple tags on some or all of the agents [4,14]. We have recently proposed installing two UWB tags on each agent [15], where we show that relative positions are observable from the range measurements alone. When combined with an inertial measurement unit (IMU) and a magnetometer, the agents' individual attitudes can be estimated relative to a world frame, allowing relative positions to be resolved in the world frame. However, magnetometer sensor measurements are substantially disturbed in the presence of metallic structures indoors [16,17], which degrades estimation accuracy. Another challenge is that there are certain formation geometries that cause the relative positions to be unobservable, such as when all the UWB tags lie on the same line [15]. This is closely tied to the well-known general dependence of positioning accuracy on the geometry of the tags, and arises even in the presence of static UWB anchors [18]. To avoid divergence of the state estimator, multi-robot missions relying on inter-robot range measurements for relative position estimation must avoid these aforementioned unobservable formation geometries. This imposes a constraint on planning algorithms. A planning solution to avoid unobservable positions is proposed in [19], where a cost function based on the Cramér-Rao bound quantifies the estimation accuracy as a function of robot positions. A similar approach is presented in [20] for multi-tag robots. Limitations of these approaches include the requirement of the presence of anchors, as well as the lack of explicit consideration of agent attitudes. This paper presents a method for computing optimal formations for relative pose estimation, and is the first to do so in the absence of anchors. Furthermore, it is shown that with two-tag agents, both the relative position and relative heading of the agents are locally observable from range measurements alone. The problem setup is deliberately formulated in the agents' body frames, thus being completely invariant to any arbitrary world frame, eliminating the need for a magnetometer. This paper further differs from [20] by using SE(n) pose transformation matrices to represent the relative poses, avoiding the complications associated with angle parameterizations of attitude. This leads to the use of an on-manifold gradient descent procedure to determine optimal formations. Simulations and experiments show that the variance of estimation error does indeed decrease as the agents approach their optimal formations. The proposed cost function is general to 2D or 3D translations, arbitrary measurement graphs, and any number of arbitrarily-located tags. Moreover, the proposed cost function goes to infinity when the agents approach unobservable configurations, meaning that its use naturally avoids such unobservable formation geometries. For these reasons, the cost function is amenable to a variety of future planning applications, such as to impose an inequality constraint on an indoor exploration planning problem. The paper is outlined as follows. The problem setup, notation, and other preliminaries are described in Section II. The optimization setup and results are described in Section III. The optimal formations are evaluated experimentally in Section IV. II. PROBLEM SETUP, NOTATION, AND PRELIMINARIES Consider N agents along with M ranging tags distributed amongst them. Let τ 1 , τ 2 , . . . , τ M consist of unique physical points collocated with the ranging tags. Let a 1 , . . . , a N represent reference points on the agents themselves. The intertag range measurements are represented by a measurement graph G = (V, E) where V = {1, . . . , M } is the set of nodes, which is equivalent to the set of tag IDs, and E is the set of edges corresponding to the range measurements. Defining the set of agent IDs as A = {1, . . . , N }, it is convenient to define a simple "lookup function" : V → A that returns the agent ID on which any particular tag is located. For example, if τ i is physically on agent α, then (i) = α. An example scenario with three agents using this notation is shown in Figure 1. A bolded 1 and 0 indicates an appropriately-sized identity and zero matrix, respectively. A. State Definition and Range Measurement Model Since the agents are rigid bodies, an orthonormal reference frame attached to their bodies can be defined. A position vector representing the position of point z, relative to point w, resolved in the body frame of agent α is denoted r zw α ∈ R n . The attitude of the body frame on agent α relative to the body frame on agent β is represented with a rotation matrix C αβ ∈ SO(n) such that r zw α = C αβ r zw β . The relative position and attitude between agents α and β, (r a β aα α , C αβ ) define the relative pose between them, and can be packaged together in a pose transformation matrix T αβ = C αβ r a β aα α 0 1 ∈ SE(n).(1) The exponential and logarithmic maps of the special Euclidean group SE(n) are denoted exp : se(n) → SE(n) and ln : SE(n) → se(n), respectively, where se(n) is the Lie algebra of SE(n). The common "wedge" operator (·) ∧ : R m → se(n) and "vee" operator (·) ∨ : se(n) → R m are also used in this paper. For a more thorough background on matrix Lie groups, including expressions for the aforementionned operators, see [21,22]. Throughout this paper, Agent 1 will be the arbitary reference agent, such that the poses of all the other agents are expressed relative to Agent 1 x = (T 12 , . . . , T 1N ).(2) A single generic range measurement between tag i and tag j is modelled as a function of the state x = (T 12 , . . . , T 1N ) with y ij (x) = ||C 1α r τiaα α + r aαa1 1 − (C 1β r τj a β β + r a β a1 1 )|| + v ij ,(3) where α = (i), β = (j), and v ij ∼ N (0, σ 2 ij ). This can be written compactly with the pose transformation matrices, y ij (x) = DT 1α r τiaα α 1 − DT 1β r τj a β β 1 + v ij DT 1α p i − DT 1β p j + v ij ,(4) where D = [1 0]. In fact, the state x = (T 12 , . . . , T 1N ), written here as a tuple of pose matrices, is an element of a Lie group of its own, x ∈ SE(n) × . . . × SE(n) SE(n) N −1 . The group operation for SE(n) N −1 is the elementwise matrix multiplication of the pose matrices in two arbitrary tuples, and the group inverse is the elementwise matrix inversion of the elements of the tuple x. The ⊕ operator is defined here as x ⊕ δx = T 12 exp(δξ ∧ 2 ), . . . , T 1N exp(δξ ∧ N ) , (5) 1 y a 2 a 3 a 1 τ 2 τ 1 τ 6 τ 3 τ 4 Fig. 1. Problem setup and notation used. Each agent possesses a reference point aα where α is the agent ID, as well as two tags τ i , τ j , where i, j are the tag IDs. 1x and 1y are vectors which represent the x and y axis of Agent 1's body frame. Throughout this paper, the red agent is the arbitrary reference agent, and it will always be Agent 1 without loss of generality. where δξ i ∈ R m , δx = [δξ T 2 . . . δξ T N ] T ∈ R m(N −1) , and will be used throughout the paper. III. OPTIMIZATION The goal is to find the relative agent poses that, with respect to some metric, provide the best relative pose estimation results if the estimation were to be done exclusively using the range measurements. The metric chosen in this paper is based on Fischer information and the Cramér-Rao bound, which will be recalled here. Definition 1 (Fischer information matrix [23]): Let y ∈ R q be a continuous random variable that is conditioned on a nonrandom variable x ∈ R n . The Fischer information matrix (FIM) is defined as I(x) = E ∂ ln p(y|x) ∂x T ∂ ln p(y|x) ∂x ∈ R n×n ,(6) where E[·] is the expectation operator and p(·) denotes a probability density function. Theorem 1 (Cramér-Rao Bound [23]): Let y ∈ R q be a continuous random variable that is conditioned on x ∈ R n . Letx(y) be an unbiased estimator of x, i.e., E[e( x)] = E[x(y) − x] = 0. The Cramér-Rao lower bound states that E e(x)e(x) T ≥ I −1 (x).(7) Theorem 2 (FIM for a Gaussian PDF): Consider the nonlinear measurement model with additive Gaussian noise, y = g(x) + v, v ∼ N (0, R).(8) The Fischer information matrix is given by I(x) = H(x) T R −1 H(x),(9) where H(x) = ∂g(x)/∂x. The Cramér-Rao bound represents the minimum variance achievable by any unbiased estimator. Hence, motivated by Theorem 1, an estimation cost function J est is defined J est (x) = − ln det I(x),(10) which will be minimized with the agent relative poses x as the optimization variables. The logarithm of the determinant of I(x) is one option amongst many choices of matrix norms, such as the trace or Frobenius norm. We have found the chosen cost function to behave well in terms of numerical optimization and, most importantly, goes to infinity when the FIM becomes non-invertible. The state x is locally observable from measurements y if the measurement Jacobian H(x) is full column rank, which also makes the FIM full rank. As will be seen in Section III-B, non-invertibility of the FIM also corresponds to formations that result in unobservable relative poses, which should be avoided. To create a measurement model in the form of (8), the range measurements are all concatenated into a single vector y(x) = [. . . y ij (x) . . .] T g(x) +v, ∀(i, j) ∈ E, v ∼ N (0, R), where R = diag(. . . , σ 2 ij , . . .). It would be possible to directly descend the cost in (10) with an optimization algorithm such as gradient descent, if not for the fact that the state x does not belong to Euclidean space R n but rather SE(n) N −1 . As such, the expression ∂g(x)/∂x is meaningless unless properly defined. A. On-manifold Cost and Gradient Descent The modification employed in this paper is to reparameterize the measurement model by defining x =x ⊕ δx, leading to y = g(x ⊕ δx) + v ḡ(δx).(11) The statex will represent the current optimization iterate, which will be updated using δx. Since the argument of the new measurement modelḡ(δx) now belongs to Euclidean space R m(N −1) , it is possible to compute the "local" approximation to the FIM [24] at x =x with I(x) = H(x) T R −1 H(x) where H(x) = ∂g(x ⊕ δx) ∂δx δx=0 , and evaluate the cost function J est (x) = − ln det I(x). Finally, an on-manifold gradient descent step can be taken withx ←x ⊕ −γ ∂J est (x ⊕ δx) ∂δx δx=0 T ,(12) where γ is a step size. The proposed gradient descent procedure is actually a standard approach to optimization on matrix manifolds [25]. From a differential-geometric point of view, an approximation to the FIM is computed in the tangent space of the current optimization iteratex, which is a familiar Euclidean vector space. A gradient descent step is computed in the tangent space, and the result is retracted back to the manifold SE(n) N −1 using the retraction Rx(δx) =x ⊕ δx. B. Cost function implementation Creating an implementable expression for the cost function J est (x) = − ln det H(x) T R −1 H(x) eventually amounts to computing the Jacobian of the range measurement model (4) with respect to δξ α and δξ β . To see this, H(x) =     . . . H ij (x) . . .     , H ij (x) = [0 . . . H ij α (x) . . . H ij β (x) . . . 0], where H ij α (x) ∂y ij (x ⊕ δx) ∂δξ α δx=0 , H ij β (x) ∂y ij (x ⊕ δx) ∂δξ β δx=0 . The row matrix H ij (x) ∈ R 1×m(N −1) represents the Jacobian of a single range measurement y ij with respect to the full state perturbation δx. This resulting matrix will be zero everywhere except for two blocks H ij α (x) ∈ R 1×m and H ij β (x) ∈ R 1×m , respectively located at the α th and β th block columns, and have closed-form expressions derived in Appendix A. The cost function J est is visualized for varying agent position in the top row of Figure 2, where the red dot shows the minimum found within that view. Looking at the topleft plot of Figure 2, there is a vertical line of high cost near the agent on the left, corresponding exactly to when all four tags line up, leading to an unobservable formation. Similarly, the three-agent scenario in the top-right plot of Figure 2 shows a high cost when the agents are nearly all on the same line, which is a situation of near-unobservability. However, as can be seen in the top-left plot, the minimum is unacceptably close to the left agent, which would cause them to collide. Indeed, we have observed that naively descending the cost J est alone leads to all the agents collapsing into each other. An explanation for this behavior is that when agents are closer together, changes in attitude result in larger changes in the range measurements, which increases Fischer information. Nevertheless, in practice, collisions must be avoided, and this is done by augmenting the cost with an additional collision avoidance term J col (x), such that the total cost J(x) is J(x) = J est (x) + J col (x), J col (x) = α,β∈A α =β J αβ col (x), where a collision avoidance cost from [26] is used, J αβ col (x) = min 0, r aαa β 1 2 − R 2 r aαa β 1 2 − d 2 2 .(13) The term R represents an "activation radius" and d is the safety collision avoidance radius. In this paper, the agent relative position is expressed as a function of pose matrices with r aαa β 1 = DT 1α b − DT 1β b, where D = [1 0], b = [0 1] T . The new cost function J is plotted on the bottom row of Figure 2, showing the effect of the collision avoidance term. Finally, one is now ready to descend the cost directly with x ←x ⊕ −γ ∂J(x ⊕ δx) ∂δx δx=0 T .(14) In this work, the Jacobian of J est is computed numerically with finite difference [27], and the optimization is only done offline for the following reasons. The solution to the optimization problem is only a function of some physical properties, the measurement graph G, and the number of robots N . For any experiments that use the same hardware, the physical properties such as the safety radius, tag locations, and measurement covariances, all remain constant. The measurement graph G can often also be assumed to be constant and fully connected. Even though full-connectedness is not necessary to find optimal formations using the proposed approach, technologies such as UWB often have a ranging limit that is well beyond the true ranges between all robots in the experiment. Hence, it is straightforward to precompute optimal formations for varying robot numbers N with fullyconnected measurement graphs, and to store the solutions in memory onboard each robot. Nevertheless, a distributed, real-time implementation is required for varying measurement graphs, which is likely to arise in the presence of obstacles that block line-of-sight. Such a scenario requires simultaneously satisfying obstacle avoidance constraints and perhaps other planning objectives, which is beyond the scope of this paper. C. Optimization results The gradient descent in (14) is performed with a step size of γ = 0.1, an activation radius of R = 2 m, and a safety radius of d = 1 m. Each agent has two tags located at Since the treatment in this paper is general to an arbitary measurement graph G, provided the FIM remains maximum rank, optimization is also performed for a non-fullyconnected measurement graph. The results for this along with a 3D scenario are shown in Figure 4. In 3D, the robot relative poses are represented with elements of SE(3). However, since the presented simulations contain only two-tag robots, relative roll and pitch between robots are unobservable, which would make the cost infinite, unless more sensors are used. Hence, roll and pitch are excluded from the optimization and their values are fixed to zero. This leaves the three translational components and heading as the four degrees of freedom available for optimization. This is easily implemented in practice with a redefinition of the "wedge" operator such that (·) ∧ : R 4 → se(3). Moreover, from an application standpoint, both ground vehicles and quadcoptertype aerial vehicles only have heading as a rotational degree of freedom available for planning. D. Validation on a least squares estimator To validate the claim that descending the cost improves the estimation performance, a non-linear least-squares estimator is used. At regular iteratesx of the optimization trajectory, a small 2000-trial Monte Carlo experiment is performed, where in each trial a set of range measurements are generated with y = g(x) + v, v = N (0, R). Then, an on-manifold Gauss-Newton procedure [22] is used to solvê x = arg min x 1 2 N α=2 ln(C T 1αČ1α ) ∨ 2 Pα + 1 2 y − g(x) 2 R ,(15) where e 2 M = e T M −1 e denotes a squared Mahalanobis distance, and an attitude prior with "mean"Č 1α and covariancě P α is also included for each agent. It turns out that minimization of only the second term in (15) yields unacceptably poor estimation performance, as the solution often converges to local minimums depending on the initial guess. The inclusion of an attitude prior, which is practically obtained by deadreckoning on-board gyroscope measurements, yields much lower overall estimation error. Figure 6 shows the value of the cost throughout the optimization trajectory, as well as the mean squared estimation error over the K = 2000 Monte Carlo trials per optimization step. The true agent poses are initialized in a near-straight line, as shown in Figure 5, and the covariances used are R = 0.1 2 1 m 2 ,P α = 0.08 2 rad 2 . The mean squared estimation error (MSE) is calculated with MSE = 1 K K k=1 δξ T δξ, δξ =    ln(T −1 12T 12 ) ∨ . . . ln(T −1 1NT 1N ) ∨    ,(16) and shows a clear correlation with the cost function. This provides evidence for the fact that descending the proposed cost function also reduces the estimation error. IV. EXPERIMENTAL EVALUATION An estimator is also run with three PX4-based Uvify IFO-S quadcopters in order to experimentally validate the claim that descending the proposed cost function results in improved estimation performance. The quadcopters start by flying in a line formation and, after 30 seconds, proceed to a triangle formation computed using the proposed framework for another 30 seconds, as shown in Figure 8. Figure 7 shows the position estimation error using the least-squares estimator presented in Section III-D. Real gyroscope measurements are used to obtain an attitude prior at all times. Range measurements are synthesized with a standard deviation of 10 cm using ground truth vehicle poses obtained from a motion capture system. The UWB tags are simulated to be 17 cm apart, corresponding to extremities of the propeller arms. As can be seen in Figure 7, moving to the optimal triangle formation, from one of the worst starting formations results in a 68% reduction in estimation variance. V. CONCLUSION This paper shows, in both simulation and experiment, that range-based relative state estimation performance can be substantially improved by a proper choice of formation geometry. The largest improvements are obtained when the robots move away from unobservable formations. The generalizability of the cost function makes it appropriate for use beyond direct minimization. For instance, consider using this function to impose an inequality constraint on an application-oriented planning problem, such as indoor exploration. Using an inequality constraint would allow robots the freedom to move within the feasible region in order to accomplish tasks such as infrastructure inspection, yet still avoid the "worst" formations with very high cost, which could cause problematically large state estimation errors. Future work may tackle a scenario similar to this, including developing a distributed computation scheme for the proposed cost function. Let y ij (x) = y ij (x) + δy ij ȳ ij + δy ij , T 1α = T 1α exp(δξ ∧ α ), T 1β =T 1β exp(δξ ∧ β ), and v ij = 0. The terms δy ij , δξ α , δξ β are assumed to be small quantities, which motivates, for example, the approximation exp(δξ ∧ α ) ≈ 1 + δξ ∧ α . Equation (4) becomes (ȳ ij +δy ij ) 2 = DT 1α (1 + δξ ∧ α )p i − DT 1β (1 + δξ ∧ β )p j T DT 1α (1 + δξ ∧ α )p i − DT 1β (1 + δξ ∧ β )p j , which, after expanding and neglecting higher-order terms, leads to (17) Next, it is straightforward to define a simple operator (·) , as per [22], such that ξ ∧ p = p ξ. Rearranging (17) yields 2ȳ ij δy ij = 2(p T iT T 1α D T − p T jT T 1β D T )DT 1α δξ ∧ α p i + 2(p T jT T 1β D T − p T iT T 1α D T )DT 1β δξ ∧ β p i .δy ij = (p T iT T 1α D T − p T jT T 1β D T ) y ij DT 1α p i δξ α − (p T iT T 1α D T − p T jT T 1β D T ) y ij ρ ij DT 1β p j δξ β . (18) The term ρ ij is the physical unit direction vector between tags i and j, resolved in Agent 1's body frame. From (18) it then follows that ∂y ij ∂δξ α = ρ ij DT 1α p i , ∂y ij ∂δξ β = −ρ ij DT 1β p j . Fig. 2 . 2All four plots show the value of the cost with varying agent position (right agent for two-agent scenario, top agent for three-agent scenario), while maintaining fixed heading. The top row shows only the estmation cost Jest, while the bottom row shows the total cost J including the collision avoidance term. α, i, j are arbitrary and the units are in meters.Figure Fig. 3 . 3Final locally optimal formations for 3, 4, 5, and 10 agents. Fig. 4 . 4(left) Optimal formation with sparse measurement graph. (right) Optimal formation with 3D position and heading as design variables. Fig. 5 . 5Trajectory taken during optimization with superimposed 1σ equalprobability contours corresponding to the Cramér-Rao bound. The ellipsoids for the starting positions are too large to fit in the figure. Fig. 6 . 6Cost function, along with various metrics of a least-squares estimator, obtained from 2000 Monte-Carlo trials at various points during the optimization. In the bottom two plots, the red squares denote the average norm of the respective estimation errors. Fig. 7 . 7Experimental results using a least squares estimator. From 0 s to 30 s, the quadcopters are in a line formation and the average positioning error is 0.77 m. From 30 s to 60 s, the quadcopters are in an optimal formation and the average positioning error is 0.22 m, a 68% reduction. Fig. 8 . 8(top) Three quadcopters in an initial straight line formation. (bottom) Quadcopters in an optimal triangle formation. APPENDIX A. Measurement model Jacobian , M. A. Shalaby, J. R. Forbes are with the Department of Mech. Engineering, McGill University. [email protected], [email protected], [email protected] D. Saussié, J. Le Ny, are with the Department of Electrical Engineering, Polytechnique Montréal. 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[ "THz Band Channel Measurements and Statistical Modeling for Urban D2D Environments", "THz Band Channel Measurements and Statistical Modeling for Urban D2D Environments" ]
[ "Naveed A Abbasi \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Jorge Gomez-Ponce \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Revanth Kondaveti \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Shahid M Shaikbepari \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Shreyas Rao \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Shadi Abu-Surra \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Gary Xu \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Charlie Zhang \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "Andreas F Molisch \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "N A Abbasi \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "J Gomez-Ponce \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "R Kondaveti \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "S M Shaikbepari \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "S Rao \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n", "A F \nDepartment of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA\n" ]
[ "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA", "Department of Electrical and Computer Engineering\nSamsung Research America and the Foreign Fulbright Ecuador SENESCYT Program\nMing Hsieh\nUniversity of Southern California\nLos AngelesCAUSA" ]
[]
THz band is envisioned to be used in 6G systems to meet the ever-increasing demand for data rate.However, before an eventual system design and deployment can proceed, detailed channel sounding measurements are required to understand key channel characteristics. In this paper, we present a first extensive set of channel measurements for urban outdoor environments that are ultra-wideband (1 GHz 3dB bandwidth), and double-directional where both the transmitter and receiver are at the same height.In all, we present measurements at 38 Tx/Rx location pairs, consisting of a total of nearly 50,000 impulse responses, at both line-of-sight (LoS) and non-line-of-sight (NLoS) cases in the 1-100 m range.We provide modeling for path loss, shadowing, delay spread, angular spread and multipath component (MPC) power distribution. We find, among other things, that outdoor communication over tens of meters is feasible in this frequency range even in NLoS scenarios, that omni-directional delay spreads of up to 100 ns, and directional delay spreads of up to 10 ns are observed, while angular spreads are also quite significant, and a surprisingly large number of MPCs are observed for 1 GHz bandwidth and 13 • beamwidth. These results constitute an important first step towards better understanding the wireless channel in the THz band.
10.1109/twc.2022.3184929
[ "https://arxiv.org/pdf/2109.13693v1.pdf" ]
238,198,356
2109.13693
00cb8942ea445ca0e13e353feb6c20d4f1c0eb2a
THz Band Channel Measurements and Statistical Modeling for Urban D2D Environments Naveed A Abbasi Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Jorge Gomez-Ponce Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Revanth Kondaveti Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Shahid M Shaikbepari Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Shreyas Rao Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Shadi Abu-Surra Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Gary Xu Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Charlie Zhang Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA Andreas F Molisch Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA N A Abbasi Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA J Gomez-Ponce Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA R Kondaveti Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA S M Shaikbepari Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA S Rao Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA A F Department of Electrical and Computer Engineering Samsung Research America and the Foreign Fulbright Ecuador SENESCYT Program Ming Hsieh University of Southern California Los AngelesCAUSA THz Band Channel Measurements and Statistical Modeling for Urban D2D Environments 1 THz band is envisioned to be used in 6G systems to meet the ever-increasing demand for data rate.However, before an eventual system design and deployment can proceed, detailed channel sounding measurements are required to understand key channel characteristics. In this paper, we present a first extensive set of channel measurements for urban outdoor environments that are ultra-wideband (1 GHz 3dB bandwidth), and double-directional where both the transmitter and receiver are at the same height.In all, we present measurements at 38 Tx/Rx location pairs, consisting of a total of nearly 50,000 impulse responses, at both line-of-sight (LoS) and non-line-of-sight (NLoS) cases in the 1-100 m range.We provide modeling for path loss, shadowing, delay spread, angular spread and multipath component (MPC) power distribution. We find, among other things, that outdoor communication over tens of meters is feasible in this frequency range even in NLoS scenarios, that omni-directional delay spreads of up to 100 ns, and directional delay spreads of up to 10 ns are observed, while angular spreads are also quite significant, and a surprisingly large number of MPCs are observed for 1 GHz bandwidth and 13 • beamwidth. These results constitute an important first step towards better understanding the wireless channel in the THz band. to 300 m maximum excess runlength of the multipath, which is sufficient for the considered environments. While VNA-based setups have a long tradition for THz channel measurements, a major challenge has been the fact that Tx and Rx are in a single casing and the maximum separation of the antennas is determined by the admissible length of the cables from the VNA to the extenders (which are placed together with the antennas), which -due to the high cable losses at mmWave frequencies -are usually less than 10 m. In order to overcome this problem, we introduced, in [20], a RF-over-fiber (RFoF) link to allows the extension of the distance to the 100 m of interest in our case and beyond. The main difference between the design in the current paper and that in our previous work [20] is that we currently use an integrated RFoF unit to improve the robustness of the design. Due to the measurement principle of mechanically rotating antennas, every measurement lasted for several hours, therefore, it was ensured that the scenarios remain static/quasi-static during this time. To to be in the zenith) is taken and a full azimuth scan is done similar to Tx. Since we have a single elevation cut at the Tx and Rx, the use of terms θ T x , θ Rx is redundant and they are not used further. Finally, both the Tx and Rx are kept at the same height (1.6 m), corresponding, e.g., to device-to-device (D2D) scenarios. The measurements extended over many days. In the beginning of each measurement day, a calibration of the VNA, as well as an over-the-air (OTA) calibration with the Tx and Rx with LoS in close proximity, was performed. More details of this setup are covered in [20]- [22]. It is pertinent to note that the current setup provides high phase stability that allows to not only conduct Fourier analysis of on our measurements but also enables high resolution parameter extraction (HRPE) algorithms to operate on the measurements. Although beyond the scope of the current paper, we note that HRPE can improve the resolution of the results and therefore provide more accurate modeling. B. Measurement locations As discussed earlier, selection of measurement locations is important for any measurement campaign and should be representative of the scenario of interest. Our investigation is focused on outdoor urban D2D scenarios, therefore, we conducted measurement campaigns in two relevant environments: (i) an outdoor courtyard surrounded by buildings and (ii) an urban outdoor crossroad. We describe both these locations in more detail in the following. Table II. III. PARAMETERS AND PROCESSING A. Data processing We obtained a collection of the frequency scans from the VNA for various Tx-Rx configurations using the measurement setup described in Section II. Each measurement is represented as a three-dimensional tensor H meas (f, φ T x , φ Rx ; d) where f represents the frequency points over the P calc (τ, φ T x , φ Rx , d) = |F −1 f {H(f, φ T x , φ Rx , d)}| 2 ,(1) where F −1 f is the inverse fast Fourier transform (IFFT) with respect to f . Finally, we apply noise and delay gating, similar to [22], [24], which is P (τ ) = [P calc (τ ) : (τ ≤ τ gate ) ∧ (P calc (τ ) ≥ P λ )](2) or 0 if it does not fulfill these conditions. Here τ gate is the delay gating value selected to avoid using long delay points and points with "wrap-around" effect of the IFFT, and P λ is the noise threshold to not count delay bins with noise which could particularly distort delay spread and angular spread. For the current measurements, τ gate is set to 833.33 ns (corresponding to 250 m excess runlength) and P λ is selected to be 6 dB above the noise floor (average noise power) of the PDP. Additionally, we analyze the channel behavior from an "omni-directional" perspective by reconstructing the omni-directional pattern from a full double-directional capture by an approach similar to similar to [21], [25], i.e., selecting the direction of the highest contribution per delay bin: P omni (τ, d) = max φ T x ,φ Rx P (τ, φ T x , φ Rx , d).(3) B. Parameter computation Using the directional and omni-directional PDPs described in the previous section, we proceed to compute several condensed parameters that are commonly used to characterize propagation channels, such as path gain, shadowing, delay spread and angular spread. 1) Path loss and shadowing: We compute path loss as the sum of the power on each delay bin in the PDP as described in [12]. P L i (d) = τ P i (τ, d),(4) where i can denote omni-directional (omni) or strongest beam (best-dir) which is selected as the beam-pair directions with the highest power: P max (τ ) = P (τ, φˆi, φĵ); (î,ĵ) = max i,j τ P (τ, φ i , φ j ).(5) Finally, we model path loss on the dB scale using the classical "power law" also know as α − β model as P L(d) = α + 10β log 10 (d) + ,(6) where α and β are the parameters of the linear model, and is the random variation of the data with respect to its mean that is commonly modeled as a zero-mean normal distribution ∼ N (0, σ), where σ represents the standard deviation of the distribution. These parameters can be obtained by techniques such as maximum likelihood estimation (MLE) or ordinary least squares (OLS) [12], [26]. As has become common in channel models, we separately extract the parameters α, β, σ for the ensemble of LoS and of NLoS measurement points. One of the challenges in evaluating a measurement campaign is that the density of distances at which measurements are done is not uniform on a logarithmic distance scale (it is generally not uniform on a linear distance scale either). For this reason, a weighted regression approach to model path loss data was developed in [26] where the measurement points are weighted (w i ) according to the density distribution along the distance in log 10 scale. In other words, this method gives more weight to points in area where the density is low. Using this strategy, we compensate the uneven distribution of points when performing the path loss modeling and give equal importance to each distance range. Even though multiple types of weights are shown in [26], we focus on the case of "equal weights to N bins over log 10 (d) (w i ∝ log 10 (d))", because it corresponds to a least square fitting of "dB vs log 10 (d)" as described in [26]. 2) Delay spread: Delay spread is another key channel parameter that is calculated as the second central moment of the PDP [12] as σ τ = τ P i (τ )τ 2 τ P i (τ ) − τ P i (τ )τ τ P i (τ ) 2 ,(7) where i can be "omni" or "max-dir". Note that long-delayed samples with small power can have a disproportionate impact on the delay spread; for this reason, noise thresholding is especially important in the assessment of delay spread. 3) Angular spread: To analyze the channel's behavior for systems that have either directional antennas or antenna arrays, we need a quantitative description of the angular dispersion. A starting point is the double-directional angular power spectrum (DDAPS) which shows the distribution of the power as a function of the angle at Tx and Rx (since our measurement setup scans only the horizontal plane, a distinction between angular and azimuthal power spectrum is moot). The DDAPS is computed as DDAP S(φ T x , φ Rx , d) = τ P (τ, φ T x , φ Rx , d).(8) The granularity of the function is directly proportional to angular captures done during the measurement. We note that it is important to perform noise thresholding and delay gating before computing the DDAPS in order to suppress accumulation of noise in the directions in which no significant MPCs occur. From the DDAPS, we can obtain the (single-directional) angular power spectrum (APS) at Tx by integrating over φ Rx , and similarly for the APS at the Rx. From these quantities, we proceed to compute the angular spread by applying Fleury's definition [27] given as σ • = φ |e jφ − µ φ | 2 AP S k (φ) φ AP S k (φ) ,(9) where k can be Tx or Rx indicating departure or arrival APS and µ φ can be computed as µ φ = φ e jφ AP S k (φ) φ AP S k (φ) .(10) This analysis is directly impacted by the finite horn antenna beamwidth, which will results in larger angular spreads than the channel inherently provides. For instance, a pure LoS case results in a non-zero angular spread due to the finite beamwidth of the horn antenna. Consequently, the results obtained from this analysis are indeed an upper bound for the real angular spreads of the channel. 4) Power distribution over MPC: A very important parameter for the channel analysis is the ratio of the strongest MPC versus the other MPCs present in the channel, as it informs the fading depth. We thus define a parameter κ 1 as κ 1 = P i (τ 1 ) τ Ñ τ =τ 2 P i (τ ) ,(11) where i can be "omni" or "max-dir", andτ k is the location of the k-th local maximum of the PDP P i (τ ), ordered by magnitude, so thatτ 1 signifies the location of the largest local maximum. Please note that κ 1 is different from "Rice Factor" because in the Fourier analysis it is not possible to differentiate between closely spaced MPCs, so that the local maximum of the PDP is not strictly identical to an MPC. Consideration of Rice factor based on HRPE will be presented in future work. We In the subsequent section, apart from other results we also perform regression analysis for the parameters σ • , σ τ , κ 1 to observe their behavior with respect to the distance between Tx and Rx. For this case, a linear regression model expressed as Z = α + β log 10 (d) is used. IV. MEASUREMENT RESULTS We now turn to the results obtained from the measurements in different scenarios. A. Power delay profiles We first show three sample PDPs, one in a LoS scenario and two for NLoS scenarios. All these cases depict both the "omni" and "max-dir" PDPs as shown in Fig. 4 and the "Childs-and Watt Way Intersection" locations, are noticeably different. One such case is shown in Fig. 5 (a) where we see a PDP not too different from a LoS case with some very strong components. The primary reason for this is the strong reflections that result from buildings in the scenarios. Finally, we note that multipath richness increases with the distance between Tx and Rx since the number of possible paths that a wave can take to get from Tx to Rx increases. This effect was also observed in [20]. Representative results for both these cases are shown in Fig. 7. Fig. 7 (a), shows the street canyon case one major cluster much like a LoS scenario is centered at φ T x = −20 • , φ Rx = −20 • . On the other hand, in Fig. 7 (b), the largest power comes from φ T x = −50 • , φ Rx = −70 • and additional MPCs clusters with comparable powers can also be observed in other directions. C. Path loss and shadowing We proceed with the analysis of the ensemble of locations, starting with path loss. The path loss plots shown in Fig. 8 (a) and Fig. 9 (a) were done using the weighting regression explained in Section III-B. However, the shadowing results shown are obtained from a regression using the OLS approach, to enforce a "zero-mean" distribution. The difference in mean between the weighted regression and OLS is fairly small; both the weighted fitting and OLS results, for both path loss and shadowing, are detailed in Tables III and VI The shadowing results shown in Fig. 8 ( Tables III and IV. We can also see from inspection that the lognormal model that is commonly used for the deviations from the linear fit model provides an excellent approximation to the measured distribution. PL [dB] Omni Max-Dir Omni Model Max-Dir Model Friis (a) Linear fitting with log 10 (d) weighting. We stress that -as in any model based on measurements -the extracted parameters and models are valid only in the range in which measurements were done, i.e., 1-100 m. Disregarding this important caveat would lead to unphysical results at distances larger than 100 m. Shadowing variations are larger in the NLoS case when compared to LoS, and the directional case shows even larger variation compared to the omni-directional case as shown in Fig. 9 (b). This is because of the changes in the strongest MPCs, and the corresponding beam directions that provide the maximum power. More specifically, the shadowing standard deviations σ Omni = 5.18 and σ M ax−Dir = 6.68 are at least four times larger than their LoS counterparts. Please see Tables III and IV for detailed results. We note that the confidence intervals (CIs) for the NLoS case span a much larger range than for the LoS case. This is to a small degree caused by a smaller collection of NLoS points when compared to LoS points (17 vs 21), but is mostly due to the larger spread of the actually measured results. We also note that the number of measurements and spreads is -from visual inspection -in line with other measurement campaigns at high frequencies in the literature, but since they generally do not provide CIs for their parameter fits, a comparison of CIs is not possible. D. RMS delay spread The RMS delay spread in the LoS case ranges between 1 and 10 ns for the max-dir case, and 10-100 ns for the omni-directional case. More precisely, the average difference between the omni-directional and directional cases is approximately 7 dBs (i.e. 5 times on a linear scale). This significant difference follows from the fact that the narrow beamwidth of the horn antennas suppresses many significant reflected components, thus making the LoS component THz, equalizers or equivalent structures will be required in many cases. When analyzing the cumulative distribution function (CDF), we find a similar behavior for both omni and max-dir cases: a lognormal distribution (i.e. Gaussian on a dB scale) as first suggested by Greenstein [28] and observed for mmWave in [25], provides an excellent fit, as shown in Fig. 10 (a). The standard deviation for the omni-directional and directional case are similar (σ τ Omni = 3.05,σ τ M ax−Dir = 3.08). Note that giving delay spreads in units of dBs is common in the channel modeling literature and is used, e.g., in the 3GPP channel model, since it is a natural fit to the lognormal model for the delay spread cdf. Delay spread shows a linear relationship with respect to the distance based on our analysis of the various measurements. In the LoS cases, the delay spread increases with the distance between the Tx and Rx for both the directional and omni-directional case; the CI in this case encompasses only positive slopes. The increase with distance occurs because at large distances, a larger number MPCs with power comparable to the LoS MPC appear. This is also consistent with observations of various measurement campaigns at lower frequencies. All of these observations can be seen in Fig. 10 (b). For the NLoS case, we expect larger delay spread compared to the LoS since the absence of a single dominant component increases the second central moment. This is indeed borne out by the Fig. 11 (a) where delay spread values of up to 30ns occur in the max-dir case. For the omni case, while the maximum observed delay spread remains at 150 ns, the average value of around 65 ns is about a factor of 3 higher than in the LoS case (20 ns). The variance observed in the directional case is larger (σ τ Omni = 1.8,σ τ M ax−Dir = 3.96). Furthermore, -in contrast to the LoS case -in the directional case the delay spread decreases with increasing distance; the CI only contains negative values, and the the absolute value of the slope is larger than in the LoS case. Given the small number of significant MPCs arriving at the Rx, we note that the variance might change considerably depending on the measurement scenario. For the omni-directional case, the delay spread also decreases with increasing distance, but in a less pronounced way, and the CI contains both positive and negative values. This can be explained by the fact that more MPCs are added at longer distances. The linear fittings are shown in Fig. 11 (b). We summarize the detailed results for various parameters in Tables V and VI. Linear fitting of angular spread versus distance is shown in Fig. 12 (b). The angular spread increases for LoS points and decreases for NLoS situations, with increasing distance. These observations and their relevant causes are similar to those from Fig. 12 (a). Detailed results for this case are presented in Tables VII and VIII. F. Power distribution of MPCs From physical considerations of the urban scenarios, we expect the values of κ 1 for directional cases to be larger than the omni-directional ones because of the large number of MPCs in the PDPs for the latter case. This is indeed borne out by the measurements: for the LoS case we find a directional mean of 18 dB, while the omni-directional mean is around 10 dB. The distributions of the κ 1 values follows a lognormal distribution, with a standard deviation is 6 dB in both cases as shown in Fig. 13a. shows an increasing trend as the distance between Tx and Rx increases because the spatial filtering provided by the horn antennas -the directional range to/from which the MPCs come is increased, but the horns filters out MPCs outside an angular range that is the beamwidth around the LoS direction. This is demonstrated in Fig. 13 (b), where we observe the linear fitting for the parameter in LoS points. G. Summary of results In this section, we summarize the key results that may be very useful from a system design perspective. Finally, please note that while a larger number of channel measurements provides better statistical validity, the time required for doing measurement campaigns like the one presented here is significant (several months). For this reason, most initial channel models are based on a small number of measurement points to satisfy the need of the wireless community for a realistic basis for system design; of course the models can be subsequently improved as more measurements are conducted. V. CONCLUSIONS In this paper, we presented the results of the first extensive wideband, double-directional THz outdoor channel measurements for urban D2D scenarios. We provided an overview of the measurement methodology and environments, as well as the signal processing to extract parameters characterizing the channels. Most importantly, we provided a parameterized statistical description of our measurement results that can be used to assess THz systems. We can draw some first conclusions about the implications on system design and deployment: Parameter µ σ negligible, they can introduce significant inter-user interference in a nearby victim receiver that receives its own desired signal from a far-away transmitter. Thus, relying on the channel to provide a "natural" separation of users in different directions might be optimistic. • We note that for even for the max-dir case, the κ 1 that is guaranteed in 90 % of all cases is only 0 dB for NLoS. Thus, even with directional antennas, it is not possible to rely on small fading depth in the design of THz transceivers. While these conclusions are mainly qualitative, they provide important insights into THz system behavior and show that some common conjectures that had been made about THz systems do not hold in the actually measured channels presented in this paper. Thus, the presented measurements should be of considerable interest not only for the propagation community, but more importantly for communication theorists and system designers. Fig. 1 : 1Channel sounding setup. determine the double-directional properties of the channel, we measured the transfer functions for different pairs of antenna orientations. Both Tx and Rx are horn antennas with a 3 dB (Full width half maximum) beamwidth of 13 degrees. Due to the equal height of Tx and Rx, it can be anticipated that propagation is happening dominantly in the horizontal plane, therefore, no elevation sweeps were performed. The positioners (rotors) were set such that the angle of 0 • for both the Tx and the Rx corresponds to the LoS for all points. The Tx scans over a single elevation cut (θ T x = 90 • ; θ T x = 0 • is defined to be in the zenith) from 0 • to 360 • with a 10 • angular resolution. For the Rx, a single elevation (θ Rx = 90 • ; θ Rx = 0 • is again defined Fig. 3 3shows the scenarios with the measurement locations. Tx6 has 3 LoS points (Rx27-29) on linear route and 3 NLoS points blocked by the building corners (Rx30-32). Tx7 was placed in the middle of the crossroad to have 4 LoS points (Rx33-36), one on each of the four sides. Finally, Tx8 has 2 LoS points (Rx37-38) diagonally with respect to its position and the crossroad. Details of all the links for both the sites are provided in Fig. 4 : 4LoS case with d T x−Rx = 98.4m (Tx6-Rx29). Fig. 7 . 7For the LoS PDP in Fig. 4, apart from a strong peak at the LoS distance (98.4 m), we observe multiple MPCs due to the interaction of the signal with environmental objects. Clearly, the multi-path structure is much more pronounced in the omni-directional case than in the directional case; this follows intuition, since reflected paths can depart and arrive in a wide variety of angles. The furthest MPCs (d ≥ 200 m) have power that is more than 40 dB lower than the strongest component, but closer components may be lower by only 20 dB. For the directional case, the extra components are diminished/filtered as a result of the use of horn antennas at both link ends. Additionally, in some cases we observe MPCs before the LoS arrives which are a product of components that arrive later than the maximum measurable distance of the system (1µs or 300 m) and become visible before the LoS component as a result of the wrap-around effect; this wrap-around is corrected in the figures. The NLoS scenarios are generally richer in terms of MPCs when compared to the LoS case, since the dominant LoS components are missing and therefore, the difference in power between various MPCs is less. Similar to the LoS case, the number of significant MPCs is larger in the omni-directional case. A high number of MPCs is observed for both the omni-directional and directional cases in Fig. 5 (b). However, the absence of a dominant component need not be the case for all the NLoS measurements. NLoS scenarios in street canyons, especially observed in case with d T x−Rx = 35.29m (Tx4-Rx20). Fig. 5 : 5PDP for two sample NLoS measurement cases. Fig. 6 : 6LoS APS for d T x−Rx = 98.4m (Tx6-Rx29). (a ) )NLoS APS for d T x−Rx = 97.59 (Tx6-Rx32). (b) NLoS APS for d T x−Rx = 35.29m (Tx4-Rx20). Fig. 7 : 7Sample NLoS APSes for two cases. B. Angular power spectrum In terms of APSs for LoS scenarios, we observe a high concentration of power around the LoS direction (i.e., Tx and Rx facing each other, φ T x = φ Rx = 0) with additional contributions from various directions corresponding to reflections. The concentration of power around the LoS decreases as the distance between Tx and Rx increases. This is demonstrated by the sample APS case shown in Fig. 6, which exhibits a large concentration of power in the LoS direction and some smaller contributions from the back of both the Tx and the Rx. These contributions are around 20 dB below the LoS components and correspond to reflections from the back -they are not antenna back lobes, which are nearly 30 dB below the main lobe for our antenna. For the NLoS case, street canyon scenarios showed peaked (concentrated) APSs, consistent with the concentration of MPCs in the PDPs discussed in the previous subsection. Most other locations showed a considerably larger angular dispersion, and more than one MPC cluster. loss modeling with log 10 (d) weighting. Fig. 8 : 8Path loss and shadowing models for LoS points. . The observations for the LoS measurements shows a close agreement with free space propagation model (FSPL) with small variations around the mean. This can be observed in Fig. 8 (a) where the omni-directional and directional path loss data exhibit similar trends. The path loss coefficients are between 1.74 and 1.86 for both the directional and the omni-directional case. This is smaller than the FSPL coefficient of 2, which is physically reasonable, as the total received signal consists of several reflected MPCs in addition to the LoS component that provides power decaying with d 2 . Fig. 9 : 9Path loss and shadowing models for NLoS points.The NLoS case exhibits additional attenuation compared to LoS points and FSPL, due to the the impact of blockage, and the losses involved in the additional propagation mechanisms such as reflection and dispersion in the various scenarios.Fig. 9(a) shows an offset between the FSPL and the measured values between 10 to 15 dB in the directional and omni-directional cases, respectively. The slopes (of path loss as a function of distance) of the directional and omni-directional cases are similar, with the directional path loss slightly larger (since it excludes some significant MPCs). It is noteworthy that the slopes in both cases are around 1.5, i.e., smaller than in the LoS case, although of course the absolute path loss values are larger than for LoS. modeling of σ τ with weighting. Fig. 10 : 10Modeling of delay spread for LoS cases. more dominant, and reducing the second central moment of the PDP. It is noteworthy that even for the most extreme case of max-dir delay spread in LoS situations, the RMS delay spread is on the order of several nanoseconds. Considering the extremely high data rates envisioned for fitting with log 10 (d) weighting. Fig. 11 : 11Modeling of delay spread for NLoS points. For the statistics of the angular spread, a distinction between Tx and Rx is moot, since both Tx and Rx are at the same height (our focus is on D2D scenarios), and the selection of a node as Tx and Rx is arbitrary. Therefore, we can analyze the ensemble of all transceiver locations, without distinction of Tx/Rx (note that this would not be correct in a scenario where one node shows distinctive features, such as an access point in an elevated location).As expected, LoS points show considerably lower values of angular spread compared to the NLoS points because of the presence of strong LoS components. However, as discussed in Sec. IV.B, there are also NLoS scenarios with low angular spread. This occurs in particular in street canyon scenarios that have concentrated beams, and therefore, a reduced angular span that leads to a decrease in the angular spread, since most of the components come from reflections inside the canyon.Fig. 12(a) demonstrates these observations. Please note that LoS values are significantly more likely -but not guaranteed -to have lower angular spread compared to the NLoS points. However, as the distance increases between the Tx and Rx, the angular spread increases as well because the difference of power between the LoS component and the rest of the MPCs is reduced. The opposite effect is observed for the NLoS points, though with larger spread values. fitting with log 10 (d) weighting. Fig. 12 : 12Modeling of σ • for all points. The omni-directional κ 1 shows a decrease as the distance increases because the reflected MPCs have powers closer to the strongest component. On the other hand, the directional cases fitting with log 10 (d) weighting. Fig. 13 : 13Modeling of κ 1 for LoS points. Fig. 14 ( 14a) and (b) show our observations for the NLoS measurements. The average κ 1 values are considerably lower, around 9 dB for the max-dir case, and 8 dB for the omni case; the standard deviation is 5 dB and 7 dB, respectively.The omni-directional case also shows an increase with respect to distance as shown inFig. fitting with log 10 (d) weighting. Fig. 14 : 14Modeling of κ 1 for NLoS points. • At 100 m distance, the THz channel exhibits a path loss of approximately 110 dB in LoS and 125 dB in NLoS. A simple link budget estimate shows that for a 1 GHz bandwidth, 23 dB gain antennas at both link ends, a required operating SNR of 5 dB, a receiver noise figure of 5 dB, and an assumed conducted transmit power of 10 dBm, communication remains possible even with a path loss of 10 + 46 + 174 − 90 − 5 − 5 = 130 dB. In other words, our results indicate that D2D communications over up to 100 m distance are feasible even TABLE I : ISetup parameters.Parameter Symbol Value Frequency points per sweep N 1001 Tx/Rx height h T x/Rx 1.6 m Start frequency fstart 145 GHz Stop frequency fstop 146 GHz Bandwidth BW 1 GHz IF bandwidth IFBW 10 KHz THz IF fT HzIF 279 MHz Antenna 3 dB beamwidth HP BW 13 • Tx rotation range φT x [0 • ,360 • ] Tx rotation resolution ∆φT x 10 • Rx Az rotation range φRx [0 • ,360 • ] Rx Az rotation resolution ∆φRx 10 • 1 ) 1Outdoor courtyard ("Epstein Family Plaza/Vivian Hall of Engineering"): This environment is a quadrangle courtyard (quad) in front of Vivian Hall of Engineering (VHE) on the USC University Park Campus, Los Angeles, CA, USA, see Fig. 2. The Eastern (rightmost on the map) part is an open area with concrete pillars having interspersed benches in the middle and building walls and doors on both sides. To the left is an L-shaped courtyard. The area is partly delimited by multi-story buildings with concrete walls and glass windows. In one part of the L, near the buildings on both sides, are trees with big canopies, as well as lampposts. The other Childs-and Watt Way Intersection"): The second scenario is on the intersection between Childs Way and Watt Way also at the USC University Park Campus, see Fig. 3. Going north on Watt Way is Cromwell Field (CFX) on the left with sparse trees and a metal fence boundary. Physical Education Building (PED), which has some trees outside it, is opposite to CFX. Looking south on Watt Way, the road lies between Wallis Annenberg Hall (ANN) and Grace Ford Salvatori Hall (GFS). Looking east on Childs Way, we have a street canyon between PED and ANN with some trees on the PED side of the road. After PED, an open area, Associate Park, can be observed with interspersed trees and benches. Finally, on the west side of Childs Way, the road goes between CFX and GFS where the GFS side has larger trees, big glass windows and glass doors whereas the CFX side has smaller sparse trees andpart of the L (left) has fewer trees and includes a large sitting area with tables, chairs, umbrellas, and a water fountain. In this scenario we used 5 different locations for the Tx and 26 locations for the Rx (12 LoS, and 14 NLoS), as shown in Fig. 2. Fig. 2: Scenario map for Epstein Family Plaza and VHE. Tx1 has 3 LoS points (Rx1-3) and 4 NLoS points (Rx4-7) at the entrance of VHE. Tx2 has 2 LoS locations (Rx8,9) and 2 NLoS points (Rx10,11). Tx3 has 3 LoS points on a linear route crossing the quad (Rx12-14) and 3 NLoS points (Rx15-17) on a linear route blocked by a building corner and foliage. Tx4 has 5 NLoS points (Rx18-22) that are all blocked by foliage, a water fountain and objects in the area. Finally, Tx5 has corresponding receiver locations on a linear route with 4 LoS points (Rx23-26) at VHE building entrance that were previously detailed in [22] 2 . 2) Outdoor crossroad ("a metal fence. For this scenario, 3 Tx locations and 12 Rx locations were selected (9 LoS, 3 NLoS) in total. TABLE II : IIDescription of Tx-Rx links and their respective direct distances.Tx identifier LoS Rx identifier d LoS (m) NLoS Rx identifier d NLoS (m) T x1 1-3 2.5, 10, 14 4-7 2.25, 7, 10, 27 T x2 8,9 24, 35 10-11 25,23 T x3 12-14 60, 80, 93 15-17 60, 80, 93 T x4 - - 18-22 25, 35, 35, 40, 40 T x5 23-26 1, 2, 5, 15 - - T x6 27-29 60, 80, 98.4 30-32 60, 80, 97.59 T x7 33-36 25, 35, 40, 40 - - T x8 37,38 28, 28 - - Fig. 3: Childs and Watt Way intersection scenario. 1 GHz bandwidth (145-145 GHz), φ T x and φ Rx represent azimuth orientation of the Tx and Rx, and d is the Tx-Rx distance. The dimensions of H meas are N ×N T x ×N Rx where N is the numberof frequency points per sweep (1001), and N T x and N Rx are the number of azimuth directions at the Tx (36) and Rx (36). In order to eliminate the effects of the system (including antennas) transfer function, we perform an OTA calibration H OT A (f ) and obtain the calibrated directional channel transfer function H(f, φ T x , φ Rx ; d) by dividing the raw data with the calibration data as H(f, φ T x , φ Rx ; d) = H meas (f, φ T x , φ Rx ; d)/H OT A (f ). From this, the directional power delay profile (PDP) is computed as b), demonstrate a fairly small shadowing effect; the shadowing standard deviation is 1.73 dB. Additionally, it can be observed that the variations from the omni-directional case are smaller compared to the directional points which is a result of the larger number of MPCs in the omni-directional PDPs. Detailed results are provided in TABLE III : IIIPath loss parameters with 95% confidence interval.Parameter Linear model parameters estimated with 95% CI α α min,95% α max,95% β β min,95% β max,95% P L LOS Omni 76.77 75.5 78.05 1.74 1.65 1.83 P L LOS M ax−Dir 76.77 75.24 78.3 1.78 1.67 1.89 P L LOS Omni OLS 76.53 74.74 78.32 1.8 1.68 1.93 P L LOS M ax−Dir OLS 76.42 74.27 78.86 1.86 1.71 2.01 P L N LOS Omni 95.45 87.29 103.61 1.49 0.96 2.03 P L N LOS M ax−Dir 100.47 89.78 111.16 1.35 0.65 2.06 P L N LOS Omni OLS 96.26 85.12 107.41 1.53 0.81 2.25 P L N LOS M ax−Dir OLS 101.03 85.97 116.08 1.45 0.48 2.42 TABLE IV : IVShadowing model parameters with 95% confidence interval.Parameter Statistical model parameters estimated with 95% CI µ µ min,95% µ max,95% σ σ min,95% σ max,95% LOS Omni 0.58 -0.08 1.24 1.45 1.11 2.09 LOS M ax−Dir 0.8 -0.02 1.63 1.81 1.38 2.61 LOS Omni OLS -0.01 -0.65 0.62 1.4 1.07 2.02 LOS M ax−Dir OLS 0.05 -0.74 0.84 1.73 1.33 2.5 N LOS Omni 1.37 -1.31 4.05 5.21 3.88 7.93 N LOS M ax−Dir 2.09 -1.37 5.54 6.72 5 10.23 N LOS Omni OLS 0.02 -2.65 2.7 5.2 3.88 7.92 N LOS M ax−Dir OLS 0 -3.43 3.44 6.69 4.98 10.18 TABLE V : VLinear model parameters for σ τ with 95% confidence interval.Parameter Linear model parameters estimated with 95% CI α α min,95% α max,95% β β min,95% β max,95% σ LOS τ Omni -82.68 -84.1 -81.26 4.63 3.62 5.64 σ LOS τ M ax−Dir -85.68 -88.2 -83.16 3.5 1.7 5.3 σ N LOS τ Omni -67.67 -68 -65.34 -3.32 -4.86 -1.79 σ N LOS τ M ax−Dir -70.43 -74.7 -66.15 -7.39 -10.21 -4.57 measurements in TABLE VI : VIStatistical model parameters for σ τ with 95% confidence interval.Parameter Statistical model parameters estimated with 95% CI µ µ min,95% µ max,95% σ σ min,95% σ max,95% σ LOS τ Omni -76.84 -78.23 -75.46 3.05 2.33 4.41 σ LOS τ M ax−Dir -82.37 -83.57 -81.16 2.64 2.02 3.81 σ N LOS τ Omni -71.97 -72.9 -71.03 1.82 1.35 2.77 σ N LOS τ M ax−Dir -80.67 -82.75 -78.58 4.06 3.02 6.17 E. Angular spread TABLE VII : VIILinear model parameters for σ • with 95% confidence interval.Parameter Linear model parameters estimated with 95% CI α α min,95% α max,95% β β min,95% β max,95% σ • LOS -0.68 -0.78 -0.58 0.12 0.05 0.19 σ • N LOS -0.02 -0.17 0.13 -0.17 -0.27 -0.08 TABLE VIII : VIIIStatistical model parameters for σ • with 95% confidence interval.Parameter Statistical model parameters estimated with 95% CI µ µ min,95% µ max,95% σ σ min,95% σ max,95% σ • LOS -0.49 -0.55 -0.43 0.19 0.16 0.25 σ • N LOS -0.23 -0.29 -0.17 0.16 0.13 0.21 TABLE IX : IXLinear model parameters for κ 1 with 95% confidence interval.Parameter Linear model parameters estimated with 95% CI α α min,95% α max,95% β β min,95% β max,95% κ LOS 1 Omni 14.95 9.94 19.96 -3.39 -7.05 0.25 κ LOS 1 M ax−Dir 11.47 4.37 18.57 6.19 1 11.38 κ N LOS 1 Omni -4.88 -11.97 -2.22 4.35 -0.33 9.03 κ N LOS 1 M ax−Dir 4.71 -3.79 13.21 2.69 -2.91 8.3 TABLE X : XStatistical model parameters for κ 1 with 95% confidence interval.Parameter Statistical model parameters estimated with 95% CI µ µ min,95% µ max,95% σ σ min,95% σ max,95% κ LOS 1 Omni 9.58 6.83 12.34 6.05 4.63 8.73 κ LOS 1 M ax−Dir 17.88 15.12 20.64 6.07 4.64 8.76 κ N LOS 1 Omni 0.54 -2 3.8 4.93 3.67 7.51 κ N LOS 1 M ax−Dir 9.84 6.27 13.42 6.95 5.18 10.58 Table XI presents the results of various linear fittings of the distance dependence of parameters for LoS and NLoS cases respectively. Table XII provides the detailed results for various statistical fits done in the paper for the LoS and NLoS cases. These results can be directly used for system simulations. TABLE XI : XILinear model parameters summary.Parameter α β P L LOS Omni 76.77 1.74 P L LOS M ax−Dir 76.77 1.78 P L LOS Omni OLS 76.53 1.8 P L LOS M ax−Dir OLS 76.42 1.86 σ LOS τ Omni -82.7 4.65 σ LOS τ M ax−Dir -86.96 4.26 κ LOS 1 Omni 14.95 -3.39 κ LOS 1 M ax−Dir 11.47 6.19 σ • LOS -0.68 0.12 σ • N LOS -0.02 -0.17 P L N LOS Omni 95.45 1.49 P L N LOS M ax−Dir 100.47 1.35 P L N LOS Omni OLS 96.26 1.53 P L N LOS M ax−Dir OLS 101.03 1.45 σ N LOS τ Omni -67.74 -3.24 σ N LOS τ M ax−Dir -70.77 -6.96 κ N LOS 1 Omni -4.88 4.35 κ N LOS 1 M ax−Dir 4.71 2.69 TABLE XII : XIIStatistical model parameters summary. 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[ "Molecular clouds and star formation toward the Galactic plane within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 •", "Molecular clouds and star formation toward the Galactic plane within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 •" ]
[ "Y Gong \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n\nUniversity of Chinese Academy of Sciences\nNo. 19A Yuquan Road100049BeijingPR China\n", "R Q Mao \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "M Fang \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "S B Zhang \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "Y Su \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "J Yang \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "Z B Jiang \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "Y Xu \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "M Wang \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "Y Wang \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "D R Lu \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n", "J X Sun \nPurple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China\n" ]
[ "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "University of Chinese Academy of Sciences\nNo. 19A Yuquan Road100049BeijingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China", "Purple Mountain Observatory & Key Laboratory of Radio Astronomy\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPR China" ]
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Context. Molecular clouds trace the spiral arms of the Milky Way and all its star forming regions. Large-scale mapping of molecular clouds will provide an approach to understand the processes that govern star formation and molecular cloud evolution. Aims. As a part of the Milky Way Imaging Scroll Painting (MWISP) survey, the aim is to study the physical properties of molecular clouds and their associated star formation toward the Galactic plane within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • , which covers the molecular cloud complex S287. Methods. Using the 3×3 Superconducting Spectroscopic Array Receiver (SSAR) at the PMO-13.7m telescope, we performed a simultaneous 12 CO (1-0), 13 CO (1-0), C 18 O (1-0) mapping toward molecular clouds in a region encompassing 3.75 square degrees. We also make use of archival data to study star formation within the molecular clouds. Results. We reveal three molecular clouds, the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud, in the surveyed region. The 50 km s −1 cloud is resolved with an angular resolution of ∼1 ′ for the first time. Investigating their morphology and velocity structures, we find that the 27 km s −1 cloud is likely affected by feedback from the stellar association Mon OB3 and the 50 km s −1 cloud is characterized by three large expanding molecular shells. The surveyed region is mapped in C 18 O (1-0) for the first time. We discover seven C 18 O clumps that are likely to form massive stars, and 15 dust clumps based on the Bolocam Galactic Plane Survey (BGPS) archive data. Using infrared color-color diagrams, we find 56 Class I and 107 Class II young stellar object (YSO) candidates toward a slightly larger region of 5.0 square degrees. Based on the distribution of YSO candidates, an overdensity is found around the HII region S287 and the intersection of two shells; this is probably indicative of triggering. The star formation efficiency (SFE) and rate (SFR) of the 27 km s −1 cloud are discussed. Comparing the observed values of the filament S287-main with fragmentation models, we suggest that turbulence controls the large-scale fragmentation in the filament, while gravitational fragmentation plays an important role in the formation of YSOs on small scales. We find that star-forming gas tends to have a higher excitation temperature, a higher 13 CO (1-0) opacity, and a higher column density than non-star-forming gas, which is consistent with the point that star formation occurs in denser gas and star-forming gas is heated by YSOs. Using the 1.1 mm dust emission to trace dense gas, we obtain a dense gas fraction of 2.7%-10.4% for the 27 km s −1 cloud.
10.1051/0004-6361/201527334
[ "https://arxiv.org/pdf/1602.00460v2.pdf" ]
119,162,779
1602.00460
461867a9e8d451648b0b5ee4f17c0f569f7295dd
Molecular clouds and star formation toward the Galactic plane within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • 3 Feb 2016 Y Gong Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China University of Chinese Academy of Sciences No. 19A Yuquan Road100049BeijingPR China R Q Mao Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China M Fang Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China S B Zhang Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China Y Su Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China J Yang Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China Z B Jiang Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China Y Xu Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China M Wang Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China Y Wang Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China D R Lu Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China J X Sun Purple Mountain Observatory & Key Laboratory of Radio Astronomy Chinese Academy of Sciences 2 West Beijing Road210008NanjingPR China Molecular clouds and star formation toward the Galactic plane within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • 3 Feb 2016Astronomy & Astrophysics manuscript no. S287 c ESO 2016 February 4, 2016surveys-ISM: clouds-stars: formation-radio lines: ISM-ISM: kinematics and dynamics Context. Molecular clouds trace the spiral arms of the Milky Way and all its star forming regions. Large-scale mapping of molecular clouds will provide an approach to understand the processes that govern star formation and molecular cloud evolution. Aims. As a part of the Milky Way Imaging Scroll Painting (MWISP) survey, the aim is to study the physical properties of molecular clouds and their associated star formation toward the Galactic plane within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • , which covers the molecular cloud complex S287. Methods. Using the 3×3 Superconducting Spectroscopic Array Receiver (SSAR) at the PMO-13.7m telescope, we performed a simultaneous 12 CO (1-0), 13 CO (1-0), C 18 O (1-0) mapping toward molecular clouds in a region encompassing 3.75 square degrees. We also make use of archival data to study star formation within the molecular clouds. Results. We reveal three molecular clouds, the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud, in the surveyed region. The 50 km s −1 cloud is resolved with an angular resolution of ∼1 ′ for the first time. Investigating their morphology and velocity structures, we find that the 27 km s −1 cloud is likely affected by feedback from the stellar association Mon OB3 and the 50 km s −1 cloud is characterized by three large expanding molecular shells. The surveyed region is mapped in C 18 O (1-0) for the first time. We discover seven C 18 O clumps that are likely to form massive stars, and 15 dust clumps based on the Bolocam Galactic Plane Survey (BGPS) archive data. Using infrared color-color diagrams, we find 56 Class I and 107 Class II young stellar object (YSO) candidates toward a slightly larger region of 5.0 square degrees. Based on the distribution of YSO candidates, an overdensity is found around the HII region S287 and the intersection of two shells; this is probably indicative of triggering. The star formation efficiency (SFE) and rate (SFR) of the 27 km s −1 cloud are discussed. Comparing the observed values of the filament S287-main with fragmentation models, we suggest that turbulence controls the large-scale fragmentation in the filament, while gravitational fragmentation plays an important role in the formation of YSOs on small scales. We find that star-forming gas tends to have a higher excitation temperature, a higher 13 CO (1-0) opacity, and a higher column density than non-star-forming gas, which is consistent with the point that star formation occurs in denser gas and star-forming gas is heated by YSOs. Using the 1.1 mm dust emission to trace dense gas, we obtain a dense gas fraction of 2.7%-10.4% for the 27 km s −1 cloud. Introduction Molecular clouds are also known as stellar nurseries because they are the densest parts of the interstellar medium that may gravitationally collapse, and thus create protostars (e.g., Shu et al. 1987;Kennicutt & Evans 2012). Molecular clouds comprise a significant fraction of the total mass of interstellar matter in the Milky Way (e.g., Tielens 2005). CO is regarded as a good tracer for molecular clouds. CO observations of the face-on spiral galaxy M51 have shown that molecular clouds correlate well with the spiral arms (e.g., Schinnerer et al. 2013). Resolving molecular clouds in the Milky way can provide information about the spiral arms, while similar physical spatial resolution cannot be easily achieved toward most of external galaxies. A very recent large-scale CO survey has shown the fact that the Scutum-Centaurus Arm might be extended to the outer second quadrant (Sun et al. 2015). Furthermore, a large-scale survey is essential to improve our understanding of the properties of molecular clouds and large-scale processes that govern star formation, molecular cloud evolution, feedback, etc. (e.g., Binney et al. 1991;Dame et al. 2001;Li et al. 2013;Zhang et al. 2014;Zhou et al. 2014;Ragan et al. 2014;Su et al. 2014Su et al. , 2015. Therefore, it is very important to study the large-scale properties of molecular clouds and their associated star-forming activities. We here present a new survey toward molecular clouds within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • in the J=1-0 rotational transitions of three main CO isotopologs ( 12 CO, 13 CO and C 18 O). This region is part of a molecular cloud complex to the north of the Maddalena cloud (Maddalena & Thaddeus 1985). Figure 1 shows an overview of our surveyed region. This region hosts the bright optical HII region S287, which was first discovered by Sharpless (1959). It is ionized by the O9.5V star LS VI −04 19 (Avedisova & Kondratenko 1984). Another four molecular clouds associated with optical HII regions BFS 56, BFS 57, BFS 58, and BFS 59, were found by Blitz et al. (1982). Of these, BFS 57, also known as the bipolar nebula NS 14 (with a size of 80 ′′ ; Neckel & Staude 1984), is illuminated by a trapezium system of four stars with spectral types B0.5, B1, B9, and A5 at a distance of 2.3 kpc (Howard et al. 1998). Moreover, the stellar association Mon OB3 lies south of the region (Mel'Nik & Efremov 1995;Mel'Nik & Dambis 2009), and the open cluster NGC 2311 is located at (l = 217.7579 • , b = −0.6916 • ) (Kharchenko et al. 2005;Buckner & Froebrich 2013). There have been some previous CO surveys toward this region. Five CO wings, indicative of molecular outflows, have also been found in the region (Fukui 1989;Yang et al. 2002). The area was first mapped by a large, unbiased 12 CO (1-0) Galactic survey with a beam width of ∼8 ′ (Dame et al. 2001). The region was further investigated by the Nagoya-4m 13 CO (1-0) survey with a beam width of 2 ′ .7 (Kim et al. 2004), and three peaks with 13 CO (1-0) integrated intensities > 1 K km s −1 were identified. An area of 1.5 square degrees associated with the HII region S287 has been mapped in 12 CO (1-0) and 13 CO (1-0) with the FCRAO-14m telescope (Lee 1994). The cloud associated with S287 was highly disturbed, and it was inferred that triggered star formation might occur in the cloud. Here, we observe not only a larger area of 3.75 square degrees with a higher sensitivity and a wider velocity coverage than the previous FCRAO survey, but we also simultaneously provide three main CO isotopologs ( 12 CO, 13 CO, and C 18 O; details are presented in Sect. 2). The three J=1-0 rotational transitions can be used to study molecular gas from low-(∼10 2 cm −3 ) to high-density regions (∼10 4 cm −3 ) because of their different abundances with respect to H 2 . Furthermore, the region is mapped in C 18 O (1-0) for the first time, and the distribution of young stellar objects (YSOs) in this region has not been studied before. Observations and data reduction 2.1. This work is based on the Milky Way Imaging Scroll Painting (MWISP 1 ) project which is dedicated to the large-scale survey of molecular gas along the northern Galactic Plane. The simultaneous observations of 12 CO (1-0), 13 CO (1-0), and C 18 O (1-0) toward molecular clouds within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • were carried out with the PMO-13.7m telescope 2 , located at Delingha (3200 m altitude) in China, from April 20 to May 9 2012. Supplementary observations were conducted on September 28 2012. The observations took about 51 hours in total to cover 15 cells (3.75 square degrees), each of which is 30 ′ × 30 ′ . The 3 × 3-beam Superconducting Spectroscopic Array Receiver (SSAR) system was used as front end. Each fast Fourier transform spectrometer (FFTS) with a bandwidth of 1 GHz provides 16384 channels and a spectral resolution of 61 kHz, equivalent to a velocity coverage of ∼2600 km s −1 and a velocity resolution of ∼0.17 km s −1 at 110 GHz. The detailed properties of this system are described in Shan et al. (2012). A specific local oscillator (LO) frequency was carefully selected so that the upper sideband is centered at the 12 CO (1-0) line and the lower sideband is able to cover the 13 CO (1-0) and C 18 O (1-0) lines. Observations are conducted in position-switch on-the-fly (OTF) mode, scanning the region at a rate of 50 ′′ per second with a dump time of 0.3 seconds. Each cell was scanned at least in two orthogonal directions, along the Galactic longitude and the Galactic latitude, to reduce scanning effects. All cells were reduced with the GILDAS 3 software including CLASS and GreG. We used the routine XY MAP in CLASS to regrid raw data and then convert them into fits files. The pixel size of these fits files is 30 ′′ ×30 ′′ . Using these standard fits files from the survey, we combined related data to make up the final region. The antenna temperature (T A ) was calibrated by the standard chopper-wheel method (Ulich & Haas 1976). All the intensities throughout the paper are converted to a scale of main beam temperatures with the relation T mb = T A /B eff , where the beam efficiency B eff is 46% at 115 GHz and 49% at 110 GHz according to the status report 4 of the PMO-13.7m telescope. The calibration accuracy is estimated to be within 10%. Typical system temperatures were 191-387 K at the upper sideband, and 142-237 K at the lower sideband. The typical sensitivity is about 0.5 K (T mb ) for 12 CO (1-0), and 0.3 K (T mb ) for 13 CO (1-0) and C 18 O (1-0) at a channel width of ∼ 0.17 km s −1 . The beam widths are about 55 ′′ and 52 ′′ at 110 GHz and 115 GHz, respectively. The pointing of the telescope has an rms accuracy of about 5 ′′ . Throughout this paper, velocities are all given with respect to the local standard of rest (LSR). Archival data We obtained infrared data from the Wide-field Infrared Survey Explorer (WISE) which mapped the full sky at 3.4, 4.6, 12, and 22 µm (W1, W2, W3, W4) (Wright et al. 2010). Using the WISE infrared data in combination with the J, H, K s band data of the Two Micron All Sky Survey (2MASS) (Skrutskie et al. 2006), we searched for the disk-bearing YSO candidates in the studied region (details are described in Sect. 3.4). We made use of the Bolocam Galactic Plane Survey (BGPS) of 1.1 mm dust continuum emission Aguirre et al. 2011;Ginsburg et al. 2013) to study dust clumps and the dense gas mass fraction of molecular clouds (details are described in Sects. 3.3 and 4.4). In the following text, the dust emission image is convolved with a Gaussian kernel of 44 ′′ to achieve a better signal-to-noise ratio, but the final angular resolution becomes 55 ′′ , coarser than the original beam size of 33 ′′ . Results Properties of molecular clouds The 12 CO (1-0) intensity-weighted velocity map (see Fig. 2a) shows multiple clouds overlapping along the light of sight toward the surveyed region. In the longitude-velocity diagram (see Fig. 2b), we identify three molecular clouds, designated as the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud. It is worth noting that the 50 km s −1 cloud has not been reported by the previous FCRAO study (Lee 1994), but was detected in previous HI and CO studies with poor angular resolution(>8 ′ , Williams & Maddalena 1996;Dame et al. 2001). Thus, the 50 km s −1 cloud is resolved with an angular resolution of ∼1 ′ for the first time. Following the method of Solomon et al. (1987), the Galactic coordinates of the three clouds were calculated with l = ΣT i l i /ΣT i and b = ΣT i b i /Σb i , where l i and b i are the Galactic coordinates of pixel i, and T i is the intensity of pixel i. The velocities were taken to be the Gaussian fit velocities of the average 13 CO (1-0) spectra. Based on the Galactic coordinates and the measured velocities of the clouds, we calculated their kinematics distances with the Galactic rotation curve (the spatial-kinematic model A5 of Reid et al. 2014). We also estimated their Galactocentric radii (D GC ) and heights (H) from the Galactic mid-plane. These results are listed in Table 1. Our distance estimate of the 27 km s −1 cloud is roughly consistent with values in previous studies (e.g., Avedisova & Kondratenko 1984;Lee 1994;Howard et al. 1998). Their small heights from the Galactic mid-plane suggest that they are within in the molecular disk of the Milky Way (the FWHM thickness of the molecular disk is larger than 100 pc at a Galactocentric radius larger than 9 kpc, Heyer & Dame 2015). Based on the fitted spiral arms of the Milky Way (Table 2 of Reid et al. 2014) and the derived positions of the three clouds (see Table 1), we find that the 15 km s −1 cloud is located at the interarm between the Local arm (predicted to be D GC =8.53±0.17 kpc at the Galactocentric azimuth, where the error value is half the width of the arm; this is the same for the following cases) and the Perseus arm (predicted to be D GC =10.17±0.19 kpc at the Galactocentric azimuth); the 27 km s −1 cloud is located at the Perseus arm (predicted to be D GC =10.07±0.19 kpc at the Galactocentric azimuth); the 50 km s −1 cloud belongs to the Outer arm (predicted to be D GC =13.24±0.32 kpc at the Galactocentric azimuth). We used the intensity-weighted radius to characterize the size of the molecular clouds. The intensity-weighted radius is defined as < r >= ΣW i r i /ΣW i , introduced by Ungerechts et al. (2000), where W i is the integrated intensity of pixel i, and r i is the measured distance relative to the centroid. Here, we took the 13 CO (1-0) intensity-weighted radius for the following analysis. The FWHM line widths of clouds were taken to be those of the 13 CO (1-0) average spectra. The line widths are supersonic, suggesting that turbulent motions are dominant in the molecular clouds. To derive the physical properties of the clouds, we employed two methods to obtain the H 2 column density. For the first method, we used the X-factor (X CO ) to convert the 12 CO (1-0) integrated intensity into the H 2 column density. Although X CO can vary from 0.9×10 20 cm −2 (K km s −1 ) −1 to 4.8×10 20 cm −2 (K km s −1 ) −1 (e.g., Heiderman et al. 2010), we chose a X CO of 2.8×10 20 cm −2 (K km s −1 ) −1 to be consistent with Kennicutt (1998) for the following discussion (see Sect. 4.1). We then derived the total cloud masses with formula (A.1) by integrating over the areas (Col. 5 in Table 2) of the 12 CO (1-0) emission with signal-to-noise ratios higher than 3. In the second method, we assumed that all molecular gas are in local thermodynamic equilibrium (LTE). We derived the excitation temperature from the optically thick 12 CO (1-0) line. With this value and the 13 CO (1-0) line parameters, we then also determined the opacity and the 13 CO column density. Milam et al. (2005) found a [ 12 C/ 13 C] isotopic ratio gradient in our Galaxy, [ 12 C/ 13 C] = 6.21D GC + 18.71 . (1) We applied this relationship to calculate the [ 12 C/ 13 C] isotopic ratios of the three clouds that were used to convert the 13 CO column density into the 12 CO column density. We then derived the H 2 column density with a constant [H 2 / 12 CO] abundance ratio of 1.1×10 4 (Frerking et al. 1982). Following the formula (A.1) and assuming that 13 CO (1-0) is optically thin (based on the result of Appendix B), we estimated the LTE masses of the clouds by integrating over the areas (Col. 8 in Table 2) of the 13 CO (1-0) emission with signal-to-noise ratios higher than 3. These properties are listed in Table 2. Taking the distance difference into account, the LTE mass of the 27 km s −1 cloud derived from 13 CO agrees with the previous FCRAO study (Lee 1994). The surface density was calculated by dividing the mass by the area. The difference of the surface densities derived from 12 CO and 13 CO is within a factor of 2. Furthermore, the derived H 2 surface densities of the 27 km s −1 cloud are similar to those (49 M ⊙ pc −2 -105 M ⊙ pc −2 ) of nearby clouds (Heiderman et al. 2010) and slightly higher than the median value (42 M ⊙ pc −2 ) of molecular clouds in the Galactic Ring Survey (Heyer et al. 2009). The 15 km s −1 cloud, which lies in the interarm, has an excitation temperature, line width, surface density, and cloud mass similar to some clouds in the spiral arms (Rathborne et al. 2009;Eden et al. 2013), indicating that the spiral-arm structure may have little effect in altering properties of molecular clouds. Figure 3a displays the three clouds that overlap along the light of sight. Figures 3b-3d show the decomposed spatial distribution of the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud. Overall, the morphologies of the 13 CO (1-0) integrated intensity resemble those of the 12 CO (1-0) integrated intensity, but the 12 CO emission is more extended, tracing more diffuse molecular gas. The three clouds are characterized by several subregions whose names are labeled in Figs. 3b-3d for further analyses of their spatial distributions and velocity structures below. Furthermore, we identify four molecular shells in the mapped regions. In this work, only large-scale shell-like structures are taken into account. Molecular shells are identified through visual investigation of their morphology and velocity structures. The shells are only identified if they show shell-like morphology in their integrated-intensity maps and show blueand red-shifted velocity components in p-v diagrams. Spatial morphology and velocity structure The 15 km s −1 cloud.-This cloud has two subregions, island I and stream (see Fig. 3b). Island I is located south of the 27 km s −1 cloud. The p-v diagram across island I (see Fig. 4) shows a velocity gradient of about 0.6 km s −1 pc −1 from north to south, which is presumably due to rotation or shear motion. On the other hand, Fig. 4b shows that the velocity dispersion becomes smaller from north to south, indicating that its southern part might be more perturbed. Stream has nearly the same velocity as the north of Island I, indicating that they are physically related and may be connected by more diffuse gas (e.g., HI gas). The 27 km s −1 cloud.-The 27 km s −1 cloud has been studied by Lee (1994). Nevertheless, we present an analysis of the cloud with a new perspective. We separated this cloud into four subregions, S287-main, island II, tail, and knots (see Fig. 3c). Its 12 CO (1-0) intensity-weighted velocity map (see Fig. 5a) presents a very complex velocity structure, which agrees with the conclusion that the cloud is highly disturbed (Lee 1994). S287-main shows a filamentary structure and is the densest of the four subregions in the 27 km s −1 cloud. Based on the H 2 column density map derived from 13 CO (1-0), we extracted the highest H 2 column density line of the filament indicated by the purple solid line in Fig. 3c. S287-main measures 71 pc in length and 1.8 pc in width, resulting in an aspect ratio of about 39:1. These properties are similar to those of the filaments which are believed to be associated with Milky Way spiral arms (Wang et al. 2015). The velocity map of S287-main shows a blueshifted velocity component in the center of the filamentary structure where the redshifted part shows a shell-like feature (shell-MonOB3, indicated by the solid black ellipse in Fig. 5a). The stellar association Mon OB3 lies inside shell-MonOB3 and has a radial velocity of ∼27 km s −1 (Mel'Nik & Efremov 1995;Mel'Nik & Dambis 2009). Mon OB3 is therefore confirmed to be associated with S287-main. Shell-MonOB3 is likely due to the formation of Mon OB3 out of S287-main and the disruption by the feedback of Mon OB3. From Fig. 6, we find that the interacting interface I (indicated in the intensity-weighted 12 CO (1-0) line width map, see Fig. 5b) is swept up to the blueshifted panels (23-25 km s −1 ), while S287-main has clearly been split into two parts in the red-shifted panels (28-30 km s −1 ). Figure 5b shows large line widths around the S287 HII region and Mon OB3, which is likely due to the perturbation by the feedback of the S287 HII region and Mon OB3. Position-velocity dia- Notes. (a) The velocity is taken to be the Gaussian fit velocity of the 13 CO average spectrum. (b) The distance is estimated with the A5 model of Reid et al. (2014). Table 2. Derived cloud properties. Cloud l b R.A.(J2000) Dec.(J2000) υ lsr a D b D GC H Location ( • ) ( • )(12 CO 13 CO Cloud T ex < r > ∆υ cloud Area Σ M Area Σ M (K) (pc) (km s −1 ) pc 2 (M ⊙ pc −2 ) (M ⊙ ) pc 2 (M ⊙ pc −2 ) (M ⊙ ) 15 km s − Notes. Column 1 gives the name of the selected clouds. Column 2 gives the median of excitation temperature (the detailed statistical results are described in Appendix B). Column 3 gives the intensity-weighted radius of the clouds. Column 4 gives the line width of the 13 CO average spectrum. The areas, surface densities, cloud masses derived from 12 CO and 13 CO are listed in Column 5-10. grams across the interacting interface I and II (indicated by the arrow in Fig. 5b) demonstrate that the line widths of the perturbed molecular gas are around 4 km s −1 (indicated by the white dashed boxes in both panels of Fig. 7), which is much larger than those (∼ 1-2 km s −1 ) of ambient gas. Two distinct velocity components at ∼22 km s −1 and ∼29 km s −1 , as shown in Fig. 7, could just be the projection of swept-up clumpy gas on either side (front and back) of Mon OB3. This would be expected in the proposed feedback scenario mentioned above. We assume that the two velocity components in the position-velocity diagrams arise from the front and backside of expanding shells, and the expansion velocity is thus half the velocity separation. With this method, the expansion velocity of shell-MonOB3 is found to be 3.5 km s −1 . The uncertainty in the estimate of the expansion velocity is approximate ±1 km s −1 . We also note that the expansion velocity is a lower limit since they are only based on the velocity along the light of sight. Position and size of shell-MonOB3 is manually fitted with the ellipse shown in Fig. 5a. The properties of shell-MonOB3 are given in Table 3. Figure 3c shows that most of the molecular gas in island II does not show much 13 CO (1-0) emission, suggestive of a low-density region. Island II might be fragmenting since its 12 CO (1-0) emission in the center is weaker than that farther outside. In the 21-22 km s −1 panels of Fig. 6, island II is connected to S287-main by weak 12 CO (1-0) emission, indicating that Island II might be affected by the feedback of Mon OB3. Tail and knots are not covered by the previous FCRAO study because of their smaller sky coverage (Lee 1994). Figure 5a shows that tail has a velocity of ∼20 km s −1 and is not aligned with the velocity gradient of the northern part of S287main, indicating that tail and S287-main have different origins. Knots also displays a velocity gradient from north to south (see Fig. 5a). The 50 km s −1 cloud.-The 50 km s −1 cloud in Fig. 3d shows many cavity-like features, indicating that the cloud is highly perturbed by past activities. Around these cavities, we reveal three large molecular shells, labeled as shell A, shell B, and shell C. Figure 8a shows that there are no red-or blue-shifted components toward the center of the three shells, which suggests that they are ring-like rather than spherical. This is readily explained if their natal molecular clouds are approximately sheet-like (e.g., Hartmann 2009;Beaumont & Williams 2010). From its 12 CO (1-0) intensity-weighted velocity map ( Fig. 8a) and p-v diagrams ( Fig. 9), we can see that there are two velocity components (blue-and red-shifted) toward the three molecular shells, which is indicative of expansion. Using the same method for shell-MonOB3, we roughly estimate the expansion velocities of shell A, shell B, and shell C to be 4.0 km s −1 , 2.5 km s −1 , and 1.5 km s −1 , respectively. The ellipses used to fit the positions and sizes of the three shells are shown in Fig. 3d. The properties of the three molecular shells are given in Table 3. The three shells have sizes larger than those driven by low-mass stars in the Perseus clouds (Arce et al. 2011) and those driven by HII regions (Beaumont & Williams 2010), but they are similar to some supernova remnants (SNRs) (Green 2014). This indicates that they are driven by energetic sources. On the other hand, the difference of expansion velocities and sizes between the three molecular shells may result from their different dynamic ages or different energy inputs of driving sources. Furthermore, we find that 13 CO (1-0) emission mainly lies in the intersections between the three molecular shells (see Fig. 3d), where molecular gas is more likely to be compressed. The intensity-weighted 12 CO (1-0) line width map (Fig. 8b) shows large velocity dispersions at the edges of the shells, supporting the assumption that the edges are highly perturbed by the expansion of the three shells. Clumps Clumps are fundamental units of star formation (e.g., Bergin & Tafalla 2007). We searched for C 18 O clumps because C 18 O can trace higher density gas than 13 CO because of its lower abundance. Moreover, the surveyed region is mapped in C 18 O (1-0) for the first time. The observations reveal seven C 18 O clumps that are all embedded in the filament S287-main (see Fig. 10). The observed properties of the seven clumps are obtained with Gaussian fits to corresponding spectra by assuming one velocity component. These results are listed in Table 4. We find that the seven clumps are all within the 27 km s −1 cloud because of the coincidence on spatial distributions along the same line of sight as well as velocities, therefore we adopted a distance of 2.42 kpc for these clumps. All clumps lie along the H 2 highest column density line, except for clump C which is offset by 1.7 pc from the line. The size of each clump was fit using the C 18 O (1-0) integrated intensity map with the task "GAUSS 2D" in GILDAS, and the fit results are shown in Figs. 10b-10d. The deconvolved radii of the clumps are defined as θ cl = ( A π − θ 2 beam 4 ) 1/2 , R cl = θ cl D ,(2) where A is the area of the fit ellipse, θ beam is the FWHM beam width, and D is the distance to the source. (Frerking et al. 1982). Since the opacities of 13 CO (1-0) are higher than 0.4 in these clumps (see Table 5), the H 2 column densities derived from 13 CO are underestimated by a factor of at least 1.2. Thus, we only discuss the properties derived from C 18 O (1-0), which has opacities lower than 0.2. Assuming that the clumps are spheres with radius R cl , we obtain the average H 2 densities of the clumps. These physical parameters of C 18 O clumps are tabulated in Table 5. From Table 4, the observed C 18 O line widths of the clumps are larger than those (0.3 km s −1 -2.1km s −1 ) of low-mass protostars and pre-stellar cores (Jørgensen et al. 2002), but similar to those clumps in infrared dark clouds (1.5 km s −1 -2.2 km s −1 , Zhang et al. 2011). From Table 5, the excitation temperatures derived from the peaks of 12 CO (1-0) spectra are found to be 16∼21 K with an average of 19 K, which is similar to the mean CO excitation temperature of ATLASGAL dust clumps dark at 8 µm (Giannetti et al. 2014) and the mean kinetic temperature of ATLASGAL dust clumps derived from low-J metastable ammonia inversion transitions (Wienen et al. 2012). The LTE masses of the clumps are similar to those of ATLASGAL dust clumps (Giannetti et al. 2014). The densities are found to be around ∼ 10 4 cm −3 . The virial parameters are estimated to be lower than 1. Thus, we suggest that these clumps are all supercritical and will collapse or must be supported against self-gravity, for instance, by magnetic fields. Furthermore, Figs. 10b-10d show that six of them are found to be associated with the Red MSX sources (RMSs) 5 which are believed to be massive young stellar object (MYSO) candidates (Lumsden et al. 2013). Clump A is 5 They are G217.3771-00.0828, G217.3020-00.0567, G217.2558-00.0306, G217.6282-00.1827, G217.0441-00.0584, G218.0230-00.3139A, G218.0230-00.3139B, G218.1025-00.3638, and G218.1472-00.5656 from the RMS catalog (Lumsden et al. 2013). According to their spatial distribution and velocity information (Urquhart et al. 2008a,b;Lumsden et al. 2013), eight of them are found to be associated with the 27 km s −1 clouds while one of them is associated with the 50 km s −1 clouds. the only C 18 O clump that is not associated with MYSO candidates. This is probably because clump A is still at an early evolutionary stage or its embedded infrared sources are not luminous enough to be detected by MSX. In the BGPS 1.1 mm source catalog (Table 1 of Rosolowsky et al. 2010), we find 15 dust clumps in the surveyed region, 12 of which match the highest H 2 column density line of the filament S287-main very well (see Fig. 11a). We made use of 13 CO spectra at the corresponding pixels of dust clump central positions to obtain the velocities and line widths of the dust clumps that are listed in Table 6. As a result, 14 dust clumps are located in the 27 km s −1 cloud while the dust clump D2 is associated with the 50 km s −1 cloud. We therefore used a distance of 5.40 kpc for dust clump D2. Assuming a gas-to-dust ratio of 100, the H 2 column densities can be calculated from flux densities of the BGPS 1.1 mm dust emission using the following formula (e.g., Hildebrand 1983;Kauffmann et al. 2008): N(H 2 ) = 100F ν µm H κ ν B ν (T d )Ω beam ,(3) where F ν denotes the flux density, µ is the mean molecular weight per hydrogen molecule which is taken to be 2.8, and m H is the mass of a hydrogen atom, κ ν represents the dust absorption coefficient at 1.1 mm, taken to be 1.14 cm 2 g −1 given by Dunham et al. (2010), who logarithmically interpolated the dust opacities (Table 1, Col. 6, dust with thin ice mantles coagulating for 10 5 yr at a gas density of 10 6 cm −3 , Ossenkopf & Henning 1994) to the BGPS 1.1 mm band with the emissivity of the graybody spectrum applied, B ν (T d ) is the Planck function for a dust temperature T D , Ω beam is the beam solid angle. The total dust clump mass is derived from the integrated flux density over the target: M dust = 100D 2 S int κ ν B ν (T d ) ,(4) where D is the distance, and S int is the integrated flux density. With a dust temperature of 20 K, the beam-averaged column density and the dust clump mass were derived. With the same method for C 18 O clumps, we also obtained the number density and virial parameters for the dust clumps. These properties are tabulated in Table 6. Seven of the dust clumps have virial parameters larger than 2, which indicates that they are subcritical. This suggests that these dust clumps might be confined by external pressure. Comparing the dust clumps and the C 18 O clumps, we find that six of the dust clumps are associated with C 18 O clumps (see Fig. 11b Following previous studies (e.g., Kauffmann et al. 2010;Urquhart et al. 2013), we also studied the mass-size relation for the C 18 O clumps and the dust clumps, which is shown in Fig. 12. All of the C 18 O clumps are found to have surface densities higher than the empirical lower bounds for massive star formation (Kauffmann et al. 2010;Urquhart et al. 2013). This further supports that the C 18 O clumps have the potential to form massive stars. The dust clumps all lie above the empirical lower bounds for massive star formation of Urquhart et al. (2013), while four of them lie in the low-mass star formation shaded region by Kauffmann et al. (2010). Consequently, at least, eleven of the dust clumps have the potential to form massive stars. Fig. 13b). In method 2, the extinction used to deredden the photometry is estimated from its location in the J − H vs. H − K s color-color diagram as described in Fang et al. (2013). The extinction law is based on the measurement of five nearby star-forming regions (Flaherty et al. 2007). The criteria for this method are also described in Koenig et al. (2012). The targets were rejected when they were associated with 2MASS extended sources. As a result, we found 163 YSO candidates, of which 56 are Class I and 107 are Class II. All Class I YSO candidates are identified with method 1. Eighty-nine Class II YSO candidates are identified with method 1, and 18 with method 2. These identified disk-bearing YSO candidates are tabulated in Table D.1. Star formation in Distribution of young stellar object candidates The spatial distribution of these YSO candidates is displayed in Fig. 14, in which nine MYSO candidates from the RMS survey are also shown. Figures 14b-d shows that most YSO candidates are distributed within molecular clouds, which is also observed in many other star-forming regions (e.g., Megeath et al. 2012;Fang et al. 2013). Comparing Fig. 14a with Fig. 14c, we find that most of bright thermal dust emission traced by 12 µm and 22 µm closely matches the morphology of the 27 km s −1 cloud, which suggests that most active star-forming activities occur in the 27 km s −1 cloud. No extended 12 µm and 22 µm emission is found toward the 15 km s −1 cloud, indicating that the dust temperature is colder in this region. Figure 14b shows YSO candidates in island I. Figure 14c shows that YSO candidates are found to lie in S287-main, tail, and knots, but are absent toward island II. This is presumably because island II is not dense enough to collapse and form stars. On the other hand, an overdensity of YSO candidates is found toward the HII region S287, indicating that triggered star formation may occur around this region. Significant YSO candidates coincide with the highest column density line of the filament. Class I YSO candidates are confined closer to the line than Class II YSO candidates. Particularly, eight MYSO candidates lie in the highest column density line. In the 50 km s −1 cloud, enhanced star formation is found around the intersection of shell B and shell C (indicated by the purple arrow in Fig. 14d). Class I YSO candidates are found to be concentrated around the shells while the distribution of Class II YSO candidates is more random. Class I YSO candidates trace more recent star-forming activities, so that the concentration of Class I YSO candidates around the shells is indicative of sequential star formation. This is probably because the region is highly compressed by the expansion of shell B and shell C, probably triggering the formation of YSOs. Discussion Star formation efficiencies and rates Star formation efficiencies (SFEs) and rates (SFRs) play an vital role in theoretical descriptions of star formation and cloud evolution (e.g., Kennicutt & Evans 2012). SFE is defined as below SFE = M * M * + M cloud ,(5) where M * is the total mass of YSOs and M cloud is the total cloud mass. SFEs vary from less than 0.1% to 50% with a median value of 2% among the molecular clouds in the inner Galaxy (Myers et al. 1986). To calculate the total stellar mass, we made the following assumptions: (1) the mass distribution of YSOs follows the initial mass function (IMF) of Chabrier (2003) ξ(m) ∼ 1/m exp[−(log(m) − log(0.22)) 2 /(2 × 0.57 2 )], m < 1 , ξ(m) ∼ m −2.3 , m > 1 ,(6) where m = M/M ⊙ . (2) According to previous studies toward Taurus and L1641, the disk fractions, defined as N(II)/N(II+III), are found to be 50%-60% (Luhman et al. 2010;Fang et al. 2013). Here, we assumed a disk fraction of 50% for our case. (3) We assumed that the lower limit of the identified YSO mass is based on 14 mag in 2MASS K s band, which corresponds to 3.3, 2.1, and 0.3 absolute magnitude in the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud. This results in lower Notes. (a) The "Y" symbol indicates that the association with MYSO candidates has been observed while the "N" symbol indicates that it has not been observed. Table 6. Properties of the dust clumps. name a R.A.(J2000) DEC.(J2000) l b b b R cl b υ lsr c ∆v c S peak b,d S int b T ex e N( Notes. (a) All dust clumps are located at a distance of 2.42 kpc except D2 which is located at 5.4 kpc. mass limits of 0.2, 0.6, and 2.7 M ⊙ for the three clouds according to the model of Siess et al. (2000), assuming that their ages are all 1 Myr. On the other hand, the upper limit in the IMF will not affect the total mass significantly and was thus simply assumed to be 80 M ⊙ . Given that we do not have spectroscopic information of YSOs, whether YSOs are associated with molecular clouds or not depends only on their spatial distribution. By cross-matching YSOs and the molecular clouds (see Fig. 14 Taking the MYSO candidates into account and assuming a disk fraction of 50%, we find that the total number of YSOs is 18, 140 and 97 in the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud. With the IMF and the assumptions described above, the total stellar mass of the three clouds is estimated to be 10 M ⊙ , 885 M ⊙ and 7288 M ⊙ . We note that the number of YSOs in the 15 km s −1 cloud is too small and statistically unimportant, therefore the derived total stellar mass may have a large uncertainty. The total stellar mass of the 50 km s −1 cloud probably is significantly overestimated since the lower limit mass (2.7 M ⊙ ) in the IMF derived from the K s band magnitude is too high and its total YSO number may be contaminated by foreground YSOs or circumstellar material such as post-AGB stars at such a far distance (5.40 kpc) (Vieira et al. 2011). The discussions below therefore only focus on the 27 km s −1 cloud. Since a part of molecular gas is detected in 12 CO, but not in 13 CO, the mass derived from 13 CO is the lower limit of the total cloud mass. As pointed out by Heiderman et al. (2010), 13 CO may underestimate the mass by a factor of about 4-5, which suggests that the total cloud mass of the 27 km s −1 cloud is about 1.0×10 5 M ⊙ (a correction factor of 4 applied for the mass derived from 13 CO). Along with the mass (8.3×10 4 M ⊙ ) derived from 12 CO, the SFEs of the 27 km s −1 cloud is thus estimated to be 1.1% and 0.9%. Therefore, the SFE of the 27 km s −1 cloud is lower than the SFEs (3.0%-6.3%) in nearby clouds (Evans et al. 2009) and the median value (2%) of the molecular clouds in the inner galaxy (Myers et al. 1986). This suggests that molecular clouds in the outer Galaxy may have lower SFEs. We also calculated the SFR of the 27 km s −1 cloud with the total mass of YSOs by SFR = M * τ ,(7) where τ is the average age of YSOs assumed to be 2 Myr (e.g., Evans et al. 2009). This results in an SFR of 443 M ⊙ Myr −1 for the 27 km s −1 cloud. The SFR for the 27 km s −1 cloud is found to be higher than individual nearby clouds such as the Perseus, Serpens, and Ophiuchus clouds. This is mainly due to its larger size, since the SFR surface density (0.28 M ⊙ yr −1 kpc −2 ) is similar to those of individual nearby clouds (Heiderman et al. 2010). On the other hand, we can estimate the SFR surface density from the Kennicutt-Schmidt law employed for other galaxies (Kennicutt 1998) with the formula Σ(SF) (M ⊙ yr −1 kpc −2 ) = (2.5 ± 0.7) × 10 −4 (Σ(gas)/1M ⊙ pc −2 ) 1.4±0.15 ,(8) where Σ(SF) is the SFR surface density, Σ(gas) is the gas surface density. To be consistent with Kennicutt (1998), we used the gas surface density of 53 M ⊙ pc −2 derived from 12 CO to calculate the expected SFR surface density for the 27 km s −1 cloud, and obtain a value of 0.065 M ⊙ Myr −1 pc −2 which is a factor of 4 lower than the observed value of 0.28 M ⊙ yr −1 kpc −2 . This is consistent with the observations toward nearby clouds (Heiderman et al. 2010;Lada et al. 2010). Fragmentation in the filament S287-main Filaments are found to be ubiquitous in molecular clouds and are thought to be going to fragment into prestellar cores (e.g., André et al. 2010). Some of them have the capability of forming massive stars and clusters (e.g., Henning et al. 2010). S287-main is such a filament, where the optical HII region S287 and the stellar association OB3 are located (see Fig. 3c and Fig. 5a), indicating that these OB stars may form out of the filament. Moreover, there are massive clumps along the highest H 2 column density line of the filament (see Figs. 10a and 11a), and these clumps are most likely to form massive stars (see Sect. 3.3). Along the highest H 2 column density line of the filament, the mean excitation temperature is 17 K. If the CO population is fully thermalized, the kinetic temperature is equal to the excitation temperature. According to an unmagnetized isothermal filament model (Ostriker 1964;Inutsuka & Miyama 1992, 1997, the critical line mass (mass per unit length) of the filament is M line,crit = 2c 2 s /G, where c s is the sound speed and G is the gravitational constant. Thus, the critical line mass of S287-main is found to be 27 M ⊙ pc −1 . Following previous studies (e.g., André et al. 2010;Arzoumanian et al. 2011;Samal et al. 2015), we first investigated the radial profile perpendicular to the highest H 2 column density line, and then used a Gaussian fit to the radial profiles. We find a characteristic width of 1.8 pc. By integrating over the characteristic radial extent of S287-main, the filament is found to harbor a mass of 1.5×10 4 M ⊙ , consist of 59% mass of the 27 km s −1 cloud derived from 13 CO (see Sect. 3.1). Dividing the length (71 pc) of S287-main by the mass of filament, we obtain an observed line mass of 211 M ⊙ pc −1 , which is much higher than the critical line mass. This suggests that S287main is supercritical. S287-main is thus unstable to perturbation and fragments into clumps. This is well consistent with the fact that clumps and YSOs lie along the highest H 2 column density line of S287-main (see Fig. 10a, 11a and 14c). To further investigate the fragmentation, we also compared observed parameters with theoretical models. In the turbulent fragmentation model of an isothermal gas cylinder (e.g., Chandrasekhar & Fermi 1953;Inutsuka & Miyama 1992, 1997Jackson et al. 2010;Wang et al. 2014), the filament will fragment into clumps separated by a typical spacing of λ t = 22H = 22δ v (4πGρ) −0.5 ,(9) where δ v is the turbulent velocity dispersion, and ρ is the gas density at the center of the filament. From Table 4 and 6, we used the line widths of 13 CO to obtain the turbulent velocity dispersion. The typical turbulent velocity dispersion is consequently about 1 km s −1 . Based on Table 5 and 6, we adopted 10 4 cm −3 as the gas density at the center of the filament. This resulted in a theoretical spacing of 4.2 pc between clumps. This is similar to the observed separations (0.8-8 pc) of clumps in the filament. This agrees with previous studies (e.g., Jackson et al. 2010;Su et al. 2015). It suggests that turbulent fragmentation plays an important role in fragmenting the filament into clumps. In a model of gravitational fragmentation (Hartmann 2002), the fragmentation length (λ c ) and the corresponding collapse timescale (τ) for the filament are λ c = 3.94 c 2 s GΣ = 1.5T 10 A −1 V pc ,(10)τ ∼ 3.7T 1/2 10 A −1 V Myr ,(11) where c s is the sound speed, Σ is the surface density at the center of the filament, A v is the visual extinction, and T 10 is the temperature in units of 10 K. Along the highest H 2 column density line of the filament, the mean column density is 9.2×10 21 cm −2 , which corresponds to a mean visual extinction of ∼10 mag (according to N(H 2 ) = 9.4 × 10 20 cm −2 (A V /mag), Bohlin et al. 1978). Hence, the fragmentation length is ∼0.3 pc, and the corresponding collapse timescale is 0.4 Myr. We explored the projected nearest neighbor separations of the Class I YSO candidates associated with the filaments, which ranges from 0.18 pc to 3.0 pc with a median of 1.05 pc. As pointed out by Hartmann (2002), the effective sound speed may have additional contributions from other complex motions which will increase the fragmentation length by a factor of a few. This could explain why significant portions of separations are larger than 0.3 pc. Furthermore, the predicted collapse timescale coincides with the lifetime of Class I YSOs (∼0.5 Myr Evans et al. 2009;Fang et al. 2013). On small-scales, our results support the hypothesis that the formation of YSOs roughly agrees with the model of gravitational fragmentation, consistent with previous studies toward other filaments (Hartmann 2002;Samal et al. 2015). Therefore, we suggest that turbulence controls large-scale fragmentation in the filament, while gravitational fragmentation plays an important role in YSOs fragmentation on small scales. Comparison of star-forming molecular gas with non-star-forming molecular gas Here, we divided molecular gas into two types, star-forming and non-star-forming, to compare their properties. Molecular gas associated with YSO candidates was assigned to be star forming while molecular gas without YSO counterparts was assigned to be non-star forming. The two types of molecular gas are only identified by the spatial distribution correlation between YSO candidates and molecular clouds. To avoid the effects that the clouds in different environments (e.g., in different arms) may have different properties, we only carried out the comparison toward the 27 km s −1 cloud. We carried out a Kolmogorov-Smirnov test (K-S test) to investigate the difference between the two types of molecular gas. Based on their cumulative distributions of physical properties (Fig. 15), the K-S test confirms that the null hypothesis that the two distributions are the same can be rejected for excitation temperature, 13 CO (1-0) opacity, and H 2 column density with greater than 3σ confidence (all the probabilities of similarity are found to be ≪0.003, to reject the null hypothesis with ≥3σ confidence the probability must be <0.003). Figure 15a shows that star-forming molecular gas tends to have a higher excitation temperature than non-starforming molecular gas. This most likely attribute to the heating of YSOs. Figures 15b and 15c demonstrate that star-forming gas has a higher 13 CO (1-0) opacity and a higher H 2 column density than non-star-forming molecular gas, suggesting that star formation occurs in denser molecular gas. This is consistent with the fact that dense gas in molecular clouds regulates star formation from Galactic star-forming regions to external galaxies (Gao & Solomon 2004;Wu et al. 2005;Lada et al. 2010). Dense gas mass fraction in the 27 km s −1 cloud At present, star formation is believed to be regulated by dense gas in molecular clouds (e.g., Gao & Solomon 2004;Wu et al. 2005). Lada et al. (2010Lada et al. ( , 2012 found that the SFR and molecular mass linearly correlate well with the same dense gas fraction ( f DG ) for both local Galactic clouds and a sample of external galaxies, and proposed f DG as a fundamental parameter of star formation. Here, we used the BGPS 1.1 mm dust emission and our CO observations to investigate the dense gas fraction of the 27 km s −1 cloud since the dust emission mainly originates in the 27 km s −1 cloud (see Fig. 16). Although the coverage of the BGPS 1.1 mm dust emission is only about 45% of our mapped region (see Fig. 11a), the dust emission covers most of the prominent 13 CO (1-0) emitting regions, and thus those regions not covered by the BGPS 1.1 mm dust emission are assumed to be free of dust emission. We took the dust emission above the threshold of a flux density of 0.09 Jy beam −1 (3σ in the convolved image, indicated by the contour in Fig. 16) into account. The dust contours above the threshold were rechecked by eye, and the emission with a low fidelity (e.g., the dust contours without 13 CO (1-0) counterparts) or not associated with the 27 km s −1 cloud was manually discarded. We used Eqs. (3) and (A.1) to estimate the total mass of dense gas in the cloud. From Eq. (3), we see that the total mass of dense gas is subject to dust temperatures. Previous surveys showed that the dust temperature is about 10-20 K in the Milky Way Planck Collaboration et al. 2011). Toward the dust clumps, the CO excitation temperature is found to be 15-29 K (see Table 6). Here, we assumed an average dust temperature from 10 K to 25 K for the dense gas in the 27 km s −1 cloud. Thus, the total mass of dense gas is estimated to be from 2700 M ⊙ to 694 M ⊙ .To be consistent with the literature, we used the cloud mass derived from 13 CO to calculate the dense gas fraction ( f DG ), which is estimated to be from 10.4% (T d = 10 K) to 2.7% (T d = 25 K). As a result, the f DG of this cloud is in the range of 2.7%-10.4%. This is similar to the 6.5% of the Gemini OB1 molecular cloud and is roughly consistent with the statistical result (7 +13 −5 )% toward giant molecular clouds of the Galactic Ring Survey (Battisti & Heyer 2014) and the 2%-12% toward giant molecular filaments (Ragan et al. 2014). Summary Based on the Milky Way Imaging Scroll Painting (MWISP) project and the archival data, we have performed a study of molecular clouds and star formation toward the region within 216.25 • ≤ l ≤ 218.75 • and −0.75 • ≤ b ≤ 1.25 • . The main results are as follows. 1. We decomposed the observed emission in this region into three molecular clouds, the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud, in the surveyed region. The 50 km s −1 cloud was resolved with an angular resolution of ∼1 ′ for the first time. The properties of the three clouds were investigated and are tabulated in Table 2. 2. By investigating the morphologies and velocity structures of the molecular clouds, we found that the 27 km s −1 cloud probably is affected by feedback from the stellar association Mon OB3, and the 50 km s −1 cloud is characterized by three large expanding molecular shells which are resolved for the first time. 3. This is the first time that the region was mapped in C 18 O (1-0). In the 27 km s −1 cloud, we found seven C 18 O clumps that are likely to form massive stars. Their masses range from 377 M ⊙ to 1165 M ⊙ , and their virial parameters are all lower than unity. Based on the Bolocam Galactic Plane Survey (BGPS) 1.1 mm dust emission, we found 15 dust clumps, which have masses ranging from 26 M ⊙ to 1093 M ⊙ . Six of these dust clumps are associated with C 18 O clumps. 4. Based on 2MASS and WISE data, we identified 56 Class I and 107 Class II young stellar object (YSO) candidates with the color-color diagrams. The YSO distribution showed enhanced star formation around the HII region S287 and the intersection of shell B and shell C, probably indicative of triggering. 5. By counting the YSOs and assuming a universal initial mass function (IMF), the star formation efficiency (SFE) of the 27 km s −1 cloud was estimated to be lower than the SFE in the nearby clouds and the median value of the molecular clouds in the inner galaxy. Assuming the average age of YSOs to be 2 Myr, the star formation rate (SFR) of the 27 km s −1 cloud was found to be 443 M ⊙ Myr −1 , and its SFR surface density of 0.28 M ⊙ yr −1 kpc −2 is similar to those in individual nearby clouds. 6. From comparing the observed values of the filament S287main with the models of fragmentation, we suggest that the turbulence controls the large-scale fragmentation in the filament while gravitational fragmentation plays an important role in the formation of YSOs on small scales. 7. From comparing star-forming molecular gas with non-starforming molecular gas, we find that star-forming gas tend to have a higher excitation temperature, higher opacity, and higher column density than non-star-forming gas, indicating that star formation occurs in denser gas and star-forming gas is heated by YSOs. 8. Using the 1.1 mm dust emission to trace dense gas, we derived a dense gas fraction ( f DG ) of 2.7%-10.4% for the 27 km s −1 cloud. Fig. 4a ( 13 CO black contours overlaid on the 12 CO image). The color bar represents 12 CO intensities in units of K. 13 CO contours start at 0.9 K and increase by 0.6 K. The white dashed line represents the velocity gradient seen in island I. Fig. 9. Fig. 9. 12 CO (1-0) position-velocity diagram along the cuts across shell A (Fig. 9a), B (Fig. 9b), and C (Fig. 9c) indicated in Fig. 8b. The white dashed lines stand for the velocity gradients across the three molecular shells. The color bars represent intensities in units of K. , which are empirical upper and lower bounds for the clump surface densities required for massive star formation. The yellow shaded region demonstrates the clouds without high-mass star formation, according to the empirical relationship by Kauffmann et al. (2010). The green shaded region represents the parameter space where young massive cluster progenitors are expected (e.g., Bressert et al. 2012). The circles and pentagrams represent the C 18 O clumps and the dust clumps, respectively. (Fig. 15a), 13 CO (1-0) opacities (Fig. 15b), and H 2 column densities ( Fig. 15c) of star-forming (dashed red lines) and non-star-forming (solid blue lines) molecular gas for the 27 km s −1 cloud. Fig. B.1. Statistics of the excitation temperature (left), 13 CO (1-0) opacity (middle), H 2 column density and surface density derived from 13 CO (1-0) and labeled at the bottom and the top side (right) for the three molecular clouds: the 15 km s −1 cloud (open blue histogram), the 27 km s −1 cloud (filled green histogram), and the 50 km s −1 cloud (line-filled red histogram). In the left panel, the black dashed line marks T ex = 4.5 K, which corresponds to a 12 CO (1-0) peak temperature of 1.5 K (∼3 σ, see Sect. 2). In the middle panel, the black dashed line marks τ 13 = 0.5. In the right panel, the black dashed line represents the median mass surface density of molecular clouds in the Galactic Ring Survey (Heyer et al. 2009). (b) These values are based on the catalog of Rosolowsky et al. (2010). (c) The velocity information is based on the corresponding 13 CO spectra. (d) The corresponding beamsize is 40 ′′ . (e) The CO excitation temperature is extracted toward the position of each dust clump. Fig. 1 .Fig. 2 . 12Overview of the surveyed region in a three-color composite image of the 22 µm (red), the 12 µm (green) and 4.6 µm emission (blue) from WISE. The optical HII region S287, four BFS sources, the stellar association Mon OB3, and the open cluster NGC 2311 are marked with symbols, near which source names are given. (Descriptions of these objects are presented in Sect. 1.) (a): 12 CO (1-0) intensity-weighted velocity map. The color bar represents the velocity in units of km s −1 . (b): The positionvelocity diagram of the 12 CO emission averaged over the observed range in latitude and plotted against its longitude. The color bar represents intensities in units of K. The lowest contour is 0.4 K, and each contour is twice the previous one. The clouds are also marked in this panel. Fig. 3 . 3(a): Three-color integrated intensity 12 CO (1-0) image of the surveyed region (red: integrated between 45 and 58 km s −1 ; green: integrated between 19 and 35 km s −1 ; blue: integrated between 10 and 19 km s −1 ). (b): 12 CO (1-0) integrated intensity map (background) overlaid with a 13 CO (1-0) integrated intensity map (contours). The two integrated velocity ranges extend from 10 to 19 km s −1 . The grayscale colors correspond to a linear stretch of 12 CO (1-0) integrated intensities in units of K km s −1 . The contours represent 13 CO (1-0) integrated intensities that start at 1.5 K km s −1 and increase by 2.5 km s −1 . (c) Same asFig. 3b, but integrated velocity ranges extend from 19 to 35 km s −1 . The contours start at 2.0 K km s −1 , and each contour is twice the previous one. The purple solid line marks the highest H 2 column density line of the filament S287-main along its long axis. (d) Same asFig. 3b, but integrated velocity ranges reach from 45 to 58 km s −1 . The contours start at 1.8 K km s −1 and increase by 1.8 K km s −1 . The subregions of the three clouds are marked and labeled in the respective panels. Fig. 4 . 4(a): 12 CO (1-0) intensity-weighted velocity map toward the 15 km s −1 cloud. The color bar represents velocities in units of km s −1 . (b): position-velocity diagram along the cut indicated in Fig. 5 . 5(a): 12 CO (1-0) intensity-weighted velocity map toward the 27 km s −1 cloud. The color bar represents velocities in units of km s −1 . The HII region S287, the stellar association Mon OB3, and the open cluster NGC 2311 are marked with a black dashed ellipse and open pentagrams. The solid black ellipse marks the cavity-like feature. (b): Same as Fig. 5a but for the intensity-weighted 12 CO (1-0) line width map of the 27 km s −1 cloud. The arrows represent the p-v cuts of the interacting interface I and II, indicated by I and II in the panel. Their corresponding position-velocity maps are shown inFig. 7 Fig. 6 . 612 CO velocity channel maps of the 27 km s −1 cloud. Each panel consists of the intensity integrated over a 1.0 km s −1 wide range, and the velocity, in km s −1 , is labeled in the upper left corner of each panel. The HII region S287, the stellar association Mon OB3, and the open cluster NGC 2311 are marked with a white ellipse and pentagrams, and their names are also given in the upper left panel. The color bar represents integrated intensities in units of K km s −1 . Fig. 7 . 7Position-velocity diagram along the cuts across the interacting interface I (a) and II (b) indicated inFig. 5b( 13 CO black contours overlaid on the 12 CO image). The color bars represent 12 CO intensities in units of K. 13 CO contours start at 0.9 K and increase by 0.6 K. The white dashed boxes display the interacting regions. Fig. 8 . 8(a): Intensity-weighted 12 CO velocity map of the 50 km s −1 cloud. Shells A, B, and C are marked with black dashed ellipses. The color bar shows the intensity-weighted velocity scale in units of km s −1 . (b): Same as Fig. 8a but for the intensity-weighted 12 CO (1-0) line width map of the 50 km s −1 cloud. The arrows show the pv-diagram cuts presented in Fig. 10 . 10(a): C 18 O intensity map (white contours) overlaid on the 13 CO line image. Both lines have been integrated over the velocity range between 22 and 29 km s −1 . Contours start from 1.2 K km s −1 (3 σ) to 3.6 K km s −1 (9 σ) by 0.8 K km s −1 (2 σ). The horizontal bar represents 15 pc at the distance of 2.42 kpc. The purple solid line marks the highest H 2 column density line of the filament S287main along its long axis. (b): Zoom in the region indicated inFig. 10a. Contours start from 1.2 K km s −1 (3 σ) to 3.6 K km s −1 (9 σ) by 0.6 K km s −1 (1.5 σ). The clumps are marked with red crosses and ellipses, and their corresponding names are also given. The RMS MYSOs are marked with blue circles. The C 18 O beam width is shown in the lower right corner of the panel. (c) Same asFig. 10b. (d) Same asFig. 10b. Fig. 11 . 11(a): Convolved BGPS 1.1 dust continuum map. The purple solid line marks the highest H 2 column density line of the filament S287-main along its long axis. (b): Zoom in the region indicated in Fig. 11a overlaid with C 18 O integrated intensity map.Contours start from 1.2 K km s −1 (3 σ) to 3.6 K km s −1 (9 σ) by 0.6 K km s −1 (1.5 σ). (c) Same asFig. 11b. (d) Same asFig. 11b. In all panels, the open red circles represent the dust clumps from the catalog ofRosolowsky et al. (2010);Ginsburg et al. (2013). Fig. 12 . 12Mass-size relationship of the clumps, modified from Fig. 15 of Urquhart et al. (2013). The upper and lower red lines represent surface densities of 1 g cm −2 (Krumholz & McKee 2008) and 0.05 g cm −2 Fig. 13 . 13(a) WISE [3.4]−[4.6] vs. [4.6]−[12] color-color diagram showing the distribution of Class I (red circles) and Class II YSOs (green circles) selected with method 1. (b) Dereddened K s −[3.4], [3.4]−[4.6] color diagram. Only 18 additional Class II YSOs (blue triangles) are selected with method 2. Fig. 14 . 14(a): Three-color composite image of the 22 µm (red), the 12 µm (green) and 4.6 µm emission (blue) from WISE overlaid with the YSO distribution. (b): Same as Fig. 3b, but overlaid with the YSO distribution. (c): Same as Fig. 3c, but overlaid with the YSO distribution. The solid red ellipse represents S287 while the purple solid line marks the highest H 2 column density line of the filament S287-main along its long axis. (d) Same as Fig. 3d, but overlaid with the YSO distribution. The dashed red ellipses represent shell A, shell B, and shell C. The purple arrow shows the overdensity of YSOs around the shells. In the four panels, the Class I and Class II YSOs are marked with white and cyan circles, while MSYOs are marked with red crosses. Fig. 15 . 15Cumulative distribution of excitation temperature Fig. 16 . 16Convolved BGPS 1.1 dust continuum emission (white contours) overlaid on the 13 CO (1-0) integrated intensity map. The 13 CO (1-0) map has been integrated over the velocity range between 22 and 29 km s −1 . The contour for the BGPS 1.1 dust continuum emission corresponds to the threshold of flux densities, 0.09 Jy beam −1 . Fig. B. 2 . 2Distribution of the excitation temperature derived from 12 CO (1-0) (Figs. B.2a-B.2c), 13 CO (1-0) opacity (Figs. B.2d-B.2f), H 2 column density ((Figs. B.2g-B.2i)) derived from 13 CO (1-0) of the three clouds. Corresponding molecular clouds and physical properties are given in the upper left corner of each panel. The color bar of the first column represents the excitation temperature in units of K. The color bar of the second column represents the 13 CO (1-0) opacity. The color bar of the third column represents the H 2 column density in units of cm −2 . Fig. C. 1 . 1Integrated intensity ratio 12 CO/ 13 CO maps of the 15 km s −1 cloud (a), the 27 km s −1 cloud (b), and the 50 km s −1 cloud (c). Table 1 . 1Molecular clouds identified in this region. Table 3 . 3Properties of molecular shells.Name l b angular size physical size PA υ exp ( • ) ( • ) ( ′ × ′ ) (pc × pc) ( • ) (km s −1 ) The 27 km s −1 cloud Shell-MonOB3 217.565 −0.150 30×35 21×25 30 3.5 The 50 km s −1 cloud Shell A 216.750 0.385 42×36 66×57 20 4.0 Shell B 217.300 0.500 30×26 47×41 0 2.5 Shell C 217.204 0.143 21×12 33×19 −60 1.5 Following the LTE methods and Appendix A, we calculated the excitation temperatures, opacities, column densities, LTE masses, and virial parameters of the clumps by assuming an isotopic ratio [ 16 O/ 18 O]=560 (Wilson & Rood 1994) and a constant [H 2 / 12 CO] abundance ratio of 1.1×10 4 The dusty circumstellar disks associated with the disk-bearing YSOs will create infrared excess, which causes their infrared colors to be different from the diskless Class III objects. On the other hand, it is impossible to distinguish the diskless YSOs and unrelated field objects only based on their infrared colors. Therefore, only the disk-bearing YSOs are investigated in this work.Two methods are employed to select YSO candidates in this work. The first is only based on the WISE data, and the details of criteria are given inKoenig et al. (2012). The selection is mainly based on the WISE [3.4]−[4.6] vs. [4.6]−[12] color-color diagram (hereafter the method 1, see Fig. 13a). The contaminations from extragalactic sources (star-forming galaxies and AGNs), shock emission blobs, and resolved PAH emission objects were removed based on their locations in the [3.4]−[4.6] vs. [4.6]−[12] color-color diagram and their WISE photometry (see details in Koenig et al. 2012). For the sources that are not detected in the WISE [12] band, their dereddened photometry in the WISE [3.4] and [4.6] bands, in combination with the dereddened 2MASS K s photometry, was used to construct the (K s −[3.4]) 0 vs. ([3.4]−[4.6]) 0 color-color diagram to find additional Class I and II YSO candidates (hereafter the method 2, seethe surveyed region 3.4.1. Identification and classification of young stellar object candidates Low-mass YSOs can be assigned into Class I, Class II and Class III objects according to their spectral indices (e.g., Evans et al. 2009). They represent different evolutionary stages throughout the whole life of low-mass YSOs. Class I and II objects are re- ferred to as disk-bearing YSOs. To select the disk-bearing YSO candidates in the studied region (216.25 • ≤ l ≤ 218.75 • , −0.75 • ≤ b ≤ 1.25 • ) 6 , we made use of the infrared data from the 2MASS and WISE surveys. Table 4 .Table 5 . 45Observed properties of the C 18 O clumps. 167 -0.667 14.8±0.5 46.9±0.5 25.6±0.1 3.0±0.1 8.0±0.3 16.9±0.3 25.6±0.1 2.0±0.1 2.1±0.3 2.7±0.2 25.3±0.1 1.2±0.1 B 218.150 -0.575 15.2±0.5 44.6±0.5 25.5±0.1 2.8±0.1 7.3±0.3 11.8±0.2 25.4±0.1 1.5±0.1 1.6±0.3 1.4±0.2 25.2±0.1 0.8±0.1 C 218.108 -0.367 15.2±0.5 70.7±0.6 26.8±0.1 4.4±0.1 5.7±0.3 17.5±0.3 26.3±0.1 2.9±0.1 1.0±0.3 2.1±0.2 25.7±0.1 1.9±0.2 D 218.017 -0.317 18.1±0.6 66.9±0.7 28.0±0.1 3.5±0.1 7.2±0.3 18.4±0.3 27.5±0.1 2.4±0.1 1.0±0.3 1.8±0.2 26.9±0.1 1.6±0.3 E 217.358 -0.067 16.4±0.5 77.6±0.6 26.4±0.1 4.5±0.1 8.1±0.3 22.3±0.3 26.1±0.1 2.6±0.1 1.5±0.3 3.5±0.3 25.9±0.1 2.2±0.2 F 217.300 -0.050 17.7±0.4 86.9±0.6 26.6±0.1 4.6±0.1 9.4±0.2 22.3±0.2 26.5±0.1 2.2±0.1 1.5±0.3 3.0±0.3 26.5±0.1 1.8±0.2 G 217.258 -0.017 12.7±0.4 55.4±0.5 27.7±0.1 4.1±0.1 5.7±0.2 16.0±0.2 27.6±0.1 2.6±0.1 0.8±0.3 1.8±0.3 27.5±0.1 2.0±0.4 Physical properties of the C 18 O clumps.12 CO 13 CO C 18 O l b T mb T mb dυ υ lsr ∆υ T mb T mb dυ υ lsr ∆υ T mb T mb dυ υ lsr ∆υ name ( • ) ( • ) (K) (K km s −1 ) (km s −1 ) (km s −1 ) (K) (K km s −1 ) (km s −1 ) (km s −1 ) (K) (K km s −1 ) (km s −1 ) (km s −1 ) A 218.13 CO C 18 O name T ex τ 13 N(H 2 ) θ cl R cl τ 18 N(H 2 ) M LTE α vir n(H 2 ) MYSO a (K) (×10 22 cm −2 ) ( ′ ) (pc) (×10 22 cm −2 ) (M ⊙ ) (cm −3 ) A 18 0.77 1.6 1.6 1.1 0.15 1.9 1165 0.3 4.3 × 10 3 N B 19 0.65 1.1 0.9 0.5 0.11 1.0 663 0.1 1.9 × 10 4 Y C 19 0.47 1.6 0.6 0.4 0.07 1.5 587 0.5 5.3 × 10 4 Y D 22 0.51 1.9 0.7 0.5 0.06 1.5 505 0.5 2.4 × 10 4 Y E 20 0.68 2.2 1.1 0.8 0.10 2.7 941 0.8 9.2 × 10 3 Y F 21 0.75 2.3 0.5 0.3 0.09 2.4 380 0.7 4.2 × 10 4 Y G 16 0.59 1.3 0.6 0.4 0.07 1.1 377 0.8 3.4 × 10 4 Y http://www.radioast.nsdc.cn/zhuangtaibaogao.php Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle The region used to classify YSO candidates covers 5 square degrees, which is larger than the area mapped in the three CO lines. Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitleTable D.1. continued. Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitleTable D.1. continued. Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitleTable D.1. continued. ACKNOWLEDGMENTSWe thank all the staff of Qinghai Station of Purple Mountain Observatory for their assistance with our observations. We would like to thank the anonymous referee for the very helpful and constructive comments that led to improvements in this paper. We are grateful to Christian Henkel for his insightful comments on the manuscript. Y. Gong warmly thanks GuangXing Li for helpful discussions. We acknowledge support by the National Natural Science Foundation of China (NSFC) (grants nos. 11127903, 11233007, and 10973040) and the Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB09000000). M.F. acknowledges the NSFC under grant 11203081. This paper made use of information from the Red MSX Source survey database at www.ast.leeds.ac.uk/RMS which was constructed with support from the Science and Technology Facilities Council of the UK. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of NASA's Astrophysics Data System.Appendix A: LTE mass, virial mass, and the virial parameterFrom the H 2 column densities under the LTE condition, the total mass of an object can be derived from the H 2 column densities bywhere µ is the mean molecular weight per hydrogen molecule which is assumed to be 2.8, m H is the mass of the atomic hydrogen, D is the distance to the object, dΩ is the solid angle element and N(H 2 ) is the column density of the molecular hydrogen. According to the definition byBertoldi & McKee (1992), the virial parameter represents the ratio between the kinetic and half gravitational potential energy, and can be estimated bywhere δ v is the one-dimensional velocity dispersion, R is the radius, G is the gravitational constant (∼ 1 232 M −1 ⊙ (km s −1 ) 2 pc), and M is the total mass. In analyses, the virial parameters are used to evaluate whether objects are subcritical or supercritical. According to the discussion about non-magnetized spheres(Kauffmann et al. 2013), the critical virial parameter is found to be 2, which is based on the isothermal hydrostatic equilibrium spheres model(Ebert 1955;Bonnor 1956). However, the magnetic field is not negligible in molecular clouds (e.g.,Li et al. 2015), which will result in a lower critical virial parameter. Thus α vir = 2 is the upper limit of the critical virial parameter(Kauffmann et al. 2013). To simplify Eq. (A.2), the virial parameter can be written as α vir = M vir M (e.g.,Kauffmann et al. 2013). Thus, the virial mass can be expressed aswhere ∆v is the FWHM line width, corresponding to √ 8ln2δ v .Appendix B: Excitation temperature, opacity, and column density toward the surveyed regionFollowing the LTE method, we made a statistical study toward the three clouds. We only took 12 CO (1-0) and 13 CO (1-0) emission detected above > 3σ (1σ = 0.5 K for 12 CO (1-0) and 1σ = 0.3 K for 13 CO (1-0) at a velocity resolution of 0.17 km s −1 ) into account.Figure B.1 shows the statistical results of excitation temperature derived from optically thick 12 CO (1-0), 13 CO (1-0) opacity, and H 2 column density toward the three clouds, whileFig. B.2 displays their distributions. We find that excitation temperatures range between 4.5 K and 14.2 K with a median temperature of 10.4 K for the 15 km s −1 cloud, 4.5 K and 29.6 K with a median temperature of 10.1 K for the 27 km s −1 cloud, 4.5 K and 20.8 K with a median temperature of 7.4 K for the 50 km s −1 cloud. Parts of the three clouds present excitation temperatures lower than typical kinetic temperatures (∼ 10 K) of molecular clouds. As pointed out byHeyer et al. (2009), this is likely because cloud densities are lower than the critical density (∼ 10 3 cm −3 at 10 K) of 12 CO (1-0) or the filling factor of 12 CO (1-0) emission is lower than unity. 13 CO (1-0) emission is found to be almost optically thin everywhere in the survey. The highest 13 CO (1-0) opacities are 0.97, 0.89 and 1.03 for the 15 km s −1 cloud, the 27 km s −1 cloud, and the 50 km s −1 cloud, respectively. The histogram of τ( 13 CO) shows 48% of all values below 0.3, 88% below 0.5, and only lower than 1% above 0.8. We also calculated surface densities by multiplying H 2 column densities by the mass of molecular hydrogen. We find that the surface densities of the three clouds (seeTable 2) are similar to those found inHeyer et al. (2009).Appendix C: 12 CO/ 13 CO line ratiosFigure C.1 displays the 12 CO/ 13 CO integrated intensity ratio maps of the three clouds, where the 13 CO (1-0) emission is detected above 3σ. We find that the 12 CO/ 13 CO integrated intensity ratios at the surface of molecular clouds have much higher values (>10) than those (∼3) of their inner parts. This might be attributed to opacity effects. Taking line broadening by opacity into account, the opacities of 12 CO and 13 CO can be estimated with the formula below (Eq. 4.20;Zeng et al. 2006),where τ 12 is the opacity of 12 CO and r is the isotopic ratio [ 12 C/ 13 C]. Adopting [ 12 C/ 13 C] isotopic ratios calculated from formula (1), we find that the opacities of 12 CO vary from lower than 18 to 135, and those of 13 CO vary from lower than 0.1 to 1.3, respectively. Consequently, the correction factors τ/[1 − exp(−τ)] range from lower than 18 to 100 for 12 CO, and vary within a factor of 2 for 13 CO. Therefore, the integrated intensity ratio gradient from the surface of molecular clouds to their inner parts is mainly due to the opacity difference of 12 CO. The derived opacities also support the assumptions used in Sect. 3.1 that 12 CO (1-0) is optically thick and 13 CO (1-0) is almost optically thin for the clouds. However, the isotopic ratio [ 12 C/ 13 C] may vary considerably, which contributes to the uncertainties of the estimated opacities. Selective photodissociation by UV photons may take effect at the very edge of the molecular clouds due to the difference in self-shielding between 12 CO and 13 CO (e.g, van Dishoeck & Black 1988), which will result in a higher isotopic ratio [ 12 C/ 13 C]. On the other hand, farther inside of molecular clouds (2 mag < A V < 5 mag), when 12 CO and 13 CO are both protected from UV photons, the charge exchange reaction with 13 C + will enrich 13 CO, leading to a lower isotopic ratio [ 12 C/ 13 C](Watson et al. 1976;Szűcs et al. 2014).Appendix D: Identified young stellar objects within the surveyed region . J E Aguirre, A G Ginsburg, M K Dunham, ApJS. 1924Aguirre, J. E., Ginsburg, A. G., Dunham, M. 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[ "Some connections between Dirac-Fock and Electron-Positron Hartree-Fock", "Some connections between Dirac-Fock and Electron-Positron Hartree-Fock" ]
[ "Jean-Marie Barbaroux [email protected] \nPlace de Lattre de Tassigny\nCPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534)\nUniversité Paris IX-Dauphine\n907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France\n", "Maria J Esteban [email protected] \nPlace de Lattre de Tassigny\nCPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534)\nUniversité Paris IX-Dauphine\n907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France\n", "Eric Sere [email protected] \nPlace de Lattre de Tassigny\nCPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534)\nUniversité Paris IX-Dauphine\n907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France\n" ]
[ "Place de Lattre de Tassigny\nCPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534)\nUniversité Paris IX-Dauphine\n907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France", "Place de Lattre de Tassigny\nCPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534)\nUniversité Paris IX-Dauphine\n907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France", "Place de Lattre de Tassigny\nCPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534)\nUniversité Paris IX-Dauphine\n907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France" ]
[]
We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a min-max problem. Then we investigate a max-min problem coming from the electron-positron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this max-min problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated selfconsistent projector.
10.1007/s00023-005-0199-7
[ "https://arxiv.org/pdf/math-ph/0402058v3.pdf" ]
15,326,115
math-ph/0402058
367881e41d156c36a728f9a555879b5081943018
Some connections between Dirac-Fock and Electron-Positron Hartree-Fock 2 Sep 2004 October 27, 2018 Jean-Marie Barbaroux [email protected] Place de Lattre de Tassigny CPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534) Université Paris IX-Dauphine 907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France Maria J Esteban [email protected] Place de Lattre de Tassigny CPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534) Université Paris IX-Dauphine 907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France Eric Sere [email protected] Place de Lattre de Tassigny CPT-CNRS Luminy Case Ceremade (UMR CNRS no. 7534) Université Paris IX-Dauphine 907 -13288, 75775Marseille Cédex 9, Paris Cédex 16France, France Some connections between Dirac-Fock and Electron-Positron Hartree-Fock 2 Sep 2004 October 27, 2018AMS classification (2000) Primary: Secondary: 1 -Introduction We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a min-max problem. Then we investigate a max-min problem coming from the electron-positron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this max-min problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated selfconsistent projector. -Introduction. The electrons in heavy atoms experience important relativistic effects. In computational chemistry, the Dirac-Fock (DF) model [1], or the more accurate multiconfiguration Dirac-Fock model [2], take these effects into account. These models are built on a multi-particle Hamiltonian which is in principle not physically meaningful, and whose essential spectrum is the whole real line. But they seem to function very well in practice, since approximate bound state solutions are found and numerical computations are done and yield results in quite good agreement with experimental data (see e.g. [3]). Rigorous existence results for solutions of the DF equations can be found in [4] and [5]. An important open question is to find a satisfactory physical justification for the DF model. It is well known that the correct theory including quantum and relativistic effects is quantum electrodynamics (QED). However, this theory leads to divergence problems, that are only solved in perturbative situations. But the QED equations in heavy atoms are nonperturbative in nature, and attacking them directly seems a formidable task. Instead, one can try to derive approximate models from QED, that would be adapted to this case. The hope is to show that the Dirac-Fock model, or a refined version of it, is one of them. Several attempts have been made in this direction (see [6,7,8,9] and the references therein). Mittleman [6], in particular, derived the DF equations with "self-consistent projector" from a variational procedure applied to a QED Hamiltonian in Fock space, followed by the standard Hartree-Fock approximation. More precisely, let H c be the free Dirac Hamiltonian, and Ω a perturbation. We denote Λ + (Ω) = χ (0,∞) (H c + Ω). The electronic space is the range H + (Ω) of this projector. If one computes the QED energy of Slater determinants of N wave functions in this electronic space, one obtains the DF energy functional restricted to (H + (Ω)) N . Let Ψ Ω be a minimizer of the DF energy in the projected space (H + (Ω)) N under normalization constraints. It satisfies the projected DF equations, with projector Λ + (Ω). Let E(Ω) := E(Ψ Ω ). Mittleman showed (by formal arguments) that the stationarity of E(Ω) with respect to Ω implies that Λ + (Ω) coincides, on the occupied orbitals, with the selfconsistent projector associated to the mean-field Hartree-Fock Hamiltonian created by Ψ Ω . From this he infers ( [6], page 1171) : "Hence, Ω is the Hartree-Fock potential when the Hartree-Fock approximation is made for the wave function". Recently rigorous mathematical results have been obtained in a series of papers by Bach et al. and Barbaroux et al. [10,11,12] on a Hartree-Fock type model involving electrons and positrons. This model (that we will call EP) is related to the works of Chaix-Iracane [9] and Chaix-Iracane-Lions [13]. Note, however, that in [10,11,12] the vacuum polarisation is neglected, contrary to the Chaix-Iracane approach. In [10], in the case of the vacuum, a max-min procedure in the spirit of Mittelman's work is introduced. In [12], in the case of N -electron atoms, it is shown that critical pairs (γ, P + ) of the electron-positron Hartree-Fock energy E EP give solutions of the self-consistent DF equations. This result is an important step towards a rigorous justification of Mittleman's ideas. All this suggests, in the case of N -electrons atoms, to maximize the minimum E(Ω) with respect to Ω. It is natural to expect that this max-min procedure gives solutions of the DF equations, the maximizing projector being the positive projector of the self-consistent Hartree-Fock Hamiltonian. We call this belief (expressed here in rather imprecise terms) "Conjecture M". In [14] and [15], when analyzing the nonrelativistic limit of the DF equations, Esteban and Séré derived various equivalent variational problems having as solution an "electronic" ground state for the DF equations. Among them, one can find min-max and max-min principles. But these principles are nonlinear, and do not solve Conjecture M. In this paper we try to give a precise formulation of Conjecture M in the spirit of Mittleman's ideas and to see if it holds true or not, in the limit case of small interactions between electrons. We prove that in this perturbative regime, given a radially symmetric nuclear potential, Conjecture M may hold or not depending on the number of electrons. The type of ions which are covered by our study are those in which the number of electrons is much smaller than the number of protons in the nucleus, with, additionally, c (the speed of light) very large. The paper is organized as follows : in §2 we introduce the notations and state our main results (Theorems 9 and 11). Sections 3 and 4 contain the detailed proofs. -Notations and main results. In the whole paper we choose a system of units in which Planck's constant, , and the mass of the electron are equal to 1 and Ze 2 = 4πǫ 0 , where Z is the number of protons in the nucleus. In this system of units, the Dirac Hamiltonian can be written as H c = −ic α · ∇ + c 2 β,(1) where c > 0 is the speed of light , β = 1 1 0 0 −1 1 , α = (α 1 , α 2 , α 3 ), α ℓ = 0 σ ℓ σ ℓ 0 and the σ ℓ 's are the Pauli matrices. The operator H c acts on 4-spinors, i.e. functions from R 3 to C 4 , and it is self-adjoint in L 2 (R 3 , C 4 ), with domain H 1 (R 3 , C 4 ) and formdomain H 1/2 (R 3 , C 4 ). Its spectrum is the set (−∞, −c 2 ] ∪ [c 2 , +∞). In this paper, the charge density of the nucleus will be a smooth, radial and compactly supported nonnegative function n, with n = 1, since in our system of units Ze 2 = 4πǫ 0 . The corresponding Coulomb potential is V := −n * (1/|x|). Then V : R 3 → (−∞, 0) is a smooth negative radially symmetric potential such that − 1 |x| ≤ V (x) < 0 (∀x) , |x| V (x) ≃ −1 for |x| large enough . Note that the smoothness condition on V is only used in step 3 of the proof of Proposition 15. Actually we believe that this condition can be removed. It is well known that H c + V is essentially self-adjoint and for c > 1, the spectrum of this operator is as follows: σ(H c + V ) = (−∞, −c 2 ] ∪ {λ c 1 , λ c 2 , . . . } ∪ [c 2 , +∞), with 0 < λ c 1 < λ c 2 < . . . and lim ℓ→+∞ λ c ℓ = c 2 . Finally define the spectral subspaces M c i = Ker(H c + V − λ c i 1 1) and let N c i denote M c i 's dimension. Since the potential is radial, it is well known that the eigenvalues λ c i are degenerate (see e.g. [16]). For completeness, let us explain this in some detail. To any A ∈ SU (2) is associated a unique rotation R A ∈ SO(3) such that ∀x ∈ R 3 , (R A x)·σ = A(x·σ)A −1 , where σ = (σ 1 , σ 2 , σ 3 ). This map is a morphism of Lie groups. It is onto, and its kernel is {I, −I}. It leads to a natural unitary representation • of SU (2) in the Hilbert spaces of 2-spinors L 2 (S 2 , C 2 ) and L 2 (R 3 , C 2 ), given by (A • φ)(x) := A φ(R −1 A x) .(2) Then, on the space of 4-spinors L 2 (R 3 , C 4 ) = L 2 (R 3 , C 2 ) ⊕ L 2 (R 3 , C 2 ), one can define the following unitary representation (denoted again by •) A • φ χ (x) := (A • φ)(x) (A • χ)(x) = Aφ(R −1 A x) Aχ(R −1 A x) .(3) The radial symmetry of V implies that H c + V commutes with •. The eigenspaces M c i are thus SU (2) invariant. Now, letĴ = (Ĵ 1 ,Ĵ 2 ,Ĵ 3 ) be the total angular momentum operator associated to the representation •. The eigenvalues ofĴ 2 =Ĵ 2 1 +Ĵ 2 2 +Ĵ 2 3 are the numbers (j 2 − 1/4) , where j takes all positive integer values. If φ is an eigenvector ofĴ 2 with eigenvalue (j 2 −1/4) , then the SU (2) orbit of φ generates an SU (2) invariant complex subspace of dimension 2j ≥ 2. This implies the following fact, which will be used repeatedly in the present paper: Lemma 1 If φ ∈ L 2 (R 3 , C 2 ) is not the zero function, then there is A ∈ SU (2) such that φ and A • φ are two linearly independent functions. Proof of the Lemma. Assume, by contradiction, that C φ is SU (2) invariant. Then φ is an eigenvector of J ℓ for ℓ = 1, 2, 3, hence it is eigenvector ofĴ 2 . But we have seen that in such a case, the SU (2) orbit of φ must contain at least two independent vectors: this is absurd. ✷ As a consequence of the Lemma, the spaces M c i have complex dimension at least 2. The degeneracy is higher in general: for each j ≥ 1 , H c + V has infinitely many eigenvalues of multiplicity at least 2j. Note that in the case of the Coulomb potential, the eigenvalues are even more degenerate (see e.g. [16]). Now, on the Grassmannian manifold G N (H 1/2 ) := {W subspace of H 1/2 (R 3 , C 4 ); dim C (W ) = N } we define the Dirac-Fock energy E c κ as follows E c κ (W ) := E c κ (Ψ) := N i=1 R 3 ((H c + V )ψ i , ψ i )dx + (4) + κ 2 R 3 ×R 3 ρ Ψ (x)ρ Ψ (y) − |R Ψ (x, y)| 2 |x − y| dxdy , where κ > 0 is a small constant, equal to e 2 /4πǫ 0 in our system of units, {ψ 1 , . . . ψ N } is any orthonormal basis of W , Ψ denotes the N -uple (ψ 1 , . . . ψ N ), ρ Ψ is a scalar and R Ψ is a 4 × 4 complex matrix, given by ρ Ψ (x) = N ℓ=1 ψ ℓ (x), ψ ℓ (x) , R Ψ (x, y) = N ℓ=1 ψ ℓ (x) ⊗ ψ * ℓ (y) .(5) Saying that the basis {ψ 1 , . . . ψ N } is orthonormal is equivalent to saying that Gram L 2 Ψ = 1 1 N .(6) We will use interchangeably the notations E c κ (W ) or E c κ (Ψ). The energy can be considered as a function of W only, because if u ∈ U (N ) is a unitary matrix, E c κ (uΨ) = E c κ (Ψ) .(7) with the notation (uΨ) k = l u kl ψ l . Note that since V is radial, the DF functional is also invariant under the representation • defined above. Its set of critical points will thus be a union of SU (2) orbits. Finally let us introduce a set of projectors as follows: Definition 2 Let P be an orthogonal projector in L 2 (R 3 , C 4 ), whose restriction to H 1 2 (R 3 , C 4 ) is a bounded operator on H 1 2 (R 3 , C 4 ). Given ε > 0, P is said to be ε-close to Λ c + := χ (0,+∞) (H c ) if and only if, for all ψ ∈ H 1 2 (R 3 , C 4 ), −c 2 ∆ + c 4 1 4 P − Λ c + ψ L 2 (R 3 ,C 4 ) ≤ ε −c 2 ∆ + c 4 1 4 ψ L 2 (R 3 ,C 4 ) . In [14] the following result is proved : Theorem 3 ([14]) Take V , N fixed. For c large and ǫ 0 , κ small enough, for all P ε 0 -close to Λ c + , c(P ) := inf W + ∈GN (P H 1/2 ) sup W ∈G N (H 1/2 ) P (W )=W + E c κ (W ) is independent of P and we denote it by E c κ . Moreover, E c κ is achieved by a solution W κ =span{ψ 1 , . . . ψ N } of the Dirac-Fock equations: H c κ,Wκ ψ i = ǫ c i ψ c i , 0 < ǫ c i < 1, Gram L 2 Ψ = 1 N (DF) with H c κ,W ϕ := (H c + V + κ ρ Ψ * 1 |x| )ϕ − κ R 3 R Ψ (x, y)ϕ(y) |x − y| dy . (MF) Remark. It is easy to verify that ε 0 > 0 given, for c large and κ small enough, χ (0,∞) (H c κ,Wκ ) is ε 0 -close to Λ + c . Corollary 4 ([14]) Take V, N fixed. Choose c large and κ small enough. If we define the projector P + κ,W = χ (0,∞) (H c κ,W ) with H c κ,W given by formula (MF), then E c κ = min W ∈Gn (H 1/2 ) P + κ,W W =W E c κ (W ) = min W ∈G N (H 1/2 ) W solution of (DF) E c κ (W ) .(2) Another variational problem was introduced in the works of Bach et al. and Barbaroux et al. ([10, 11, 12]) : define P κ = {P + κ, W = χ [0,∞) (H c κ, W ) ; W ∈ G N (H 1/2 )},(3) and S N κ, W := {γ ∈ S 1 (L 2 ) , γ = γ * , H c κ, W γ ∈ S 1 , P + κ, W γP − κ, W = 0 , −P − κ, W ≤ γ ≤ P + κ, W , tr γ = N }, with the notation P − κ, W := 1 I − P + κ, W , and S 1 being the Banach space of trace-class operators on L 2 (R 3 , C 4 ). For all γ ∈ S N κ, W , let F c κ (γ) = tr ((H c + V )γ) + κ 2 ρ γ (x)ρ γ (y) |x − y| dx dy − κ 2 |γ(x, y)| 2 |x − y| dx dy. Here, ρ γ (x) := 4 s=1 γ s,s (x, x) = n w n |ψ n (x)| 2 , with w n the eigenvalues of γ and ψ n the eigenspinors of γ, and γ(x, y) = n w n ψ n (x) ⊗ ψ n (y), i.e., γ(x, y) is the kernel of γ. In [12] it has been proved that for every P + κ, W ∈ P κ , the infimum of F c κ on the set S N κ, W is actually equal to the infimum defined in the smaller class of Slater determinants. More precisely, with the above notations, Theorem 5 ( [12]) For κ small enough and for all P + κ, W ∈ P κ , one has inf γ∈S N κ, W F c κ (γ) = inf W ∈GN (P + κ, W H 1/2 ) E c κ (W )(4) Moreover, the infimum is achieved by a solution of the projected Dirac-Fock equations, namely γ min = N i=1 ψ i , . ψ i with P + κ, W ψ i = ψ i (i = 1, . . . N ) , and for W min := span(ψ 1 , . . . , ψ N ) , P + κ, W H c κ,Wmin P + κ, W ψ i = ǫ i ψ i , 0 < ǫ i < 1, Gram L 2 Ψ = 1 N(5) Let us now define the following sup-inf: e c κ := sup P + κ, W ∈ P inf W ∈GN (P + κ, W H 1/2 ) E c κ (W ) .(6) Then, Theorem 5 has the following consequence: Corollary 6 If κ is small enough, e c κ = sup P + κ, W ∈ Pκ inf γ∈S N κ, W F c κ (γ) From the above definitions, Theorem 3, Corollary 4 and the remark made after Theorem 3, we clearly see that for all κ small and c large, E c κ ≥ e c κ .(7) One can hope more: Conjecture M: The energy levels E c κ and e c κ coincide, and there is a solution W c κ of the DF equations such that E c κ (W c κ ) = e c κ = inf V ∈GN (P + κ,W c κ H 1/2 ) E c κ (V ) . In other words, the max-min level e c κ is attained by a pair (W, P + κ, W ) such that W = W . This paper is devoted to discussing this conjecture, which, if it were true, would allow us to interpret the Dirac-Fock model as a variational approximation of QED. In order to study the different cases that can appear when studying the problems E c κ and e c κ for κ small, we begin by discussing the case κ = 0. Proposition 7 Conjecture M is true in the case κ = 0. Proof. The case κ = 0 is obvious. Indeed, all projectors P + 0, W coincide with the projector χ [0,∞) (H c + V ). The level E c 0 , seen as the minimum of Corollary 2, is achieved by any N -dimensional space W min spanned by N orthogonal eigenvectors of H c + V whose eigenvalues are the N first positive eigenvalues of H c + V , counted with multiplicity. Then E c 0 is the sum of these N first positive eigenvalues. Clearly, (W min , χ [0,∞) (H c + V )) realizes e c 0 . ✷ The interesting case is, of course, κ > 0 , when electronic interaction is taken into account. For κ > 0 and small two very different situations occur, depending on the number N of electrons. The first situation (perturbation from the linear closed shell atom) corresponds to N = I i=1 N c i , I ∈ Z +(8) is treated in detail in §3. We recall that N c i is the dimension of the eigenspace M c i = Ker(H c + V − λ c i 1 1) already defined. Under assumption (8), for κ = 0, there is a unique solution, W c 0 , to the variational problems defining E c 0 and e c 0 , W c 0 = I i=1 M c i . The "shells" of energy λ c i , 1 ≤ i ≤ I , are "closed": each one is occupied by the maximal number of electrons allowed by the Pauli exclusion principle. The subspace W c 0 is invariant under the representation • of SU (2). We are interested in solutions W c κ of the Dirac-Fock equations lying in a neighborhood Ω ⊂ G N (H 1/2 ) of W c 0 , for κ small. Using the implicit function theorem, we are going to show that for each κ small, W c κ exists, is unique, and is a smooth function of κ. Information about the properties enjoyed by W c κ is given by Proposition 8 Fix c large enough. Under assumption (8), for κ small enough, E c κ = E c κ (W c κ ) = inf W ∈GN (P + κ,W c κ H 1/2 ) E c κ (W ),(9) and W c κ is the unique solution of this minimization problem. This proposition will be proved in §3. Our first main result follows from it : Theorem 9 Under assumption (8), for c > 0 fixed and κ small enough, E c κ = e c κ and both variational problems are achieved by the same solution W c κ of the self-consistent Dirac-Fock equations. For e c κ , the optimal projector in P κ is P + κ,W c κ . Proof. The above proposition implies that for κ small, e c κ ≥ inf W ∈GN (P + κ,W c κ H 1/2 ) E c κ (W ) = E c κ (W c κ ) = E c κ .(10) Therefore, e c κ = E c κ . Moreover, by Proposition 8, e c κ is achieved by a couple (W c κ , P ) such that P = P + N = I i=1 N c i + k, I ∈ Z + , 0 < k < N c I+1 .(11) It is treated in detail in §4. When (11) holds and when κ = 0, there exists a manifold of solutions, S 0 , whose elements are the spaces I i=1 M c i ⊕ W c I+1,k , for all W c I+1,k ∈ G ℓ (M c I+1 ). These spaces are all the solutions of the variational problems defining E c 0 and e c 0 . The (I + 1)-th "shell" of energy λ c I+1 is "open": it is occupied by k electrons, while the Pauli exclusion principle would allow N c I+1 − k more. Note that we use the expression "open shell" in the linear case κ = 0 only : indeed, adapting an idea of Bach et al. [17], one can easily see that for κ positive and small, the solutions to (DF) at the minimal level E c κ have no unfilled shells. For κ > 0 and small we look for solutions of the DF equations near S 0 (see §4). We could simply quote the existence results of [15], and show the convergence of solutions of (DF) at level E c κ , towards points of S 0 , as κ goes to 0. But we prefer to give another existence proof, using tools from bifurcation theory. This approach gives a more precise picture of the set of solutions to (DF) near the level E c κ (Theorem 12). In particular, we obtain in this way all the solutions of (DF) with smallest energy E c κ (Proposition 13). We now choose one of these minimizers, and we call it W c κ . We have P − κ,W c κ (W c κ ) = 0 . Since V is radial, W c κ belongs to an SU (2) orbit of minimizers. We are interested in cases where this orbit is not reduced to a point. Then the mean-field operator H c κ,W c κ should not commute with the action • of SU (2), and one expects the following property to hold: (P) : Given c large enough, if κ is small, then for any solution W c κ of (DF) at level E c κ , there is a matrix A ∈ SU (2) such that P − κ,W c κ (A • W c κ ) = 0 .(12) Let us explain why (P) contradicts Conjecture M: Proposition 10 If (P) is satisfied, then for c large enough and κ small, given any solution W c κ of the nonlinear Dirac-Fock equations such that E c κ (W c κ ) = E κ , we have E c κ = E c κ (W c κ ) > inf W ∈G N (H 1/2 ) P − κ,W c κ W =0 E c κ (W ).(13) This proposition will be proved in §4. Moreover, we verify (see Proposition 15) that (P) holds when I ≥ 1 and k = 1, i.e. when in the linear case there is a single electron in the highest nonempty shell. Our second main result follows directly from Propositions 10 and 15. Theorem 11 Take N = I i=1 N c i + 1, I ≥ 1 . -Perturbation from the linear closed shells case. Let us recall that we are in the case N = I i=1 N c i , I ∈ Z + , N c i being the dimension of the eigenspace M c i = Ker(H c + V − λ c i 1 1). We want to apply the implicit function theorem in a neighborhood of W c 0 , for κ small. For this purpose, we need a local chart near W c 0 . Take an orthonormal basis (ψ 1 , · · · , ψ N ) of W c 0 , whose elements are eigenvectors of H c + V , the associated eigenvalues being µ 1 ≤ · · · ≤ µ N (i.e. λ c 1 , . . . , λ c I counted with multiplicity). Let Z be the orthogonal space of W c 0 for the L 2 scalar product, in H 1/2 (R 3 , C 4 ). Then Z is a Hilbert space for the H 1/2 scalar product. The map C : χ = (χ 1 , · · · , χ N ) → span(ψ 1 + χ 1 , · · · , ψ N + χ N ) , defined on a small neighborhood O of 0 in Z N , is the desired local chart. Denote G χ the N × N matrix of scalar products (χ l , χ ℓ ) L 2 . Then E c κ • C(χ) = E c κ (I + G χ ) −1/2 (ψ + χ) . The differential of this functional defines a smooth map F κ : O ⊂ Z N → (Z ′ ) N , where Z ′ ⊂ H −1/2 is the topological dual of Z for the H 1/2 topology, identified with the orthogonal space of W c 0 for the duality product in H −1/2 × H 1/2 . Note that F κ depends smoothly on the parameter κ. A subspace C(χ) is solution of (DF) if and only if F κ (χ) = 0. To apply the implicit function theorem, we just have to check that the operator L := D χ F 0 (0) is an isomorphism from Z N to its dual (Z ′ ) N . This operator is simply the Hessian of the DF energy expressed in our local coordinates: Lχ = (H c + V − µ 1 )χ 1 , · · · , (H c + V − µ N )χ N .(14) Under assumption (8), the scalars µ k , k = 1, . . . N , are not eigenvalues of the restriction of H c +V to the L 2 -orthogonal subspace of W c 0 . This implies that L is an isomorphism. As a consequence, there exists a neighborhood of W c 0 × {0} in G N (H 1/2 ) × R, Ω × (−κ 0 , κ 0 ) and a smooth function h c : (−κ 0 , κ 0 ) → Ω such that for κ ∈ (−κ 0 , κ 0 ), W c κ := h c (κ) is the unique solution of the Dirac-Fock equations in Ω. Moreover, for all κ ∈ (−κ 0 , κ 0 ), the following holds: u(W c κ ) = W c κ , ∀u ∈ SU (2) .(15) Indeed, the subset A of parameters κ such that (15) holds is obviously nonempty (it contains 0) and closed in (−κ 0 , κ 0 ). Now, for κ in a small neighborhood of A, the SU (2) orbit of W c κ stays in Ω. But this orbit consists of solutions of the Dirac-Fock equations, so, by uniqueness in Ω, it is reduced to a point. This shows that A is also open. A is thus the whole interval of parameters (−κ 0 , κ 0 ). Now we are in the position to prove Proposition 8. Proof of Proposition 8. Remember that for κ = 0, P + 0,W c V of W c 0 in G N (H 1/2 ), there is a constant δ = δ(V) > 0 such that E c 0 (W ) ≥ E c 0 (W c 0 ) + δ , ∀ W ∈ G + 0 ∩ (G N (H 1/2 ) \ V) .(16) Moreover, looking at formula (14), one easily sees that the Hessian of E c 0 on G + 0 is positive definite at W c 0 . We now take κ > 0 small, and we consider again the chart C constructed above. We define the submanifold G + κ := G N (P + κ,W c κ H 1/2 ). Then the restriction C + κ of C to (P + κ,W c κ Z) N is a local chart of G + κ near W c κ . For κ small enough, there is a neighborhood U of 0 in Z N such that the second derivative of E c κ • C + κ is positive definite on U + κ := U ∩ (P + κ,W c κ Z) N . The functional E c κ • C + κ is thus strictly convex on U + κ . Now, for κ small, there is a unique χ κ ∈ U + κ such that C + κ (χ κ ) = W c κ . Then the derivative of E c κ • C + κ vanishes at χ κ . As a consequence W c κ = C + κ (χ κ ) is the unique minimizer of E c κ on V + κ := C + κ (U + κ ). Now, we choose, as neighborhood of W c 0 in G N (H 1/2 ), the set V := C(U), and we consider the constant δ > 0 such that (16) is satisfied. Taking κ > 0 even smaller, we can impose min V + κ E c κ + δ/2 ≤ inf G + κ \V + κ E c κ . Hence, W c κ is the unique solution to the minimization problem (9). ✷ -Bifurcation from the linear open shell case. Recall that here we are in the case N = I i=1 N c i + k, I ∈ Z + , 0 < k < N c I+1 . For κ = 0, there exists a manifold of solutions, S 0 , whose elements are the spaces I i=1 M c i ⊕ W c I+1,k , for all W c I+1,k ∈ G ℓ (M c I+1 ). These spaces are all the solutions of the variational problems defining E c 0 and e c 0 . For κ > 0 and small we want to find solutions of the DF equations near S 0 , by using tools from bifurcation theory. If λ I+1 has only multiplicity 2, then (11) implies k = 1 and by Lemma 1 of §2, S 0 is an SU (2) orbit. Then, as in §3, one can find, in a neighborhood of S 0 , a unique SU (2) orbit S κ of solutions of (DF). But there are also more degenerate cases in which λ I+1 has a higher multiplicity, and S 0 contains a continuum of SU (2) orbits. In such situations, κ = 0 is a bifurcation point, and one expects, according to bifurcation theory, that the manifold of solutions S 0 will break up for κ = 0, and that there will only remain a finite number of SU (2) orbits of solutions. To find these orbits, one usually starts with a Lyapunov-Schmidt reduction: one builds a suitable manifold S κ which is diffeomorphic to S 0 (see e.g. [18]). When S 0 contains several SU (2) orbits, the points of S κ are not necessarily solutions of (DF), but S κ contains all the solutions sufficiently close to S 0 . Moreover, all critical points of the restriction of E c κ to S κ are solutions of (DF). The submanifold S κ is constructed thanks to the implicit function theorem. More precisely, we consider the projector Π : L 2 → I+1 i=1 M c i . To each point z ∈ S 0 we associate the submanifold F z := {w ∈ G N (H 1/2 ) : Πw = z}. For w a point of F z , let ∆ w := T w F z ⊂ T w G N (H 1/2 ). Then the following holds: Theorem 12 Under the above assumptions, there exist a neighborhood Ω of S 0 in G N (H 1/2 ), a small constant κ 0 > 0, and a smooth function h : S 0 × (−κ 0 , κ 0 ) → Ω such that (a) h(z, 0) = z ∀z ∈ S 0 (b) Denoting S κ := h(S 0 , κ), S κ is also the set of all points w in Ω such that < (E c κ ) ′ (w), ξ >= 0, ∀ξ ∈ ∆ w (17) (c) h(z, κ) ∈ F z , ∀(z, κ) ∈ S 0 × (−κ 0 , κ 0 ). Proof. We first fix a point z in S 0 . Let N be the orthogonal space of I+1 i=1 M c i in H 1/2 for the L 2 scalar product. As in §3, we can define a local chart C z : O ⊂ (N ) N → F z near z, by the formula C(χ) = span(ψ +χ), where ψ = (ψ 1 , · · · , ψ N ) is an orthonormal basis of z consisting of eigenvectors of H c +V , with eigenvalues µ 1 ≤ · · · ≤ µ N (i.e. λ c 1 , . . . , λ c I counted with multiplicity). The Hessian of E c 0 • C z at χ = 0 is given once again by formula (14). It is an isomorphism between (N ) N and its dual. So, arguing as in §3, we find, by the implicit function theorem, a small constant κ z > 0, a neighborhood ω z of z in F z and a functionh z : (−κ z , κ z ) → Ω z such that: (i)h z (0) = z (ii)h z (κ) is the unique point w in Ω z such that < (E c κ ) ′ (w), ξ >= 0, ∀ξ ∈ ∆ w(18) Since S 0 is compact and E c κ (w) a smooth function of (w, κ), it is possible to choose κ z , Ω z such that κ 0 := inf z∈S0 κ z > 0, with Ω := z∈S0 Ω z a neighborhood of S 0 , and h(z, κ) :=h z (κ) a smooth function on S 0 × (−κ 0 , κ 0 ) with values in Ω. This function satisfies (a,b,c). ✷ From (b) any critical point of E c κ in Ω must lie on S κ . From (c) it follows that S κ is a submanifold diffeomorphic to S 0 , and transverse to each fiber F z in G N (H 1/2 ). If z ∈ S 0 is a critical point of E c κ • h(·, κ), then, taking w = h(z, κ), the derivative of E c κ at w vanishes on T w S κ . From (b), it also vanishes on the subspace ∆ w which is transverse to T w S κ in T z G N (H 1/2 ), hence (E c κ ) ′ (w) = 0. This shows that the set of critical points of E c κ in Ω coincides with the set of critical points of the restriction of E c κ to S κ . Arguing as in the proof of Proposition 8, one gets more: Proposition 13 For κ > 0 small, the solutions of (DF) of smallest energy E c κ are exactly the minimizers of E c κ on S κ . We are now ready to prove Proposition 10. Proof of Proposition 10 . Since κ is small, for any matrix A ∈ SU (2) the map P + κ,A•W c κ induces a diffeomorphism between the submanifolds G N (P + κ,W c κ H 1/2 ) and G N (P + κ, A•W c κ H 1/2 ) . Now, we fix A ∈ SU (2) such that (12) holds. Then there exists a unique point W + ∈ G N (H 1/2 ) such that P − κ,W c κ W + = 0, P + κ, A•W c κ W + = A • W c κ(19) By (12), we have W + = A • W c κ . On the other hand, in [14] it was proved that E c κ (A • W c κ ) = sup W ∈G N (H 1/2 ) P + κ,A•W c κ W =A•W c κ E c κ (W )(20) and A • W c κ is the unique solution of this maximization problem. Therefore, E c κ (A • W c κ ) > E c κ (W + ) . But E c κ (W + ) ≥ inf W ∈GN (P + κ,W c κ H 1/2 ) E c κ (W ) , hence, by invariance of E c κ under the action of SU (2), We now exhibit a case where (P) holds. E c κ = E c κ (A • W c κ ) > inf W ∈GN (P + κ,W c κ H 1/2 ) E c κ (W ) ,Proposition 15 Assume that N = I i=1 N c i + 1, I ≥ 1. Then (P) is satisfied. Proof. Step 0. Fix c large enough and take a sequence of positive parameters (κ ℓ ) ℓ≥0 converging to 0. Let (W c ℓ ) ℓ≥0 be a sequence in G N (H 1/2 ), with W c ℓ a minimizer of E c κ ℓ on S κ ℓ . Let ψ c ℓ ∈ W c ℓ be an eigenvector of the mean-field Hamiltonian H c κ ℓ ,W c ℓ , normalised in L 2 and corresponding to the highest occupied level. Extracting a subsequence if necessary, we may assume that ψ c ℓ → ψ c ∈ M c I+1 = Ker(H c + V − λ c I+1 ). Moreover, from Theorem 12 we have W c ℓ → W c 0 = I i=1 M c i ⊕ C ψ c . Step 1. Fix c ≥ 1 . Since P − κ ℓ ,W c ℓ ψ c ℓ = 0, we can write, by a classical result due to Kato, P − κ ℓ ,A•W c ℓ ψ c ℓ = 1 2π +∞ −∞ (H c κ ℓ ,W c ℓ − iη) −1 −(H c κ ℓ ,A•W c ℓ − iη) −1 ψ c ℓ dη (21) = 1 2π +∞ −∞ (H c κ ℓ ,W c ℓ − iη) −1 (H c κ ℓ ,A•W c ℓ −H c κ ℓ ,W c ℓ )(H c κ ℓ ,A•W c ℓ − iη) −1 ψ c ℓ dη = κ ℓ 2π +∞ −∞ (H c + V − iη) −1 (Ω A•W c 0 − Ω W c 0 )(H c + V − iη) −1 ψ c dη + o(κ ℓ ) , where by Ω W we denote the nonlinear part of H c κ,W : H c κ,W = H c + V + κ Ω W . But note that since the space (2), I i=1 M c i is invariant under the action of SUΩ A•W c 0 − Ω W c 0 = Ω A•ψ c − Ω ψ c . So, we just have to prove that for c sufficiently large and for all ψ c ∈ M c I+1 , there exists A ∈ SU (2) such that +∞ −∞ (H c + V − iη) −1 (Ω A•ψ c − Ω ψ c )(H c + V − iη) −1 ψ c dη = 0 .(22) Since (H c + V − iη) −1 ψ c = ψ c λ c I+1 − iη and Ω ψ c ψ c = 0 , what we need to prove is that for all nonzero ψ c ∈ M c I+1 , there exists A ∈ SU (2) such that L c (Ω A•ψ c ψ c ) = 0, with L c := +∞ −∞ (H c + V − iη) −1 dη λ c I+1 − iη . Step 2. We give an asymptotic expression for L c when c → +∞: L c = 1 c 2 +∞ −∞ 1 c 2 (H c + V ) − i η c 2 −1 d(η/c 2 ) λ c I+1 c 2 − i η c 2 = 1 c 2 L c + O 1 c 2 ,(23) where L c , in the Fourier domain, is the operator of multiplication by the matrix Hence, by the residues theorem, 2 πL c (p) = β − 1 + (α · p) c + O |p| 2 c 2 . Step 3. It is well known (see [16]) that ψ c can be written as ψ c = φ −i(σ·∇)φ 2c + O 1 c 2 , φ ∈ L 2 (R 3 , C 2 ) being an eigenstate of ( −∆ 2 + V ), with eigenvalue µ = lim c→+∞ (λ c I+1 − c 2 ). Since we have assumed that V is smooth, this asymptotic result holds for the topology of the Schwartz space S(R 3 ). So, 2c 2 π L c (Ω A•ψ c ψ c ) = i c 0 f (A, φ) + O 1 c 2 , where f (A, φ) := |A • φ| 2 * x · σ |x| 3 φ − < A • φ, φ > C 2 * x · σ |x| 3 (A • φ) .(25) What remains to prove is : Step 4. For any eigenvector φ of the Schrödinger operator − ∆ 2 + V , there exists an A ∈ SU (2) such that f (A, φ) ≡ 0 . Proof of Step 4. We consider the integral I A,φ (r) := S 2 < (x · σ)φ, f (A, φ) > C 2 (r ω)dω . Since φ has exponential fall-off at infinity, the electrostatic field |A • φ| 2 * x |x| 3 takes the asymptotic form R 3 |A • φ| 2 x |x| 3 + O 1 |x| 3 when |x| is large. The same phenomenon holds for the convolution product < A • φ, φ > C 2 * x |x| 3 . As a consequence, for r large, r I A,φ (r) = R 3 |A • φ| 2 S 2 |φ| 2 (r ω) dω − R 3 < A • φ, φ > C 2 S 2 < φ, A • φ > C 2 (r ω) dω +O 1 r S 2 |φ| 2 (r ω) dω . Since • is unitary, the Cauchy-Schwartz inequality gives S 2 |φ| 2 (r ω) dω = S 2 |A • φ| 2 (r ω) dω ≥ S 2 < A • φ, φ > C 2 (r ω) dω . By Lemma 1 of §1, we can choose A such that φ and A • φ are not colinear. Then R 3 |A • φ| 2 = R 3 |φ| 2 > R 3 < A • φ, φ > C 2 . So there is a constant δ > 0 such that, for r large enough, |r I A,φ (r)| ≥ δ R 3 |φ| 2 S 2 |φ| 2 (r ω) dω .(26) Being an eigenvector of the Schrödinger operator − ∆ 2 + V , the function φ cannot have compact support. So the lower estimate (26) implies that the function I A,φ (r) is not identically 0, hence f (A, φ) ≡ 0 . Step 4 is thus proved, and (P) is satisfied. ✷ Aknowledgement. The authors wish to thank the referee for useful comments on the first version of this paper. a solution of the Dirac-Fock equations. This ends the proof. ✷ The second situation (perturbation from the linear open shell case) occurs when For c large and κ > 0 small, there is no solution W * of the nonlinear Dirac-Fock equations with positive Lagrange multipliers, such that the couple (W * , P + κ,W * ) realizes the max-min e c κ . So Conjecture M is wrong. 0 coincides with χ (0,∞) (H c + V ). Now, W c0 is clearly the unique minimizer of E c 0 on the Grassmannian submanifold G + 0 := G N precisely, in topological terms, for any neighborhood ( −iu + β + (α · p)/c) −1 = 1 −iu + ω c (p)Λ c + (p) + 1 −iu − ω c (p)Λ c − (p) with ω c (p) := 1 + |p| 2 /c 2 ,Λ c ± (p) = ω c (p) ± (β + (α · p)/c) 2ω c (p). and the Proposition is proved. ✷ Since there are no solutions of (DF) under level E c κ , and e c κ ≤ E c κ , Proposition 10 has the following consequence: Corollary 14 If (P) is satisfied, then for c large enough and κ small, there is no solution W * of the nonlinear Dirac-Fock equations with positive Lagrange multipliers, such that the couple (W * , P + κ,W * ) realizes the max-min e c κ . So Conjecture M is wrong when (P) holds. The relativistic self-consistent field. B Swirles, Proc. Roy. Soc. A. 152B. Swirles. The relativistic self-consistent field. Proc. Roy. Soc. A 152 (1935), p. 625-649. Relativistic self-consistent field calculations. Case Stud. I Lindgren, A Rosen, At. 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Variational stability of the vacuum of quantum electrody- namics in the mean-field approximation. J. Phys. B 22 (23), 3815-3828 (De- cember 1989). Nonrelativistic limit of the Dirac-Fock equations. M J Esteban, E Séré, A.H.P. 2M.J. Esteban, E. Séré. Nonrelativistic limit of the Dirac-Fock equations. A.H.P. 2 (2001), p. 941-961 A max-min principle for the ground state of the Dirac-Fock functional. M J Esteban, E Séré, Contemp. mathem. 307M.J. Esteban, E. Séré. A max-min principle for the ground state of the Dirac-Fock functional. Contemp. mathem. 307 (2002), p. 135-139. The Dirac Equation. B Thaller, Springer-VerlagB. Thaller. The Dirac Equation. Springer-Verlag, 1992. There are no unfilled shells in unrestricted Hartree-Fock theory. V Bach, E H Lieb, M Loss, J P Solovej, Phys. Rev. Lett. 7219V. Bach, E.H. Lieb, M. Loss, J.P. Solovej. There are no unfilled shells in unrestricted Hartree-Fock theory. Phys. Rev. Lett. 72(19) (1994), p. 2981- 2983. 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[ "Network Inference from a Mixture of Diffusion Models for Fake News Mitigation", "Network Inference from a Mixture of Diffusion Models for Fake News Mitigation", "Network Inference from a Mixture of Diffusion Models for Fake News Mitigation", "Network Inference from a Mixture of Diffusion Models for Fake News Mitigation", "Network Inference from a Mixture of Diffusion Models for Fake News Mitigation", "Network Inference from a Mixture of Diffusion Models for Fake News Mitigation" ]
[ "Karishma Sharma [email protected] \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Xinran He \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Sungyong Seo [email protected] \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Yan Liu \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Karishma Sharma [email protected] \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Xinran He \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Sungyong Seo [email protected] \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Yan Liu \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Karishma Sharma [email protected] \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Xinran He \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Sungyong Seo [email protected] \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n", "Yan Liu \nDepartment of Computer Science\nUniversity of Southern California\nLos AngelesUSA\n" ]
[ "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA", "Department of Computer Science\nUniversity of Southern California\nLos AngelesUSA" ]
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The dissemination of fake news intended to deceive people, influence public opinion and manipulate social outcomes, has become a pressing problem on social media. Moreover, information sharing on social media facilitates diffusion of viral information cascades. In this work, we focus on understanding and leveraging diffusion dynamics of false and legitimate contents in order to facilitate network interventions for fake news mitigation. We analyze real-world Twitter datasets comprising fake and true news cascades, to understand differences in diffusion dynamics and user behaviours with regards to fake and true contents. Based on the analysis, we model the diffusion as a mixture of Independent Cascade models (MIC) with parameters θT , θF over the social network graph; and derive unsupervised inference techniques for parameter estimation of the diffusion mixture model from observed, unlabeled cascades. Users influential in the propagation of true and fake contents are identified using the inferred diffusion dynamics. Characteristics of the identified influential users reveal positive correlation between influential users identified for fake news and their relative appearance in fake news cascades. Identified influential users tend to be related to topics of more viral information cascades than less viral ones; and identified fake news influential users have relatively fewer counts of direct followers, compared to the true news influential users. Intervention analysis on nodes and edges demonstrates capacity of the inferred diffusion dynamics in supporting network interventions for mitigation.
10.1609/icwsm.v15i1.18093
[ "https://arxiv.org/pdf/2008.03450v1.pdf" ]
221,132,266
2008.03450
8e1de70623675b3dc6c8ef623d47965f3a933ccc
Network Inference from a Mixture of Diffusion Models for Fake News Mitigation Karishma Sharma [email protected] Department of Computer Science University of Southern California Los AngelesUSA Xinran He Department of Computer Science University of Southern California Los AngelesUSA Sungyong Seo [email protected] Department of Computer Science University of Southern California Los AngelesUSA Yan Liu Department of Computer Science University of Southern California Los AngelesUSA Network Inference from a Mixture of Diffusion Models for Fake News Mitigation The dissemination of fake news intended to deceive people, influence public opinion and manipulate social outcomes, has become a pressing problem on social media. Moreover, information sharing on social media facilitates diffusion of viral information cascades. In this work, we focus on understanding and leveraging diffusion dynamics of false and legitimate contents in order to facilitate network interventions for fake news mitigation. We analyze real-world Twitter datasets comprising fake and true news cascades, to understand differences in diffusion dynamics and user behaviours with regards to fake and true contents. Based on the analysis, we model the diffusion as a mixture of Independent Cascade models (MIC) with parameters θT , θF over the social network graph; and derive unsupervised inference techniques for parameter estimation of the diffusion mixture model from observed, unlabeled cascades. Users influential in the propagation of true and fake contents are identified using the inferred diffusion dynamics. Characteristics of the identified influential users reveal positive correlation between influential users identified for fake news and their relative appearance in fake news cascades. Identified influential users tend to be related to topics of more viral information cascades than less viral ones; and identified fake news influential users have relatively fewer counts of direct followers, compared to the true news influential users. Intervention analysis on nodes and edges demonstrates capacity of the inferred diffusion dynamics in supporting network interventions for mitigation. Introduction Falsified information, that is generally intended to deceive people, influence public opinion and manipulate social outcomes has become a prominent topic of discussion. In 2013, the World Economic Forum regarded fake news as a rising global risk in the report entitled 'Digital Wildfires in a Hyper-connected World'. Even though deception through falsified information has existed in the past, the increasing use and nature of social media, has made the problem much more intense and difficult to combat. The risks associated with fake news are more significant due to the scale and reach of social media; the last decade Copyright c 2020, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. itself has seen more than a ten-fold increase in social media usage (Perrin 2015). The major impacts of fake news have been in social, economic and political issues around the world such as the 2016 US Presidential Elections (Allcott and Gentzkow 2017). Besides that, misleading stories discrediting the severity of climate change (Roozenbeek and van der Linden 2018), and recurrent attempts to promote fear and confusion during natural disasters (Gupta et al. 2013;Takayasu et al. 2015) cannot be neglected. Fake news mitigation has been largely studied from the perspective of detection using content analysis, social bots analysis, and analysis of user responses/engagements to the content on social media (Sharma et al. 2019). As compared to traditional media, online social media allows decentralized dissemination and sharing of content, that can rapidly result in viral information cascades, and widespread impact of misinformation. Therefore, research in intervention strategies to mitigate fake news by monitoring or limiting such diffusions were developed in (Farajtabar et al. 2017;Goindani and Neville 2019). Farajtabar et al. derived optimal intervention intensities required at nodes in the network to accelerate diffusion of true news through external stimulation. However, facilitating network interventions such as this, requires learning diffusion dynamics of fake and true contents from observed user engagements. Here, we consider the problem of learning diffusion dynamics from observed, but unlabeled cascades of fake and true news; and leveraging the inferred dynamics to facilitate network interventions for fake news mitigation. Contributions and Outline In this work, we address the phenomenon of diffusion of fake and true contents on social media, using two real-world datasets comprising false and legitimate content cascades collected from user engagements on Twitter. Based on user behaviours in fake and true cascades, we propose a diffusion mixture model (MIC) with parameters θ T , θ F as a generative model of the diffusion process; and derive unsupervised inference techniques for parameter estimation. Unsupervised estimation is important in this domain, since the cost of acquiring labeled (fake/true) cascades is higher due to reliance on expert verification. Using the inferred param-eters, we evaluate the role of different users and the network in the propagation of misinformation, and provide analysis for network interventions. The following is an outline of the contributions: • We investigate the nature of user behaviours in response to fake and true news on Twitter datasets. Our findings indicate statistical differences in diffusion patterns of fake and true news with non-homogeneous sharing behaviours. • We propose an unsupervised method to learn the diffusion dynamics from observed, unlabeled information cascades, under the diffusion mixture model MIC, and are the first to examine learnability guarantees in the same. • We evaluate if fake and true cascades are separable using inferred dynamics, compared to unsupervised clustering methods based on text, user and propagation features. • We examine characteristics of users identified as influential in spreading legitimate and fake contents using the inferred diffusion dynamics. Inferred influential fake news users have positive correlation with relative appearance in fake cascades; have relatively fewer counts of direct followers compared to influential true news users; and inferred influential users tend to have engagements in topics of more viral/larger cascades, than smaller ones. • Intervention analysis demonstrates reduction in fake cascade size compared to other unsupervised methods. The learned diffusion dynamics are useful towards actively limiting or mitigating misinformation. Related Work Fake news mitigation is largely addressed as a detection (classification) task in existing literature. Sharma et al. (2019) classified approaches for fake news detection based on the features used for classification. Broadly, the methods focus on content or writing style analysis (Wang et al. 2014b;Khattar et al. 2019), source or bot analysis (Ferrara et al. 2016), and features from user responses/engagements on social media (Qian et al. 2018;Ma, Gao, and Wong 2017). The features from user responses are found to be informative and complementary to content or source analysis. In this work, our focus is on information diffusion on social media, to understand how fake and true contents are propagated, and learn a generative model of propagation. Farajtabar et al.;Goindani and Neville (2017; studied intervention strategies based on reinforcement learning for accelerating or limiting diffusions. However, their focus is not on learning diffusion dynamics from observations; and they assume random or known diffusion parameters. Our work on learning diffusion dynamics is therefore complementary to it, and supports different intervention strategies including these. Network inference refers to the problem of inferring the diffusion process, under a mathematical model of propagation, from observed information cascades. It is studied under different models of propagation (Rodriguez, Balduzzi, and Schölkopf 2011;Gomez-Rodriguez, Leskovec, and Krause 2012;Zhou, Zha, and Song 2013). The objective of network inference is to estimate the parameters of a diffusion model from observed information cascades; which might entail inferring the edges of the diffusion network, or both the edges and the strength of influence (or weights) on the edges. For instance, in the Independent Cascade model (Kempe, Kleinberg, and Tardos 2003), for every pair of users u and v, there is a parameter p u,v which represents the probability with which u activates v, that is information successfully propagates from u to v. In other words it is the strength of influence between u and v. In the multivariate Hawkes process model, parameters α u,v ≥ 0 model mutually-exciting nature of network activities, with conditional intensity functions capturing the instantaneous rate of future events conditioned on past events. Most works in network inference do not address heterogeneity in strength of influence between a pair of users. Furthermore, none of them examine whether the influence is heterogeneous with regards to legitimate and fake contents. Earlier works only considered topic or time specific networks (Yang and Zha 2013;Wang et al. 2014a;He and Liu 2017); such as MultiCascades (He and Liu 2017) wherein heterogeneous diffusion models are tied together with joint network priors, and inferred from observed but labeled cascades. Our method considers a heterogeneous diffusion model for true and fake news propagation, and in contrast, we propose an unsupervised method for inference that does not require labeled cascades. Diffusion Mixture Model Information propagation or diffusion is widely studied using probabilistic models, in domains related to viral marketing (Domingos and Richardson 2001), and disease and epidemics (Newman 2002). Diffusion models provide a way to solve important computational problems in each domain. For instance, Domingos and Richardson addressed an important question in viral marketing, that is -to trigger a large cascade of product adoptions, who are the most influential users to target in ad campaigns? Such problems can be efficiently solved using submodular optimization under certain diffusion models such as the Independent Cascade model (Kempe, Kleinberg, and Tardos 2003). The choice of model dictates how efficient it is to optimize for impor- tant problems such as this. It also affects whether it is possible to derive analytical solutions for learning algorithms in order to infer the parameters of the diffusion model from real observed cascades. Here, we introduce the Independent Cascade Model, followed by our extension of the diffusion model to legitimate and fake cascades. Cascade: A cascade is defined as a time-ordered sequence of user responses/ engagements that a piece of information (content) receives, when it is circulated on a social network. It can be labeled as a true or fake news cascade, in accordance with the veracity of the content (eg. Fig. 1). Independent Cascade (IC) Model: First, we discuss the formulation of the Independent Cascade Model studied in Kempe, Kleinberg, and Tardos. G = (V, E) is the directed graph with n = |V | number of nodes (users) and m = |E| edges. A node is activated in an information cascade, if its user has an engagement with the content being propagated. Each edge (u, v) ∈ E is associated with a parameter p u,v ∈ [0, 1]. The diffusion process starts with an initial set of seed nodes assumed to be activated at the first timestep. At each following time step of the diffusion process, a node u activated at the previous time step t, independently makes a single activation attempt on each inactive neighbor v. The activation succeeds with probability p u,v and a node once activated remains activated in the diffusion process. The influence function σ is a function of the seed set S and σ θ (S) is defined as the expected number of nodes activated by the end of the diffusion process starting at seeds S, where θ = {p u,v |(u, v) ∈ E} refers to the parameter set. Mixture of Independent Cascade (MIC) Given a social network G = (V, E), we extend the IC model to include the diffusion of both legitimate (true) and misinformation (fake) contents using separate sets of parameters θ T = {p T uv |(u, v) ∈ E} and θ F = {p F uv |(u, v) ∈ E}, i.e., both types of contents share the same network skeleton G but with separate parameters for activation probabilities on the edges. Based on this parameterization (illustrated in Fig 2), we study the inference of the proposed diffusion mixture model parameters from observed, unlabeled cascades.. First, we formally define the inference problem for the proposed diffusion mixture model, formulated as a mixture of independent cascade models (MIC). We assume that the observed set of diffusion cascades C contains a mix-ture of unlabeled true and fake cascades. We study whether the diffusion process of true and fake contents can be learned directly from C, without requiring cascade labels ∈ {true/fake}. This makes the inference problem more challenging, but practically more useful when collection of labeled cascades requires expert human verification. Problem Formulation: We assume π T is the probability with which a true news cascade emerges, and π F = 1 − π T is the probability with which a fake news cascade emerges. Let π = [π T , π F ] be the mixing weights of the diffusion mixture model, then each cascade c i ∈ C is assumed to be generated independently under MIC as follows: 1. Generated seed set S ⊆ V is sampled from some unknown distribution P over V . 2. Generated cascade corresponds to true or fake news based on the outcome of the random variable h i ∼ Bernoulli (π T ); Cascade labels and mixing weights π unobserved. 3. Generated cascade is drawn from the diffusion mixture model with c i ∼ IC(θ T ) if h i = 1 and c i ∼ IC(θ F ) otherwise; diffusion parameters θ T , θ F are unobserved. The objective of the network inference problem thereby is to infer θ T and θ F and π from unlabeled cascades C. Real Datasets and Diffusion Analysis In the previous section, we proposed the diffusion mixture model with separate sets of parameters θ T = {p T uv |(u, v) ∈ E} and θ F = {p F uv |(u, v) ∈ E} for legitimate and fake contents. In this section we first answer two important questions • Are the diffusion patterns of fake cascades significantly different from true cascades? • Are user behaviours with respect to fake and true contents non-homogeneous? We first analyze the diffusion patterns and investigate user behaviours in fake and true cascades. Significant differences between fake and true cascades would mean that the diffusion of fake and true contents are non-homogeneous with respect to user behaviours and should be modeled with separate parameters θ T and θ F . For the purpose of our analysis, we consider real world Twitter datasets described in the following section, followed by statistical hypothesis testing to analyze their diffusion characteristics. A few earlier studies such as (Kwon et al. 2013;Castillo, Mendoza, and Poblete 2011;Liu et al. 2017) identified which features of a set of hand-crafted features were most discriminative in training classifiers for detecting fake from legitimate contents. There findings indicate that features with high predictive power include -fraction of information flow from low to high-degree nodes which is higher for fake contents, multiple periodic spikes that are particular to fake contents, and greater depth to breadth ratio in the diffusion trees of fake cascades. In our analysis, we consider temporal and structural differences in diffusion cascades of fake/true news that investigate how user behaviours towards different types of contents differ. C i = [(u 1 , t 1 ), (u 2 , t 2 ), . . . ] where u j , t j corresponds to the engagement of user u j at time stamp t j with content corresponding to cascade C i . For temporal diffusion analysis, we report statistical tests on each dataset based on the observed cascades. For structural diffusion analysis, we consider only Twitter-1, since it additionally provides follower links from which we can construct retweet structure, similar to (Kwon et al. 2013). The follower graph represents whether user A follows user B. Diffusion of a content from B to A can occur if A follows B, and B posts before A in that cascade. Therefore, we can construct the retweet graph of each cascade, from the cascade engagement sequence and the follower graph. In case A has multiple parents, the edge from the latest parent is retained. Studying User Behaviours in Fake/True Cascades In this subsection, we study the diffusion patterns and investigate how these patterns reflect user behaviours. We conduct statistical tests to determine temporal and structural characteristics of fake and true cascades. First, we perform a two-sample t-test to verify whether the average time delay between engagements (posts) is higher in fake cascades v/s in true cascades. The first group of samples S 1 consists of the fake cascades in the datasets. The second group S 2 comprises the true cascades. The log-transform of the data is normally distributed. The null hypothesis is that there (p-values) to verify that average time between engagements is higher in fake news cascades than true news cascades (temporal); and to verify that ratio of # of connected components to total engagements is higher in fake news cascades (structural). is no significant difference between the average time delay between engagements in cascades from the two groups H 0 : µ f = µ t . The alternate hypothesis is the average time delay between engagements is higher for fake cascades H 1 : µ f > µ t . The p-value is shown in Table 2. The null hypothesis is rejected at significance level α = 0.01 which suggests that there is statistically significant difference between the temporal characteristics of fake and true cascades. Second, we perform statistical significance test to examine differences in structural characteristics of the cascades. We compute the number of connected components (cc) in the retweet graph of each cascade, constructed as mentioned in the previous subsection. Then we define the proportion of connected components in a cascade r = number (cc) number of engagements . The null hypothesis is that there is no significant difference in the proportion of connected components in the two groups of fake and true cascades H 0 : r f = r t . The alternate hypothesis is that it is higher for fake cascades H 1 : r f > r t . The data is not normally distributed, so we compute the non-parametric MannWhitney U test and report the z-score and p-value in Table 2. The null hypothesis is rejected at α = 0.05 which suggests that the proportion of connected components in fake cascades is higher than in true cascades. Both statistical tests confirm that diffusion patterns differ based on the type of cascade and the user behaviours towards fake and true contents are non-homogeneous. The distribution of average time between engagements and proportion of connected components is provided in Figure 3 for Twitter-1. The distribution of avg. time between engagements for Twitter-2 cascades is similar to Twitter-1, and structural follower graph is unavailable in Twitter-2; therefore omitted. Unsupervised Diffusion Network Inference In this section, we consider a reduction of the problem of learning the diffusion mixture under the MIC model to the problem of learning a mixture of product distributions over a discrete domain; followed by an EM algorithm for parameter estimation of the mixture model. PAC-Learnability Reduction Each edge e = (u, v) in the diffusion model is associated with the parameter p T e and p F e as stated earlier. Each true content cascade can be alternately represented in terms of a 'live-edge' graph, such that each edge e ∈ E is independently declared as live with probability p T e and included in the graph or blocked with probability 1 − p T e and not included in the graph. The cascade is then defined by the reachability from seed set S over this graph i.e. a node is activated in the cascade iff there is a directed live edges path from S to the node. Similarly, for fake content cascades. Therefore, each edge can be represented by a random variable x T e and x F e indicating its live-edge status under the diffusion mixture model, i.e. representing whether the edge e is live or blocked in a given diffusion cascade. Naturally, x T e ∼ Bernoulli (p T e ) and x F e ∼ Bernoulli (p F e ) Let X be a vector of random variables indicating liveedge status of each edge under the mixture diffusion model. According to the generative process of the diffusion mixture model, X is a mixture of k = 2 components X T and X F with mixing weights π. Therefore, X T is then a discrete distribution over {0, 1} m where m is the number of edges in G and all the x T e are independent. Therefore X which is the mixture distribution of X T and X F is simply a mixture of discrete product distributions with mixing weights π. The problem is therefore reduced to learning a mixture of discrete product distributions given the live-edge graphs of the observed cascades. Mixture distributions are more generally used in recommendations systems, medicine and other applications (Feldman, O'Donnell, and Servedio 2008) and different algorithms can be used to learn the parameters of the mixture distributions, which in our case are p T e , p F e for all edges in G and mixing weights π by definition. Theorem 1. Given a mixture of unlabeled cascades with completely observed live-edge graphs, with diffusion parameters θ M = {p M e |e ∈ E} with M ∈ {T, F } and any , δ > 0, with mixing weight π M ≥ mn we can recover in time poly (m 2 n/ ) · log (1/δ), a list of poly (m 2 n/ ) many candidates, at least one of which satisfies the following bound on the influence function σ θ M (S) and its estimatê σ θ M (S) learned from the observed cascades for seed set S drawn from any distribution P over nodes in G, P S∼P (|σ θ M (S) − σ θ M (S)| > ) ≤ δ with sample complexity O ( n 4 m 8 4 ) 3 ln m δ (Proof in Appx). Parameter Estimation We can estimate the parameters of the diffusion mixture model θ T , θ F and mixing weights π from unlabeled cascades, by deriving a maximum likelihood based estimation procedure. We assume that the observed cascades record the sequence of user engagements, and the order or timestamps of user engagements (activations) are known. Notation: We use a general notation M ∈ {T, F } to denote a component in the mixture model, wherein {T, F } refer to the true and fake components of the k = 2 component mixture model MIC. θ T , θ F are the set of edge influence parameters for each component IC model in the diffusion mixture model with mixing weights π = [π T , π F ]. That is for graph G = (V, E), θ T = {p T uv |u, v ∈ E} and θ F = {p F uv |u, v ∈ E}. We use the notation s to specify an observed sample cascade belonging to the set of cascades C. where Z s indicates the component to which the cascade s belongs and θ is the complete set of parameters θ T , θ F , and π. Applying Bayes' rule, ( γ M s = P (Z s = M ; θ) = π M P (s; θ M ) k i=1 π i P (s; θ i ) (1) Let P a(v) represent parents of v in G that is, u ∈ P a(v) if and only if (u, v) ∈ E. Similarly, Ch(v) is the children of v. Let p M s (v) be1 − p M u,v )(2) In addition, let A u,v ⊆ C be the subset of cascades in which both u and v are activated and t s (v) = t s (u)+1 and B u,v be the subset of cascades in which u is activated at some time t and v is not activated up to and including time t + 1. Derivation and algorithm: We derive an expectation maximization based maximum likelihood estimation procedure. The joint log probability of cascade labels and cascades under the mixture model is, log P (C, Z; θ) = s∈C M ∈k 1 {Zs=M } log(π M P (s; θ M )) Our goal is to maximize the expected joint log probability, Q = E [log P (C, Z; θ)] = s∈C M ∈k γ M s log π M + s∈C M ∈k γ M s log P (s; θ M ) The maximization of Q with respect to π subject to constraints M π M = 1, π M ≥ 0, we get, π M = 1 |C| s γ M s from the first term of Q containing π M . Now to update the estimates for edge probabilities p M u,v , we need to maximize the second term of Q by differentiating Q with respect to p M u,v . Letp M u,v be the current estimates of edge influence parameters of edge (u, v) for component M . As stated in the notations, C s (t) is the set of activated nodes at times step t in cascade s and t s (v) is the time of activation of node v in cascade s. p M s (v) is the probability with which v is activated in cascade s under diffusion model M . The second term of Q involves the product terms of Equation 2 which cannot be solved analytically. However, based on the definition of the IC model, it is possible to approximate Q based on the current estimates of the parameters (Gruhl et al. 2004;Saito, Nakano, and Kimura 2008). We utilize the linear approximation chosen in (Saito, Nakano, and Kimura 2008). Primarily Q can be decomposed in terms of nodes activated in a cascade and nodes not activated in a cascade. For the second case of inactive nodes, we will not need any approximation as the likelihood involves log(1 − p M s (v)) which eliminates the product form of Equation 2. For the first case of active nodes, the form is complex because we do not know which active parent was responsible in activating a given node v. This is because, by the definition of IC, activation attempts of all parents of v activated at a given time step are arbitrarily sequenced. Therefore, following (Saito, Nakano, and Kimura 2008), we can instead approximate p M s (v) for this case in terms ofp M u,v p M s (v) for every active parent u sincethe probability that v was activated by u should be proportional to the current estimatep M u,v of the strength of influence of u on v. Therefore, the second term of Q is as follows, where X = C s (t+1)∩Ch(u) and X = Ch(u)\D s (t+1), s∈C M ∈k γ M s T −1 t=0 u∈Cs(t) v∈X p M u,v p M s (v) log p M u,v + 1 −p M u,v p M s (v) log(1 − p M u,v ) + v ∈X log(1 − p M u,v ) Differentiating the above with respect to p M u,v and setting it to zero, and considering P a(v) represents parents of v in base graph G, A u,v is the subset of samples in which both u and v are activated and t s (v) = t s (u) + 1 and B u,v is the subset of samples in which u is activated at some time t and v is not activated up to and including time t + 1 we get, p M u,v = 1 s∈Au,v γ M s + s∈Bu,v γ M s s∈Au,v γ M sp M u,v p M s (v) This completes the derivation of the EM procedure with iterative updates in E and M-steps shown in Alg 1. Relaxation: Since observed cascades only contain the order of activations or time stamps at which users are activated, rather than discrete timesteps, and the edges in G are unobserved; we relax Equation 2 to deal with continuous time and let p M s (v), the probability that v is active in s under component M equal 1− u∈Cs(ts(v)−W ≤τ <ts(v))) (1−p M u,v ) where W is a lookback window and hyperparameter of the algorithm. Thus, any u activated in W before t v (s) is considered a potential parent and influencer (that can activate) v. W can be set in unit of time or in terms of number of past events. Algorithm 1 MIC: Diffusion Mixture Parameter Estimation Input: observed, unlabeled cascades C Output: estimateθ M ,π M , γ M s ; ∀s ∈ C and M ∈ {T, F } 1:π T ,θ T ,θ F ← init ∈ [0, 1];π F ← 1 −π T . 2: while not converged do 3: // E-Step 4: γ M s ←π M P (s;θ M ) k i=1π i P (s;θi) 5: p M s (v) ← 1 − u∈P a(v) ∩ Cs(ts(v)−1) (1 −p M u,v ) 6: // M-step 7:π M ← 1 |C| s∈C γ M s ; 8:p M u,v ← 1 s∈Au,v γ M s + s∈Bu,v γ M s s∈Au,v γ M sp M u,v p M s (v) Experimental Analysis on Real Datasets Using the parameter estimation algorithm, we infer diffusion mixture MIC parameters for the real Twitter datasets described earlier. From the inferred parameters, we evaluate if fake and true cascades are separable based on inferred diffusion dynamics, compared to unsupervised baseline methods for clustering cascades. Next, we identify users that are influential in the propagation of true and fake contents, from the inferred parameters and learned diffusion model; and investigate their characteristic features from the data. Lastly, we demonstrate node and edge interventions based on the inferred diffusion dynamics and show reduction in fake cascade size compared to other baselines. Clustering Cascades From the inferred parameters, we can determine if an observed cascade is more likely to be considered fake or true based on the posterior probability of the cascade under each component of the mixture MIC. The predicted component for each cascade is thus obtained as argmax M γ M s . This will result in two clusters of cascades. Each cluster is assigned fake or true label based on a held out one-fifth set of cascades with known labels. In Table 3, we evaluate if the fake and true cascades in the datasets are separable based on inferred diffusion dynamics, compared to unsupervised baseline methods for clustering cascades. The implemented baselines are as follows -TruthFinder (TF) (Yin, Han, and Philip 2008) is a credibility propagation algorithm that exploits conflicting sentiments between user comments to the same content. StanceEval (SE) exploits the average sentiment of users tweet texts in a cascade as a measure of its type; as it is found that fake cascades tend to elicit negative and questioning responses (Qian et al. 2018;Zhao, Resnick, and Mei 2015). K-Means (KM) clustering based on temporal and propagation features identified in (Ma et al. 2015;Kwon et al. 2013;Castillo, Mendoza, and Poblete 2011) namely, number of posts in a cascade, time length of cascade, average time gap between posts in the cascade, and fraction of most active users in the cascade. SEIZ (SZ) (Jin et al. 2013) is a rumor model proposed for unsupervised rumor detection. It partitions users as either "susceptible", "infected", "exposed" or "skeptic" with regards to the content and models state transitions between them. The model is fit to each cascade separately by solving differential equations. They define a ratio based on the learned parameters of the rumor model for each cascade to classify it as fake or true. Lastly, we include HIC, where we assume a homogeneous IC model, with a single parameter value (f) shared over all edges for the fake component and another single parameter value (t) for the true component i.e. p T uv = t ∀(u, v). Summary: In comparison with MIC where differences in inferred dynamics are exploited for separation; TF and SE utilize aggregate sentiments of user responses which are relatively noisy signals of veracity. We find that KM was biased towards producing a single cluster without being able to effectively separate them; HIC does not model heterogeneous influence across user pairs which limits expressibility of the model; and SZ cannot capture common patterns across cascades, as it fits separate parameters per cascade. Analysis: In terms of the distribution π estimated in MIC, we report the Mean Absolute Error (MAE) between the estimated value and the true data distribution. The data is near balanced, and the estimated π = [0.44, 0.56] in Twitter-1 is close to the true distribution, with Mean Absolute Error (MAE) of 0.04. Twitter-2 estimated π = [0.47, 0.53] with MAE of 0.058. Therefore, MIC outputs balanced clusters of cascade types. KMeans (KM) on the other hand produces unequal sized, biased clusters, resulting in close to random accuracy predicting most cascades to one type, with low f1. The sentiment analysis methods like StanceEval (SE) make mistakes in cases where true content evokes negative sentiments such as "Is horrified to read about the missing Air France plane" and also due to sentiment lexicons that map certain words like "missing" to negative, such as in "Air France jet missing with 228 people over Atlantic after running into thunderstorms". This results in negatively correlated predictions below 50% depending on the sentiment patterns and content in the data. TruthFinder (TF) also utilizes user sentiments but is more robust as it accounts for conflicting relationships between users. SEIZ (SZ) is the better baseline based on rumor modeling. But it does depend on an estimated threshold for the ratio per cascade used to determine if the cascade is fake. (Jin et al. 2013) use median ratio over the set of cascades as the threshold, and any observed ratio above this threshold is considered as fake. This can be result in lower quality estimates of the threshold in datasets like Twitter-1 with fewer cascades. We additionally compare the IC model with MIC. IC model does not have the proposed parameterization for different cascade types, and hence cannot be compared in clustering. Therefore, we report the Average Negative Loglikelihood (NLL) or loss per cascade instead, after parameter estimation using IC and MIC in the datasets. Lower NLL indicates better fit to the observed cascades. Average NLL per cascade on a 20% held-out set of cascades in Twitter-1 for MIC is 9.035 (train cascades 6.29), and for IC, it is 3757.84 (train cascades 58.45). Average NLL per cascade in Twitter-2, for MIC is 8.45 (train cascades 8.35), for IC, it is 1414.25 (train cascades 1052.97). Therefore, separate parameters to represent cascade types allows better diffusion modeling. Influential Users Test In this subsection, we first identify users that are influential in the propagation of true and fake contents (i.e. users that if selected as seed sets would trigger the largest cascades), using the inferred diffusion model. Selecting and analyzing the top 100 influential users identified for fake and true news, we can further evaluate the quality of inferred parameters. Selection of influential users: Influential users for each component IC model with inferred paramsθ T ,θ F , are selected using greedy maximization algorithm implemented based on Goyal, Lu, and Lakshmanan 2011. Result: In Fig 4, we report the box-plot for inferred influential users based on % relative appearance in fake vs true cascades. Inferred users identified for fake news (Inf(F)) have high positive correlation with relative appearance in fake news cascades, as seen from the figure, for both datasets, in comparison with influential true news users (Inf(T)), and a uniform random sample of users (Unif.). Degree of separation: Uniform random sample (Unif.) of users, provides insights into the degree of separation between true and fake cascade clusters in the two datasets. Compared to Twitter-1, uniformly sampled users in Twitter-2 are more likely to engage with both contents, whereas in Twitter-1 the median of the uniformly sampled users interact purely with true contents; showing potentially larger separation in Twitter-1 between fake and true cascade users. Table 4 lists the features of users identified as most influential for true and fake news under estimated parameters; Table 4: Characteristics of most influential users inferred for propagation of fake and true news in Twitter-1. Characteristics of Influential Users Comp # Followers # Following # Posts Description Tags True 54418 1157 24182 HuffPost real life is news, and news is personal. Features of Inf(T): Inferred influential users identified for true news, as seen, correspond largely to accounts of known credible news and opinion websites and blogs. In terms of topic distribution, the dominant types of influential users include accounts disseminating news related to politics, entertainment, infotainment, technology updates, and local news; and tend to have large number of direct followers. Features of Inf(F): Top influential users identified for fake news include accounts with relatively fewer counts of direct followers, compared to those for true news users. For some of these the screen name and description is unavailable from TwitterAPI (reported as 'unk.' in the table). Several of these accounts also do not have a listed description along with their screen name, unlike in the previous case of influential true news users. Therefore, we list the topic of the fake news cascades in which the users appear and their total count of engagements/appearances in the fake cascades. The accounts influential in fake news propagation also appear among a diverse range of topics; similarly dominated by politics, technology, entertainment, and news or trending topics such as SwineFlu and current events. Interestingly, the identified top influential users appear among the larger and more viral fake cascades in the dataset such as ones corresponding to SwineFlu, Obama's citizenship status, LadyGaga's gender identity and technology rumors like launch of Xbox720. BuzzFeed interestingly has been historically linked to unreliable journalism, especially before 2014. It appears in connection with false stories related to man-eating catfish, BigFoot and other viral false stories. Intervention Analysis on Nodes and Edges In Fig 5 we investigate different intervention mechanisms (mechanisms to monitor or intercept the propagation paths of fake news) leveraging the inferred diffusion dynamics, so as to limit the spread of fake news on a network. Node Intervention: In node intervention, we determine which nodes can be monitored, in order to block false contents from spreading in the network. The inferred influential users for fake news identified earlier are chosen candidates for node intervention under MIC, ranked by influence. For offline evaluation of the intervention strategy, we utilize the available fake cascades in the datasets. First, we consider that K users are selected for intervention/monitoring. If a fake news cascade reaches any of the monitored users, it can be intercepted and removed from the network, thereby limiting its future spread. The effectiveness of the interception can be evaluated based on the % reduction in fake cascade size due to the intervention. In Fig 5a and 5b, we evaluate the proposed MIC intervention against the previously considered baselines; and we include an additional baseline TopU that intercepts users ranked by their total engagement count in the set of observed cascades. For the other baselines, the selection of K users is as follows: rank users by their total engagement count in the cascades predicted as fake news cascades by the baseline method. Edge Intervention: In edge intervention, we select K edges in the network in order to intercept the propagation of fake cascades. The edges are ranked by the weight (strength of influence) p F e under the inferred fake component of MIC. These are the identified high transmission paths for fake news cascades and thus removed/blocked. For offline evaluation, we again compare the percentage reduction in fake cascade size due to edge intervention with MIC, against a Random strategy that selects edges uniformly at random from the network for intervention, as shown in Fig 5c and 5d. Here the reduction is calculated over the size of fake cascades simulated over 1000 rounds under the fake component with and without the K edges removed/intercepted for intervention. The simulations are triggered from seeds sampled from users at the head of the sequence of observed fake cascades in the datasets. Conclusion In this work, we proposed a mixture of independent cascade models (MIC) to express and infer the diffusion dynamics of false and legitimate contents. With statistical analysis on real datasets, we confirmed notable differences in user behaviours towards fake and true contents in temporal and structural aspects of diffusion, that can be expressed with MIC. Based on that, we derived an unsupervised inference method for parameter estimation from observed unlabeled cascades, and conducted experiments on Twitter datasets with fake/ true news cascades. The experiments revealed interesting analysis of the characteristics of users identified as influential in true and fake content propagation, under the inferred diffusion dynamics; and their effectiveness towards node and edge interventions to limit fake news. Discussion and Future Work We assumed two sets of parameters θ T , θ F to differentiate fake from true cascades, based on verifying that (i) differences in diffusion patterns of the two types are statistically significant in the datasets, and (ii) the datasets are built from collections of events reported during a specific period, with samples across types collected from the same data source; and no known collection biases across types. In order to account for multiple types (such as satire, differences in political stance, source credibility or content), the mixture model easily generalizes to multiple types of cascades, when k > 2 components are initialized in Algorithm 1 (the derivation is written for the general case k). A limitation of the current work is that it assumes a fixed number of components k, which need not be known a priori. In future work, this can be addressed to adaptively split and merge components starting with a large initial k, while optimizing for likelihood of the cascades. In the experiments on influential users identified based on the inferred diffusion parameters, we find that the inferred set of parameters are correlated with the two types assumed in this work, in terms of engagements with fake and true cascades (Fig 4) and reduction in fake cascade size (Fig 5). However, although the proposed model directly generalizes to k > 2, we consider evaluating the model on multiple types with k unknown a priori for future work with multi-label datasets. The runtime analysis details of the inference algorithm are provided in the Appendix. The runtime scales in the order of O(k|C|V 2 ) which is reduced to O(k|C|V W ) by setting a constant window W smaller than V , where W is the window size described in Relaxation section under Parameter Estimation, V is the number of users, k is the number of components, and C is the set of cascades. This is a limitation of applying the algorithm to large-scale graphs. In future work, we can integrate dimensionality reduction techniques to reduce the number of unique user representations. There are other possible directions of future work. The first is to provide online estimation of parameters for time evolving networks; to allow for changing dynamics due to social bots and fake accounts with manufactured and evolving social connections. A second interesting direction is to leverage diffusion network inference to better understand polarization and existence of echo chambers, and its impact on the spread of misinformation -whether polarization fuels misinformation, and can interventions to mitigate one phenomenon support the other phenomenon. Appendix Proof for Theorem 1. Proof. Each edge (coordinate) j has associated bernoulli variables x i j with parameter p i j for component i in the kcomponent mixture distribution. The pairwise coordinate means then are defined as follows, corr(j, j ) = E[x j x j ] = k i=1 π i p i j p i j , 1 ≤ j < j ≤ m (3) The sample estimate of corr(j, j ) can be obtained directly from the observed live-edge graphs of unlabeled cascades. By the reduction to learning mixtures of discrete product distributions, given the sample estimates of the pairwise coordinate means, the parameters p i j and π can be estimated using algorithm Weights and Means (WAM) (Feldman, O'Donnell, and Servedio 2008) for learning mixture distributions. We restate lemmas in (Feldman, O'Donnell, and Servedio 2008;Chen et al. 2016) used in the proof for completeness, with notations used in the reduction. Lemma 2 ( (Feldman, O'Donnell, and Servedio 2008)). For k = O(1) and any , δ > 0, WAM runs in time poly (m/ ) · log(1/δ ) and outputs a list of poly (m/ ) many candidates, at least one of which (with probability at least 1 − δ ) satisfies the following, |π i − π i | ≤ , ∀i and |p i e − p i e | ≤ , ∀π i ≥ Lemma 3 (Lemma 4 in (Chen et al. 2016)). Given graph G and parameter space ϑ such that ∀θ 1 , θ 2 ∈ ϑ , ||θ 1 − θ 2 || ∞ ≤ 0 , then, ∀S ⊆ V , |σ θ1 (S) − σ θ2 (S)| ≤ mn 0 (a) Runtime analysis of minutes vs. # of active users considered in Twitter-2. Figure 6: Runtime Analysis. Using the above lemmas and setting 0 = mn , δ = δ and = mn , the sample complexity for the desired influence function estimate is obtained. WAM requires sample estimates for E[x j x j ] for all 1 ≤ j < j ≤ m to be within an additive accuracy of matrix = 2 m 2 (k+1) . x j x j ∈ {0, 1} and therefore is Bernoulli distributed with some parameter say p jj equal to E[x j x j ]. Letp jj be the sample estimate for E[x j x j ] calculated from the observed cascades. Since each observed cascade is independently generated, we can compute the sample complexity of estimating E[x j x j ] = p jj within additive accuracy of matrix given the observed cascades. Applying chernoff bounds, we get P (|p jj − p jj | ≥ matrix ) ≤ δ matrix with number of observed samples being at least 2+ matrix 2 matrix ln 2 δmatrix . Applying union bound, we get P (|p jj − p jj | ≥ matrix ) ≤ δ matrix m(m − 1)/2 for all j, j ∈ [m]. Setting δ matrix = 2δ m(m−1) , we get with probability at least 1 − δ, |p jj − p jj | is within additive accuracy of matrix for all j, j and the sample complexity is O ( n 4 m 8 4 ) k+1 ln m δ . Runtime Analysis In Fig 6, the runtime analysis of MIC vs. baseline SEIZ (SZ) on Twitter-2 are provided. The baseline SEIZ is run with time interval of 24hours and cut-off time of 10K hours, and it runs differential equation solvers for each cascade, to fit the data with parameters specific to each cascade. The runtimes are evaluated and compared on Intel(R) Xeon(R) CPU E5-2630 v3 @ 2.40GHz on single thread in python. Multithreading, parallelization is left to future implementations. The runtime analysis is conducted on Twitter-2, since is the larger of the two datasets (with more users and more cascades), so that runtime can be analyzed with respect to different user sizes. We implemented vectorized computations and pre-computed users and cascades needed in the likelihood computation at the start of the EM iterations which improves computational efficiency, and reduces impact of number of cascades on runtime due to vectorization. The EM estimation in Algorithm 1 is trained till convergence, i.e. the change in likelihood is smaller than 0.01. The lookback window W , discussed in Section Relaxation under Parameter Estimation, is set to 10 past events. The value of W impacts computational time and should be set to a constant smaller than V , that is the number of users. In the experiments, we set W with line search in the range {5, 10, 15} based on cross validation for computational efficiency. The EM converges within few iterations. The worst-case runtime complexity per EM iteration is O(k|C|V 2 ) where V is the number of users and C is the set of cascades and k is the number of components, and by setting W to a constant smaller than V , the complexity reduces to O(k|C|V W ). Figure 1 : 1Example of diffusion cascades on Twitter (# tweets per day) for (a) emergency landing of an airliner in Hudson river in 2009 (b) information suggesting that the combination of Coke and Mentos can lead to death in 2006. Figure 2 : 2Diffusion mixture model (MIC). Figure 3 : 3Statistical tests distributions. We define C s (t) as the set of nodes activated at time step t in cascade s and t s (v) as the time of activation of node v in cascade s. Also, we define D s (t) as all activated nodes up to and including time t. Let p M u,v andp M u,v as the actual and estimated edge activation parameter in component M . In addition, we represent γ M s as the posterior probability that cascade s is generated under diffusion component M i.e. γ M s = P (Z s = M ; θ) the probability with which v is activated in cascade s under diffusion component M . By the definition of the IC model, v is activated at time step t s (v) in cascade s iff at least one activation attempt of an active parent of v in s is successful. Therefore, p M s (v) = 1 − u∈P a(v)∩Cs(ts(v)−1) Figure 4 : 4Results on quality of influential users selected based on the estimated diffusion parameters for true and fake news in (a) Twitter-1 and (b) Twitter-2. Inf(T) and Inf(F) are inferred influential users for true and fake news. Figure 5 : 5Intervention Analysis on Twitter-1 and Twitter-2 (a, b) Node Interventions (c, d) Edge Interventions. reported # of followers, posts from 2009 Twitter-1 snapshot. Figure 7 :Figure 8 : 78Parameter Recoverability. Mean absolute error on estimated diffusion mixture model (MIC) parameters; variation with # of cascades at different mixture distributions. Cascade Separability. Clustering accuracy and f1 results with estimated diffusion mixture model (MIC); variation with # of cascades at different mixture distributions. Table 1 : 1Data statistics for Twitter-1 and Twitter-2. hrs and Twitter-2 of 1983 hrs. The datasets contain cascades in the form of time-stamped sequences of user engagements, for example, cascadeDataset Twitter-1 Twitter-2 # Users 117,824 233,719 # Engagements 192,350 529,391 # Fake Cascades 60 498 # True Cascades 51 494 Avg T length per cascade (hr) 8,177 1,983 Avg T interval per cascade (hr) 80 65 Avg # engagements in cascade 1,733 597 Real-World Datasets We utilize two publically available Twitter datasets which we refer to as Twitter-1 (Kwon, Cha, and Jung 2016) 1 and Twitter-2 (Ma et al. 2016) 2 . Twitter-1 was collected dur- ing 2006-2009 and Twitter-2 from March-Dec 2015. In both datasets, topics (contents) are identified as false or legitimate from fact-checking websites like Snopes, and corresponding engagements on Twitter are obtained by keyword search re- lated to the content. The dataset statistics are summarized in Table 1. For analysis we retain all users, and for inference, we retain users that have at least five engagements in the cas- cade set, resulting effectively in 3K and 7K users in the two datasets. The former contains 111 cascades and the later 992 cascades; with Twitter-1 cascades of average time length of 8177 Table 2 : 2Hypothesis testing results (a) Distribution of avg. time delay after log-transform to reduce skewness for t-test in Twitter-1.Temporal Structural t-statistic p-value z-score p-value Twitter-1 4.9975 ≤ .00001 1.87577 .03005 Twitter-2 12.760 ≤ .00001 NA NA 5 0 5 10 Log (Avg Time Delay in Eng.) 0.00 0.05 0.10 0.15 0.20 0.25 PDF Fake True Table 3 : 3Clustering (Cascade Separability) Results.Twitter-1 Twitter-2 F1-Score Accuracy F1-Score Accuracy TF 0.576 0.522 0.573 0.536 SE 0.535 0.531 0.388 0.469 KM 0.253 0.522 0.312 0.490 SZ 0. Table 5 : 5Follower Graph Statistics in Twitter-1. https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi\ %3A10.7910\%2FDVN\%2FBFGAVZ 2 https://www.dropbox.com/s/46r50ctrfa0ur1o/rumdect.zip?dl= 0 Additional Data StatisticsWe provide the follower graph statistics inTable 5available in Twitter-1(Kwon et al. 2013). The follower graph was used for structural diffusion analysis; and is a directed graph between the active users considered in the dataset, which appear at least five times in the cascades set; as described in the Section on real-world datasets. The direction of the edge from A to B indicates that A follows B. The table provides degree distribution and connected components statistics.Experimental Results on Synthetic DatasetsWe construct observed cascades at different mixture distributions π i.e. (0.5, 0.5), (0.2, 0.8) and (0.35, 0.65). on a random graph with 512 nodes and 1024 edges and uniform [0,1] edge probabilities. 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[]
[ "D R A F T Origin of filaments in finite-time in Newtonian and non-Newtonian thin-films", "D R A F T Origin of filaments in finite-time in Newtonian and non-Newtonian thin-films" ]
[ "Saksham Sharma \nDepartment of Chemical Engineering and Biotechnology\nUniversity of Cambridge\nPhilippa Fawcett DriveCB3 0ASCambridgeUK\n", "D Ian Wilson \nDepartment of Chemical Engineering and Biotechnology\nUniversity of Cambridge\nPhilippa Fawcett DriveCB3 0ASCambridgeUK\n" ]
[ "Department of Chemical Engineering and Biotechnology\nUniversity of Cambridge\nPhilippa Fawcett DriveCB3 0ASCambridgeUK", "Department of Chemical Engineering and Biotechnology\nUniversity of Cambridge\nPhilippa Fawcett DriveCB3 0ASCambridgeUK" ]
[ "PNAS" ]
The sticky fluids found in pitcher plant leaf vessels can leave fractallike filaments behind when dewetting from a substrate. To understand the origin of these filaments, we investigate the dynamics of a retreating thin-film of aqueous polyethylene oxide (PEO) solutions which partially wet polydimethyl siloxane (PDMS) substrates. Under certain conditions the retreating film generates regularly-spaced liquid filaments. The early-stage thin-film dynamics of dewetting are investigated to identify a theoretical criterion for liquid filament for-mation. Starting with a linear stability analysis of a Newtonian or simple non-Newtonian (power-law) thin-film, a critical film thickness is identified which depends on the Hamaker constant for the fluidsubstrate pair and the surface tension of the fluid. When the measured film thickness is smaller than this value, the film is unstable and forms filaments as a result of van der Waals forces dominating its behaviour. This critical film-height is compared with experimental measurements of film thickness obtained for receding films of Newtonian (glycerol-water mixtures) and non-Newtonian (PEO) solutions generated on substrates inclined at angles0 • , 30 • , and 60 • to the vertical. The observations of filament and its absence show good agreement with the theory. Further analysis of the former case, involving a stability analysis of the contact line, yields a prediction of the spacing (wavelength)λ f between filaments asλ fη /γ ∝ Ca, whereĈa is the capillary number for contact line motion: our experiments yieldλ fη /γ ∝ Ca 1.08 and earlier studies in the literature reportedλ fη /γ ∝ Ca 0.945 . The evolution of the thin-film shape is modelled numerically to show that the formation of filaments arises because the thin-film equation features a singular solution after a finite-time, hence termed a "finite-time singularity". Thin-film | Filaments | Finite-time singularity | Pitcher plants
null
[ "https://export.arxiv.org/pdf/2304.07902v1.pdf" ]
258,180,307
2304.07902
8b70e85d3a574677e632fe3f89f5e9db84f9716c
D R A F T Origin of filaments in finite-time in Newtonian and non-Newtonian thin-films April 18. 2023 Saksham Sharma Department of Chemical Engineering and Biotechnology University of Cambridge Philippa Fawcett DriveCB3 0ASCambridgeUK D Ian Wilson Department of Chemical Engineering and Biotechnology University of Cambridge Philippa Fawcett DriveCB3 0ASCambridgeUK D R A F T Origin of filaments in finite-time in Newtonian and non-Newtonian thin-films PNAS April 18. 202310.1073/pnas.XXXXXXXXXXThis manuscript was compiled on April 18, 2023arXiv:2304.07902v1 [cond-mat.soft] 2 To whom correspondence should be addressed. The sticky fluids found in pitcher plant leaf vessels can leave fractallike filaments behind when dewetting from a substrate. To understand the origin of these filaments, we investigate the dynamics of a retreating thin-film of aqueous polyethylene oxide (PEO) solutions which partially wet polydimethyl siloxane (PDMS) substrates. Under certain conditions the retreating film generates regularly-spaced liquid filaments. The early-stage thin-film dynamics of dewetting are investigated to identify a theoretical criterion for liquid filament for-mation. Starting with a linear stability analysis of a Newtonian or simple non-Newtonian (power-law) thin-film, a critical film thickness is identified which depends on the Hamaker constant for the fluidsubstrate pair and the surface tension of the fluid. When the measured film thickness is smaller than this value, the film is unstable and forms filaments as a result of van der Waals forces dominating its behaviour. This critical film-height is compared with experimental measurements of film thickness obtained for receding films of Newtonian (glycerol-water mixtures) and non-Newtonian (PEO) solutions generated on substrates inclined at angles0 • , 30 • , and 60 • to the vertical. The observations of filament and its absence show good agreement with the theory. Further analysis of the former case, involving a stability analysis of the contact line, yields a prediction of the spacing (wavelength)λ f between filaments asλ fη /γ ∝ Ca, whereĈa is the capillary number for contact line motion: our experiments yieldλ fη /γ ∝ Ca 1.08 and earlier studies in the literature reportedλ fη /γ ∝ Ca 0.945 . The evolution of the thin-film shape is modelled numerically to show that the formation of filaments arises because the thin-film equation features a singular solution after a finite-time, hence termed a "finite-time singularity". Thin-film | Filaments | Finite-time singularity | Pitcher plants Receding thin films arise in a range of fields and are important in coating, printing and drying applications, where the stability of the thin film near the moving contact line will determine the uniformity of the product. Our interest in this topic arises from observation of residual filaments left by the evaporation of sessile droplets of the sticky digestive fluid secreted by N. Rafflesiana pitcher plants. Fig. 1 shows that these filaments are formed in the early stages of evaporation, where the shrinking drop forms a receding thin film at the contact line, and can be generated artificially by sucking liquid from the drop ( Fig. 1(c)). The filaments exhibit regular spacing on relatively smooth surfaces, indicating that the filaments arise from an instability in the thin film rather than contact line pinning. Pitcher plant fluids are non-Newtonian aqueous solutions containing long-chain polysaccharides (1). Deblais et al. (2) observed similar filaments with thin films of viscous Newtonian and non-Newtonian liquids (glycerine and synthetic polymer solutions, respectively) generated by a blade arrangement which allowed the initial height and contact line velocityÛ to be controlled independently. As the contact line velocityÛ decreased, uniform films with a straight contact line were replaced by ones with regularly spaced cusps and rivulets: for Newtonian liquids the rivulets were unstable and gave rise to droplets whereas the higher extensional viscosity of the polymer solutions stabilised the filaments and gave patterns analogous to those in Fig. 1(c)). They reported that the threshold of cusp (and filament) formation corresponded to a critical value of the capillary number, Ca =ηÛ /γ, wherê η is the apparent viscosity andγ the surface tension, but did not provide a theoretical treatment. In the present work, we provide a theoretical explanation for the observed filament formation. It draws on recent experimental work by Xue & Stone on a liquid film draining down a glass slide under the action of gravity. The thin-film non-linear PDE used to model the film considered three forces: viscous resistance, gravity, and surface tension. A major assumption there was the consideration of perfectly wetting (zero degree contact angle) fluids, arising from the difficulty of incorporating partial wetting in the model, as remarked by co-author Stone when presenting this work at the GKB 100 symposium (3). Partial wetting is included in the present study, following the work by Witelski and coworkers on the stability analysis and evolution of thin-films ((4), (5), (6) and (7)) by employing expressions for the van der Waals forces' dependency on the film thickness. We demonstrate that the formation of cusps and filaments observed during the dewetting of pitcher plant fluids has its roots in the inherent instability of the thin-film. Strictly speaking, this instability gives rise to a 'finite-time singularity'. The article is organised as follows. The hydrodynamic equations for the thin-film are presented in Sec. 1 and stability analysis is then performed to find the critical criterion for filament formation (Sec. 2). The rationale behind the finite-time singularity feature of the thin-film equation is discussed in Sec. 3 along with some numerical investigations. The criterion is compared with experimental results in Sec. 4. A scaling law for the spacing between filaments is derived and compared with experimental data in Sec. 5. The key findings and potential further directions for this work are discussed in Sec. 6. D R A F T D R A F T Theory Consider a thin liquid film of local thicknessĥ(x,ŷ,t) on the plane (Oxy) as shown in Fig. 2(a)), with heightĥ pointing towards Oz. Ox points horizontally along the liquid-substrateair contact line and Oy points directly down the slope. A standard thin-film equation for a non-Newtonian liquid (8) exhibiting power-law behaviour with exponent n is 3η ∂ĥ ∂t = ∂ ∂x m(ĥ) ∂ ∂x (p) − 3ρg ′ĥ2 ∂ĥ ∂ŷ 1/n [1] Hereη is short-form for apparent viscosityη(γ), whereγ is the shear-rate of the thin-film defined at its free surface and given asÛ /h (see Fig. 2(b.ii)),ρ the density,m(ĥ) the mobility coefficient,p is the hydrodynamic pressurep(x,t) of the thin-film, and g ′ = g cos α; hat denotes that the term is dimensional. The boundary condition at the film-substrate interface (on surface Oxy) givesm(ĥ) =ĥ k where k ∈ [1,3]. The pressurep(x,t) in the thin-film (6, see Eq. (1.3)) is given byp =Π(ĥ) −γ ∂ 2ĥ ∂x 2 [2] whereΠ(ĥ) is the disjoining pressure which accounts for the van der Waals interactions between the thin-film and the substrate. The second term on the RHS accounts for the effect of surface tensionγ and curvature of the thin-film. The disjoining pressureΠ(ĥ) can be written in the form Π(ĥ) = h 3 1 −ĥ U T F h [3] where van der Waals forces are characterised by the Hamaker constant ( > 0 means the interaction is hydrophobic and A < 0 hydrophilic) andĥUT F is the height of the adsorbed precursor film in Fig.2(b)) (6). There are two timescales in Eq. (1): i)Tx, when the pressure term (first term on the RHS) is dominant, and ii)Ty, when the gravity term (second term on the RHS) dominates. A scaling analysis gives these timescales asT x = 3ηLx 4 h 3γ ;Ty =ηL ŷ ρg ′h2 [4] withLx =h γĥ 2 U T F / ((see 6, p. 016301-2)), and length scaleLy the length of the glass substrate in the Oy direction This means that early-stage dynamics of the thin-film, when surface tension and van der Waals interactions dominate, are characterised by a time scale of 10 −8 s, compared to the intermediate stage where gravity is important. Similar arguments about time scales have been reported previously (6, p. 016301-2). It means that, as soon as the thin film is deposited on the substrate (or, in these experiments, the substrate is withdrawn from the liquid pool), an interplay between surface tension and van der Waals interaction forces begins. The experiments reported here employed Newtonian solutions (mixtures of glycerol and water) of different viscosity as well as non-Newtonian ones (mixtures of polyethylene oxide, PEO, in water). Scaled viscosity is used to label the liquids when presenting results, given byη/(ργ 3 /g ′ ) 1/4 . The PEO solutions exhibit shear-thinning, which can be described by the Cross model (see Supplementary Material). The shear rate in the experimental films lie in the power law regime for this model (see Supp. Fig. S1), which is why this model is used in Eq. (1). Hence, in the next section, we focus on the early-stage dynamics of the thin-film by ignoring the gravity term in Eq. (1), viz. 3η ∂ĥ ∂t = ∂ ∂x ĥ k ∂ ∂x  h 3 1 −ĥ U T F h −γ ∂ 2ĥ ∂x 2 1/n . [5] Eq. (5) is non-dimensionalised by introducing scaleŝ h =hh,x =Lxx,t =Txt,ĥUT F =hζ [6] where the variables without hats are dimensionless and ζ = hUT F /h < 1. This yields ∂h ∂t = ∂ ∂x h k ∂ ∂x Γ(h) − ∂ 2 h ∂x 2 1/n [7] where Γ(h) is the dimensionless form ofΠ(h) and can be written as Γ(h) = ζ 2 h 3 1 − ζ h [8] and the dimensionless pressure term is p = Γ(h) − ∂ 2 h ∂x 2 [9] which will be used in the next section to simplify the analysis of this PDE. Stability analysis of the steady state solution of the thin-film PDE We investigate the tendency of a thin-film to maintain (or lose its) stability at an early stage. Since the formation of filaments happens almost instantaneously, the phenomenon is primarily linked to the interplay of surface tension and van der Waals interaction forces. A linear stability analysis is performed with respect to arbitrary infinitesimal perturbations. We start by finding a Lyapunov function for Eq. (7) -in effect, modelling it as a dynamic system -because the existence of such a function provides an indication of the nature of stability of the system (9). Consider the following integral I[h] = 1 0 1 2 ∂h ∂x 2 − ς(h) dx [10] where ∂ς(h)/∂h = Γ(h). The reason behind choosing this integral is that its first derivative w.r.t h, has the property ∂I ∂h = ∂ ∂h 1 2 ∂h ∂x 2 − ς(h) = ∂ 2 h ∂x 2 − Γ(h) = −p. [11] D R A F T g F S (b) (d) (a) (i) (ii) (i) (ii) D R A F T Such an integral is referred to as an energy functional of the system. The rate of energy dissipation in a control volume dV for this energy functional is given by ∂I ∂t = ∂I ∂h ∂h ∂t = −p ∇. h 3 ∇p [12] where the derivative is written in terms of operator ∇ to simplify further analysis. Total energy dissipation over volume V is dI dt = V −p ∇. h 3 ∇p dV [13] Integrating by parts gives [14] and applying the divergence theorem gives dI dt = −p V ∇. h 3 ∇p dV − V ∇p ∇.(h 3 ∇p)dVdI dt = −p S h 3 (∇p.n)dS − S h 3 (|∇p| 2 .n) dS. [15] The first term on the RHS is zero because of the boundary condition which assumes no flux of the liquid normal to the boundary, i.e. ∇p.n = 0 on surface dS (the liquid/vapour interface). As a result, the expression for net energy dissipation is dI dt = − S h 3 |∇p| 2 dS [16] which is always ≤ 0 because h ≥ 0. The non-increasing nature of Lyapunov function I[h] confirms the direction of stability of the steady state solution of Eq. (7), which is towards the higher value of h. This approach allows one to find all possible time-independent solutions of Eq. (7) which obey Eq. (16), and these solutions are: (i) h = 0 and (ii) ∇p = 0. The former corresponds to the absence of any fluid on the surface and the latter means that a uniform pressure field exists for the steady-state solution. The steady-state solution with uniform pressurep is selected to proceed further, such that Γ(h) − ∂ 2 h ∂x 2 =p,[17] which has a uniform solution given by h =h andp = Γ(h) = ζ 2 h 3 1 − ζ h > 0. It will be demonstrated in the next Section that such a solution can evolve to approach the first solution, h = 0, for certain parametric conditions. For now, we are interested in finding families of solutions (apart from h = h) of Eq. (7) which can also exist if the original equation is perturbed. To do this, we expand the pressure and the film thickness asp =pc + δ [18a] h(x) =h + ǫh1(x) + ǫ 2 h2(x) + ... [18b] where ǫ, δ << 1 are small perturbation parameters. At O(ǫ), Eq. (17) becomes 3ζ 2 h1 h 4 1 − ζ h + ∂ 2 h1 ∂x 2 = 0 [19] which is a second-order ODE when higher order ǫ 2 terms are ignored. The equation is made closed-form by considering it over a periodic domain, 0 ≤ x ≤ 1, with Neumann boundary conditions: hx(0) = hx(1) = 0. Using the periodic boundary condition allows one to explain the formation of a periodic array of filaments in the Ox direction. Eq. (19) can be written as a harmonic oscillator equation of the form ∂ 2 h1 ∂x 2 + Λ 2 h1 = 0 [20] where Λ 2 = 3ζ 2 [1 − ζ/h]/h 4 . The eigenvalues of this equation are Λi = iπ for i ∈ Z + . Solving forh, one obtains the quintic relationship 3 ζ 2 h 4 1 − ζ h = i 2 π 2 [21] with the approximate solution hi = 3 1/4 ζ iπ [22] The solutionh1 for i = 1 is the primary bifurcation point which demarcates a transition between stable and unstable solutions: above the primary bifurcation point (h >h1), the solution is stable, and below it (h <h1), it is unstable (see 4, p. 159). Higher order bifurcations for i ≥ 2 are not considered because they involve only unstable modes (see 10, Theorem 5). To find a dimensional version of this solution, dimensional analysis of Eq. (5) is required, which yields the scaling relation h ∽γTx/η. UsingTx from Eq. (4) andLx =h 2 ζ γ/Â, the critical height of the thin-film is then h f = 3 −5/16 ζ π −1/8 ζ −1 γ A −1/2 [23] such that ifh <ĥ f , the film is unstable, and ifh >ĥ f , it is stable. It should be noted that this theoretical treatment does not predict the shape or behaviour of the unstable film: it provides a criterion to compare the steady state film thickness with a theoretical limit, above which the film is stable and below which it is unstable. This analysis is independent of the rheology of the thin-film and the shear rate associated with the motion. The physical reasoning behind this is that the formation of filaments happens over a timescale significantly faster (theoretically, eight orders of magnitude) than the timescale of receding motion, and thus, the shear rate or rheology of the thin-film does not play any role in the critical film thicknessĥ f . The reader is referred to the work of Xue and Stone (11) where the evolution of the shape of the thin-film is considered. An unstable filament evolves to yield the steady-state solution h ≈ 0 (as discussed above). The dynamics of this transition are considered in the next Section, where it is shown that the evolution of a thin-film to approach zero film thickness in finite-time is the reason for the origin of the filament. Finite-time filament formation We start with a simplified form of Eq. (7) in this section, where n = 1, to study the bounds or limits of the modified equation which has surface tension and van der Waals force terms. The motivation in the section is to understand the behaviour of the solutions and thereby predict the conditions necessary for the solution to remain bounded or to blowup. A general form of Eq. (7) with n = 1 (see (12)) is D R A F T ∂h ∂t = − ∂ ∂x M (h) ∂ ∂x −Q(h) + R(h) ∂ 2 h ∂x 2 [24] with boundary condition: h(x + L) = h(x). If M (h) = f ( h), R(h) = 1 and dQ(h)/dh = −g(h)/f (h), then the equation becomes ht = −(f (h)hxxx)x − (g(h)hx)x [25] where subscripts denote differentiation. For (25) is equivalent to Eq. (7). It was proved in (13) and when s 2 f (s)/g(s) → ∞, as s → ∞, the classical-solution is uniformly bounded and positive. In the current work, i.e. Eq. (7) with n = 1, s 2 f (s)/g(s) = s 6+k /(3ζ 2 (1 − 4ζ/3s)), which grows to ∞ as s → ∞. Hence, the classical-positive solution of Eq. (7) (for n = 1) is bounded and finite. Having established the bounds of the equation, we next explore how the solution evolves over time. f (h) = h k (1 ≤ k ≤ 3) and g(h) = 3 ζ 2 h 4 (1 − 4ζ 3h ), Eq. Eq. (5) states that the height of the film h is determined by two forces: van der Waals interactions and surface tension. For wetting liquids the former is attractive in nature and brings the fluid in contact with the substrate; being inversely proportional to h, it increases in magnitude as h → 0. The latter, on the other hand, tries to minimise the surface energy of the fluid by reducing the area of the liquid-vapour interface; it prefers a non-planar film to a planar one. It is expected that the combination of these will lead to the presence of both features, namely reduced film-height and non-planar film-geometry, in the system. To focus on the highly nonlinear terms (van der Waals and surface tension forces) which grow very fast, we takeĥ U T F = 0 (as the order ofĥ U T F is signficantly lower than h) in Eq. (5), so that only the first order term in van der Waals interaction and the surface tension is compared and evaluated in the numerical study. The numerical solution is evaluated using Wolfram Mathematica. A finite domain with width (in the x direction) 2L d is considered with the boundary condition h(−L d , t) = h(L d , t) and the initial condition h(x, 0) =h − δ cos πx L d such thath = 1 is the average film height and δ = 0.1 is the amplitude of perturbation (0.1 being chosen to ease graphical visualisation). As derived previously in (7, see Eq. (3.3)), there is a critical thickness L d /π (following a similar stability analysis to that in the previous Section) below which the uniform film is unstable and above which the uniform film is stable to small perturbations. Fig. 3(a) shows the former case, for L d = 4.34, > π, where the film-height h approaches 0 at t ≈ 4.79 (in finite-time). Near this time, the Mathematica software suspects a singularity to exist, hence it is not possible to find the solution at longer times. Fig. 3(b) shows an example of the latter case, for L d = 2.94, where no such singularity is found even at larger times; the initial sinusoidal perturbation decays to the average film-heighth = 1, suggesting that the film remains stable. Results In the experiments a receding liquid film is generated on a static coated glass slide by moving the liquid reservoir downwards. The film thickness is measured and the initial behaviour of the receding film is recorded as outlined in Fig. 2(b). Fig. 4 plots the ratioh/ĥ f , which determines whether the film will remain stable or not, against the dimensionless viscosity, η scaled (scaled with (ργ 3 /g sin α) 1/4 ) for glycerol-water mixtures and aqueous PEO solutions. Fig. 4(a) shows good agreement with the stability criterion for the Newtonian liquids. Ifh/ĥ f < 1, the film becomes unstable because the thinner film gives rise to larger van der Waals forces. A more viscous liquid, i.e. larger η scaled , results in a larger film thicknessh (measured experimentally -see Supplementary Tables S3-S5), whileĥ f does not depend on viscosity: less viscous Newtonian films are more likley to be unstable. An unstable film touches the surface to create a dry spot which grows in the region (7, see Fig.5) and gives rise to satellite beads. Satellite beads have been reported previously, e.g. (14), where drops sliding down on a partially wetting substrate emitted smaller drops from their cusped tail. For the non-Newtonian case, Fig. 4(b), whenh/ĥ f > 1, the thin-film again remains stable and does not yield a singularity. However, forh/ĥ f < 1, the thin-film yields a singular solution and filaments are observed. Experimentally observed zones of filament formation and non-formation are predicted very well by the stability theory. The crossover between the zones occurs at different η scaled values for the two fluid types: elucidation of this aspect requires further work, as does testing on pitcher fluids. D R A F T It is to be noted that the current theory has not been tested for the pitcher fluids. The critical requirements to perform these experiments on pitcher fluids or any other biological fluids, are as follows: (i) ample amount of fluid should be available (approximately 3-5 mL) for enough number of tests across a wide range of parameters; (ii) measurements of wetting and dispersive properties of the fluid. We invite future studies in this direction, wherein, the experts of pitcher fluids and that of thin-film physics could join forces to further test this theory against pitcher fluids, belonging to different species and locations. Stability analysis of moving contact line In previous sections, the condition for formation of droplets or filaments was determined using stability analysis of a young thin-film. We now seek to determine how the spacing between such filaments depends on the contact line velocity. The theory developed in (15) is used to derive the scaling-relation for the size and spacing of filaments, which is then compared with our experiments and experimental data reported in the literature. The dimensionless lubrication equation for a moving contact line of a thin-film of height h(x, y, t) with slip at the fluid-solid interface is given by (16) ∂th + ∇.(h U) = 0 [27a] ∇κ + ey + 3(−Ca ey − U) h(h + 3ls) = 0 [27b] where U(x, y, t) = Uxex + Uyey is the depth-averaged fluid velocity inside the film, ls is the slip length to ensure Navierslip along the substrate, Ca is the capillary number given bŷ η(γ)Û /γ,Û is the contact line velocity, and ∇ = ex∂x + ey∂y. All the terms are dimensionless. A linear stability analysis of Eq. (27) yields h(x, y, t) = h0(y) + ǫh;1 (y)e −σt+iqx [28a] κ(x, y, t) = κ0(y) + ǫκ1(y)e −σt+iqx [28b] where κ is two times the mean curvature of the interface (nondimensionalised by 1/h). h0(z) and κ0(z) denote the shape of the film as a result of balance between capillary forces and gravity;σ is the growth rate of perturbations;q is the wavenumber of the disturbance in the x direction ( Fig. 5(a)). The detailed stability analysis of Eq. (28) is given in the Appendix of (15) where θr is the receding contact angle of the thin film, estimated using the Cox-Voinov law (17), θ 3 r = θ 3 e − 9 Ca ln(y/ls) [30] where θe is the equilibrium contact angle, y is the distance from the contact line ( Fig. 5(a)) and ls is the slip length. with the first term on the RHS given by 6πγ σT Yη Ca = 6π σT YÛ . This predicts thatλ is dependent on the first power ofÛ for a given growth rateσ. It should be noted that growth rate in Eq. (29) is positive, because receding angle θr increases with Ca (15). Hence, any small perturbation ǫ on h will eventually die with a relaxation rateσ, which can be assumed to be constant for a given disturbance ǫ. Interestingly, a similar kind of stability analysis is performed while deriving the growth rate of a Plateau-Rayleigh instability; the major difference being that the wavelength of the most unstable mode is analytically derived in that case, because of imaginary values of growth rateσ. In the present case, we have shown that wavelengthλ is dependent on the first power ofÛ . In §3, it was shown that the thin-film, with a thickness lower than the critical film thicknessĥ f , can break apart, self-similarly, to form filaments. The origin of filament formation is then followed by the motion of contact line and formation of fluctuation-induced crests and troughs of wavelengthλ (Fig. 5(a)). Filaments are experimentally observed to have a regular spacing between them, given byλ f . We now compare the above theory with: (i) experimental data collected using PEO solutions in the setup in Figure 2, and (ii) the experimental data reported for aqueous polyacrylamide solutions in (2). For the former, the spacing between filamentsλ f varies withÛ as: λ f ∝Û 0.72 (Fig. 5(b)) , whereas, for the latter, it varies asλ f ∝Û 0.80 (Fig. 5(c)). In terms of Ca, for the former (i.e. our experiments), plot ofλ fη /γ versus Ca shows the following relation:λ fη /γ ∝ Ca 1.08 in our experiments and λ fη /γ ∝ Ca 0.945 in the literature. There is therefore good agreement between the experiments reported here, those of Deblais et al. (2), result obtained here from stability analysis. Discussion and conclusions The ability to predict regimes of filament/drop formation by a thin-film is likely the result of simplifying the geometry of this problem and focusing on one single question: what drives the formation of these thin-film instabilities? The answer to this question inherently lies in the finite-time singularity feature of thin-film PDE which is a 4th order parabolic degenerate PDE. The stability analysis of this equation allows us to find an analytical criterion to predict the conditions which makes the thin-film unstable. Once it happens, the thin-film dynamically evolves to touch the substrate and creates a dry spot region which grows in size and divides the film into multiple filaments. The spacing between filaments is explained by performing stability analysis on the moving D R A F T contact line, to find out howλ f depends onÛ . After numerically indicating the finite-time rupture feature of thin-film using sinusoidal perturbation. By applying sinusoidal perturbation to the initial film height in the thin-film equation, the numerical results show that the thin-film tends towards a rupture (h = 0) in a finite-time. After this, an important question to consider is whether this rupture occurs for any arbitrary perturbation, which is smooth and belongs to C ∞ class of function (i.e., can be differentiated infinite times). Pinching-singularities for different thin-film equations have been proposed previously, but only some have been proven to date. One recent example is the Hele-Shaw case, with f (h) = h, g(h) = 0 in Eq. (25), for which it was numerically shown by (19) that h → 0 in finite-time, and was proved in 2018 by (20) that the thin-film height h reaches a minimum after a finite-time, which is zero. In the similar spirit, the following question remains unsolved: • For Eq. (25) with f (h) = h 3 and g(h) = h −1 , prove that for all profiles of initial data, the thin-film height h after a finite-time reaches a minima, i.e. zero, giving rise to finite-time singularity. One possible way to prove finite-time singularity for the above problem is to embed this form of Eq. (25) (hereafter, referred to as the filament equation) on a Turing machine. The techniques on how to do this for Euler equations were reported recently (21). If the same can be done for the filament equation, then it means that for a general initial perturbation, the solution to the equation is computable or, in other words, Turing-complete. A Turing-complete solution can be used to program the fluid flow according to the requirement of the problem, as mentioned in (22). The physical interpretation of a Turing-complete flow is that the path of particles are undecidable * in a sense that nothing a priori can be said about their future trajectory; this feature is also known as universality of thin-film equation. Dimensional analysis of the filament equation (4) shows that the solution behaves in a self-similar way, because of its scalar-invariant nature. Combining the university feature with the scalar-invariant feature, the computable solution of the thin-film equation should evolve in a self-similar fashion for an arbitrary initial datum. The height of the thin-film reduces over time, following the trend shown in Supplementary Figure S4, and then it eventually blows up in a finite-time t f at height h = 0, for any arbitrary initial datum. The physical manifestation of this singularity is in the origin of the filament formation. Detailed discussion of this point is given in Supplementary Note 2. D R A F T as shown in the schematic, Fig. 2(d). The surface tensionγ of the liquids was measured using pendant droplet tensiometry (see (26) (25 mm in our tests). Withh = O(10 −4 ) m, = O(10 −17 ) J,ĥUT F = O(10 −9 ) m andγ = O(10 −2 ) N/m, this giveŝ Tx = O(10 −8 ) s andTy = O(1) so the first term on the RHS in Eq. (1) is expected to be dominant. Fig. 2 . 2(a) Schematic of experimental setup showing the coated glass microscope slide on which a thin-film dewets as it drains downwards. Cameras F (xiQ USB3.0) and S (acA1300-200um) record the dewetting process from the front and side, respectively. Cartesian co-ordinates indicated by axes. (b) Side view of the receding film -(i) schematic and (ii) camera image -with receding contact angle θr, film heightsĥ U T F andh (scale: width of glass slide is 1 mm shown by the shaded grey region). (c) Microscope images of the substrate (i) with and (ii) without filaments during dewetting of PEO solution. (d) Front (camera F ) view of substrate (i) with and (ii) without droplet formation during dewetting of aqueous glycerol solution. Fig. 3 . 3Numerical solution of Eq. (5) (forĥ U T F = 0), over domain −L d ≤ x ≤ L d . (a) For L d = 4.34, heighth approaches zero at time t f ≈ 4.79 (finite-time singularity) after which the dry region grows in the direction of the arrows, and the film splits to form two filaments (not shown). (b) For L d = 2.94, height h remains stable with time and does not tend towards zero. Fig. 4 . 4Comparison of experimental results with theory: effect of dimensionless viscosity, ηscaled, onh/ĥ f for (a) glycerol-water mixtures, and (b) PEO solutions. Solid symbols denote unstable film, forming singularities in the form of drops or filaments; hollow symbols denote a stable film. Symbol colour indicates the angle of plate inclination: black for α = 0 • ; blue for α = 30 • ; red for α = 60 • . Here,h is the experimentally measured film thickness andĥ f is the theoretical critical film thickness. Fig. 5 . 5d(y/ls) dCa ∝TY because ls is dependent only on molecular factors (self-diffusion coefficient and molecular size)(18).Since θr is small, tan θr ≈ θr and Eq.(29) becomeŝ σ = −|q|γ η θr 1 − θ 2 r /2 Comparison of the experimental data from literature and experiments in the current work on filament spacing,λ f with predictions. (a) Schematic of a moving contact line with disturbance wavelengthλ in the x direction. g ′ is the component of gravity acting along the plane of the substrate; (b) Effect of receding contact line velocityÛ on spacing between filamentsλ f for experiments performed by us and reported in the literature (2); (c) Effect of capillary number Ca onλ fη γ for PEO solutions; symbol colour indicates angle of inclination: black for α = 0 • ; blue for α = 30 • ; red for α = 60 • . Star symbols denote the experimental values reported in the literature (2). Fig. 1. Filaments formed by overnight evaporation of ground pitcher N. Rafflesiana fluid on (a) polystyrene and (b) borosilicate glass surfaces. (c) Forced shrinkage of a sessile drop of N. Rafflesiana on borosilicate glass, caused by withdrawal of liquid via pipette labelled P. Evaporation in (a) and (b) results in concentration of dissolved species in the drop, slowing evaporation and triggering the transition from a receding film to a pinned state and 'coffee ring effect' features. Liquid removal in (c) is faster and there is little evaporation: filaments are observed once the contact line starts to recede.1 mm t=0 s t=5 s t=7 s t=10 s t=12 s t=16 s (a) (c) (b) P P P P P P P 300 µm Table S1 . S1). The values obtained are reported in Supplementary Author Affiliations. Department of Chemical Engineering and Biotechnology, University of Cambridge, Philippa Fawcett Drive, Cambridge CB3 0AS, UK To whom correspondence should be addressed. E-mail: [email protected] www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | April 18, 2023 | vol. XXX | no. XX | 1-10 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Sharma & Wilson | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Sharma & Wilson | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Sharma & Wilson | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Sharma & Wilson Sharma & Wilson PNAS | April 18, 2023 | vol. XXX | no. XX | 9 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Sharma & Wilson Bibliography.Apart from the theoretical-computational treatment as the next step in this work, it is also possible to solve Eq.(25)using an operator-learner based PDE solver. A physicsinformed neural network (PINN) technique was recently employed to find a smooth self-similar solution (scalar invariant) for Boussinesq equations (waves of wavelength λ in a shallow water pool of depth d; λ ≫ d)(23). Whilst the PINN technique can be used to solve Eq.(25)numerically, the problem of dealing with a singularity requires a resolution-invariant * An undecidable problem is a problem for which it is proved to be impossible construct a function/algorithm that always yield a correct 'yes' or 'no' answer.tool. Attention-based operator learners are reported to have this capability, hence any blowup of the solution, if it occurs, will not be dependent on the resolution of the software(24). This is essential if one needs to use a numerically-computed self-similar solution of thin-film PDE, which blows up in a resolution-invariant way, to devise a computer-assisted proof of finite-time singularity formation from an initial datum (25).Materials and MethodsA. Experimental setup. The experimental setup consists of a coated glass slide, placed on a stand as shown inFig. 2(a). A bath consisting of the test liquid is raised vertically upwards to wet the initially dry glass slide, and then brought downwards rapidly to its initial position to generate a thin-film which drains downwards under the action of gravity g. Camera F is used to capture the thin-film in the xy plane, and side camera S is used to capture the macroscopic receding contact angle θr of the thin-film in the yz plane. Movies of the receding film are recorded using a Ximea xiQ USB3.0 (F ) camera at 17 fps and a Basler acA1300-200um USB3.0 (S) camera at 50 fps. Subsequent image analysis is performed using ImageJ software. All experiments are performed at a lab temperature of 21 • C.Borosilicate glass slides with dimensions 25 × 75 mm 2 and thickness 1 mm were coated with PDMS (polydimethylsiloxane) solution. The PDMS solution was prepared by dissolving Sylgard TM 184 silicone elastomer and curing agent in the weight ratio 10:1. Slides were prepared using the following protocol: cleansing with surface active cleaning agent (Decon ® 90 at 2%), rinsing with pure water, drying with an air gun, and further cleaning using an air plasma cleaner. PDMS solution was then coated on the glass slide by spin-coating at 3000 rpm for 30 s.B. Solution preparation. PEO solutions (3500 to 11000 ppm) were prepared by adding the required amount of PEO powder (mol. wt. 8×10 6 , Acros Organics) to deionised water at room temperature and stirred at 800 rpm for 48 hours to obtain a uniform solution. Glycerol-water mixtures (80-100 % v/v) were stirred for shorter periods.C. Material characterisation. This section describes the characterisation of the liquids employed in the current work. These can be broadly classified as (i) Newtonian, glycerolwater mixtures, (ii) non-Newtonian, aqueous PEO solutions. C.1. Shear rheology. The shear rheology of the liquids was studied using a Kinexus controlled shear rheometer (Malvern Instruments, UK) fitted with a Couette cell. The apparent viscosity, ηapp, was measured using controlled stress ramps. SupplementaryFig. S1(a)shows that the glycerol-water mixtures were Newtonian. The PEO solutions exhibit shear-thinning behaviour, particularly in the experimentally-relevant shear rate range of 0.1 − 60 s −1 .Supplementary Fig. S1(b)shows the fit of the PEO data to the Crosss model,η −η∞ η 0 −η∞ = 1 1+(λcγ) m . The model parameters are reported inTable S1and the values ofη0 and λc for different PEO concentrations are plotted inSupplementary Fig. 2.C.2. Receding contact angle and surface tension.The receding angle θr of the thin-film was measured using the side camera S, Deadly glue-adhesive traps of carnivorous plants. W Adlassnig, M Lendl, Peroutka, Lang, SpringerW Adlassnig, T Lendl, M Peroutka, I Lang, Deadly glue-adhesive traps of carnivorous plants. 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[ "Bianchi-I cosmology with generalized Chaplygin gas and periodic deceleration parameter", "Bianchi-I cosmology with generalized Chaplygin gas and periodic deceleration parameter" ]
[ "N Hulke \nDepartment of Applied Mathematics\nShri Ramdeobaba College of Engineering and Management\nNagpurIndia\n", "G P Singh \nDepartment of Mathematics\nVisvesvaraya National Institute of Technology\nNagpur-440010India\n", "Binaya K Bishi \nDepartment of Mathematical Sciences\nUniversity of Zululand\nPrivate Bag X1001, Kwa-Dlangezwa 3886South Africa\n\nDepartment of Mathematics\nLovely Professional University\nPunjab-144401Phagwara, JalandharIndia\n" ]
[ "Department of Applied Mathematics\nShri Ramdeobaba College of Engineering and Management\nNagpurIndia", "Department of Mathematics\nVisvesvaraya National Institute of Technology\nNagpur-440010India", "Department of Mathematical Sciences\nUniversity of Zululand\nPrivate Bag X1001, Kwa-Dlangezwa 3886South Africa", "Department of Mathematics\nLovely Professional University\nPunjab-144401Phagwara, JalandharIndia" ]
[]
We investigate the Bianchi-I cosmological model in presence of generalized Chaplygin gas (GCG), variable gravitational and cosmological constant. The exact solutions of Einstein field equations are obtained with time varying periodic deceleration parameter. The graphical representation method has been used to discuss the physical and dynamical behaviour of the model. Further, the stability and physical acceptability of the obtained solutions have been investigated. Most of the parameters shows periodic behaviour in this study due to the presence of cosine function in the deceleration parameter. In all the cases, pressure is negative, which leads us to late time expansion of the universe. The considered models are found to be stable.
10.1007/s12648-022-02315-1
[ "https://arxiv.org/pdf/2009.00460v3.pdf" ]
246,706,428
2009.00460
af0bd7806f13b5a23704bc0431da1dcb19367cc0
Bianchi-I cosmology with generalized Chaplygin gas and periodic deceleration parameter February 11, 2022 N Hulke Department of Applied Mathematics Shri Ramdeobaba College of Engineering and Management NagpurIndia G P Singh Department of Mathematics Visvesvaraya National Institute of Technology Nagpur-440010India Binaya K Bishi Department of Mathematical Sciences University of Zululand Private Bag X1001, Kwa-Dlangezwa 3886South Africa Department of Mathematics Lovely Professional University Punjab-144401Phagwara, JalandharIndia Bianchi-I cosmology with generalized Chaplygin gas and periodic deceleration parameter February 11, 2022Bianchi-Iperiodic deceleration parameterdark energycosmological constant We investigate the Bianchi-I cosmological model in presence of generalized Chaplygin gas (GCG), variable gravitational and cosmological constant. The exact solutions of Einstein field equations are obtained with time varying periodic deceleration parameter. The graphical representation method has been used to discuss the physical and dynamical behaviour of the model. Further, the stability and physical acceptability of the obtained solutions have been investigated. Most of the parameters shows periodic behaviour in this study due to the presence of cosine function in the deceleration parameter. In all the cases, pressure is negative, which leads us to late time expansion of the universe. The considered models are found to be stable. Introduction Cosmological and astronomical data [1][2][3][4][5][6][7] reveals that the universe is currently undergoing accelerating expansion and it has been originated with a bang from phase of very high density and temperature. For a long time, it was believed that either the universe will expand eternally or the inward pull of gravity, gradually slows down the expansion of the universe and would ultimately came to a halt and contract into a big crunch singularity. At the end of twentieth century, it has been discovered that the universe might be expanding with acceleration. The fact of accelerating universe surprised the cosmologist as the idea of cosmic acceleration was against the standard predictions of decelerating expansion caused by gravity. The universe expansion is accelerating due to some exotic stuff termed as 'Dark energy' (DE) having highly negative pressure. The successive disclosure in this direction gave more and more documentation for a flat, dark energy dominated accelerating universe [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. However, by modifying the theory of gravity one can find the alternative way to explain the acceleration of expanding universe [26]. In General theory of relativity (GTR), cosmological constant Λ had been introduced by Einstein, which now is one of the feasible candidates of dark energy [27][28][29]. The homogeneous and isotropic model with cosmological constant in general relativistic framework is also known as Lambda cold dark matter (ΛCDM) model. However, the ΛCDM model faces some serious issues [8,30] such as the fine tuning (a typical small value), cosmic coincidence problems (although the universe is in accelerated phase of expansion, why the dark matter and the dark energy are of the same order?). To resolve these issues, various DE models with quintessence, k-essence, phantom, tachyon and so on [26], were analysed in the literature. In spite of these attempts, the perfect DE model is still lacking [26]. Chaplygin gas (CG) [31] with equation of state (EoS) p = − B ρ [32] where, ρ and p are energy density and pressure respectively, is one of the most explored candidate of DE. Among the various class of dark energy models, CG is a simple characterisation in order to understand the cosmic acceleration and yields an unified scenario for dark energy and dark matter (DM) [9][10][11][12][13]. The remarkable feature of this unified model is that, at early times the CG behaves as a dust-like matter and at late times it behaves like cosmological constant. CG also plays a fascinating part in holography [33] and string theory [34] inspired models. CG depicts a transition to the present cosmic acceleration from a decelerated cosmic expansion and conceivably submit a deformation of ΛCDM model. CG model fails the tests connected with structure formation and observed strong oscillations of matter power spectrum [35]. To overcome this failure of CG model, the generalised Chaplygin gas (GCG) with EoS p = − B ρ α with 0 α 1 has been proposed [9,10]. GCG model also acts like CG model i.e., at early times GCG mimic a dust-like matter and at late times it mimic a cosmological constant. With the aim of describing the unification of DE and DM, Zhang et al. [11] have proposed another version of GCG with EoS p = − A(R) ρ α (R is a scale factor) termed as new generalised Chaplygin gas (NGCG). By using various combinations of latest observational data sample including SNIa and CMB, Salahedin et al. [12] have analysed for the constraints on free parameters of NGCG model by using the statistical Markov Chain Monte Carlo method. Mamon et al. [13] have studied an extended GCG model with EoS of the NGCG and found that the NGCG model is consistent with gravitational thermodynamics. In literature, the GCG model is again modified as modified generalised Chaplygin gas (MGCG) model with EoS p = µρ − B ρ α , where µ is a positive constant [11]. This modification of GCG to MGCG has been done because the inferences from GCG models are almost similar to the CDM models [14]. In general, above examples of cosmic fluid may written as p = f (ρ) and we term these formulations as barotropic fluid. Variable Chaplygin gas (VCG) p = − Ba −n ρ is one of the form among the class of modified forms of Chaplygin gas. A flat Friedmann-Lemaitre-Robertson-walker (FLRW) cosmological model with perfect fluid comprising of VCG yields ΛCDM model like behaviour at late-times [15]. In anisotropic Bianchi-I background, MCG equation of state and the barotropic fluid model satisfying general linear EoS may also characterize different phase of the universe and approach to ΛCDM model at late-times [16,19]. Non-linear electrodynamics model may induce barotropic fluid like scenario and results into varying deceleration parameter which is decreasing function of time [20]. In cosmological modelling, explanation of current accelerating stage and transition from decelerating past to accelerating stage of universe evolution are important and interesting ingredients. In order to explain the phase transition of the universe from decelerating to accelerating phase, one may also use varying deceleration parameter for anisotropic or isotropic universe model [17,18]. Deceleration parameter denotes the rate at which the universe expansion is slowing down. The model of oscillating universe with quintom matter in FRW framework illustrate that the universe undergoes decelerating and accelerating expansion alternately and Hubble parameter oscillates and keeps positive for the time periodic varying deceleration parameter (TPVDP) [21]. The field equations of Barber's second self-creation theory with TPVDP yield the values of state finder parameters r and s into state-finder parameters of standard ΛCDM model r = 1 and s = 0 for parameter m = 3 2 [36]. TPVDP model with string cloud in context of f (R, T ) gravity yield vanishing string tension density in the considered universe scenario [37]. In the f (R, T ) gravity model, the cosmological solutions with TPVDP have been obtained for the flat homogenous and isotropic space-time and these solutions are consistent with observations for different values of model parameters [38]. Extension of FRW results in LRS Bianchi-I space-time with f (R, T ) gravity yields the evolution of universe from decelerated phase to super exponential expansion [39]. The late time acceleration of the universe may be caused in the presence of negative cosmological constant as a non conventional mechanism with TPVDP [40]. Deceleration parameter has been used in modified gravity theories to investigate the cosmological implications along with other measurements in isotropic as well as anisotropic backgrounds [41][42][43][44][45]. The current accelerating expansion phase of universe may be explained in GTR framework with positive cosmological constant. The presence of cosmological constant in models may also affect the violation of energy conditions [46,47]. In GTR framework, the Einstein field equations with the cosmological constant using variable redshift and shape functions have been solved for wormhole solutions [46]. The spherical regions for the wormhole may also satisfy the energy conditions for positive cosmological constant [46]. Energy conditions are coordinate invariant restrictions on energy momentum tensor and force various linear combinations of energy density and pressure of model to be positive [46][47][48][49][50][51]. The energy conditions have been used in literature, to derive many theorems such as the singularity theorems, black hole area increase theorem as well as the positive mass theorem [48][49][50]. The introduction of anisotropies in the cosmological modelling may give rise to a richer dynamical structure, yet the model remains simple enough to provide numerical and/or analytical results. These anisotropic models allow us to analyse the problems regarding behaviour of models on the approach to space-time singularities like why do the present day universe appear highly isotropic, the effects of anisotropy on astronomical observables, etc [16,19,26,39,51]. By considering the features of periodic deceleration parameter, role of Chaplygin gas and its modifications in cosmological modelling, it is worthwhile to investigate the dark energy cosmological model with time periodic varying deceleration parameter in Bianchi-I geometrical background. The paper is organized as follows: In section-2, we write the field equations of general relativity with Bianchi-I space-time metric. In section-3, we present the cosmological solutions by assuming the time periodic varying deceleration parameter and generalised Chaplygin gas. In section-4, we investigate the stability and acceptability of the solution using classical stability criterion, using the square sound speed and energy conditions. In section-5, we summarize our obtained result and concluding remarks. Field equations We take Bianchi-I space-time metric as, ds 2 = dt 2 − R 2 1 (t)dx 2 + R 2 2 (t)dy 2 + R 2 3 (t)dz 2 ,(1) where R 1 , R 2 , R 3 are directional scale factors. Einstein's field equations with gravitational and cosmological constant for perfect fluid distribution are given as R ij − 1 2 Rg ij = −8πGT ij + Λg ij ,(2) where R ij , g ij , R, G(t) and Λ(t) are Ricci tensor, metric tensor, Ricci scalar, gravitational constant and cosmological constant respectively. T ij is energy momentum tensor and is given as T ij = (ρ + p)u i u j − pg ij ,(3) where ρ is energy density, p is perfect fluid pressure and u i represents the four velocity vector such that u i u i = −1. For the Bianchi-I space time metric represented by equation (1), the Einstein's field equation (2) yields the following equationsR 2 R 2 +R 3 R 3 +Ṙ 2 R 2Ṙ 3 R 3 = −8πGp + Λ,(4)R 1 R 1 +R 3 R 3 +Ṙ 1 R 1Ṙ 3 R 3 = −8πGp + Λ,(5)R 1 R 1 +R 2 R 2 +Ṙ 1 R 1Ṙ 2 R 2 = −8πGp + Λ,(6)R 1 R 1Ṙ 2 R 2 +Ṙ 2 R 2Ṙ 3 R 3 +Ṙ 3 R 3Ṙ 1 R 1 = 8πGρ + Λ,(7) where overhead dot denotes derivative with respect to cosmic time t. From equations (4)-(7) one can easily obtainρ + (ρ + p) Ṙ 1 R 1 +Ṙ 2 R 2 +Ṙ 3 R 3 + ρĠ G +Λ 8πG = 0.(8) The energy momentum conservation equation (T ij ;j = 0) suggestṡ ρ + (ρ + p) Ṙ 1 R 1 +Ṙ 2 R 2 +Ṙ 3 R 3 = 0.(9) From equations (8) and (9) we have ρĠ G +Λ 8πG = 0.(10) Further, equations (4)-(6) yield the solutions R i = c i Re k i 3 1 R 3 dt ,(11) where k i and c i (i = 1, 2, 3) are constants which satisfies 3 i=1 k i = 0 and 3 i=1 c i = 1. From the above results, one can notice that the metric potential can be explicitly expressed in terms of scale factor R = (R 1 R 2 R 3 ) 1 3 represented by Bianchi-I space-time. Cosmological Solutions There are four linearly independent equations (4)-(7) with six unknowns in the form of R 1 , R 2 , R 3 , ρ, G and Λ. Hence to solve the system of equations completely, two additional physically plausible relations among these variables are required. To obtain the cosmological solution we considered the time periodically varying deceleration parameter (TPVDP) of the form [21] q = m cos(nt) − 1,(12) where m and n are positive constants. This type of deceleration parameter is known as TPVDP. The deceleration parameter play a crucial role in determining the nature of the constructed models of the universe i.e. decelerating or accelerating in nature. According to the value/ranges of q the universe exhibits the expansion in the following ways [15,53]: • q > 0 : Decelerating expansion • q = 0 : Expansion with constant rate • −1 < q < 0 : Accelerating power law expansion • q = −1 : Exponential expansion/de Sitter expansion • q < −1 : Super exponential expansion From the considered form of q in equation (12), the deceleration parameter shows periodic nature due to the presence of cos(nt). The deceleration parameter lies in the interval −(m + 1) ≤ q ≤ m − 1. Here we observed that: 1. For m = 0, the deceleration parameter q is equal to −1 and the universe exhibits exponential expansion/de Sitter expansion. 2. For m ∈ (0, 1), the deceleration parameter q becomes negative and leads to accelerated expansion in a periodic way. 3. For m = 1, q lies in the interval [−2, 0], this shows that the universe evolves from expansion with constant rate to super exponential expansion in a periodic way followed by accelerating power law expansion to de Sitter expansion. 4. For m > 1, phase transition takes place from decelerating phase to accelerating phase in a periodic way where the universe starts with a decelerating expansion and evolves to super exponential expansion. The constraints on generlised deceleration parameter from cosmic chronometers were investigated and specifies the range as q 0 = −0.53 +0.17 −0.13 [52]. In the study of particle creation mechnism in higher derivative theory, Singh et al. [53] examine the values of free parameters using the observatinal range of q. The present observational limit of the considered deceleration parameter q 0 = −0.53 +0. 17 −0.13 [52], suggests the following values of m and n, which is given in Table 1. (2) represents the behaviour of deceleration parameter for different values of m and n. One can easily analyse the behaviour of q from the plots. It can be seen that, (i) for n = 0.01 and 0.34 ≤ m ≤ 0.64, considered models are accelerating in nature i.e q < 0 and (ii) for n = 0.10 and 1.70 ≤ m ≤ 3.20, models show phase transition from decelerating phase to accelerating phase. In this situation q take values from positive to negative. We are mainly focus to investigate the phase transition scenario so in all discussed models, as a representative case we have considered the value of m in the range 1.70 ≤ m ≤ 3.20 and n = 0.10. In order to obtain the Hubble parameter from equation (12), we used the relation between Hubble parameter and deceleration parameter as q = d dt ( 1 H ) − 1, which leads to H = n m sin(nt) + nc 4 , c 4 is the constant of integration.(13) The value of integration constant c 4 does not affect the qualitative behaviour of the Hubble parameter but it affect the scale factor. We classify the considered cosmological model in three different cases as per c 4 = 0, c 4 > 0 and c 4 < 0, i.e. c 4 = −c 5 , c 5 > 0. Case-I: c 4 = 0 In this case the Hubble parameter in equation (13) leads to H = n m sin(nt) . We know the relation between Hubble parameter and scale factor as H =Ṙ R , which along with equation (14) leads to the scale factor of the form R = c 6 1 − cos(nt) sin(nt) 1 m = c 6 tan nt 2 1 m , c 6 is the constant of integration.(15) Further, equations (11) and (15) yields the following metric potentials as R i = c i1 tan nt 2 1 m exp c i2 tan nt 2 − 3 m dt ,(16) where c i1 = c i c 6 and c i2 = ki for i = 1, 2, 3. We take the generalized Chaplygin gas, described by EoS [9,10] p = − A ρ α , A > 0 and 0 ≤ α ≤ 1.(17) The expression for the energy density can be obtained from equations (9), (14) and (17) as ρ = A + c 7 R 3(1+α) 1 1+α = A + ρ 0 (sin(nt)) − 3(1+α) m (1 + cos(nt)) 3(1+α) m 1 1+α ,(18)where ρ 0 = c 7 c −3(1+α) 6 and c 7 is the constant of integration. The directional Hubble parameter H i are given by, H i =Ṙ i R i = 3c 3 6 n + (sin(nt)) m−3 m (1 + cos(nt)) 3 n k i m 3c 3 6 m sin(nt) , i = 1, 2, 3.(19) From Equations (4), (7) , (16), (17) and (18), one can get the expression for Gravitational constant as G = 1 8π(ρ + p) k 4 (sin(nt)) − 6 m (cos(nt) − 1) (cos(nt) + 1) m+6 m − 18c 6 6 n 2 cos(nt) 9c 6 6 m(cos 2 (nt) − 1) ,(20) where k 4 = m(k 1 k 2 + k 1 k 3 − k 2 2 − k 2 3 ) . With the help of equations (4) and (7), one can obtain the expression for cosmological constant as Λ = 1 ρ + p m 2 (sin(nt)) − 6 m (cos(nt) − 1) (cos(nt) + 1) m+6 m (k 5 ρ + k 6 p) − 27c 6 6 n 2 (ρ + p) + 18c 6 6 n 2 m cos(nt)ρ 9c 6 6 m 2 (cos 2 (nt) − 1) ,(21) where k 5 = k 2 k 3 + k 2 2 + k 2 3 and k 6 = k 1 k 2 + k 1 k3 + k 2 k 3 . The physical quantities of the observational interest are expansion scalar (Θ), shear scalar (σ 2 ) and the anisotropic parameter (A m ), which are defined as follows: Θ = 3H = H 1 + H 2 + H 3 ,(22)σ 2 = 1 2 H 2 1 + H 2 2 + H 2 3 − Θ 2 6 ,(23)A m = 1 3 3 i=1 H i − H H 2 .(24) In this case, the physical quantities are obtained as Θ = 3n m sin(nt) ,(25)σ 2 = (k 2 1 + k 2 2 + k 2 3 ) [1 + cos(nt)] 6 m [sin(nt)] −6+2m m 18c 6 6 (1 − cos 2 (nt)) = k 2 1 + k 2 2 + k 2 3 18c 6 6 1 + cos(nt) sin(nt) 6 m ,(26)A m = (k 2 1 + k 2 2 + k 2 3 )m 2 [1 + cos(nt)] 6 m [sin(nt)] −6+2m m 27c 6 6 n 2 .(27) In this case the state finder parameters are defined and expressed as r = ... R RH 3 = m 2 cos 2 (nt) − 3m cos(nt) + 1 + m 2 ,(28)s = r − 1 3(q − 0.5) = 2m(m cos 2 (nt) − 3 cos(nt) + m) 3(2m cos(nt) − 3) .(29) Relation between r and s is given by r = 1 + 9 2 s 2 ± 3 2 s 9 + 9s 2 − 4m 2 .(30) In terms of the deceleration parameter the state finder parameters are given as The energy density, pressure, cosmological constant and gravitational constant are periodic in nature, which is noticed from the Figure (4) to Figure (7) respectively due to the presence of cos(nt) and sin(nt) terms in the expressions of these physical quantities. Here one can note that, ρ, p, Λ, G → ∞ at t = n2π n , ∀ n 2 ∈ Z. The energy density is positive where as pressure is negative for different values of m with evolvement of time. In this case, the qualitative behavior of energy density follow the pattern of higher energy density value to lower energy density value (approaching to zero) to higher energy density value. This process will continue due to the periodic nature of the terms cos(nt) and sin(nt) involve in the expression of energy density (18). Again it is also pointed out that, cosmological constant is positive for some values of m and positive to negative values for some m (See Figure 6). Gravitational constant takes values from negative to positive values and positive to negative values in a periodic way for different values of m (See Figure 7). Further, it is noticed that the expansion scalar, shear scalar and anisotropy parameter also have singularity at t = n2π n , ∀ n 2 ∈ Z and they behaves periodically. r = −1 − q + q 2 + m 2 ,(31)s = 2(−2 − q + q 2 + m 2 ) 3(−1 + 2q) .(32) The profile of scale factor R, energy density ρ, pressure p, cosmological constant Λ and gravitational constant G against time is presented in the Figure ( Further, equations (11) and (34) yields the following metric potentials as R i = c i3 e 2 arctan    nc 4 tan ( nt 2 ) +m √ n 2 c 2 4 −m 2    √ n 2 c 2 4 −m 2 exp       c i4 e − 6 arctan    nc 4 tan ( nt 2 ) +m √ n 2 c 2 4 −m 2    √ n 2 c 2 4 −m 2 dt       , i = 1, 2, 3,(35) where c i3 = c i c 8 and c i4 = ki . The expression for the energy density can be obtained from equations (9) and (34) as H i =      ρ = A + c 7 R 3(1+α) 1 1+α =       A + ρ 1 e − 6(1+α) arctan    nc 4 tan ( nt 2 ) +m √ n 2 c 2 4 −m 2    √ n 2 c 2 4 −m 2       1 1+α ,(37) where ρ 1 = c 7 c In this case, the physical quantities are obtained as Θ = 3n m sin(nt) + nc 4 ,(40)σ 2 = (k 2 1 + k 2 2 + k 2 3 ) 18c 6 8 e − 12 arctan    nc 4 tan ( nt 2 ) +m √ n 2 c 2 4 −m 2    √ n 2 c 2 4 −m 2 ,(41)A m = − (k 2 1 + k 2 2 + k 2 3 )(m 2 cos 2 (nt) − 2nmc 4 sin(nt) − m 2 − n 2 c 2 4 ) 27n 2 c 6 8 e − 12 arctan    nc 4 tan ( nt 2 ) +m √ n 2 c 2 4 −m 2    √ n 2 c 2 4 −m 2 .(42) In this case the state finder parameters are defined and expressed as s = r − 1 3(q − 0.5) = − 2 3(2m cos(nt) − 3) ×        (43) (m sin(nt) + nc 4 ) 3 1 + 3m + 2m 2 + nmc 4 sin(nt) + 4m 2 cos 4 nt 2 − (4m 2 + 6m) cos 2 nt 2 12nm 2 c 4 + 4m 3 sin(nt) cos 4 nt 2 − 12nm 2 c 4 + 4m 3 sin(nt) cos 2 nt 2 − 3n 2 mc 2 4 sin(nt) − n 3 c 3 4 + 1         .(44) In terms of the deceleration parameter the state finder parameters are given as Figure (11) and Figure (12) shows the qualitative behaviour of gravitational constant and cosmological constant with respect to cosmic time. We noticed that, both behaves in a periodic way with time and also Λ > 0 and G < 0 with the evolution of cosmic time. Λ > 0 and G < 0 are free from the initial singularity. r = m 2 − 1 − 2q − q 2 + nc 4 3 m 2 − 1 − q + q 2 + nc 4 m 2 − 1 − 2q − q 2 (m 2 − 1 − 2q − q 2 + 3n 2 c 2 4 ) m 2 − 1 − 2q − q 2 + 3nc 4 m 2 − 1 − 2q − q 2 + n 2 c 2 4 3 ,(45)s = 2 m 2 − 1 − 2q − q 2 + nc 4 3 m 2 − 1 − q + q 2 + nc 4 m 2 − 1 − 2q − q 2 3(2q − 1) (m 2 − 1 − 2q − q 2 + 3n 2 c 2 4 ) m 2 − 1 − 2q − q 2 + 3nc 4 m 2 − 1 − 2q − q 2 + n 2 c 2 4 3 − 2 3(2q − 1) .(46) Further, equations (11) and (48) yields the following metric potentials as R i = c i5 e − 2 arctan    nc 5 tan ( nt 2 ) −m √ n 2 c 2 5 −m 2    √ n 2 c 2 5 −m 2 exp       c i6 e 6 arctan    nc 5 tan ( nt 2 ) −m √ n 2 c 2 5 −m 2    √ n 2 c 2 5 −m 2 dt       ,(49) where c i5 = c i c 9 and c i6 = ki . The expression for the energy density can be obtained from equations (9) and (21) as ρ = A + c 7 R 3(1+α) 1 1+α =       A + ρ 2 e 6(1+α) arctan    nc 5 tan ( nt 2 ) −m √ n 2 c 2 5 −m 2    √ n 2 c 2 5 −m 2       1 1+α ,(51) where ρ 2 = c 7 c In this case, the physical quantities are obtained as Θ = 3n m sin(nt) − nc 5 ,(54)σ 2 = (k 2 1 + k 2 2 + k 2 3 ) 18c 6 9 e − 12 arctan    m−nc 5 tan ( nt 2 ) √ n 2 c 2 5 −m 2    √ n 2 c 2 5 −m 2 ,(55)A m = 27n 2 m 2 c 6 9 −2 tan nt 2 + tan 2 nt 2 sin(nt) + sin(nt) 2 e − 12 arctan    nc 5 tan ( nt 2 ) −m √ n 2 c 2 5 −m 2    √ n 2 c 2 5 −m 2 + 4 − nc5 2 − nc5 tan( nt 2 ) 2 + m tan nts = r − 1 3(q − 0.5) = 2 3(2m cos(nt) − 3) ×     (m sin(nt) − nc 5 ) 3 −4m 2 cos 4 nt 2 + (6m + 4m 2 ) cos 2 nt 2 + nmc 5 sin(nt) − 3m − 2m 2 − 1 4m 2 (m sin(nt) − 3nc 5 ) cos 2 nt 2 cos 2 nt 2 − 1 + n 3 c 3 5 − 3mn 2 c 2 5 sin(nt) − 1     .(57) In terms of the deceleration parameter the state finder parameters are given as Table 2 to Table 4 r = m 2 − 1 − 2q − q 2 − nc 5 3 m 2 − 1 − q + q 2 − nc 5 m 2 − 1 − 2q − q 2 (m 2 − 1 − 2q − q 2 + 3n 2 c 2 5 ) m 2 − 1 − 2q − q 2 − 3nc 5 m 2 − 1 − 2q − q 2 + n 2 c 2 5 3 ,(59)s = 2 3(2q − 1) ×    m 2 − 1 − 2q − q 2 − nc 5 3 m 2 − 1 − q + q 2 − nc 5 m 2 − 1 − 2q − q 2 (m 2 − 1 − 2q − q 2 + 3n 2 c 2 5 ) m 2 − 1 − 2q − q 2 − 3nc 5 m 2 − 1 − 2q − q 2 + n 2 c 2 5 3 − 1    .(60) The squared sound speed We determine the classical stability of considered models on the basis of an adiabatic squared sound speed. It is one of the important quantity in cosmology. Adiabatic squared sound speed for system is defined [54], as c 2 s = ∂p ∂ρ .(61) Here c 2 s has three possibilities i.e c 2 s < 0 or c 2 s = 0 or c 2 s > 0. The sign of c 2 s is very important to investigate as it leads to the instability of the cosmological models through which one can reject or accept the constructed cosmological models. The case when, c 2 s < 0, leads to classical instability of the cosmological models due to the uncontrolled grow of the energy density perturbation. The case when c 2 s > 0 may leads to the issue of occurrence of casuality. As a matter of fact, it is usually considered as c s ≤ 1 and the bound on c 2 s is 0 ≤ c 2 s ≤ 1. In addition to that, the complementary bound c s > 1 is used as a condition for rejecting the theories. The details regarding c 2 s can be found in [21]. In present study, c 2 s is obtained as c 2 s = ∂p ∂ρ = Aα ρ 1+α =                                Aα A+ρ0(sin(nt)) − 3(1+α) m (1+cos(nt)) 3(1+α) m , for c 4 = 0 Aα A+ρ1e − 6(1+α) arctan    nc 4 tan ( nt 2 ) +m √ n 2 c 2 4 −m 2    Energy conditions The pointwise energy conditions (ECs) depends on energy momentum tensor at a given point in a space time. These conditions are contractions of time like or null vector with the Einstein's tensor and energy momentum tensor coming from Einstein's field equations. ECs can be imposed in order to investigate the constraints on the free parameters involved in the cosmological models. For example the evolution of acceleration or deceleration of the universe and the emergence of Big Rip singularity, can be related to the constraints imposed by the ECs. For energy momentum tensor T ij = (ρ + p)u i u j − pg ij , the standard pointwise energy conditions are defined as follows [48][49][50] • Null energy condition (NEC): ρ + p ≥ 0 • Weak energy condition (WEC): ρ ≥ 0 and ρ + p ≥ 0 • Dominant energy condition (DEC): ρ ≥ 0 and ρ ± p ≥ 0 • Strong energy condition (SEC): ρ + p ≥ 0 and ρ + 3p ≥ 0 We use the ECs to investigate the stability and physical acceptibility of the solutions in the present models. In general the universe in cosmological model should satisfy WEC and DEC and violates SEC for late time accelerated expansion of the universe. For each of the three cases, Figures (24), (25) and (26) represents the profile of WEC, DEC and SEC respectively versus cosmic time. From these figures it can seen that profile of energy conditions follows periodic variation for each of the cases. Considered model satisfies WEC (ρ + p) (except for m=1.7 in case − I) and DEC (ρ − p) whereas SEC is satisfied and violated periodically throughout the evolution of the universe. Conclusion In this work, we have considered Bianchi-I cosmological models with generalised Chaplygin gas represented by equation (17) and cosmological solutions are obtained by using time periodic deceleration parameter represented by equation (12). By fixing the value of constant parameter n, we have computed the range of parameter m (see Table-1 for present observational value (range) of deceleration parameter). Figures (1) and (2) shows the variation of deceleration parameter with respect to cosmic time for n = 0.01 and n = 0.10 and corresponding range of m. For n = 0.01 deceleration parameter is negative throughout the evolution of the universe whereas for n = 0.10 it shows transition from decelerated phase to accelerated phase periodically throughout the evolution of the universe. Hubble parameter represented by (13) is computed by using it's relation with deceleration parameter. The integration constant c 4 affects the scale factor so we have discussed three different cases for the considered model depending on the positive negative and neutral nature of c 4 . In each of the cases we have analyzed the solutions and physical quantities of the observational interest (such as expansion scalar (Θ), shear scalar (σ 2 ), anisotropy parameter (A m ), state finder parameters (r and s)). Gravitational constant G varies from positive to negative in the present cosmological framework. With positive and negative G, we will have attractive and repulsive nature of gravity respectively. Positive and negative cosmological constant also yields repulsive and attractive gravity respectively. The dark energy is responsible for accelerating universe and positive cosmological constant acts as a mechanism for dark energy. The periodic nature of Λ and G also highlight the attractive as well as repulsive gravity era in the model. For more details one can refer Ayuso et al. [55]. In bouncing model, the scale factor is having its minima at the bounce instant. The universe should be contracting before the bounce and expanding after the bounce. The solutions given in Eq. (34) and (48) may lead to asymmetric bounce for different values of model parameters. On the other hand, the solution (15) will not exhibit bouncing scenario, however, the toy model will exhibit periodic behaviour of scale factor but the behaviour is not of bouncing type. In the recent, Sahoo et al. [38] have discussed the periodic varying deceleration parameter in f (R, T ) = R + 2λT gravity. For λ = 0, this reduces to the result in general relativity. In our investigation, we noticed that for all the models SEC is violated but WEC and SEC are violated in their study. For stability of the solutions we both have different approaches, they have used linear homogeneous perturbations in the FRW background whereas we used the square sound speed. Our solutions are stable but their stability of solution depends on the parameter k and λ. Further, all other physical parameters we both have the same periodic qualitative behaviour. In all the figures, we have taken the cosmic time t along the horizontal axis which is measured in giga years (1Gyr = 10 9 years). The conclusion of our study are as follows: • Almost all the parameters which we have discussed in our study shows periodic behaviour due to the choice of deceleration parameter. • For each of the cases energy ρ is positive and p is negative throughout the evolution of the universe and this negative pressure guarantees the late time expansion of the universe. • Models discussed in Case-I and Case-II, tends to ΛCDM model whereas model in Case-III fails for the value of n and m provided in Table 1. The expressions obtained for expansion scalar, shear scalar, and anisotropic parameter bears singularities in Case-I whereas these are free from singularities in Case-II and Case-III. • The square sound speed satisfies the bounds 0 ≤ c 2 s ≤ 1 for each of the cases. So the considered model is stable and physically acceptable. • For each of the cases considered model satisfies WEC (except for m=1.7 in case − I) and DEC whereas it violate SEC periodically throughout the evolution of the universe. So using the results of energy conditions one can conclude that the violation of SEC may leads to the accelerating universe. 1 : 1Computational range of the parameter m for fixed values of n, obtained from the present value of the deceleration parameter. n Interval of m 0.01 0.34 ≤ m ≤ 0.64 0.02 0.35 ≤ m ≤ 0.66 0.03 0.37 ≤ m ≤ 0.69 0.04 0.39 ≤ m ≤ 0.74 0.05 0.43 ≤ m ≤ 0.82 0.06 0.49 ≤ m ≤ 0.94 0.07 0.59 ≤ m ≤ 1.11 0.08 0.74 ≤ m ≤ 1.39 0.09 1.02 ≤ m ≤ 1.93 0.10 1.70 ≤ m ≤ 3.20 Figures (1) and Figure 1 : 1Deceleration parameter q against cosmic time for 0.34 ≤ m ≤ 0.64 and n = 0.01. Figure 2 : 2Deceleration parameter q against cosmic time for 1.70 ≤ m ≤ 3.20 and n = 0.10. 3) to Figure (7) respectively for fix n and different values of m with suitable choice of arbitrary constant involve in the expressions of the physical quantities. The scale factor is increasing with the evolvement of time in the provided range of time, which can be seen from the Figure (3).However, the qualitative behaviour is similar to tan function due to the presence of tan nt 2 in equation(15) and R → ∞ at t = (2n1+1)π n , n 1 ∈ Z. 3. 2 Figure 3 :R, c 8 238Case-II: c 4 > 0 In this case the Hubble parameter in equation (13) takes the form H = n m sin(nt) + nc 4 . (33) Scale factor parameter R against cosmic time for different m and n.The relation H =Ṙ R , which along with(33) leads to the scale factor of the form = is the constant of integration. . The directional Hubble parameters are expressed as + nc 4 k i + mk i sin(nt) nc 4 + m sin(nt)) Figure 4 : 4Energy density ρ against cosmic time for different m, n = 0.1, α = 0.5, ρ 0 = 1 and A = 1. Figure 5 : 5Pressure p against cosmic time for different m, n = 0.1, α = 0.5, ρ 0 = 1 and A = 1. Figure 6 : 6Cosmological constant Λ against cosmic time for different m, n = 0.1, α = 0.5, and A = 1. Figure 7 : 7Gravitational constant G against cosmic time for different m, n = 0.1, α = 0.5 and A = 1. − m 2 cos 2 nt 2 − nmc 4 sin nt 2 cos nt 2 − cos 4 nt 2 − m 2 cos 2 nt 2 − nmc 4 sin nt 2 cos nt 2 − 4 , 4sin(nt) + nc 4 ) 3 1 + 3m + 2m 2 + nmc 4 sin(nt) + 4m 2 cos 4 nt 2 − (4m 2 + 6m) cos 2 nt 2 12nm 2 c 4 + 4m 3 sin(nt) cos 4 nt 2 − 12nm 2 c 4 + 4m 3 sin(nt) cos 2 nt 2 − 3n 2 mc 2 4 sin(nt) − n 3 c 3 Figure ( 8 8) indicates the profile of scale factor against the cosmic time. Here we noticed that, the scale factor is increasing with respect to cosmic time in a periodic way.Figure (9) and Figure (10) are representing the profile of pressure and energy density. As per the evolution of time, energy density and pressure are positive and negative respectively in a periodic way for different values of m. Figure ( 13 ) 13and Figure(14) shows the qualitative behaviour of shear scalar and anisotropic parameter with respect to cosmic time. Both the parameters behave in a periodic way with time and positive with the evolution of cosmic time. Figure 8 :Figure 9 : 89Scale factor R against cosmic time for different m 3.3 Case-III: c 4 < 0 i.e. c 4 = −c 5 , c 5 > 0 In this case the Hubble parameter in equation (13) takes the form H = n m sin(nt) − nc 5 . (47) Pressure p against cosmic time for 1.70 ≤ m ≤ 3.20 and n = 0.10. Figure 10: Energy density ρ against cosmic time for 1.70 ≤ m ≤ 3.20 and n = 0.10. Figure 11 : 11Gravitational constant G against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1, α = 0.5,c 4 = 55 and n = 0.10. Figure 12 : 12Cosmological constant Λ against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1,α = 0.5,c 4 = 55 and n = 0.10. Figure 13 : 13Shear scalar σ 2 against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1,α = 0.5, c 4 = 55 and n = 0.10. Figure 14 : 14Anisotropic parameter A m against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1,α = 0.5,c 4 = 55 and n = 0.10. , c 9 9H =Ṙ R along with equation(47) leads to the scale factor of the form is the constant of integration. − nc 5 k i + mk i sin(nt) m sin(nt) − nc 5 ) − m 2 cos 2 nt 2 + nmc 5 sin nt 2 cos nt 2 − cos 4 nt 2 − m 2 cos 2 nt 2 + nmc 5 sin nt 2 cos nt 2 − sin(nt) − nc 5 ) 3 −4m 2 cos 4 nt 2 + (6m + 4m 2 ) cos 2 nt 2 + nmc 5 sin(nt) − 3m − 2m 2 − 1 4m 2 (m sin(nt) − 3nc 5 ) cos 2 nt 2 cos 2 nt 2 − 1 + n 3 c 3 5 − 3mn 2 c 2 5 sin(nt) , Figure ( 15 ) 15and Figure (16) indicates the profile of Hubble parameter against cosmic time for different values of c 4 . Similar qualitative behaviour as that of case-II is observed for physical parameters likes energy density[Figure (17)], pressure[Figure (18)], gravitational constant [Figure (19)], cosmological constant [Figure (20)], shear scalar [Figure (21)] and anisotropic parameter [Figure (22)].Further, we have analyzed whether these derived models are approaching ΛCDM model or not for the computational range of n and m(Table 1). From equations(31),(32),(45),(46), (59) and (60), we have evaluated the {r, s} pair for the different model parameters, which are presented in respectively for three cases. Here one can notice that, model discussed in case-I approaches to ΛCDM model for different value of q and m. At current value of q = − 1 2 , the model in case-I approaches to ΛCDM model for m = √ 5 2 . The models in case-II and Case-III, the {r, s} pair depends on the model parameters n, m, c 4 , c 5 and q. At current value of q = − 1 2 , the model in case-II approaches to ΛCDM model whereas model in Case-III does not approaches to ΛCDM model in view of the values of n and m presented inTable 1. Figure 15 : 15Hubble parameter H against cosmic time for m = 2.2, n = 0.1 and different c 4 . Figure 16 : 16Hubble parameter H against cosmic time for m = 0.5, n = 0.01 and different c 4 . Figure 17 : 174: {r, s} pair for different model parameters with fixed q Energy density ρ against cosmic time for 1.70 ≤ m ≤ 3.20 and n = 0.10. Figure 18 : 18Pressure p against cosmic time for 1.70 ≤ m ≤ 3.20 and n = 0.10. Figure 19 : 19Profile of Gravitational constant G against cosmic time for 1.70 ≤ m ≤ 3.20, c 7 = c 8 = A = 1,α = 0.5,c 4 = 55 and n = 0.10. Figure 20 : 20Cosmological constant Λ against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1,α = 0.5,c 4 = 55 and n = 0.10. 4 Stability and physical acceptability of the solutions 4. Figure 21 : 21Shear scalar σ 2 against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1, α = 0.5,c 4 = 55 and n = 0.10. Figure 22 : 22Anisotropic parameter A m against cosmic time for 1.70 ≤ m ≤ 3.20,c 7 = c 8 = A = 1,α = 0.5,c 4 = 55 and n = 0.10. Figure (23), we have presented the profile of squared sound speed against time for different values of m. It is observed from the Figure (23) that, c 2 s is periodic in nature and 0 ≤ c 2 s < 0.6 in all the cases. Thus in account of the prescribe range of c 2 s , the solution presented here are stable. Figure 23 : 23Squared sound speed against cosmic time for three different cases: c 4 = 0 (Left),c 4 > 0 (Right) and c 4 < 0 (Middle). Figure 24 : 24Energy conditions (WEC(Left), DEC (Middle), SEC (Right)) against time for case-I. Figure 25 : 25Energy conditions (WEC(Left), DEC (Middle), SEC (Right)) against time for case-II. Figure 26 : 26Energy conditions (WEC(Left), DEC (Middle), SEC (Right)) against time for case-III. 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[ "ARTICLE Orbital disproportionation of electronic density is a universal feature of alkali-doped fullerides", "ARTICLE Orbital disproportionation of electronic density is a universal feature of alkali-doped fullerides" ]
[ "Naoya Iwahara \nTheory of Nanomaterials Group\nKatholieke Universiteit Leuven\nCelestijnenlaan 200FB-3001Heverlee, LeuvenBelgium\n", "Liviu F Chibotaru \nTheory of Nanomaterials Group\nKatholieke Universiteit Leuven\nCelestijnenlaan 200FB-3001Heverlee, LeuvenBelgium\n" ]
[ "Theory of Nanomaterials Group\nKatholieke Universiteit Leuven\nCelestijnenlaan 200FB-3001Heverlee, LeuvenBelgium", "Theory of Nanomaterials Group\nKatholieke Universiteit Leuven\nCelestijnenlaan 200FB-3001Heverlee, LeuvenBelgium" ]
[]
Alkali-doped fullerides show a wide range of electronic phases in function of alkali atoms and the degree of doping. Although the presence of strong electron correlations is well established, recent investigations also give evidence for dynamical Jahn-Teller instability in the insulating and the metallic trivalent fullerides. In this work, to reveal the interplay of these interactions in fullerides with even electrons, we address the electronic phase of tetravalent fulleride with accurate many-body calculations within a realistic electronic model including all basic interactions extracted from first principles. We find that the Jahn-Teller instability is always realized in these materials too. In sharp contrast to the correlated metals, tetravalent system displays uncorrelated band-insulating state despite similar interactions present in both fullerides. Our results show that the Jahn-Teller instability and the accompanying orbital disproportionation of electronic density in the degenerate lowest unoccupied molecular orbital band is a universal feature of fullerides.
10.1038/ncomms13093
null
15,799,749
1605.00175
bd5628d69837df524e9c021f90c731df3731a795
ARTICLE Orbital disproportionation of electronic density is a universal feature of alkali-doped fullerides Published 7 Oct 2016 Naoya Iwahara Theory of Nanomaterials Group Katholieke Universiteit Leuven Celestijnenlaan 200FB-3001Heverlee, LeuvenBelgium Liviu F Chibotaru Theory of Nanomaterials Group Katholieke Universiteit Leuven Celestijnenlaan 200FB-3001Heverlee, LeuvenBelgium ARTICLE Orbital disproportionation of electronic density is a universal feature of alkali-doped fullerides Published 7 Oct 201610.1038/ncomms13093Received 4 May 2016 | Accepted 1 Sep 2016 |OPEN Correspondence and requests for materials should be addressed to L.F.C. (email: [email protected]). Alkali-doped fullerides show a wide range of electronic phases in function of alkali atoms and the degree of doping. Although the presence of strong electron correlations is well established, recent investigations also give evidence for dynamical Jahn-Teller instability in the insulating and the metallic trivalent fullerides. In this work, to reveal the interplay of these interactions in fullerides with even electrons, we address the electronic phase of tetravalent fulleride with accurate many-body calculations within a realistic electronic model including all basic interactions extracted from first principles. We find that the Jahn-Teller instability is always realized in these materials too. In sharp contrast to the correlated metals, tetravalent system displays uncorrelated band-insulating state despite similar interactions present in both fullerides. Our results show that the Jahn-Teller instability and the accompanying orbital disproportionation of electronic density in the degenerate lowest unoccupied molecular orbital band is a universal feature of fullerides. T he understanding of electronic phases of alkali-doped fullerides A n C 60 is a long-standing and challenging task for material scientists 1 . The prominent feature of these narrow-band molecular materials is the coexistence of strong intra-site Jahn-Teller (JT) effect with strong electron correlation, which underlies the unconventional superconductivity in A 3 C 60 (refs 2-9) and a broad variations of electronic properties in this series of materials in function of the size of alkali ions and the degree of their doping [10][11][12][13][14] . External pressure and insertion of neutral spacers add new possibilities for the engineering of their electronic phases [15][16][17] . This was recently demonstrated for the Cs 3 C 60 fulleride, which undergoes transitions from Mott-Hubbard (MH) antiferromagnet to a high-temperature superconductor (T c ¼ 38 K) and then to strongly correlated metal under external pressure 3,4,[6][7][8] . Signs of the JT effect in alkali-doped fullerides were inferred from nuclear magnetic resonance 18,19 , infared 20,21 , electron energy loss 22,23 spectra, and scanning tunnelling microscopy 24,25 in various compounds. Recently, the parameters governing the complex JT interaction on fullerene anions have been firmly established [26][27][28] , which opened the way for accurate theoretical investigation of the electronic states in fullerides. It was found that in the MH insulating phase of cubic fullerides such as Cs 3 C 60 at ambient pressure, the para-dynamical JT effect is realized as independent pseudorotations of JT deformations at each C 60 site 29 . The same para dynamical JT effect was found in the metallic phase of A 3 C 60 close to MH transition, whereas the pseudorotation of JT deformation at different sites are expected to be correlated with further departure from the MH transition due to the increase of the band energy 30 . These findings have found confirmation in a very recent investigation of Cs 3 C 60 fulleride, showing an almost unchanged infrared spectrum on both sides in the vicinity of MH metalinsulator transition, whereas displaying its significant variation when the material was brought deeper into the metallic phase 31 . Moreover, our calculations have also shown that the metallic phase in these systems exhibits an orbital disproportionation of electronic density as a result of the dynamical JT instability 30 . This successful theoretical approach is applied here for the investigation of the electronic phase in the A 4 C 60 fullerides, containing an even number of doped electrons per site. We find that these materials exhibit a dynamical JT instability too. As in A 3 C 60 , the ground state of A 4 C 60 displays again the orbital disproportionation of electronic density, thus identifying it as a universal key feature of the electronic phases of alkali-doped fullerides. Results Diagram of JT instability in A 4 C 60 . It is well established that the t 1u lowest unoccupied molecular orbital (LUMO) band mainly defines the electronic properties of fullerides 1 . Following the recent treatment of A 3 C 60 (ref. 30), we consider all essential interaction in this band including the one-electron, the bielectronic and the vibronic contributions: H ¼Ĥ t þĤ bi þĤ JT ; H t ¼ X m;Dm X ll 0 s t Dm ll 0ĉ y m þ Dmlsĉml 0 s ; H bi ¼ 1 2 X m X ls U knmlsnml À s þ U ? X l 0 ð 6 ¼ lÞs 0n mlsnml 0 s 0 2 4 À J X l 0 ð 6 ¼ lÞn mlsnml 0 s Àĉ y mlsĉml 0 sĉ y ml À sĉml 0 À s Àĉ y mlsĉml 0 sĉ y ml 0 À sĉ ml À s 3 5 ; H JT ¼ X m ' o X g 1 2 p 2 mg þ q 2 mg þ g X ll 0 s X g G g ll 0ĉ y mlsĉml 0 s q mg " # ;ð1Þ where, m denote the fullerene sites, Dm the neighbours of site m, l,l 0 the t 1u LUMO orbitals (x,y,z) on each C 60 (Supplementary Fig. 1), s,s 0 the spin projections,ĉ mls andĉ y mls are annihilation and creation operators of electron, respectively,n mls ¼ĉ y mlsĉ mls , q mg and p mg are the normal vibrational coordinate for the g component of the h g mode (g ¼ y,e,x,Z,z) and its conjugate momentum, respectively, and G g ll 0 is Clebsch-Gordan coefficient. The transfer parameters t Dm ll 0 ofĤ t have been extracted from density functional theory (DFT) calculations (see ref. 30 for K 3 C 60 , Methods, Supplementary Methods and Supplementary Table 1 for K 4 C 60 ). The frequency o and the orbital vibronic coupling constant g for an effective single-mode JT model of C n À 60 have been calculated in ref. 29. The phonon dispersion was neglected, because it is weak in fullerides 1 . The projection of the bielectronic interaction in the t 1u LUMO band onto intra-site Hamiltonian (Ĥ bi ) is an adequate approximation due to strong molecular character of fullerides 1 . The intra-site repulsion q ¼ 0, E g (0), is subtracted from E g (q) for each U || . parameters U || and U > , obeying the relation U || À U > ¼ 2J, are strongly screened: first, by high-energy interband electron excitations reducing their value from 3 eV to ca 1 eV 32 and, second, by intra t 1u -band excitations. The latter can further reduce U || and U > several times 32 ; however, the extent of this screening strongly depends on the character of the correlated t 1u band and can, therefore, be assessed only in a self-consistent manner. On the other hand, the vibronic coupling to the h g modes, representing a quadrupolar perturbation, is hardly screened. It is the same for the Hund's rule coupling J, for which we take the calculated molecular value 29 . We leave U || as the only free parameter of the theory. The ground state has been calculated within a self-consistent Gutzwiller approach (see Methods), which proved to be successful for the investigation of A 3 C 60 (ref. 30). To unravel the role played by JT interactions in the ground electronic phase in A 4 C 60 , we first consider the case of a face-centred cubic (fcc) H t as in A 3 C 60 , the corresponding bands being populated by four electrons per site. Figure 1a shows the calculated total energy as function of the amplitude q of static JT distortions 33,34 of h g y type on fullerene sites. As in the case of A 3 C 60 (ref. 30), the energy curve E g (q) has two minima, one at the undistorted configuration q ¼ 0 and the other at a value q 0 approximately corresponding to the equilibrium distortion in an isolated C 4 À 60 (see the Supplementary Fig. 2, Supplementary Table 2 and Supplementary Note 1). For U || smaller than the critical value U c E0.64 eV, the static JT distortion is quenched, q ¼ 0. At U || 4U c , the JT distortion reaches its equilibrium value, q 0 . The full diagram of the total energy E g (q, U || ) is shown in Fig. 1b (for E g of A 3 C 60 , see Supplementary Fig. 3). The character of the electronic phase differs drastically in the two domains of U || . The difference is clearly seen in the electron population in the LUMO orbitals n l and the Gutzwiller's reduction factor q ll . The evolution of the population n l with respect to U || (Fig. 2a) shows that for U || oU c the phase corresponds to equally populated LUMO bands. This equally populated phase gradually becomes strongly correlated with increasing U || , which is testified by the accompanying decrease of the Gutzwiller's reduction factors for these bands (Fig. 2c). On the contrary, for U || 4U c , it exhibits orbital disproportionation of electronic density among the LUMO orbitals ( Fig. 2a) with a sudden jump of the Gutzwiller factor (Fig. 2c). The existence of the two kinds of phases with and without the JT deformation is explained by the competition between the band energy hĤ t i and the JT stabilization energy in the presence of the strong electron repulsion U || . The former stabilizes the system the most when the splitting of the orbital is absent, whereas the JT effect does by lowering the occupied orbitals. On the other hand, the bielectronic energy is reduced by the quenching of the charge fluctuation (localization of the electrons), which results in the decrease of the band energy and the relative enhancement of the JT stabilization. Therefore, when U || is small (U || oU c ), the homogeneous (with equal orbital populations) band state is favoured and the JT distortion is quenched. With the increase of U || over U c , the band energy is reduced to the extent that the JT stabilization on C 60 sites is favoured, resulting in orbitally disproportionated ground state. We note that these results are general, which neither depends on the form of the JT distortion on sites nor on the uniformity of these distortions, which can also be dynamical as in A 3 C 60 (ref. 30; vide infra). Band-insulating state induced by strong electron repulsion. To better understand the physics of the obtained orbitally disproportionated electronic phase, first consider a simplified model forĤ t , which includes only the diagonal electron transfers after orbital indices, t Dm ll 0 ¼ d ll 0 t Dm ll 0 (a widely used approximation for the study of multiorbital correlation effects [35][36][37]. Figure 3a shows the total energies for the two phases with and without JT distortion in function of U || . We see again an evolution of the ground state with the stabilization of orbitally disproportionated electronic phase in the large U || domain. We find this behaviour pretty similar to the case when the fullĤ t for fcc lattice is considered (Fig. 3b). Owing to the simplification, we can fully identify the orbitally disproportionated phase, because of its exact solution. Indeed, in terms of band solutionŝ a y kas j 0i ¼ 1= ffiffiffiffi N p P m e ikÁmĉy mas j 0i, where N is the number of sites, we obtain for the orbitally disproportionated phase (see Supplementary Methods): F 0 j i ¼ Y all ksâ y kxsâ y kys 0 j i;ð2Þ that is, a pure band state with occupied x and y, and empty z band. In the case of a JT distortion different from the h g y type, the solution will be identical to equation (2) but involving band orbitals that are linear combinations of x, y and z orbitals. The solution F 0 is exact in the whole domain of U || . However, owing to its fully disproportionated character, always corresponding to the orbital populations (2,2,0), it becomes ground state, that is, intersects the correlated homogeneous solution (Fig. 3a), only under the opening of the gap between occupied degenerate orbitals x, y and the empty orbital z. This means that the orbitally disproportionated phase in Fig. 3a is nothing but conventional band insulator. The obtained result is not specific to the simplified model. In the case of fullĤ t (Fig. 3b), the orbitally disproportionated state differs only slightly from F 0 in equation (2), which is seen from the population of the orbital components of the LUMO band n l that are close to (2,2,0) (Fig. 2a) and the jump of the Gutzwiller factor to its uncorrelated value 1 (Fig. 2c). Thus, we encounter here a counterintuitive situation: with the increase of the electron repulsion on sites, the system passes from a strongly correlated metal to an uncorrelated band insulator. To get further insight into the correlated metal to band insulator transition, we compare the electronic state of A 4 C 60 with that of the correlated A 3 C 60 , which turns into MH insulator for large U || . In both fullerides, the transition from the orbitally degenerate phase to the disproportionated phase is observed with the increase of U || ; however, the nature of the latter phases is significantly different. As orbital disproportionation is indissolubly linked to JT distortions on fullerene sites, either static or dynamic, the LUMO band in A 3 C 60 will be split in three orbital subbands. Figure 2b,d show that the lowest orbital subband in A 3 C 60 becomes fully occupied and practically uncorrelated (q ll E1) with increase of U || in very close analogy with the behaviour of the two lowest subbands in A 4 C 60 (Fig. 2c). At the same time the electron correlation in the middle half-occupied subband gradually increases, implying that the MH transition basically occurs in this subband 30 . Indeed, the bielectronic energy is reduced by quenching the charge fluctuations in the half-filled middle subband. This is seen as the decrease of the Gutzwiller's factor with the increase of U || (Fig. 2d), testifying about suppression of the intersite electron hopping. On the contrary, the doubly occupied orbitals are not subject to electron correlation (Gutzwiller's factor becomes close to 1; Fig. 2d). In the case of A 4 C 60 , the LUMO orbitals split into two doubly filled orbitals and non-degenerate empty orbital by the JT interaction (see the inset of Fig. 2a). The fully occupied orbitals are similar in nature to those of A 3 C 60 , being basically uncorrelated, the same for the empty orbital (all Gutzwiller's factors are close to 1; Fig. 2c). Stabilization of orbitally disproportionated phase. The necessary condition for achieving the band-insulating state is that in the atomic limit of large U || , the orbitally disproportionated molecular state (S ¼ 0) has lower energy than the homogeneous S ¼ 1 Hund state on each C 60 . Consider the t 1u orbital shell of one single fullerene site. Owing to the Hund's rule coupling, the high-spin configurations (S ¼ 1), for example, (2,1,1), are stabilized by 3J with respect to the low-spin configurations (S ¼ 0), for example, (2,2,0). The high-spin (Hund) state always contains half-filled orbitals and leads, therefore, to MH insulator in the limit of large U || . On the other hand, in the presence of a relatively strong static JT effect, the low-spin state is stabilized by E JT ¼ 4E ð1Þ JT , where E ð1Þ JT ¼ ' og 2 =2 is the JT stabilization energy in C À 60 (refs 33,34). Thus, the low-spin state and, consequently, the band-insulating state are realized as the ground state when the condition E JT 43J is fulfilled. With the estimate E JT ¼ 50 meV and J ¼ 44 meV 26,29 , we conclude that all A 4 C 60 with hypothetical cubic structure will be band insulators in the static JT limit at sufficiently large U || . This condition is modified when there is an intrinsic orbital gap D 0 at fullerene sites, which arises due to the lowering of the symmetry of the crystal field (CF) in non-cubic fullerides ( Table 1). Band structure calculations of A 4 C 60 with body-centred Fig. 4) show that the lowsymmetry CF is weak and does not admix the excited electronic states on fullerene sites. Accordingly, the strength of the JT coupling is not modified by this CF splitting. When one of the t 1u orbitals is destabilized by the CF splitting D 0 (Table 1, B), the Hund configuration (2,1,1), with S ¼ 1, is also destabilized by D 0 , whereas the energy of the low-spin configuration (2,2,0), with S ¼ 0, remains unchanged, because the destabilized orbital is not populated (n ¼ 0). The orbitally disproportionated state becomes the ground one when E JT þ D 0 43J, which means that the low-symmetry CF splitting enhances the tendency towards disproportionation. Moreover, if the CF splitting D 0 is larger than the Hund's rule energy 3J, the system becomes band insulator for sufficiently large U || even in the absence of the JT effect (E JT ¼ 0). tetragonal (bct) lattice (Supplementary On the contrary, if two t 1u orbitals are equally destabilized by D 0 (Table 1,C), both the high-spin and the low-spin configurations are destabilized by 2D 0 ; thus, the system does never become band insulator only due to CF splitting. The band insulator is achieved in this case only when the JT stabilization in the low-spin state is stronger than the Hund energy 3J, which results in the same criterion as for the degenerate case (Table 1, A). We stress that the amplitude of the CF splitting does not play a role in this case. It only plays a role when the destabilizations of the low-and high-spin configurations are different, such as in the case of the second scenario (Table 1,B) or the last one (Table 1,D) corresponding to complete CF lift of degeneracy. In the latter case, on the argument given above, only the CF splitting between the highest two orbitals adds to the criterion, which looks now as intermediate (0o1 À mo1, see Table 1) to the previous scenarios, (Table 1, B and C). According to the tight-binding simulations of the DFT LUMO band (Fig. 4a), the pattern of the orbital splitting for the bct K 4 C 60 corresponds to the third scenario of the CF splitting (Table 1, C) with a gap D 0 of ca. 130 meV. Given a similar lattice structure, the same situation is expected also for Rb 4 C 60 . Therefore, according to the criterion in Table 1, no band-insulating state can arise in these two fullerides, unless the JT stabilization energy exceeds the Hund energy (3J). Following the estimations of E ð1Þ JT and J (see above), we conclude that the uncorrelated band-insulating phase is stabilized in A 4 C 60 with A ¼ K, Rb, in agreement with experiment. In body-centred orthorhombic (bco) Cs 4 C 60 , the low-symmetric CF will completely lift the degeneracy of the t 1u orbitals, leading to a scenario D in Table 1. The splitting between the highest and the middle t 1u orbitals will enhance the tendency towards the stabilization of the band-insulating state, according to the criterion in Table 1. Finally, we consider the effect of the JT dynamics on the stabilization of the orbitally disproportionated phase. In the cubic A 4 C 60 , due to a perfect disproportionation (2,2,0) of the occupation of orbital subbands, the dynamical JT effect on the fullerene sites will be unhindered by hybridization of orbitals between sites pretty much as in metallic A 3 C 60 close to MH transition 30 . The pseudorotation of JT deformations in the trough of the ground adiabatic potential surface of fullerene anion gives a gain in nuclear kinetic energy of ' o=2 % 30 meV per dimension of the trough 29 . The gain amounts to ' o in the case of two-dimensional trough in C 4 À 60 (ref. 33,34). This will enhance the criterion for band insulator by ' o in the case of cubic lattice (Table 1). For relatively large intrinsic CF gap, D 0 4' o=2, one of the rotational degrees of freedom in the trough will be quenched and the JT dynamics will reduce to a one-dimensional pseudorotation of JT deformations entraining only the two degenerate orbitals in the scenarios of splitting shown in Table 1, B and C. This is apparently the case of bct K 4 C 60 and Rb 4 C 60 at ambient pressure. In the case of last scenario ( Table 1, D) of CF splitting, the JT pseudorotational dynamics will be completely quenched if the separations between the three orbitals exceed much ' o=2. Whether this is the case of Cs 4 C 60 with a relevant bco lattice remains to be answered by a DFT-based analysis similar to one done here for K 4 C 60 (Fig. 4a,c). + − E JT > 3J + + (B) Δ 0 − − Δ 0 > 3J + − Δ 0 + E JT > 3J + + (C) Δ 0 − − Never + − E JT > 3J + + (D) μΔ 0 (1−μ)Δ 0 − − (1− )Δ 0 > 3J + − (1− )Δ 0 + E JT > 3J + + (1− )Δ 0 + E JT > 3J Δ 0 + E JT + 1 2h > 3J E JT + 1 2h > 3J E JT +h > 3J þ / À , presence/absence; D 0 , the (non-JT) crystal-field splitting of the t 1u LUMO shell on one fullerene site; 0rmr1; E JT , JT stabilization energy for C 4 À 60 ; h o=2, energy gain due to the JT dynamics per dimension of the trough; JTE, Jahn-Teller effect. The criterion for correlated metal to band insulator transition in threefold degenerate band system with four electrons per site*. *The Hund's rule energy 3 J will be slightly modified by taking into account the multiplet structure due to the presence of two low-spin terms in C 4 À 60 . Another ingredient defining the transition from the correlated metal to band insulator is the bielectronic interaction U || . The value of U || at which the band-insulating state is stabilized (the crossing point of the two phases in Fig. 3) depends on the relation between the band energy in the homogeneous correlated metal phase hĤ t i and the gain of intra-site energy due to disproportionated orbital occupations (static and dynamic JT stabilization energies). The calculations (Fig. 3) show that in the cubic model of A 4 C 60 , the band-insulating state arises already at modest values of U || , which means that it is always achieved in these fullerides (cf. experimental Hubbard UE0.4-0.6 eV for K 3 C 60 (refs 38,39)). As the necessary conditions for the cubic and bct A 4 C 60 are the same (Table 1), the band-insulating state seems to be well achieved in the bct K 4 C 60 and Rb 4 C 60 . The stabilization of the band-insulating state in the bco Cs 4 C 60 seems to be facilitated by a larger U || expected due to the larger distance between C 60 sites. This is in line with the experimental observation of insulating non-magnetic state in all A 4 C 60 at ambient pressure 13,14,40 . We want to emphasize that the intrinsic CF splitting of the t 1u LUMO orbitals on C 60 sites in fullerides does not render them automatically band insulators. Thus, the DFT calculations of K 4 C 60 (Fig. 4a,c) do not give a band insulator, but rather a metal despite the intrinsic CF splitting of 130 meV (see Supplementary Fig. 5 for Brillouin zone). The same situation is realized in Cs 4 C 60 and any other fulleride in which the intrinsic CF splitting is significantly smaller than the uncorrelated bandwidth. The bandinsulating state (Fig. 4b,d) only arises due to JT distortions on fullerene sites and due to the effects of electron repulsion in the t 1u shell reducing much the band energy of the homogeneous metallic state. In general, the band-insulating state will be achieved at any value of the gap between the highest and the middle LUMO orbitals D (a sum of CF and JT splittings) at C 60 sites, which fulfills the necessary condition in Table 1. The only difference is that smaller D will require larger U || for achieving the intersection with the homogeneous correlated metal phase (Fig. 5). One should note that the band-insulating state arises not only three-orbital systems such as fullerides, but also in other orbitally degenerate systems with even numbers of electrons per site when both D and U || are sufficiently large. Thus, the scenario B without JT effect in Table 1 was considered for a one-third-filled three-orbital model with infinite-dimensional Bethe lattice 37 . Universality of orbital disproportionation in fullerides. Given the established orbital disproportionation of the LUMO electronic density in A 3 C 60 (refs 29,30), its persistence in A 4 C 60 found in the present work makes the orbital disproportionation a universal feature of electronic phases in alkali-doped fullerides. Indeed, the same electronic phase is expected also for A 2 C 60 fullerides 13,18 , which are described by essentially the same interactions as A 4 C 60 . The only difference will be the inversion of the intrinsic CF and JT orbital splittings on the fullerene sites. The existence of the orbital disproportionation in fullerides is imprinted on their basic electronic properties. As discussed in 'Band-insulating state induced by strong electron repulsion' and ref. 30, in the disproportionated phase of metallic A 3 C 60 the orbital degeneracy is lifted and the electron correlation develops in the middle subband, whereas it does not play a role in other subbands. Therefore, the MH transition also mainly develops in the middle subband 30 and, hence, one has no ground whatsoever to claim strong effects of orbital degeneracy on the MH transition in these materials as was done repeatedly in the past 35 disproportionation is the similar JT dynamics corresponding to independent pseudorotation of JT deformations on different fullerene sites in both MH phase 29 and strongly correlated metallic phase 30 of A 3 C 60 . This has recently found a firm experimental confirmation in the equivalence of infrared spectra of the corresponding materials 31 . In A 4 C 60 , the experimental evidence for the (2,2,0) orbital disproportionated phase comes, first of all, from the observed non-magnetic insulating ground state. Moreover, as implied by the intersection picture of the two ground phases (Fig. 3), the correlated metal to band insulator transition could be observed by the decrease of the bielectronic interaction U || with respect to the band energy. This seems to be realized as the metal-insulator transition in Rb 4 C 60 under pressure 43 , where the electron transfer (band energy) is enhanced by the decrease of the distance between the sites and U || is concomitantly reduced by the enhanced screening. Further evidence for the orbitally disproportionated phase comes from spectroscopy. In the case of static JT distortions of h g y type on fullerene sites, the single-particle excitations are exactly described by the uncorrelated band solutions, jF e zks i ¼â y zks jF 0 i for electron and jF h aks i ¼â aks jF 0 i, a ¼ x,y, for hole quasiparticles, respectively (see Supplementary Note 2). Figure 4 shows that the dispersion of electron-and hole-like excitation basically corresponds to the decoupled z and (x,y) bands due to practically suppressed hybridization between occupied and unoccupied LUMO orbitals ( Supplementary Figs 6 and 7) when the band gap opens. The hole-like excitations (Fig. 4d) show the density of states closely resembling the width and the shape of the LUMO feature in the photoemiossion spectrum 44 . Discussion In this study, we investigated theoretically the ground electronic phase of A 4 C 60 fullerides. It is found that the relatively strong electron repulsion on C 60 sites stabilizes the uncorrelated bandinsulating state in these materials. A particular conclusion of the present study is that the widely used term 'Jahn-Teller-Mott insulator' 20,23,45,46 is not appropriate here, because it involves mutually excluding phenomena. A 4 C 60 or any similar multi-orbital system with even number of electrons per sites can be either a correlated metal with no JT distortions, high-spin (Hund) MH insulator, or uncorrelated band insulator stabilized by static or dynamic JT distortions. We prove here that the latter is the case in the fullerides due to a weaker Hund's rule interaction compared with JT stabilization energy, which is ultimately due to relatively large radius of C 60 . Similar situation should arise in other crystals with large unit cells with local orbital degeneracy, the first candidate being the molecular crystals of K 4 clusters 47 . The present demonstration of the persistence of band-insulating phase in A n C 60 with even n identifies the orbital disproportionation of the LUMO electronic density as a universal key feature of all alkali-doped fullerides, which undoubtly has a strong effect on their electronic properties. We would like to emphasize that the ultimate reason of orbital disproportionation in fullerides is the existence of equilibrium JT distortions, static or dynamic, on fullerene sites. These are always present in fullerides due to the crucial effect of electron correlation on the JT instability of C n À 60 sites. Methods Self-consistent Gutzwiller's approach. The ground states of A 4 C 60 were calculated using the self-consistent Gutzwiller's approach developed for the JT system 30 . Within this approach, both the JT effect and the electron correlation are simultaneously treated by introducing the orbital specific Gutzwiller's variational parameter in the Gutzwiller's wave function, C G j i¼P G F S j i, where F S is a Slater determinant andP G is the Gutzwiller's projector. Besides the Gutzwiller's parameters inP G , the orbital coefficients in the Slater determinant F S are also treated as variational parameters. The total energy was minimized with respect to both Gutzwiller's parameter and the orbital coefficients self-consistently (see Supplementary Methods and ref. 30 for detail). DFT calculations. The transfer parameters t Dm ll 0 were taken from ref. 30 for fcc K 3 C 60 and derived from the DFT calculations for bct K 4 C 60 . The DFT calculations were performed within the generalized gradient approximation with the pseudopotentials C.pbe-mt_fhi.UPF and K.pbe-mt_fhi.UPF of QUANTUM Evolution of the the Gutzwiller reduction factors q ll for A 4 C 60 within the cubic model used in Fig. 1 in function of U || for different orbital gaps D, which are sums of JT and CF splittings (the former is considered arbitrary now). The monotonously decreasing line corresponds to a correlated metal, which for DoD c (D c E100) evolves into a MH insulator. The jumps to q ll E1 for values D4D c correspond to onsets of band insulator. Figure 1 | 1Ground-state energy in function of q and U || . (a) Total energy E g (q) of the ground electronic phase of a A 4 C 60 with cubic band dispersion (see the text) as a function of amplitude of static JT distortion for several values of U || . (b) A two-dimensional plot of E g (q, U || ). Red and blue regions stand for positive and negative values proportional to the intensity of the colour. The red points show the amplitude of spontaneous static JT distortion in function of U || . In both figures, the energy at UFigure 2 | 2II (eV) U II (eV) U II (eV) U II (eV) Occupation numbers and Gutzwiller reduction factors. (a) Occupation numbers per electron spin of LUMO orbitals n l and (c) Gutzwiller reduction factors in the corresponding bands for a model A 4 C 60 with cubic band dispersion (see the text) subject to static JT interaction as function of U || . (b,d) Same as a and c, respectively, for fcc A 3 C 60 . Figure 3 | 3Crossing of the correlated metallic and the band-insulating states. (a) Total energy of the ground electronic phase of A 4 C 60 with cubic band dispersion and suppressed interband electron transfer (t Dm ll 0 ¼ d ll 0 t Dm ll 0 ) as function of U || . The red and the blue lines indicate the correlated band solution (q ¼ 0) and band-insulating solution with JT splitting, respectively, and the solid and dashed lines indicate the ground and the excited states, respectively, for each U || . The bielectronic energy (6U || À 10J) is subtracted from E g . (b) The same for the model of A 4 C 60 with full transfer Hamiltonian used in equation (1). Figure 4 | 4LUMO band and density of states. (a) LUMO band dispersion and (c) corresponding density of states of K 4 C 60 calculated by DFT (generalized gradient approximation) for experimental structure. (b) Dispersion of single-particle excitations and (d) the corresponding density of states corresponding to the orbitally disproportionated ground state at U || ¼ 0.5 eV and without hybridization between the occupied and empty band orbitals. The blue dots in a and c show the tight-binding simulation, and red dots in (a) the DFT calculations. The black line in c and d corresponds to a full DOS, whereas the coloured lines the orbitally projected DOS. Figure 5 | 5ESPRESSO 5.1 (ref. 48). The nuclear positions were relaxed, whereas the lattice constants from ref. 20 were fixed. The tight-binding parameters were obtained by fitting the DFT band to the model transfer Hamiltonian (Ĥ t ) including the nearest-neighbour and next nearest-neighbour terms. The results are shown in Fig. 4a. For the model transfer Hamiltonian and the obtained 18 parameters, see the Supplementary Methods and Supplementary Table 1, respectively.Data availability. The data in this manuscript are available from the authors on request. Transition from correlated metal to band insulator induced by orbital splitting. Table 1 | 1Criterion for the transition from correlated metal to band insulator.Intrinsic orbital splitting Static JTE Dynamical JTE Band insulator (A) − − Never NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13093 NATURE COMMUNICATIONS | 7:13093 | DOI: 10.1038/ncomms13093 | www.nature.com/naturecommunications AcknowledgementsWe acknowledge useful discussions with Dennis Arčon, Katalin Kamarás, Erio Tosatti and Martin Knupfer. N.I. is an overseas researcher under Postdoctoral fellowship of Japan Society for the Promotion of Science. N.I. also acknowledge the financial support from the Flemish Science Foundation (FWO) and the GOA grant from KU Leuven.Author contributionsN.I. made the calculations. L.F.C. conveived the idea and guided the work. Both authors discussed the results and wrote the manuscript.Additional informationSupplementary Information accompanies this paper at http://www.nature.com/ naturecommunications Alkali-Doped Fullerides: Narrow-Band Solids with Unusual Properties. 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[ "High Frequency Waves in Chromospheric Spicules", "High Frequency Waves in Chromospheric Spicules" ]
[ "W Bate \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n", "D B Jess \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n\nDepartment of Physics and Astronomy\nCalifornia State University Northridge\n18111 Nordhoff Street91330NorthridgeCAUSA\n", "V M Nakariakov \nCentre for Fusion\nSpace and Astrophysics\nPhysics Department\nUniversity of Warwick\nCV4 7ALCoventryUK\n\nSt. Petersburg Branch\nSpecial Astrophysical Observatory\nRussian Academy of Sciences\n196140St. PetersburgRussia\n", "S D T Grant \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n", "S Jafarzadeh \nLeibniz Institute for Solar Physics (KIS)\nSchöneckstr. 679104FreiburgGermany\n\nRosseland Centre for Solar Physics\nUniversity of Oslo\nBlindernP.O. Box 1029NO-0315OsloNorway\n", "M Stangalini \nASI\nItalian Space Agency\nVia del Politecnico snc00133RomeItaly\n", "P H Keys \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n", "D J Christian \nDepartment of Physics and Astronomy\nCalifornia State University Northridge\n18111 Nordhoff Street91330NorthridgeCAUSA\n", "F P Keenan \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n" ]
[ "Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK", "Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK", "Department of Physics and Astronomy\nCalifornia State University Northridge\n18111 Nordhoff Street91330NorthridgeCAUSA", "Centre for Fusion\nSpace and Astrophysics\nPhysics Department\nUniversity of Warwick\nCV4 7ALCoventryUK", "St. Petersburg Branch\nSpecial Astrophysical Observatory\nRussian Academy of Sciences\n196140St. PetersburgRussia", "Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK", "Leibniz Institute for Solar Physics (KIS)\nSchöneckstr. 679104FreiburgGermany", "Rosseland Centre for Solar Physics\nUniversity of Oslo\nBlindernP.O. Box 1029NO-0315OsloNorway", "ASI\nItalian Space Agency\nVia del Politecnico snc00133RomeItaly", "Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK", "Department of Physics and Astronomy\nCalifornia State University Northridge\n18111 Nordhoff Street91330NorthridgeCAUSA", "Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK" ]
[]
Using high cadence observations from the Hydrogen-alpha Rapid Dynamics camera imaging system on the Dunn Solar Telescope, we present an investigation of the statistical properties of transverse oscillations in spicules captured above the solar limb. At five equally separated atmospheric heights, spanning approximately 4900 − 7500 km, we have detected a total of 15 959 individual wave events, with a mean displacement amplitude of 151±124 km, a mean period of 54±45 s, and a mean projected velocity amplitude of 21 ± 13 km s −1 . We find that both the displacement and velocity amplitudes increase with height above the solar limb, ranging from 132 ± 111 km and 17.7 ± 10.6 km s −1 at ≈ 4900 km, and 168 ± 125 km and 26.3 ± 14.1 km s −1 at ≈ 7500 km, respectively. Following the examination of neighboring oscillations in time and space, we find 45% of the waves to be upwardly propagating, 49% to be downwardly propagating, and 6% to be standing, with mean absolute phase velocities for the propagating waves on the order of 75−150 km s −1 . While the energy flux of the waves propagating downwards does not appear to depend on height, we find the energy flux of the upwardly propagating waves decreases with atmospheric height at a rate of −13 200 ± 6500 W m −2 /Mm. As a result, this decrease in energy flux as the waves propagate upwards may provide significant thermal input into the local plasma.
10.3847/1538-4357/ac5c53
[ "https://arxiv.org/pdf/2203.04997v1.pdf" ]
247,363,016
2203.04997
6072621979efd201a48684c563ddee3d061eee91
High Frequency Waves in Chromospheric Spicules W Bate Astrophysics Research Centre School of Mathematics and Physics Queen's University Belfast BT7 1NNBelfastUK D B Jess Astrophysics Research Centre School of Mathematics and Physics Queen's University Belfast BT7 1NNBelfastUK Department of Physics and Astronomy California State University Northridge 18111 Nordhoff Street91330NorthridgeCAUSA V M Nakariakov Centre for Fusion Space and Astrophysics Physics Department University of Warwick CV4 7ALCoventryUK St. Petersburg Branch Special Astrophysical Observatory Russian Academy of Sciences 196140St. PetersburgRussia S D T Grant Astrophysics Research Centre School of Mathematics and Physics Queen's University Belfast BT7 1NNBelfastUK S Jafarzadeh Leibniz Institute for Solar Physics (KIS) Schöneckstr. 679104FreiburgGermany Rosseland Centre for Solar Physics University of Oslo BlindernP.O. Box 1029NO-0315OsloNorway M Stangalini ASI Italian Space Agency Via del Politecnico snc00133RomeItaly P H Keys Astrophysics Research Centre School of Mathematics and Physics Queen's University Belfast BT7 1NNBelfastUK D J Christian Department of Physics and Astronomy California State University Northridge 18111 Nordhoff Street91330NorthridgeCAUSA F P Keenan Astrophysics Research Centre School of Mathematics and Physics Queen's University Belfast BT7 1NNBelfastUK High Frequency Waves in Chromospheric Spicules Draft version March 11, 2022 Typeset using L A T E X twocolumn style in AASTeX631Sun: atmosphere -Sun: chromosphere -Sun: oscillations -Sun: solar spicules Using high cadence observations from the Hydrogen-alpha Rapid Dynamics camera imaging system on the Dunn Solar Telescope, we present an investigation of the statistical properties of transverse oscillations in spicules captured above the solar limb. At five equally separated atmospheric heights, spanning approximately 4900 − 7500 km, we have detected a total of 15 959 individual wave events, with a mean displacement amplitude of 151±124 km, a mean period of 54±45 s, and a mean projected velocity amplitude of 21 ± 13 km s −1 . We find that both the displacement and velocity amplitudes increase with height above the solar limb, ranging from 132 ± 111 km and 17.7 ± 10.6 km s −1 at ≈ 4900 km, and 168 ± 125 km and 26.3 ± 14.1 km s −1 at ≈ 7500 km, respectively. Following the examination of neighboring oscillations in time and space, we find 45% of the waves to be upwardly propagating, 49% to be downwardly propagating, and 6% to be standing, with mean absolute phase velocities for the propagating waves on the order of 75−150 km s −1 . While the energy flux of the waves propagating downwards does not appear to depend on height, we find the energy flux of the upwardly propagating waves decreases with atmospheric height at a rate of −13 200 ± 6500 W m −2 /Mm. As a result, this decrease in energy flux as the waves propagate upwards may provide significant thermal input into the local plasma. INTRODUCTION Spicules are dynamic plasma jets that are prevalent within the solar chromosphere, and which generally have diameters on the order of hundreds of km. They are relatively short-lived features, typically having a lifetime of less than 10 minutes (Pereira et al. 2012). When viewed in the visible and UV bands at the solar limb, spicules appear ubiquitously as a dense forest of narrow, straw-like features (Sterling 1998(Sterling , 2000. Secchi (1877) was the first to observe solar spicules, and they have remained a focal point of solar physics research over the last 140 years. Transverse oscillations in spicules were first identified in the 1960s (Pasachoff et al. 1968), utilizing ground-based observations obtained at the Sacramento Peak Observatory (for a comprehensive review see Zaqarashvili & Erdélyi 2009). The magnetic cylinder model is generally accepted as being the most applicable to spicules, allowing their oscillatory behavior to be interpreted and modeled as magnetohydrodynamic (MHD; Alfvén 1942) modes (Edwin & Roberts 1983). Sterling (2000) highlighted that high-resolution observations, due to the small width of the structures, are vital for a complete description of the spicule wave phenomena. Wedemeyer-Böhm et al. (2007) also note that the ability to detect oscillatory power at higher frequencies is influenced by the spatial resolution of the observations (see also the discussions provided by Jess et al. 2020Jess et al. , 2021. One of the major focuses of current solar physics research is the so-called 'coronal heating paradox'. One of the proposed theoretical mechanisms to explain the source of this heating is linked to the propagation and dissipation of wave phenomena, commonly referred to as the 'AC' heating mechanism (Alfvén & Lindblad 1947). Spicules are of particular interest when attempting to explain the heating of the solar atmosphere due to their potential to facilitate the transfer of mass and energy between the photosphere and corona. They are often categorized by their properties into two types, type i and type ii (De Pontieu et al. 2007a;Pereira et al. 2012), although such distinct classifications are still under debate (e.g., Zhang et al. 2012;Pereira et al. 2013). Observations of transverse oscillations of spicules, fibrils, and mottles in the upper chromosphere typically find mean periods on the order of 80 − 300 s, often with ∼ 50−1000 examples found during the course of an individual data sequence (Nikol'Skii & Sazanov 1967;Nikolsky & Platova 1971;De Pontieu et al. 2007b;Kuridze et al. 2012;Morton et al. 2012Morton et al. , 2013Morton et al. , 2014. A major exception are the short period (45 ± 30 s) transverse oscillations found in 89 type ii spicules within a coronal hole identified by Okamoto & De Pontieu (2011). However, Okamoto & De Pontieu (2011) suggest that this short average period is likely due to them utilizing a methodology which did not allow for the measurement of the properties of the longer period waves (> 100 s). Another exception is the more recent work of Shetye et al. (2021), which found 30 examples of transverse spicule oscillations with periods ranging from 16 − 100 s. These authors also note a selection effect due to only choosing an event if its oscillation period is less than its lifetime, resulting in longer period waves not being considered. Finding the energy flux of spicule oscillations is an important step in investigating their contribution to the heating of the chromosphere and corona. It is estimated that an energy flux of 10 3 − 10 4 W m −2 is required to heat the chromosphere. The energy required to heat the corona is around an order of magnitude less than that required to heat the chromosphere (Withbroe & Noyes 1977). This suggests that accounting for chromospheric heating is a challenge of equal or greater magnitude than for coronal heating when investigating solar atmospheric heating mechanisms. Energy flux estimations are based on the interpretation of these transverse oscillations as MHD wave modes. De Pontieu et al. (2007b) interpreted the transverse oscillations of spicules as bulk Alfvén waves and assumed a filling factor of unity. Using this interpretation, an energy flux estimate of 4000 − 7000 W m −2 was suggested. However, this bulk Alfvén interpretation has attracted criticism, with Erdélyi & Fedun (2007) and Van Doorsselaere et al. (2008) pointing out that Alfvén waves do not result in the bulk transverse motions observed, and instead proposing that these transverse oscillations are best interpreted as kink modes. The filling factor, a measure of what fraction of the total volume is occupied by oscillating spicules, is another extremely important consideration for energy flux calculations (Van Doorsselaere et al. 2014). An equivalent interpretation, assuming that spicules have an approximately constant width across varying heights, would be the ratio of the area of the solar surface covered by spicules to the total solar surface area. Makita (2003) found a spicule filling factor of 5% at a height of 4000 km using Ca ii H & K line observations taken during a solar eclipse. This suggests that a filling factor of 0.05 is more appropriate than unity. Using the revised interpretations of the most realistic MHD mode and associated filling factor, Van Doorsselaere et al. (2014) found that the energy flux estimates by De Pontieu et al. (2007b) were reduced from 4000 − 7000 W m −2 to 200 − 700 W m −2 , a difference exceeding one order of magnitude. Furthermore, Morton et al. (2012) used high-resolution Hα observations taken by the Dunn Solar Telescope to find a similar upper limit for the filling factor (4 − 5%) for open chromospheric structures that connected to higher layers of the solar atmosphere. By interpreting the transverse oscillations of fibrils as kink modes, the authors estimated the energy flux as 170 ± 110 W m −2 , similar to that derived by Van Doorsselaere et al. (2014). However, in addition to the 4−5% filling factors commonly used in modern literature, lower estimates have also been put forward, with Beckers (1972) suggesting a filling factor of 0.6%. As a result, it is generally believed that the spicule filling factor spans an approximate order-of-magnitude (ranging between ≈ 0.5−5%), with differing values being applicable depending on factors such as the atmospheric height sampled and the degree of solar activity (i.e., it is not a quantity that can be applied universally across all observations). The influence of the chosen filling factor on energy flux calculations is discussed further in Section 3. An important caveat when interpreting these energy flux estimates is that they are only based on resolved transverse oscillations. Waves with amplitudes too small to be spatially resolved or periods too short to be temporally resolved are not included in these estimations, leaving the possibility that a significant amount of wave energy may be unaccounted for (Verth & Jess 2016). Another aspect contributing to the underestimation of the total energy flux may be the presence of kink motions along the observer's line-of-sight, which will not manifest as visible transverse oscillations. Examples of this have been documented by Sharma et al. (2018) and Shetye et al. (2021), who measured helical motions of spicules through Doppler measurements (see also the modeling work by Zaqarashvili & Skhirtladze 2008). If these line-of-sight motions are not taken into account when calculating the energy flux, it may result in an underestimation of the true value. The aim of the current study is to identify the properties of spicule oscillations across a statistically significant sample that is extracted from different chromospheric heights. With oscillation characteristics measured across a range of atmospheric layers, we calculate the energy flux carried by these waves as a function of geometric height. To achieve this objective, we utilize ground-based instrumentation with high spatial and temporal resolutions, providing unprecedented data products that are ideally suited for this study. OBSERVATIONS Our analysis employs data collected on 2015 July 27 from 13:52 -15:29 UT using the Dunn Solar Telescope (DST; Dunn 1969) at the National Solar Observatory in New Mexico, USA. The Rapid Oscillations in the Solar Atmosphere (ROSA; Jess et al. 2010b) and Hydrogenalpha Rapid Dynamics camera (HARDcam; Jess et al. 2012a) imaging systems were used to observe a large sunspot, which was part of NOAA AR12391, close to the solar limb at N07.8E73.6 in the conventional heliographic coordinate system. Seeing conditions remained excellent throughout the first hour of the observing period, gradually worsening towards the latter stages of the observing window. HARDcam observations employed a narrow (0.25Å FWHM) bandpass filter centered on the Hα line core (6562.8Å), while the ROSA camera system observed the same region through G-band (10Å FWHM centered at 4305Å) and broadband 4170Å continuum filters. The HARDcam data have a pixel size of 0.092 (66.5 km), providing a 180 × 180 field-of-view, while the ROSA system was slightly undersampled (0.180 per pixel) to provide an identical field-of-view size to that of the HARDcam observations. To correct for wavefront deformations in real time, higher-order adaptive optics (AO) were used during the observations (Rimmele 2004; Rimmele & Marino 2011). Original data from both ROSA and HARDcam were taken at a frame rate of 30.3 s −1 , with the images synchronized by way of a master trigger with microsecond precision (Jess et al. 2010b). The resulting HARDcam Hα images were then improved using speckle reconstruction algorithms (Wöger et al. 2008), utilizing a 30 → 1 restoration, resulting in a final reconstructed cadence of 0.990 s. ROSA continuum observations were coaligned using cross-correlation techniques (see, e.g., Jess et al. 2010a) with contemporaneous continuum images from the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012), providing sub-arcsecond pointing accuracy for the field-of-view covered by the DST. Following this, the HARDcam field was aligned with the master ROSA images using sequences of targets acquired during the calibration procedures at the DST, resulting in Hα observations that have precise pointing metadata that is consistent with modern space-based observatories. Contextual images from SDO/HMI, ROSA, and HARDcam, following the processing steps outlined above, are shown in Figure 1. ANALYSIS AND DISCUSSION During the course of the observations, the DST's AO system was locked onto the high-contrast sunspot structure that was very close to the limb. As a result, limb spicules close to the central portion of the field-of-view were accurately corrected from atmospheric seeing effects by the AO. Hence, the current HARDcam Hα dataset offers an unprecedented opportunity to examine limb spicules at extremely high time cadence (0.990 s) and spatial resolution (133 km two-pixel resolution). A sub-field, spanning approximately 70 Mm along the central portion of the field-of-view, where the AO corrections were operating optimally, was isolated for further study. As the DST was tracking the sunspot contained within the field-of-view, over time the pixel coordinates corresponding to the limb position change as a result of the sunspot rotating on to the disk. The image sequence was hence stabilized with respect to the limb, which was achieved by first choosing a reference frame towards the beginning of the dataset. Next, a small area of the limb image with high contrast was selected, with subsequent images compared and shifted using twodimensional cross correlation techniques. Pixel shift values that produced the highest cross correlation coefficients were selected and applied to each image in the time series iteratively. The resulting shifted images lead to the limb remaining stationary at the same pixel location throughout the dataset, providing a robust base- line from which to examine spicule oscillations above the fixed limb. Multiscale Gaussian Normalization (MGN; Morgan & Druckmüller 2014) was applied to each image in the dataset in order to more easily identify each spicule and its associated motion. It must be noted that MGN does not preserve photometric accuracy. However, this is not an issue when mapping the transverse oscillations of features since we are not concerned with comparisons of relative intensities. For the application of MGN, we employed the convolution of HARDcam images with Gaussian kernels with one-sigma widths of w = 1.25, 2.5, 5, 10, 20, 40 pixels, followed by the production of gamma-transformed images with a γ value of 3.2 (Poynton 2003). Five slits were placed at equally spaced, constant radial heights above the limb, spanning approximately 4900 km to 7500 km in steps of ≈ 650 km. These slits were curved in nature in order to maintain a constant radial height above the limb, and the highest and lowest slits are shown by the white lines in Figure 2. When taking this approach, it is important to note that superposition along the line of sight of spicules anchored behind the limb, in front of the limb, and on the limb is unavoidable. As a result of the slit heights being based on a geometric distance above the limb, this will result in the foreground/background spicules being sampled further along their lengths than those precisely located on the limb. We have carefully selected the minimum and maximum heights of the slits to be in the range of 4890 − 7500 km (see Figure 2), which is towards the upper end of the 'dense forest' of spicules, hence minimizing the degree of feature superposition. Due to the (minimized) spicule superposition affecting each of the slits in a similar way, and considering the large numbers of spicules observed at each height, comparisons between wave properties taken with different slits will still be valid. However, it is still important to consider this effect when examining wave properties taken from a single slit in isolation, since the chosen slit height will be a minimum value of the distance sampled along the spicule due to these geometric considerations. Timedistance diagrams were then produced from each of the slits, with an example shown in Figure 3. The Automatic Northumbria University Wave Tracking (Auto-NUWT; Morton et al. 2013;Weberg et al. 2018) code was utilized in order to identify the location of the spicules as a function of time, track their transverse motion, and extract the properties of their oscilla- tions. Features are identified by fitting a sum of Gaussians to each time slice in the time-distance diagrams, enabling the determination of sub-pixel values for the location of the center of the feature. The transverse oscillatory behavior of these features is probed through the application of Fourier analysis to the position of the center of the feature as a function of time. At each of the five heights considered, over 3000 spicule features are detected in the time-distance diagrams. Employing Fourier analysis, the properties of the waves present in the transverse motions of these features were determined. As a representative example, the averages and deviations of wave properties found at a height of 6850 km above the limb are displayed in Figure 4, where the distributions of the displacement amplitudes, periods, and calculated velocity amplitudes of these waves are plotted as histograms. These properties follow approximate log-normal distributions, which are shown by the solid green lines in Figure 4. Log- Average wave properties for each height are presented in Table 1. Importantly, the averages of the displacement and velocity amplitudes appear to be consistent with those found in previous studies of transverse waves in spicules. However, the average period of the waves in the current study are shifted to lower values than those found previously (see the summary provided by Jess et al. 2015). Specifically, the majority of earlier studies found average periods on the order of 80 − 300 s (e.g., Nikolsky & Platova 1971;De Pontieu et al. 2007b;Kuridze et al. 2012;Morton et al. 2012Morton et al. , 2013Morton et al. , 2014, while we find the average period to be 53 ± 45 s (middle panel of Figure 4). It should be noted that the mean has been chosen for comparison with previous studies here. However, as the wave properties approximately follow log-normal distributions, the modal value represents a more useful statistic in understanding the peak of this distribution. As a point of comparison, Nikolsky & Platova (1971) found modal and mean periods of 60 s and 85 s, respectively, for their observed spicule oscillations, whereas example modal and mean periods found at a height of 6850 km above the solar limb in this study are ≈24 s and ≈54 s, respectively, as shown in Figure 4. We consider the detection of these shorter period waves likely due, at least in part, to the unprecedented ∼ 1 s time cadence of the dataset utilized. For comparison, previous investigations using data from the Swedish Solar Telescope provided cadences on the order of 5 s, which would make it very difficult for the lowest oscillation periods (< 10 s) identified here to be detected. Across all five defined slits, over 16 600 spicular threads were identified, of which 15 959 (95.9%) exhibit at least one complete wave cycle. Of these examples, 8568 (51.5%) threads exhibit a single wave, 5770 (34.7%) consist of two superposed waves, and 1621 (9.2%) have three (or more) superposed waves. These proportions are similar to those found in transverse oscillations of coronal plumes using Auto-NUWT by Weberg et al. (2018). Two examples of the identified waves are shown in Figure 5. These are chosen as they have radically different periods and displacement amplitudes, consisting of ≈138 s and ≈358 km, respectively, for the top panel, with ≈20 s and ≈79 km, respectively, for the lower panel. Both waves are observed for longer than one full period. The wave identified in the top panel of Figure 5 has properties consistent with those found in previous studies of transverse spicule oscillations (see the review by Jess et al. 2015), highlighting that these longer period (> 50 s) waves are also present within our data and are fitted well using our techniques. However, due to the high spatial and temporal resolutions provided by HARDcam, much shorter period waves are able to be identified, including the example shown in the lower panel of Figure 5. As the wave properties have been determined for each of the five equally-spaced slits above the solar limb, we are able to compare and study characteristics as a function of atmospheric height. The mean values for displacement amplitude and velocity amplitude are shown in Figure 6, where both parameters can be seen to increase with height. By fitting a linear line of best fit through the corresponding data points (see the dashed red lines in Figure 6), the displacement amplitude increases at a rate of 14.6 ± 0.8 km/Mm, and the veloc- Grant et al. 2018;Houston et al. 2018Houston et al. , 2020Riedl et al. 2021). Employing spectropolarimetric inversions of the Ca ii spectral line, revealed evidence that the mass density of spicules decreases exponentially with height, requiring the velocity amplitude to similarly increase to conserve energy flux. Hence, a quadratic fit is presented in the lower panel of Figure 6 using a blue dashed line to show the potential synergy between expected mass density and velocity amplitude. However, we note that this is presented only for completeness, since it is difficult to infer the true nature of the relationship with only five data points. The average periods do not show a similar trend with atmospheric height, instead ranging within the same interval of 48.8 − 57.0 s for the five heights considered. In order to measure the phase velocity of these waves, it was necessary to identify the same feature across different heights. This was achieved by extracting individual wave properties from a certain atmospheric height and searching through the wave catalog for waves with similar properties identified at an adjacent height. The properties considered for this study were the equilibrium x-position of the spicule (±5 pixels or ±330 km), the midpoint time (whether-or-not the next spicule feature lay between the start and end times of the wave being compared), duration of the oscillation (±50%), and the frequency (±10%). Based on these criteria, around 140 waves were found to be suitably similar between each set of adjacent heights, providing large number statistics with a similar proportion (tracked waves in relation to total identified waves) to that documented by Jafarzadeh et al. (2017). The phase difference between all sets of waves identified at adjacent heights was calculated using Fourier phase lag analysis. The cross-power spectrum was cal- culated using the representative Fourier spectra of the two waves found at adjacent heights (Bendat & Piersol 2000). The real part of the cross-power spectrum (cospectrum) was used to verify that each original Fourier spectrum had a peak at the same frequency. The phase of the cross-power spectrum was then computed at the same frequency to determine the phase lag between the two heights (Vaughan & Nowak 1997). Finding this phase lag, φ (in degrees), allows for the calculation of the phase velocity, v ph (in km/s), v ph = 360d T φ ,(1) where d is the height difference between the two slits in km, and T is the period of the wave in seconds (Jess et al. 2012b;González Manrique et al. 2020). Importantly, calculation of the phase velocities of the waves embedded within the spicules allows for the eventual calculation of their energy fluxes. The distribution of phase velocities for the 135 propagating waves iden-tified traveling between the heights of 6200 → 6850 km are shown in Figure 7, where waves propagating in both the upward and downward directions are identified. Any waves displaying zero phase lags (i.e., providing infinite phase speeds in Equation 1) were classified as standing modes. Due to the relatively small number of standing modes present in our dataset, this type of wave is not included in the histogram depicted in Figure 7. We must highlight that the distribution of upwardly and downwardly propagating waves shown in Figure 7 appear to originate from the same population, with an approximately Gaussian distribution encompassing waves propagating both upwards and downwards. However, when examining the energy flux carried by these waves it is important to examine the direction of energy propagation, which is determined by the sign of their associated phase speed. Thus, distinctions are made between upwardly and downwardly propagating waves for the sake of further analysis, but it must be emphasized that there do not seem to be two distinct populations present in Figure 7. It should be noted that any plasma flows within the spicules will affect the apparent phase velocities of the measured kink oscillations. In the case of upflowing plasma, the apparent (i.e., measured) phase velocities of the upwardly propagating waves will be related to (v ph + U ), while the downwardly propagating waves will have apparent phase velocities equal to (−v ph + U ), where v ph is the true phase velocity and U is the velocity of the upflow (Nakariakov & Roberts 1995). This is similar to observations put forward by Grant et al. (2015), who examined the bulk plasma upflow within a magnetic pore and the subsequent effect this had on the apparent wave speeds of sausage mode oscillations. Strong upflows are typically associated with type ii spicules. However, the spicules observed in this study are likely not best characterized by this classification (De Pontieu et al. 2007a). This effectively means that the apparent phase velocities of the upwardly propagating waves can be considered an upper limit to their true phase velocities. Conversely, in the case of the downwardly propagating waves, this can be considered as a lower limit. As the velocity of any possible upflows are not known, the measured phase velocities have been used in all further calculations, but it is important to note that this will result in the calculated energy fluxes being upper/lower limits for the upwardly/downwardly propagating kink waves. Across the four sets of adjacent heights, the occurrence rates of upwardly propagating, downwardly propagating, and standing mode waves were found to be 45%, 49%, and 6%, respectively. This is in contrast to (2011) were observed within a coronal hole, so may have different properties to those examined here. Importantly, our present study highlights a more equal balance of upward/downward propagation, with fewer examples categorized as standing modes. The lack of standing mode detections may also be a consequence of the incredibly high spatial and temporal resolutions of the HARDcam dataset, since phase precision is drastically improved as a result of the sub-1 second cadence. It might initially be assumed that a roughly equal balance of upward/downward propagation should be expected, due to the high reflection coefficient of the transition region (Hollweg et al. 1982). Liu et al. (2014) also observed downwardly propagating transverse waves within solar spicules and note that low-frequency (periods of ≈ 100 s) are expected to reflect strongly in the transition region (Suzuki & Inutsuka 2005). However, Okamoto & De Pontieu (2011) argue that such reflection would result in more standing modes being observed due to the superposition of upwardly and downwardly propagating waves. This would create an imbalance in observations, with more upward than downward propagations detected. This superposition is, however, heavily dependent on the height of the reflecting boundary, the phase velocity of the upward wave, the lifetime of the spicule, and the time that the wave persists for. If there is insufficient time for the reflected wave to interact with the upward wave, due to any combination of the aforementioned criteria, then wave superposition (and hence a standing wave) will not be observed. Although a full characterization of the driving mechanisms behind the downwardly propagating waves, as well as the clear domination of these waves with a phase speed around zero (see Figure 7), is beyond the scope of the present work, these are important questions to be investigated in future studies. For each of the four sets of adjacent heights, waves identified as upwardly propagating were segregated from their downwardly propagating counterparts. It was hence possible to calculate the rate of change of phase velocity as a function of atmospheric height for both the upwardly and downwardly propagating waves. Average phase velocities for each set of adjacent heights are shown in Table 2 and plotted in Figure 8, where the upper panel corresponds to the average phase velocity of the upwardly propagating waves, while the lower panel depicts the average phase velocity of the downward propagation. The uncertainties shown in Figure 8 have been calculated following the 'bootstrap' methodologies described by Efron et al. (1979). Due to the combined presence of traditional (periods ≥ 50 s) and high-frequency (periods < 50 s) spicule oscillations, it is challenging to assign basic standard errors to the derived phase velocities, especially since the equivalence (or lack thereof) of the driving mechanisms responsible for these characteristics have yet to be observationally and/or theoretically verified. As such, we apply bootstrapping techniques to better constrain the confidence intervals of data following non-normal or unknown distributions (similar to that presented by Simpson & Mayer-Hasselwander 1986;Desmars et al. 2009;Yao et al. 2017). The upward phase velocities appear to increase with atmospheric height at a rate of approximately 10 ± 15 km s −1 /Mm. However, due to the size of the associated uncertainties (see the error bars in the upper panel of Figure 8), it is difficult to unequivocally stipulate the precise relationship. A more pronounced trend is present in the downward phase velocities (lower panel of Figure 8), which appear to decrease (as the height sampled decreases) at a rate of approximately 24 ± 11 km s −1 /Mm, implying that the wave slows as it travels down the spicule and encounters more dense layers of the lower solar atmosphere. With the velocity amplitudes and phase velocities of the oscillations measured, it was possible to estimate the energy flux associated with both the upwardly and downwardly propagating waves. In order to calculate the energy flux, a model for the density of the spicules with height is required. observed a limb spicule and derived a model of its density using the Non-LTE Inversion Code using the Lorien Engine (NICOLE; Socas-Navarro et al. 2015) inversion code. The final density model takes the form, where y is the height above the solar limb, ρ(y) is the spicule density as a function of height, h 0 is the base height, ρ 0 is the density at the base height, and Λ is the density scale height. Values for our energy flux calculations were taken directly from , where ρ 0 ≈ 6 × 10 −7 kg m −3 , h 0 = 2000 km, and Λ = 1500 km. The energy flux, F , from transverse waves in a multiple flux tube system can be calculated as, ρ(y) = ρ 0 e (y−h0)/Λ ,(2)F ≈ f 1 2 (ρ i + ρ e )v 2 v gr ,(3) where f is the density filling factor, ρ i is the density inside the flux tube filled in by the spicule, ρ e is the density outside the spicule, v is the velocity amplitude, and v gr is the group speed (Van Doorsselaere et al. 2014). For propagating kink waves, the group velocity can be approximated by the phase speed, v ph , as they are only weakly dispersive (Terradas et al. 2010;Nakariakov et al. 2021). The internal density for spicules can be assumed to be much larger than the external density, i.e., ρ i ρ e (Uchida 1961), providing a simplified equation for the energy flux, F ≈ f 1 2 ρ i v 2 v ph .(4) Taking the upper limit of the spicule density filling factor as 5% ) allowed the energy fluxes to be calculated for each adjacent set of heights, which are displayed individually for all propagating waves (top panel) alongside upwardly (middle) and downwardly (bottom) propagating waves in Figure 9. For all waves examined, it can clearly be seen that there is a decrease in energy flux with height, indicated using solid black data points in the upper panel of Figure 9. A linear line of best fit is presented using a dashed black line in the upper panel of Figure 9, with a gradient of −12 600 W m −2 /Mm. However, an exponential fit would perhaps be more appropriate, since the main factor for the energy flux decrease is expected to be density stratification, which is typically represented by a decaying exponential profile with height (e.g., Verth et al. 2011). Due to the relatively small number of data points under consideration, a linear fit has been chosen for simplicity. Regardless of the fitting function employed, the important message is that the energy flux of the propagating transverse waves clearly decreases with atmospheric height, hinting at some sort of damping and/or dissipation process. It is important to consider the effect of using a filling factor of 5%. This means that Equation 4 estimates the energy flux under the assumption that the waves are omnipresent, i.e., does not take into account the sporadic nature of the observed wave motion. In addition, as the waves are not seen to exist in all spicules, the actual filling factor, f , should be reduced to account for this effect. Thus, the estimation based on Equation 4 gives us the upper limit of the energy flux in the waves. However, as the filling factor is a multiplicative term, this only affects the magnitude of any energy flux estimations. The trends in energy flux examined with respect to height are independent of any adjustment to the filling factor. For example, using the relatively low filling factor of 0.6% suggested by Beckers (1972) will simply lower all energy flux and rate of change of flux values by a linear factor of 0.12 when compared to those values calculated with a filling factor of 5%. The values presented in the text and within Figure 9 utilize a filling factor of 5%, unless stated otherwise, and should therefore be taken as upper limits. For all upwardly propagating waves, we observe the energy flux to decrease as a function of height at a rate of −13 200 ± 6500 W m −2 /Mm, which is indicated in the middle panel of Figure 9 using a dashed black line derived from a linear least-squares fit. For complete-ness, it is estimated that energy fluxes in the range of 10 3 − 10 4 W m −2 are required to heat the chromosphere (Withbroe & Noyes 1977). Hence, the total energy flux, in addition to the measured rate of energy flux decay with height, are on the same order as the total energy input required to provide basal heating to the solar chromosphere. Even considering the relatively low filling factor of 0.6%, as suggested by Beckers (1972), the rate of energy flux decrease would be −1580 ± 780 W m −2 /Mm, still within the range that is needed to balance the radiative losses of the chromosphere. By contrast, the energy flux for all of the waves propagating downwards does not appear to depend on the height sampled (black data points in the lower panel of Figure 9), with the energy flux estimates remaining consistent (∼ 4 × 10 4 W m −2 ) across the height range of approximately 7500 → 4900 km above the solar limb. The similarity in the rate of energy flux drop off in height is consistent between the full set of waves (upper panel of Figure 9) and the upwardly propagating ones (middle panel of Figure 9). This is to be expected, since the downward energy flux remains approximately constant with atmospheric height. The decrease in upward energy flux with atmospheric height may be due to at least three different factors: (1) physical thermalization of wave energy into localized heat via dissipation mechanisms (e.g., Hollweg 1986;He et al. 2009;Antolin et al. 2015Antolin et al. , 2018Okamoto et al. 2015, to name but a few examples), (2) damping of detectable transverse waves through the process of mode conversion, where kink mode amplitudes decay as a result of the transfer of energy from transverse kink oscillations to azimuthal Alfvén motions (Pascoe et al. 2010(Pascoe et al. , 2012(Pascoe et al. , 2013, and/or (3) reflection of the waves downward at varying heights above the solar limb (Hollweg et al. 1982;Suzuki & Inutsuka 2005). Tentative observational evidence has shown that torsional Alfvén and kink waves may exist concurrently in spicules, providing credence for the applicability of mode conversion processes . Previous modeling work by Sterling & Hollweg (1984) has shown that Alfvén waves within spicules can produce high-frequency signatures, including periodicities of 112, 37, and 22 s for the fundamental, first, and second harmonic resonant periods, respectively, which are similar to the periodicities found in our current work. Employing simultaneous plane-ofsky imaging and line-of-sight Doppler measurements will allow more precise definitions of the embedded spicule wave modes, which will allow the high-frequency Alfvén modes to be examined and compared to the models put forward by Sterling & Hollweg (1984). In order to establish if the wave energy is dissipated in the form of localized heating, measurements of thermal processes in the vicinity of these spicules are necessary. This may be achieved using differential emission measures of optically thin coronal EUV observations directly above the spicules (McIntosh 2012; Vanninathan et al. 2012). An alternative approach would be to use the Atacama Large Millimeter/Submillimeter Array (ALMA; Wootten & Thompson 2009;Wedemeyer et al. 2016) to find the temperature of the spicules and the surround-ing plasma (Chintzoglou et al. 2021;Henriques et al. Preprint;Jafarzadeh et al. 2021). Importantly, the timing information related to the decay of the spicule oscillations would need to be harnessed to provide both spatial and temporal information to examine localized temperature fluctuations that may be a result of thermalization mechanisms. While this is beyond the scope of the present work, it will form the basis of a follow-up study over the coming months. The downwardly propagating waves maintain an approximately constant energy flux through a reduction in both velocity amplitude and phase velocity as they travel down the spicule, visible in Figures 6 and 8, respectively. It is likely that this is due to the wave interacting with the denser plasma at lower heights above the solar limb, resulting in a slower Alfvén speed in these regions (Okamoto & De Pontieu 2011). This is not unexpected, as the theoretical modeling of propagating kink waves in longitudinally stratified waveguides found that phase velocities and velocity amplitudes decrease with height (Soler et al. 2011). It has been proposed that in order to supply the quasisteady effects needed to heat the solar atmosphere, the dissipation of short period waves is of paramount importance (Hasan et al. 2005;Hasan & Van Ballegooijen 2008;Van Ballegooijen et al. 2011). The energy flux carried by both short period (< 50 s) and long period (≥ 50 s) waves between each set of adjacent heights is shown in Figure 9 using red and blue data points, respectively. In order to calculate the associated energy flux for the propagating wave modes, new filling factors were calculated by combining the previously used spicule density filling factor (5%; Morton et al. 2012) with the fraction of waves which were found to fall into each relevant category (i.e., < 50 s or ≥ 50 s). The new filling factors were approximately 2.5%, which is a result of the 50 s boundary being very close to the average period found at each height (see Table 1). It can be seen from Figure 9 that the energy flux of the short period waves is greater than that of the long period waves for the full set of propagating waves (upper panel), and both the upwardly propagating (middle panel) and downwardly propagating (lower panel) waves. For the full set of propagating waves and the upwardly propagating waves, both the short and long period waves show a similar energy flux decrease with height as that for the total energy flux values. The energy flux of both the short and long period downwardly propagating waves show a similar lack of dependence on atmospheric height, which is consistent with the total energy flux measurements. This suggests that both short and long period upwardly propagating waves have the potential to heat the solar atmosphere, although the short period waves have a larger energy flux across all heights, giving them a greater potential capacity for heating. CONCLUSIONS The results presented here represent a sizable increase in the statistical population of examined transverse spicule oscillations. Our use of data with a time cadence of ∼ 1 s also allowed for the identification of high frequency waves, similar to those found by Okamoto & De Pontieu (2011), with periods as short as 10 − 20 s, only now with a significant increase in the examined population size. Observations with even higher spatial and temporal resolutions may allow for the detection of even shorter period and smaller-scale oscillations, and further extend the statistical distributions (see, e.g., Figure 4) down to even smaller values. We examined the wave properties of spicule oscillations across multiple atmospheric heights, which facilitated the calculation of associated phase speeds, hence allowing us to categorize the waves as either being upwardly/downwardly propagating or standing. Almost an equal balance was found between upwardly (45%) and downwardly (49%) propagating waves, in contrast to the earlier study by Okamoto & De Pontieu (2011), who found that upwardly propagating waves were dominant in their time series. However, the observations presented here are in close proximity to the solar active region NOAA AR12391 and may therefore have distinctly different properties to the coronal hole observations examined by Okamoto & De Pontieu (2011). Directional information for the spicule waves allowed the calculation of their associated energy flux as a function of upwardly and downwardly propagating waves across a number of atmospheric heights. Energy flux estimates are relatively consistent across all heights for the waves propagating in a downwards direction. However, for the upwardly propagating waves, a negative correlation with height is demonstrated, with the overall energy flux decreasing at a rate of −13 200 ± 6500 W m −2 /Mm calculated with a spicule filling factor of 5% (or at a rate of −1580 ± 780 W m −2 /Mm using a lower-limit filling factor of 0.6%). The mechanism responsible may either be due to thermalization of the mechanical wave energy or mode coupling, although investigation of the proportional contributions of each mechanism are beyond the scope of this study. If even a small fraction of the wave energy carried in the transverse waves of the spicules examined is deposited as thermal energy, then it may significantly contribute to the 10 3 − 10 4 W m −2 requirements needed to balance the radiative losses of the chromosphere (Withbroe & Noyes 1977 (grant No. 18-29-21016). SJ acknowledges support from the European Research Council under the European Union Horizon 2020 research and innovation program (grant agreement No. 682462) and from the Research Council of Norway through its Centres of Excellence scheme (project No. 262622). The Dunn Solar Telescope at Sacramento Peak/NM was operated by the National Solar Observatory (NSO). NSO is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under cooperative agreement with the National Science Foundation (NSF). The authors wish to acknowledge scientific discussions with the Waves in the Lower Solar Atmosphere (WaLSA; www.WaLSA.team) team, which is supported by the Research Council of Norway (project number 262622), and The Royal Society through the award of funding to host the Theo Murphy Discussion Meeting "High-resolution wave dynamics in the lower solar atmosphere" (grant Hooke18b/SCTM). Auto-NUWT (Morton et al. 2013;Weberg et al. 2018) Figure 1 . 1Contextual SDO/HMI continuum (left), ROSA G-band (middle), and Hα line core (right) images acquired at 13:59:09 UT. The area imaged by ROSA and HARDcam is marked by the red square in the full disk SDO/HMI continuum image. Numerous spicules are clearly visible above the solar limb as narrow, straw-like structures in the corresponding Hα image. Figure 2 . 2An Hα core sub-field (67 × 16 Mm 2 ) image acquired using HARDcam at 14:49:45 UT. Numerous spicules are clearly visible above the solar limb as narrow, straw-like structures. The two most extreme slits used to take the time distance diagrams are shown by the white lines, at heights of 4890 and 7500 km. The axes are shown using different scales to aid with visual clarity. Figure 3 . 3A time-distance diagram captured using a curved slit at a height of 6850 km above the solar limb. Each bright streak is a feature passing through the slit, with the clear oscillatory features representative of transverse motions displaying a range of amplitudes and periods. Figure 4 . 4Histograms of the wave properties identified at a height of 6850 km above the solar limb. The upper, middle, and lower panels display information related to the displacement amplitudes, periods, and velocity amplitudes, respectively. Measurements of the corresponding averages and deviations are displayed on the right of each histogram. MAD denotes the median absolute deviation. Figure 5 . 5The displacement curves, corresponding to two of the waves identified in our dataset, are shown (in their raw form) using the solid black lines. The dashed red lines highlight the fitted waves with properties derived using Fourier analysis. The top panel shows a wave with a period of ≈ 138 s and a displacement amplitude of ≈ 358 km, while the bottom panel shows a wave with a period of ≈ 20 s and a displacement amplitude of ≈ 79 km. Figure 6 . 6Mean values of spicule displacement amplitudes (top panel) and velocity amplitudes (lower panel) plotted against height above the solar limb. Linear lines of best fit are shown in both panels using a dashed red line. In the bottom panel a quadratic fit is shown using a dashed blue line. Errors associated with each data point represent the standard error on the mean. normal distributions for these properties are consistent with those found in previous studies (e.g., De Pontieu et al. 2007b; Okamoto & De Pontieu 2011; Pereira et al. 2012). Figure 7 . 7A histogram showing the calculated phase velocities of the 135 propagating waves identified between the heights of 6200 and 6850 km above the solar limb. The solid red line denotes the cumulative probability function. A bin width of 20 km s −1 was used for the creation of this histogram. Figure 8 . 8Mean values of phase velocity shown with height above the solar limb. The values for upwardly and downwardly propagating waves are shown in the top and bottom graphs respectively. Errors are calculated using bootstrapping. Figure 9 . 9Energy flux estimations as a function of atmospheric height for all propagating waves (upper panel), upwardly propagating waves (middle panel), and downwardly propagating waves (lower panel). The total energy flux provided by short/long period waves is shown in black, while the energy fluxes for short-(< 50 s) and long-period (> 50 s) waves are shown in red and blue, respectively. The energy fluxes provided by the full set of waves (including upwardly and downwardly propagating) and for all upwardly propagating waves are depicted, using a linear line of best fit, as a dashed black line in the upper and middle panels, with gradients equal to −12 600 W m −2 /Mm and −13 200 W m −2 /Mm, respectively. Facilities: DST (HARDcam; Jess et al. 2012a) Software: MGN (Morgan & Druckmüller 2014), Table 1. Mean wave properties and their standard deviations at each sampled height.Height Number of Waves Displacement Amplitude Period Velocity Amplitude (km) (km) (s) (km s −1 ) 4890 4880 132.4±111.2 55.1±45.0 17.7±10.6 5550 4920 139.6±118.8 57.0±48.1 18.3±10.7 6200 5022 152.4±128.5 55.8±48.1 20.6±11.6 6850 5209 163.7±128.6 53.4±45.4 23.4±13.1 7500 5298 168.1±125.4 48.9±39.9 26.3±14.1 ity amplitude at 3.33 ± 0.08 km s −1 /Mm. The conser- vation of energy flux requires a reciprocal relation be- tween density and velocity amplitude (see, e.g., the dis- cussions in Stein & Leibacher 1974; Ebadi et al. 2012; de la Cruz Rodríguez et al. 2013; Khomenko & Colla- dos 2015; ) . )WB, DBJ, and FPK acknowledge support from the Leverhulme Trust via the Research Project Grant RPG-2019-371. DBJ and SDTG wish to thank Invest NI and Randox Laboratories Ltd. for the award of a Research & Development Grant (059RDEN-1), in addition to the UK Science and Technology Facilities Council (STFC) for the consolidated grant ST/T00021X/1. DBJ also acknowledges funding from the UK Space Agency via the National Space Technology Programme (grant SSc-009). VMN acknowledges support from STFC grant ST/T000252/1 and Russian Foundation for Basic Research (RFBR) grant . H Alfvén, 10.1038/150405d0Nature. 150Alfvén, H. 1942, Nature, 150, doi: 10.1038/150405d0 . H Alfvén, B Lindblad, 10.1093/mnras/107.2.211Monthly Notices of the Royal Astronomical Society. 107211Alfvén, H., & Lindblad, B. 1947, Monthly Notices of the Royal Astronomical Society, 107, 211, doi: 10.1093/mnras/107.2.211 . P Antolin, T J Okamoto, B De Pontieu, 10.1088/0004-637X/809/1/72Astrophysical Journal. 809Antolin, P., Okamoto, T. J., De Pontieu, B., et al. 2015, Astrophysical Journal, 809, doi: 10.1088/0004-637X/809/1/72 . 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